---
title: "R Manuals :: An Introduction to R"
toc-depth: 2
number-sections: true
---
 
# An Introduction to R {.unnumbered}

This is an introduction to R ("GNU S"), a language and environment for
statistical computing and graphics. R is similar to the
award-winning[^1] S system, which was developed at Bell
Laboratories by John Chambers et al. It provides a wide variety of
statistical and graphical techniques (linear and nonlinear modelling,
statistical tests, time series analysis, classification, clustering,
\...).

This manual provides information on data types, programming elements,
statistical modelling and graphics.

This manual is for R, version 4.3.0 Under development.

Copyright © 1990 W. N. Venables\
Copyright © 1992 W. N. Venables & D. M. Smith\
Copyright © 1997 R. Gentleman & R. Ihaka\
Copyright © 1997, 1998 M. Maechler\
Copyright © 1999--2022 R Core Team

> Permission is granted to make and distribute verbatim copies of this
> manual provided the copyright notice and this permission notice are
> preserved on all copies.
>
> Permission is granted to copy and distribute modified versions of this
> manual under the conditions for verbatim copying, provided that the
> entire resulting derived work is distributed under the terms of a
> permission notice identical to this one.
>
> Permission is granted to copy and distribute translations of this
> manual into another language, under the above conditions for modified
> versions, except that this permission notice may be stated in a
> translation approved by the R Core Team.


[^1]: ACM Software Systems award, 1998:
<https://awards.acm.org/award_winners/chambers_6640862.cfm>.
 
# Preface {.unnumbered}

This introduction to R is derived from an original set of notes
describing the S and [S-PLUS]{.small} environments written in 1990--2 by
Bill Venables and David M. Smith when at the University of Adelaide. We
have made a number of small changes to reflect differences between the R
and S programs, and expanded some of the material.

We would like to extend warm thanks to Bill Venables (and David Smith)
for granting permission to distribute this modified version of the notes
in this way, and for being a supporter of R from way back.

Comments and corrections are always welcome. Please address email
correspondence to <R-help@R-project.org>.

### Suggestions to the reader {.unnumbered}

Most R novices will start with the introductory session in Appendix A.
This should give some familiarity with the style of R sessions and more
importantly some instant feedback on what actually happens.

Many users will come to R mainly for its graphical facilities. See
@sec-graphical-procedures, which can be read at almost any time and need
not wait until all the preceding sections have been digested.

 
# Introduction and preliminaries {#sec-introduction-and-preliminaries}

 
## The R environment {#sec-the-r-environment}

R is an integrated suite of software facilities for data manipulation,
calculation and graphical display. Among other things it has

-   an effective data handling and storage facility,
-   a suite of operators for calculations on arrays, in particular
    matrices,
-   a large, coherent, integrated collection of intermediate tools for
    data analysis,
-   graphical facilities for data analysis and display either directly
    at the computer or on hardcopy, and
-   a well developed, simple and effective programming language (called
    'S') which includes conditionals, loops, user defined recursive
    functions and input and output facilities. (Indeed most of the
    system supplied functions are themselves written in the S language.)

The term "environment" is intended to characterize it as a fully planned
and coherent system, rather than an incremental accretion of very
specific and inflexible tools, as is frequently the case with other data
analysis software.

R is very much a vehicle for newly developing methods of interactive
data analysis. It has developed rapidly, and has been extended by a
large collection of *packages*. However, most programs written in R are
essentially ephemeral, written for a single piece of data analysis.
 
## Related software and documentation {#sec-related-software-and-documentation}

R can be regarded as an implementation of the S language which was
developed at Bell Laboratories by Rick Becker, John Chambers and Allan
Wilks, and also forms the basis of the [S-PLUS]{.small} systems.

The evolution of the S language is characterized by four books by John
Chambers and coauthors. For R, the basic reference is *The New S
Language: A Programming Environment for Data Analysis and Graphics* by
Richard A. Becker, John M. Chambers and Allan R. Wilks. The new features
of the 1991 release of S are covered in *Statistical Models in S* edited
by John M. Chambers and Trevor J. Hastie. The formal methods and classes
of the **methods** package are based on those described in *Programming
with Data* by John M. Chambers. See @sec-appendix-f-references, for
precise references.

There are now a number of books which describe how to use R for data
analysis and statistics, and documentation for S/[S-PLUS]{.small} can
typically be used with R, keeping the differences between the S
implementations in mind. See [What documentation exists for R](https://cran.r-project.org/doc/FAQ/R-FAQ.html#What-documentation-exists-for-R_003f)
in the R statistical system FAQ.

 
## R and statistics {#sec-r-and-statistics}

Our introduction to the R environment did not mention *statistics*, yet
many people use R as a statistics system. We prefer to think of it of an
environment within which many classical and modern statistical
techniques have been implemented. A few of these are built into the base
R environment, but many are supplied as *packages*. There are about 25
packages supplied with R (called "standard" and "recommended" packages)
and many more are available through the CRAN family of Internet sites
(via <https://CRAN.R-project.org>) and elsewhere. More details on
packages are given later (see @sec-packages).

Most classical statistics and much of the latest methodology is
available for use with R, but users may need to be prepared to do a
little work to find it.

There is an important difference in philosophy between S (and hence R)
and the other main statistical systems. In S a statistical analysis is
normally done as a series of steps, with intermediate results being
stored in objects. Thus whereas SAS and SPSS will give copious output
from a regression or discriminant analysis, R will give minimal output
and store the results in a fit object for subsequent interrogation by
further R functions.
 
## R and the window system {#sec-r-and-the-window-system}

The most convenient way to use R is at a graphics workstation running a
windowing system. This guide is aimed at users who have this facility.
In particular we will occasionally refer to the use of R on an X window
system although the vast bulk of what is said applies generally to any
implementation of the R environment.

Most users will find it necessary to interact directly with the
operating system on their computer from time to time. In this guide, we
mainly discuss interaction with the operating system on UNIX machines.
If you are running R under Windows or macOS you will need to make some
small adjustments.

Setting up a workstation to take full advantage of the customizable
features of R is a straightforward if somewhat tedious procedure, and
will not be considered further here. Users in difficulty should seek
local expert help.
 
## Using R interactively {#sec-using-r-interactively}

When you use the R program it issues a prompt when it expects input
commands. The default prompt is `>`, which on UNIX might be the same
as the shell prompt, and so it may appear that nothing is happening.
However, as we shall see, it is easy to change to a different R prompt
if you wish. We will assume that the UNIX shell prompt is `$`.

In using R under UNIX the suggested procedure for the first occasion is
as follows:

1.  Create a separate sub-directory, say `work`, to hold data
    files on which you will use R for this problem. This will be the
    working directory whenever you use R for this particular problem.
    ``` R
    $ mkdir work
    $ cd work
    ```

2.  Start the R program with the command
    ``` R
    $ R
    ```

3.  At this point R commands may be issued (see later).

4.  To quit the R program the command is

    ``` R
    > q()
    ```

    At this point you will be asked whether you want to save the data
    from your R session. On some systems this will bring up a dialog
    box, and on others you will receive a text prompt to which you can
    respond [yes]{.kbd}, [no]{.kbd} or [cancel]{.kbd} (a single letter
    abbreviation will do) to save the data before quitting, quit without
    saving, or return to the R session. Data which is saved will be
    available in future R sessions.

Further R sessions are simple.

1.  Make `work` the working directory and start the program as
    before:
    ``` R
    $ cd work
    $ R
    ```
2.  Use the R program, terminating with the `q()` command at the end of
    the session.

To use R under Windows the procedure to follow is basically the same.
Create a folder as the working directory, and set that in the
`Start In` field in your R shortcut. Then launch R by double
clicking on the icon.

## 1.6 An introductory session {#sec-1.6-an-introductory-session}

Readers wishing to get a feel for R at a computer before proceeding are
strongly advised to work through the introductory session given in
@sec-appendix-a-a-sample-session.
 
## Getting help with functions and features {#sec-getting-help-with-functions-and-features}

R has an inbuilt help facility similar to the `man` facility of UNIX. To
get more information on any specific named function, for example
`solve`, the command is

``` R
> help(solve)
```

An alternative is

``` R
> ?solve
```

For a feature specified by special characters, the argument must be
enclosed in double or single quotes, making it a "character string":
This is also necessary for a few words with syntactic meaning including
`if`, `for` and `function`.

``` R
> help("[[")
```

Either form of quote mark may be used to escape the other, as in the
string `"It's important"`. Our convention is to use double quote marks
for preference.

On most R installations help is available in HTML format by running

``` R
> help.start()
```

which will launch a Web browser that allows the help pages to be browsed
with hyperlinks. On UNIX, subsequent help requests are sent to the
HTML-based help system. The 'Search Engine and Keywords' link in the
page loaded by `help.start()` is particularly useful as it is contains a
high-level concept list which searches though available functions. It
can be a great way to get your bearings quickly and to understand the
breadth of what R has to offer.

The `help.search` command (alternatively `??`) allows searching for help
in various ways. For example,

``` R
> ??solve
```

Try `?help.search` for details and more examples.

The examples on a help topic can normally be run by

``` R
> example(topic)
```

Windows versions of R have other optional help systems: use

``` R
> ?help
```

for further details.
 
## R commands, case sensitivity, etc. {#sec-r-commands-case-sensitivity-etc.}

Technically R is an *expression language* with a very simple syntax. It
is *case sensitive* as are most UNIX based packages, so `A` and `a` are
different symbols and would refer to different variables. The set of
symbols which can be used in R names depends on the operating system and
country within which R is being run (technically on the *locale* in
use). Normally all alphanumeric symbols are allowed[^2]
(and in some countries this includes accented letters) plus `.` and
`_`', with the restriction that a name must start with '`.` or a
letter, and if it starts with `.` the second character must not be a
digit. Names are effectively unlimited in length.

Elementary commands consist of either *expressions* or *assignments*. If
an expression is given as a command, it is evaluated, printed (unless
specifically made invisible), and the value is lost. An assignment also
evaluates an expression and passes the value to a variable but the
result is not automatically printed.

Commands are separated either by a semi-colon (`;`), or by a newline.
Elementary commands can be grouped together into one compound expression
by braces (`{`' and '`}`). *Comments* can be put
almost[^3] anywhere, starting with a hashmark (`#`),
everything to the end of the line is a comment.

If a command is not complete at the end of a line, R will give a
different prompt, by default

``` R
+
```

on second and subsequent lines and continue to read input until the
command is syntactically complete. This prompt may be changed by the
user. We will generally omit the continuation prompt and indicate
continuation by simple indenting.

Command lines entered at the console are limited[^4] to
about 4095 bytes (not characters).

[^2]: For portable R code (including that to be used in R packages) only
A--Za--z0--9 should be used.

[^3]: **not** inside strings, nor within the argument list of a function
definition

[^4]: some of the consoles will not allow you to enter more, and amongst
those which do some will silently discard the excess and some will use
it as the start of the next line.
 
## Recall and correction of previous commands {#sec-recall-and-correction-of-previous-commands}

Under many versions of UNIX and on Windows, R provides a mechanism for
recalling and re-executing previous commands. The vertical arrow keys on
the keyboard can be used to scroll forward and backward through a
*command history*. Once a command is located in this way, the cursor can
be moved within the command using the horizontal arrow keys, and
characters can be removed with the DEL key or added with the other keys.
More details are provided later: see @sec-appendix-c-the-command-line-editor
.

The recall and editing capabilities under UNIX are highly customizable.
You can find out how to do this by reading the manual entry for the
**readline** library.

Alternatively, the Emacs text editor provides more general support
mechanisms (via ESS, *Emacs Speaks Statistics*) for working
interactively with R. See [R and Emacs](https://cran.r-project.org/doc/FAQ/R-FAQ.html#R-and-Emacs)
 in The R statistical
system FAQ.
 
## Executing commands from or diverting output to a file {#sec-executing-commands-from-or-diverting-output-to-a-file}

If commands[^5] are stored in an external file, say
`commands.R` in the working directory `work`, they may
be executed at any time in an R session with the command

``` R
> source("commands.R")
```

For Windows **Source** is also available on the **File** menu. The
function `sink`,

``` R
> sink("record.lis")
```

will divert all subsequent output from the console to an external file,
`record.lis`. The command

``` R
> sink()
```

restores it to the console once again.

[^5]: of unlimited length.
 
## Data permanency and removing objects {#sec-data-permanency-and-removing-objects}

The entities that R creates and manipulates are known as *objects*.
These may be variables, arrays of numbers, character strings, functions,
or more general structures built from such components.

During an R session, objects are created and stored by name (we discuss
this process in the next section). The R command

``` R
> objects()
```

(alternatively, `ls()`) can be used to display the names of (most of)
the objects which are currently stored within R. The collection of
objects currently stored is called the *workspace*.

To remove objects the function `rm` is available:

``` R
> rm(x, y, z, ink, junk, temp, foo, bar)
```

All objects created during an R session can be stored permanently in a
file for use in future R sessions. At the end of each R session you are
given the opportunity to save all the currently available objects. If
you indicate that you want to do this, the objects are written to a file
called `.RData`[^6] in the current directory,
and the command lines used in the session are saved to a file called
`.Rhistory`.

When R is started at later time from the same directory it reloads the
workspace from this file. At the same time the associated commands
history is reloaded.

It is recommended that you should use separate working directories for
analyses conducted with R. It is quite common for objects with names `x`
and `y` to be created during an analysis. Names like this are often
meaningful in the context of a single analysis, but it can be quite hard
to decide what they might be when the several analyses have been
conducted in the same directory.

[^6]: The leading "dot" in this file name makes it *invisible* in normal
file listings in UNIX, and in default GUI file listings on macOS and
Windows.
 
# Simple manipulations; numbers and vectors {#sec-simple-manipulations-numbers-and-vectors}

 
## Vectors and assignment {#sec-vectors-and-assignment}

R operates on named *data structures*. The simplest such structure is
the numeric *vector*, which is a single entity consisting of an ordered
collection of numbers. To set up a vector named `x`, say, consisting of
five numbers, namely 10.4, 5.6, 3.1, 6.4 and 21.7, use the R command

``` r
> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)
```

This is an *assignment* statement using the *function* `c()` which in
this context can take an arbitrary number of vector *arguments* and
whose value is a vector got by concatenating its arguments end to
end.[^7]

A number occurring by itself in an expression is taken as a vector of
length one.

Notice that the assignment operator (`<-`), which consists of the two
characters `<`' ("less than") and '`-` ("minus") occurring strictly
side-by-side and it 'points' to the object receiving the value of the
expression. In most contexts the `=` operator can be used as an
alternative.

Assignment can also be made using the function `assign()`. An equivalent
way of making the same assignment as above is with:

``` R
> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))
```

The usual operator, `<-`, can be thought of as a syntactic short-cut to
this.

Assignments can also be made in the other direction, using the obvious
change in the assignment operator. So the same assignment could be made
using

``` R
> c(10.4, 5.6, 3.1, 6.4, 21.7) -> x
```

If an expression is used as a complete command, the value is printed
*and lost*[^8]. So now if we were to use the command

``` R
> 1/x
```

the reciprocals of the five values would be printed at the terminal (and
the value of `x`, of course, unchanged).

The further assignment

``` r
> y <- c(x, 0, x)
```

would create a vector `y` with 11 entries consisting of two copies of
`x` with a zero in the middle place.

[^7]: With other than vector types of argument, such as `list` mode
arguments, the action of `c()` is rather different. See [Concatenating
lists](Concatenating-lists.html).

[^8]: Actually, it is still available as `.Last.value` before any other
statements are executed.
 
## Vector arithmetic {#sec-vector-arithmetic}

Vectors can be used in arithmetic expressions, in which case the
operations are performed element by element. Vectors occurring in the
same expression need not all be of the same length. If they are not, the
value of the expression is a vector with the same length as the longest
vector which occurs in the expression. Shorter vectors in the expression
are *recycled* as often as need be (perhaps fractionally) until they
match the length of the longest vector. In particular a constant is
simply repeated. So with the above assignments the command

``` r
> v <- 2*x + y + 1
```

generates a new vector `v` of length 11 constructed by adding together,
element by element, `2*x` repeated 2.2 times, `y` repeated just once,
and `1` repeated 11 times.

The elementary arithmetic operators are the usual `+`, `-`, `*`, `/` and
`^` for raising to a power. In addition all of the common arithmetic
functions are available. `log`, `exp`, `sin`, `cos`, `tan`, `sqrt`, and
so on, all have their usual meaning. `max` and `min` select the largest
and smallest elements of a vector respectively. `range` is a function
whose value is a vector of length two, namely `c(min(x), max(x))`.
`length(x)` is the number of elements in `x`, `sum(x)` gives the total
of the elements in `x`, and `prod(x)` their product.

Two statistical functions are `mean(x)` which calculates the sample
mean, which is the same as `sum(x)/length(x)`, and `var(x)` which gives

``` R
sum((x-mean(x))^2)/(length(x)-1)
```

or sample variance. If the argument to `var()` is an *n*-by-*p* matrix
the value is a *p*-by-*p* sample covariance matrix got by regarding the
rows as independent *p*-variate sample vectors.

`sort(x)` returns a vector of the same size as `x` with the elements
arranged in increasing order; however there are other more flexible
sorting facilities available (see `order()` or `sort.list()` which
produce a permutation to do the sorting).

Note that `max` and `min` select the largest and smallest values in
their arguments, even if they are given several vectors. The *parallel*
maximum and minimum functions `pmax` and `pmin` return a vector (of
length equal to their longest argument) that contains in each element
the largest (smallest) element in that position in any of the input
vectors.

For most purposes the user will not be concerned if the "numbers" in a
numeric vector are integers, reals or even complex. Internally
calculations are done as double precision real numbers, or double
precision complex numbers if the input data are complex.

To work with complex numbers, supply an explicit complex part. Thus

``` R
sqrt(-17)
```

will give `NaN` and a warning, but

``` R
sqrt(-17+0i)
```

will do the computations as complex numbers.

 
## Generating regular sequences {#sec-generating-regular-sequences}

R has a number of facilities for generating commonly used sequences of
numbers. For example `1:30` is the vector `c(1, 2, …, 29, 30)`. The
colon operator has high priority within an expression, so, for example
`2*1:15` is the vector `c(2, 4, …, 28, 30)`. Put `n <- 10` and compare
the sequences `1:n-1` and `1:(n-1)`.

The construction `30:1` may be used to generate a sequence backwards.

The function `seq()` is a more general facility for generating
sequences. It has five arguments, only some of which may be specified in
any one call. The first two arguments, if given, specify the beginning
and end of the sequence, and if these are the only two arguments given
the result is the same as the colon operator. That is `seq(2,10)` is the
same vector as `2:10`.

Arguments to `seq()`, and to many other R functions, can also be given
in named form, in which case the order in which they appear is
irrelevant. The first two arguments may be named
`from=``value`{.variable} and `to=``value`{.variable}; thus `seq(1,30)`,
`seq(from=1, to=30)` and `seq(to=30, from=1)` are all the same as
`1:30`. The next two arguments to `seq()` may be named
`by=``value`{.variable} and `length=``value`{.variable}, which specify a
step size and a length for the sequence respectively. If neither of
these is given, the default `by=1` is assumed.

For example

``` R
> seq(-5, 5, by=.2) -> s3
```

generates in `s3` the vector `c(-5.0, -4.8, -4.6, …, 4.6, 4.8, 5.0)`.
Similarly

``` r
> s4 <- seq(length=51, from=-5, by=.2)
```

generates the same vector in `s4`.

The fifth argument may be named `along=``vector`{.variable}, which is
normally used as the only argument to create the sequence
`1, 2, …, length(``vector`{.variable}`)`, or the empty sequence if the
vector is empty (as it can be).

A related function is `rep()` which can be used for replicating an
object in various complicated ways. The simplest form is

``` r
> s5 <- rep(x, times=5)
```

which will put five copies of `x` end-to-end in `s5`. Another useful
version is

``` r
> s6 <- rep(x, each=5)
```

which repeats each element of `x` five times before moving on to the
next.
 
## Logical vectors {#sec-logical-vectors}

As well as numerical vectors, R allows manipulation of logical
quantities. The elements of a logical vector can have the values `TRUE`,
`FALSE`, and `NA` (for "not available", see below). The first two are
often abbreviated as `T` and `F`, respectively. Note however that `T`
and `F` are just variables which are set to `TRUE` and `FALSE` by
default, but are not reserved words and hence can be overwritten by the
user. Hence, you should always use `TRUE` and `FALSE`.

Logical vectors are generated by *conditions*. For example

``` r
> temp <- x > 13
```

sets `temp` as a vector of the same length as `x` with values `FALSE`
corresponding to elements of `x` where the condition is *not* met and
`TRUE` where it is.

The logical operators are `<`, `<=`, `>`, `>=`, `==` for exact equality
and `!=` for inequality. In addition if `c1` and `c2` are logical
expressions, then `c1 & c2` is their intersection (*"and"*), `c1 | c2`
is their union (*"or"*), and `!c1` is the negation of `c1`.

Logical vectors may be used in ordinary arithmetic, in which case they
are *coerced* into numeric vectors, `FALSE` becoming `0` and `TRUE`
becoming `1`. However there are situations where logical vectors and
their coerced numeric counterparts are not equivalent, for example see
the next subsection.
 
## Missing values {#sec-missing-values}

In some cases the components of a vector may not be completely known.
When an element or value is "not available" or a "missing value" in the
statistical sense, a place within a vector may be reserved for it by
assigning it the special value `NA`. In general any operation on an `NA`
becomes an `NA`. The motivation for this rule is simply that if the
specification of an operation is incomplete, the result cannot be known
and hence is not available.

The function `is.na(x)` gives a logical vector of the same size as `x`
with value `TRUE` if and only if the corresponding element in `x` is
`NA`.

``` r
> z <- c(1:3,NA);  ind <- is.na(z)
```

Notice that the logical expression `x == NA` is quite different from
`is.na(x)` since `NA` is not really a value but a marker for a quantity
that is not available. Thus `x == NA` is a vector of the same length as
`x` *all* of whose values are `NA` as the logical expression itself is
incomplete and hence undecidable.

Note that there is a second kind of "missing" values which are produced
by numerical computation, the so-called *Not a Number*, `NaN`, values.
Examples are

``` R
> 0/0
```

or

``` R
> Inf - Inf
```

which both give `NaN` since the result cannot be defined sensibly.

In summary, `is.na(xx)` is `TRUE` *both* for `NA` and `NaN` values. To
differentiate these, `is.nan(xx)` is only `TRUE` for `NaN`s.

Missing values are sometimes printed as `<NA>` when character vectors
are printed without quotes.
 
## Character vectors {#sec-character-vectors}

Character quantities and character vectors are used frequently in R, for
example as plot labels. Where needed they are denoted by a sequence of
characters delimited by the double quote character, e.g., `"x-values"`,
`"New iteration results"`.

Character strings are entered using either matching double (`"`) or
single (`'`) quotes, but are printed using double quotes (or sometimes
without quotes). They use C-style escape sequences, using `\` as the
escape character, so `\` is entered and printed as `\\`, and inside
double quotes `"` is entered as `\"`. Other useful escape sequences are
`\n`, newline, `\t`, tab and `\b`, backspace---see `?Quotes` for a full
list.

Character vectors may be concatenated into a vector by the `c()`
function; examples of their use will emerge frequently.

The `paste()` function takes an arbitrary number of arguments and
concatenates them one by one into character strings. Any numbers given
among the arguments are coerced into character strings in the evident
way, that is, in the same way they would be if they were printed. The
arguments are by default separated in the result by a single blank
character, but this can be changed by the named argument,
`sep=``string`{.variable}, which changes it to `string`{.variable},
possibly empty.

For example

``` r
> labs <- paste(c("X","Y"), 1:10, sep="")
```

makes `labs` into the character vector

``` R
c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10")
```

Note particularly that recycling of short lists takes place here too;
thus `c("X", "Y")` is repeated 5 times to match the sequence `1:10`.
[^9]

[^9]: `paste(..., collapse=``ss`{.variable}`)` joins the arguments into a
single character string putting `ss`{.variable} in between, e.g.,
`ss <- "|"`. There are more tools for character manipulation, see the
help for `sub` and `substring`.
 
## Index vectors; selecting and modifying subsets of a data set {#sec-index-vectors-selecting-and-modifying-subsets-of-a-data-set}

Subsets of the elements of a vector may be selected by appending to the
name of the vector an *index vector* in square brackets. More generally
any expression that evaluates to a vector may have subsets of its
elements similarly selected by appending an index vector in square
brackets immediately after the expression.

Such index vectors can be any of four distinct types.

1.  **A logical vector**. In this case the index vector is recycled to
    the same length as the vector from which elements are to be
    selected. Values corresponding to `TRUE` in the index vector are
    selected and those corresponding to `FALSE` are omitted. For example

    ``` r
    > y <- x[!is.na(x)]
    ```

    creates (or re-creates) an object `y` which will contain the
    non-missing values of `x`, in the same order. Note that if `x` has
    missing values, `y` will be shorter than `x`. Also

    ``` R
    > (x+1)[(!is.na(x)) & x>0] -> z
    ```

    creates an object `z` and places in it the values of the vector
    `x+1` for which the corresponding value in `x` was both non-missing
    and positive.

2.  **A vector of positive integral quantities**. In this case the
    values in the index vector must lie in the set {1, 2, ...,
    `length(x)`}. The corresponding elements of the vector are selected
    and concatenated, *in that order*, in the result. The index vector
    can be of any length and the result is of the same length as the
    index vector. For example `x[6]` is the sixth component of `x` and

    ``` R
    > x[1:10]
    ```

    selects the first 10 elements of `x` (assuming `length(x)` is not
    less than 10). Also

    ``` R
    > c("x","y")[rep(c(1,2,2,1), times=4)]
    ```

    (an admittedly unlikely thing to do) produces a character vector of
    length 16 consisting of `"x", "y", "y", "x"` repeated four times.

3.  **A vector of negative integral quantities**. Such an index vector
    specifies the values to be *excluded* rather than included. Thus

    ``` r
    > y <- x[-(1:5)]
    ```

    gives `y` all but the first five elements of `x`.

4.  **A vector of character strings**. This possibility only applies
    where an object has a `names` attribute to identify its components.
    In this case a sub-vector of the names vector may be used in the
    same way as the positive integral labels in item 2 further above.

    ``` r
    > fruit <- c(5, 10, 1, 20)
    > names(fruit) <- c("orange", "banana", "apple", "peach")
    > lunch <- fruit[c("apple","orange")]
    ```

    The advantage is that alphanumeric *names* are often easier to
    remember than *numeric indices*. This option is particularly useful
    in connection with data frames, as we shall see later.

An indexed expression can also appear on the receiving end of an
assignment, in which case the assignment operation is performed *only on
those elements of the vector*. The expression must be of the form
`vector[``index_vector`{.variable}`]` as having an arbitrary expression
in place of the vector name does not make much sense here.

For example

``` r
> x[is.na(x)] <- 0
```

replaces any missing values in `x` by zeros and

``` r
> y[y < 0] <- -y[y < 0]
```

has the same effect as

``` r
> y <- abs(y)
```
 
## Other types of objects {#sec-other-types-of-objects}

Vectors are the most important type of object in R, but there are
several others which we will meet more formally in later sections.

-   *matrices* or more generally *arrays* are multi-dimensional
    generalizations of vectors. In fact, they *are* vectors that can be
    indexed by two or more indices and will be printed in special ways.
    See @sec-arrays-and-matrices.
-   *factors* provide compact ways to handle categorical data. See @sec-ordered-and-unordered-factors.
    
-   *lists* are a general form of vector in which the various elements
    need not be of the same type, and are often themselves vectors or
    lists. Lists provide a convenient way to return the results of a
    statistical computation. See @sec-lists.
-   *data frames* are matrix-like structures, in which the columns can
    be of different types. Think of data frames as 'data matrices' with
    one row per observational unit but with (possibly) both numerical
    and categorical variables. Many experiments are best described by
    data frames: the treatments are categorical but the response is
    numeric. See @sec-data-frames.
-   *functions* are themselves objects in R which can be stored in the
    project's workspace. This provides a simple and convenient way to
    extend R. See @sec-writing-your-own-functions
    .
 
# Objects, their modes and attributes {#sec-objects-their-modes-and-attributes}

 
## Intrinsic attributes: mode and length {#sec-intrinsic-attributes:-mode-and-length}

The entities R operates on are technically known as *objects*. Examples
are vectors of numeric (real) or complex values, vectors of logical
values and vectors of character strings. These are known as "atomic"
structures since their components are all of the same type, or *mode*,
namely *numeric*[^10], *complex*, *logical*,
*character* and *raw*.

Vectors must have their values *all of the same mode*. Thus any given
vector must be unambiguously either *logical*, *numeric*, *complex*,
*character* or *raw*. (The only apparent exception to this rule is the
special "value" listed as `NA` for quantities not available, but in fact
there are several types of `NA`). Note that a vector can be empty and
still have a mode. For example the empty character string vector is
listed as `character(0)` and the empty numeric vector as `numeric(0)`.

R also operates on objects called *lists*, which are of mode *list*.
These are ordered sequences of objects which individually can be of any
mode. *lists* are known as "recursive" rather than atomic structures
since their components can themselves be lists in their own right.

The other recursive structures are those of mode *function* and
*expression*. Functions are the objects that form part of the R system
along with similar user written functions, which we discuss in some
detail later. Expressions as objects form an advanced part of R which
will not be discussed in this guide, except indirectly when we discuss
*formulae* used with modeling in R.

By the *mode* of an object we mean the basic type of its fundamental
constituents. This is a special case of a "property" of an object.
Another property of every object is its *length*. The functions
`mode(``object`{.variable}`)` and `length(``object`{.variable}`)` can be
used to find out the mode and length of any defined structure
[^11].

Further properties of an object are usually provided by
`attributes(``object`{.variable}`)`, see 
@sec-getting-and-setting-attributes. Because of this,
*mode* and *length* are also called "intrinsic attributes" of an object.

For example, if `z` is a complex vector of length 100, then in an
expression `mode(z)` is the character string `"complex"` and `length(z)`
is `100`.

R caters for changes of mode almost anywhere it could be considered
sensible to do so, (and a few where it might not be). For example with

``` r
> z <- 0:9
```

we could put

``` r
> digits <- as.character(z)
```

after which `digits` is the character vector `c("0", "1", "2", …, "9")`.
A further *coercion*, or change of mode, reconstructs the numerical
vector again:

``` c
> d <- as.integer(digits)
```

Now `d` and `z` are the same.[^12] There is a large
collection of functions of the form `as.``something`{.variable}`()` for
either coercion from one mode to another, or for investing an object
with some other attribute it may not already possess. The reader should
consult the different help files to become familiar with them.

[^10]: *numeric* mode is actually an amalgam of two distinct modes, namely
*integer* and *double* precision, as explained in the manual.

[^11]: Note however that `length(``object`{.variable}`)` does not always
contain intrinsic useful information, e.g., when `object`{.variable} is
a function.

[^12]: In general, coercion from numeric to character and back again will
not be exactly reversible, because of roundoff errors in the character
representation.
 
## Changing the length of an object {#sec-changing-the-length-of-an-object}

An "empty" object may still have a mode. For example

``` r
> e <- numeric()
```

makes `e` an empty vector structure of mode numeric. Similarly
`character()` is a empty character vector, and so on. Once an object of
any size has been created, new components may be added to it simply by
giving it an index value outside its previous range. Thus

``` r
> e[3] <- 17
```

now makes `e` a vector of length 3, (the first two components of which
are at this point both `NA`). This applies to any structure at all,
provided the mode of the additional component(s) agrees with the mode of
the object in the first place.

This automatic adjustment of lengths of an object is used often, for
example in the `scan()` function for input. (see @sec-the-scan-function.)


Conversely to truncate the size of an object requires only an assignment
to do so. Hence if `alpha` is an object of length 10, then

``` r
> alpha <- alpha[2 * 1:5]
```

makes it an object of length 5 consisting of just the former components
with even index. (The old indices are not retained, of course.) We can
then retain just the first three values by

``` r
> length(alpha) <- 3
```

and vectors can be extended (by missing values) in the same way.
 
## Getting and setting attributes {#sec-getting-and-setting-attributes}

The function `attributes(``object`{.variable}`)` returns a list of all
the non-intrinsic attributes currently defined for that object. The
function `attr(``object`{.variable}`, ``name`{.variable}`)` can be used
to select a specific attribute. These functions are rarely used, except
in rather special circumstances when some new attribute is being created
for some particular purpose, for example to associate a creation date or
an operator with an R object. The concept, however, is very important.

Some care should be exercised when assigning or deleting attributes
since they are an integral part of the object system used in R.

When it is used on the left hand side of an assignment it can be used
either to associate a new attribute with `object`{.variable} or to
change an existing one. For example

``` r
> attr(z, "dim") <- c(10,10)
```

allows R to treat `z` as if it were a 10-by-10 matrix.
 
## The class of an object {#sec-the-class-of-an-object}

All objects in R have a *class*, reported by the function `class`. For
simple vectors this is just the mode, for example `"numeric"`,
`"logical"`, `"character"` or `"list"`, but `"matrix"`, `"array"`,
`"factor"` and `"data.frame"` are other possible values.

A special attribute known as the *class* of the object is used to allow
for an object-oriented style[^13] of programming in
R. For example if an object has class `"data.frame"`, it will be printed
in a certain way, the `plot()` function will display it graphically in a
certain way, and other so-called generic functions such as `summary()`
will react to it as an argument in a way sensitive to its class.

To remove temporarily the effects of class, use the function
`unclass()`. For example if `winter` has the class `"data.frame"` then

``` c
> winter
```

will print it in data frame form, which is rather like a matrix, whereas

``` c
> unclass(winter)
```

will print it as an ordinary list. Only in rather special situations do
you need to use this facility, but one is when you are learning to come
to terms with the idea of class and generic functions.

Generic functions and classes will be discussed further in [Object
orientation](Object-orientation.html), but only briefly.

[^13]: A different style using 'formal' or 'S4' classes is provided in
package `methods`.
 
# Ordered and unordered factors {#sec-ordered-and-unordered-factors}

A *factor* is a vector object used to specify a discrete classification
(grouping) of the components of other vectors of the same length. R
provides both *ordered* and *unordered* factors. While the "real"
application of factors is with model formulae (see
[Contrasts](Contrasts.html)), we here look at a specific example.


## 4.1 A specific example {#sec-4.1-a-specific-example}

Suppose, for example, we have a sample of 30 tax accountants from all
the states and territories of Australia[^14] and
their individual state of origin is specified by a character vector of
state mnemonics as

``` r
> state <- c("tas", "sa",  "qld", "nsw", "nsw", "nt",  "wa",  "wa",
             "qld", "vic", "nsw", "vic", "qld", "qld", "sa",  "tas",
             "sa",  "nt",  "wa",  "vic", "qld", "nsw", "nsw", "wa",
             "sa",  "act", "nsw", "vic", "vic", "act")
```

Notice that in the case of a character vector, "sorted" means sorted in
alphabetical order.

A *factor* is similarly created using the `factor()` function:

``` r
> statef <- factor(state)
```

The `print()` function handles factors slightly differently from other
objects:

``` R
> statef
 [1] tas sa  qld nsw nsw nt  wa  wa  qld vic nsw vic qld qld sa
[16] tas sa  nt  wa  vic qld nsw nsw wa  sa  act nsw vic vic act
Levels:  act nsw nt qld sa tas vic wa
```

To find out the levels of a factor the function `levels()` can be used.

``` R
> levels(statef)
[1] "act" "nsw" "nt"  "qld" "sa"  "tas" "vic" "wa"
```

  ------------------------------------------------------------------------------------------------------------------ ---- --
  • [The function tapply() and ragged arrays](The-function-tapply_0028_0029-and-ragged-arrays.html){accesskey="1"}        
  • [Ordered factors](Ordered-factors.html){accesskey="2"}                                                                
  ------------------------------------------------------------------------------------------------------------------ ---- --

[^14]: Readers should note that there are eight states and territories in
Australia, namely the Australian Capital Territory, New South Wales, the
Northern Territory, Queensland, South Australia, Tasmania, Victoria and
Western Australia.
 
## The function `tapply()` and ragged arrays {#sec-the-function-tapply-and-ragged-arrays}

To continue the previous example, suppose we have the incomes of the
same tax accountants in another vector (in suitably large units of
money)

``` r
> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56,
               61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46,
               59, 46, 58, 43)
```

To calculate the sample mean income for each state we can now use the
special function `tapply()`:

``` r
> incmeans <- tapply(incomes, statef, mean)
```

giving a means vector with the components labelled by the levels

``` R
   act    nsw     nt    qld     sa    tas    vic     wa
44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250
```

The function `tapply()` is used to apply a function, here `mean()`, to
each group of components of the first argument, here `incomes`, defined
by the levels of the second component, here
`statef`[^15], as if they were separate vector
structures. The result is a structure of the same length as the levels
attribute of the factor containing the results. The reader should
consult the help document for more details.

Suppose further we needed to calculate the standard errors of the state
income means. To do this we need to write an R function to calculate the
standard error for any given vector. Since there is an builtin function
`var()` to calculate the sample variance, such a function is a very
simple one liner, specified by the assignment:

``` r
> stdError <- function(x) sqrt(var(x)/length(x))
```

(Writing functions will be considered later in @sec-writing-your-own-functions
. Note that R's a builtin
function `sd()` is something different.) After this assignment, the
standard errors are calculated by

``` r
> incster <- tapply(incomes, statef, stdError)
```

and the values calculated are then

``` R
> incster
act    nsw  nt    qld     sa tas   vic     wa
1.5 4.3102 4.5 4.1061 2.7386 0.5 5.244 2.6575
```

As an exercise you may care to find the usual 95% confidence limits for
the state mean incomes. To do this you could use `tapply()` once more
with the `length()` function to find the sample sizes, and the `qt()`
function to find the percentage points of the appropriate
*t*-distributions. (You could also investigate R's facilities for
*t*-tests.)

The function `tapply()` can also be used to handle more complicated
indexing of a vector by multiple categories. For example, we might wish
to split the tax accountants by both state and sex. However in this
simple instance (just one factor) what happens can be thought of as
follows. The values in the vector are collected into groups
corresponding to the distinct entries in the factor. The function is
then applied to each of these groups individually. The value is a vector
of function results, labelled by the `levels` attribute of the factor.

The combination of a vector and a labelling factor is an example of what
is sometimes called a *ragged array*, since the subclass sizes are
possibly irregular. When the subclass sizes are all the same the
indexing may be done implicitly and much more efficiently, as we see in
the next section.

[^15]: Note that `tapply()` also works in this case when its second
argument is not a factor, e.g., `tapply(incomes, state)`, and this is
true for quite a few other functions, since arguments are *coerced* to
factors when necessary (using `as.factor()`).
 
## Ordered factors {#sec-ordered-factors}

The levels of factors are stored in alphabetical order, or in the order
they were specified to `factor` if they were specified explicitly.

Sometimes the levels will have a natural ordering that we want to record
and want our statistical analysis to make use of. The `ordered()`
function creates such ordered factors but is otherwise identical to
`factor`. For most purposes the only difference between ordered and
unordered factors is that the former are printed showing the ordering of
the levels, but the contrasts generated for them in fitting linear
models are different.
 
# Arrays and matrices {#sec-arrays-and-matrices}

 
## Arrays {#sec-arrays}

An array can be considered as a multiply subscripted collection of data
entries, for example numeric. R allows simple facilities for creating
and handling arrays, and in particular the special case of matrices.

A dimension vector is a vector of non-negative integers. If its length
is *k* then the array is *k*-dimensional, e.g. a matrix is a
*2*-dimensional array. The dimensions are indexed from one up to the
values given in the dimension vector.

A vector can be used by R as an array only if it has a dimension vector
as its *dim* attribute. Suppose, for example, `z` is a vector of 1500
elements. The assignment

``` r
> dim(z) <- c(3,5,100)
```

gives it the *dim* attribute that allows it to be treated as a *3* by
*5* by *100* array.

Other functions such as `matrix()` and `array()` are available for
simpler and more natural looking assignments, as we shall see in [The
array() function](The-array_0028_0029-function.html).

The values in the data vector give the values in the array in the same
order as they would occur in FORTRAN, that is "column major order," with
the first subscript moving fastest and the last subscript slowest.

For example if the dimension vector for an array, say `a`, is `c(3,4,2)`
then there are 3 \* 4 \* 2 = 24 entries in `a` and the data vector holds
them in the order `a[1,1,1], a[2,1,1], …, a[2,4,2], a[3,4,2]`.

Arrays can be one-dimensional: such arrays are usually treated in the
same way as vectors (including when printing), but the exceptions can
cause confusion.
 
## Array indexing. Subsections of an array {#sec-array-indexing.-subsections-of-an-array}

Individual elements of an array may be referenced by giving the name of
the array followed by the subscripts in square brackets, separated by
commas.

More generally, subsections of an array may be specified by giving a
sequence of *index vectors* in place of subscripts; however *if any
index position is given an empty index vector, then the full range of
that subscript is taken*.

Continuing the previous example, `a[2,,]` is a 4 \* 2 array with
dimension vector `c(4,2)` and data vector containing the values

``` R
c(a[2,1,1], a[2,2,1], a[2,3,1], a[2,4,1],
  a[2,1,2], a[2,2,2], a[2,3,2], a[2,4,2])
```

in that order. `a[,,]` stands for the entire array, which is the same as
omitting the subscripts entirely and using `a` alone.

For any array, say `Z`, the dimension vector may be referenced
explicitly as `dim(Z)` (on either side of an assignment).

Also, if an array name is given with just *one subscript or index
vector*, then the corresponding values of the data vector only are used;
in this case the dimension vector is ignored. This is not the case,
however, if the single index is not a vector but itself an array, as we
next discuss.

 
## Index matrices {#sec-index-matrices}

As well as an index vector in any subscript position, a matrix may be
used with a single *index matrix* in order either to assign a vector of
quantities to an irregular collection of elements in the array, or to
extract an irregular collection as a vector.

A matrix example makes the process clear. In the case of a doubly
indexed array, an index matrix may be given consisting of two columns
and as many rows as desired. The entries in the index matrix are the row
and column indices for the doubly indexed array. Suppose for example we
have a *4* by *5* array `X` and we wish to do the following:

-   Extract elements `X[1,3]`, `X[2,2]` and `X[3,1]` as a vector
    structure, and
-   Replace these entries in the array `X` by zeroes.

In this case we need a *3* by *2* subscript array, as in the following
example.

``` r
> x <- array(1:20, dim=c(4,5))   # Generate a 4 by 5 array.
> x
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    5    9   13   17
[2,]    2    6   10   14   18
[3,]    3    7   11   15   19
[4,]    4    8   12   16   20
> i <- array(c(1:3,3:1), dim=c(3,2))
> i                             # i is a 3 by 2 index array.
     [,1] [,2]
[1,]    1    3
[2,]    2    2
[3,]    3    1
> x[i]                          # Extract those elements
[1] 9 6 3
> x[i] <- 0                     # Replace those elements by zeros.
> x
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    5    0   13   17
[2,]    2    0   10   14   18
[3,]    0    7   11   15   19
[4,]    4    8   12   16   20
>
```

Negative indices are not allowed in index matrices. `NA` and zero values
are allowed: rows in the index matrix containing a zero are ignored, and
rows containing an `NA` produce an `NA` in the result.

As a less trivial example, suppose we wish to generate an (unreduced)
design matrix for a block design defined by factors `blocks` (`b`
levels) and `varieties` (`v` levels). Further suppose there are `n`
plots in the experiment. We could proceed as follows:

``` r
> Xb <- matrix(0, n, b)
> Xv <- matrix(0, n, v)
> ib <- cbind(1:n, blocks)
> iv <- cbind(1:n, varieties)
> Xb[ib] <- 1
> Xv[iv] <- 1
> X <- cbind(Xb, Xv)
```

To construct the incidence matrix, `N` say, we could use

``` r
> N <- crossprod(Xb, Xv)
```

However a simpler direct way of producing this matrix is to use
`table()`:

``` r
> N <- table(blocks, varieties)
```

Index matrices must be numerical: any other form of matrix (e.g. a
logical or character matrix) supplied as a matrix is treated as an
indexing vector.
 
## The `array()` function {#sec-the-array-function}

As well as giving a vector structure a `dim` attribute, arrays can be
constructed from vectors by the `array` function, which has the form

``` r
> Z <- array(data_vector, dim_vector)
```

For example, if the vector `h` contains 24 or fewer, numbers then the
command

``` r
> Z <- array(h, dim=c(3,4,2))
```

would use `h` to set up *3* by *4* by *2* array in `Z`. If the size of
`h` is exactly 24 the result is the same as

``` r
> Z <- h ; dim(Z) <- c(3,4,2)
```

However if `h` is shorter than 24, its values are recycled from the
beginning again to make it up to size 24 (see [The recycling
rule](The-recycling-rule.html)) but `dim(h) <- c(3,4,2)` would signal an
error about mismatching length. As an extreme but common example

``` r
> Z <- array(0, c(3,4,2))
```

makes `Z` an array of all zeros.

At this point `dim(Z)` stands for the dimension vector `c(3,4,2)`, and
`Z[1:24]` stands for the data vector as it was in `h`, and `Z[]` with an
empty subscript or `Z` with no subscript stands for the entire array as
an array.

Arrays may be used in arithmetic expressions and the result is an array
formed by element-by-element operations on the data vector. The `dim`
attributes of operands generally need to be the same, and this becomes
the dimension vector of the result. So if `A`, `B` and `C` are all
similar arrays, then

``` r
> D <- 2*A*B + C + 1
```

makes `D` a similar array with its data vector being the result of the
given element-by-element operations. However the precise rule concerning
mixed array and vector calculations has to be considered a little more
carefully.

 
### Mixed vector and array arithmetic. The recycling rule {#sec-mixed-vector-and-array-arithmetic.-the-recycling-rule}

The precise rule affecting element by element mixed calculations with
vectors and arrays is somewhat quirky and hard to find in the
references. From experience we have found the following to be a reliable
guide.

-   The expression is scanned from left to right.
-   Any short vector operands are extended by recycling their values
    until they match the size of any other operands.
-   As long as short vectors and arrays *only* are encountered, the
    arrays must all have the same `dim` attribute or an error results.
-   Any vector operand longer than a matrix or array operand generates
    an error.
-   If array structures are present and no error or coercion to vector
    has been precipitated, the result is an array structure with the
    common `dim` attribute of its array operands.
 
## The outer product of two arrays {#sec-the-outer-product-of-two-arrays}

An important operation on arrays is the *outer product*. If `a` and `b`
are two numeric arrays, their outer product is an array whose dimension
vector is obtained by concatenating their two dimension vectors (order
is important), and whose data vector is got by forming all possible
products of elements of the data vector of `a` with those of `b`. The
outer product is formed by the special operator `%o%`:

``` r
> ab <- a %o% b
```

An alternative is

``` r
> ab <- outer(a, b, "*")
```

The multiplication function can be replaced by an arbitrary function of
two variables. For example if we wished to evaluate the function f(x; y)
= cos(y)/(1 + x\^2) over a regular grid of values with *x*- and
*y*-coordinates defined by the R vectors `x` and `y` respectively, we
could proceed as follows:

``` r
> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)
```

In particular the outer product of two ordinary vectors is a doubly
subscripted array (that is a matrix, of rank at most 1). Notice that the
outer product operator is of course non-commutative. Defining your own R
functions will be considered further in @sec-writing-your-own-functions
.

### An example: Determinants of 2 by 2 single-digit matrices {#sec-an-example:-determinants-of-2-by-2-single-digit-matrices}

As an artificial but cute example, consider the determinants of *2* by
*2* matrices *\[a, b; c, d\]* where each entry is a non-negative integer
in the range *0, 1, ..., 9*, that is a digit.

The problem is to find the determinants, *ad - bc*, of all possible
matrices of this form and represent the frequency with which each value
occurs as a *high density* plot. This amounts to finding the probability
distribution of the determinant if each digit is chosen independently
and uniformly at random.

A neat way of doing this uses the `outer()` function twice:

``` r
> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(fr, xlab="Determinant", ylab="Frequency")
```

Notice that `plot()` here uses a histogram like plot method, because it
"sees" that `fr` is of class `"table"`. The "obvious" way of doing this
problem with `for` loops, to be discussed in [Loops and conditional
execution](Loops-and-conditional-execution.html), is so inefficient as
to be impractical.

It is also perhaps surprising that about 1 in 20 such matrices is
singular.
 
## Generalized transpose of an array {#sec-generalized-transpose-of-an-array}

The function `aperm(a, perm)` may be used to permute an array, `a`. The
argument `perm` must be a permutation of the integers *{1, ..., k}*,
where *k* is the number of subscripts in `a`. The result of the function
is an array of the same size as `a` but with old dimension given by
`perm[j]` becoming the new `j`-th dimension. The easiest way to think of
this operation is as a generalization of transposition for matrices.
Indeed if `A` is a matrix, (that is, a doubly subscripted array) then
`B` given by

``` r
> B <- aperm(A, c(2,1))
```

is just the transpose of `A`. For this special case a simpler function
`t()` is available, so we could have used `B <- t(A)`.
 
## Matrix facilities {#sec-matrix-facilities}

As noted above, a matrix is just an array with two subscripts. However
it is such an important special case it needs a separate discussion. R
contains many operators and functions that are available only for
matrices. For example `t(X)` is the matrix transpose function, as noted
above. The functions `nrow(A)` and `ncol(A)` give the number of rows and
columns in the matrix `A` respectively.

 
### Matrix multiplication {#sec-matrix-multiplication}

The operator `%*%` is used for matrix multiplication. An *n* by *1* or
*1* by *n* matrix may of course be used as an *n*-vector if in the
context such is appropriate. Conversely, vectors which occur in matrix
multiplication expressions are automatically promoted either to row or
column vectors, whichever is multiplicatively coherent, if possible,
(although this is not always unambiguously possible, as we see later).

If, for example, `A` and `B` are square matrices of the same size, then

``` R
> A * B
```

is the matrix of element by element products and

``` R
> A %*% B
```

is the matrix product. If `x` is a vector, then

``` R
> x %*% A %*% x
```

is a quadratic form.[^16]

The function `crossprod()` forms "crossproducts", meaning that
`crossprod(X, y)` is the same as `t(X) %*% y` but the operation is more
efficient. If the second argument to `crossprod()` is omitted it is
taken to be the same as the first.

The meaning of `diag()` depends on its argument. `diag(v)`, where `v` is
a vector, gives a diagonal matrix with elements of the vector as the
diagonal entries. On the other hand `diag(M)`, where `M` is a matrix,
gives the vector of main diagonal entries of `M`. This is the same
convention as that used for `diag()` in [MATLAB]{.small}. Also, somewhat
confusingly, if `k` is a single numeric value then `diag(k)` is the `k`
by `k` identity matrix!

[^16]: Note that `x %*% x` is ambiguous, as it could mean either x'x or x
x', where x is the column form. In such cases the smaller matrix seems
implicitly to be the interpretation adopted, so the scalar x'x is in
this case the result. The matrix x x' may be calculated either by
`cbind(x) %*% x` or `x %*% rbind(x)` since the result of `rbind()` or
`cbind()` is always a matrix. However, the best way to compute x'x or x
x' is `crossprod(x)` or `x %o% x` respectively.
 
### Linear equations and inversion {#sec-linear-equations-and-inversion}

Solving linear equations is the inverse of matrix multiplication. When
after

``` r
> b <- A %*% x
```

only `A` and `b` are given, the vector `x` is the solution of that
linear equation system. In R,

``` R
> solve(A,b)
```

solves the system, returning `x` (up to some accuracy loss). Note that
in linear algebra, formally `x = A^{-1} %*% b` where `A^{-1}` denotes
the *inverse* of `A`, which can be computed by

``` R
solve(A)
```

but rarely is needed. Numerically, it is both inefficient and
potentially unstable to compute `x <- solve(A) %*% b` instead of
`solve(A,b)`.

The quadratic form  `x %*% A^{-1} %*% x`   which is used in multivariate
computations, should be computed by something
like[^17] `x %*% solve(A,x)`, rather than computing
the inverse of `A`.

[^17]: Even better would be to form a matrix square root B with A = BB' and
find the squared length of the solution of By = x , perhaps using the
Cholesky or eigen decomposition of A.
 
### Eigenvalues and eigenvectors {#sec-eigenvalues-and-eigenvectors}

The function `eigen(Sm)` calculates the eigenvalues and eigenvectors of
a symmetric matrix `Sm`. The result of this function is a list of two
components named `values` and `vectors`. The assignment

``` r
> ev <- eigen(Sm)
```

will assign this list to `ev`. Then `ev$val` is the vector of
eigenvalues of `Sm` and `ev$vec` is the matrix of corresponding
eigenvectors. Had we only needed the eigenvalues we could have used the
assignment:

``` r
> evals <- eigen(Sm)$values
```

`evals` now holds the vector of eigenvalues and the second component is
discarded. If the expression

``` R
> eigen(Sm)
```

is used by itself as a command the two components are printed, with
their names. For large matrices it is better to avoid computing the
eigenvectors if they are not needed by using the expression

``` r
> evals <- eigen(Sm, only.values = TRUE)$values
```
 
### Singular value decomposition and determinants {#sec-singular-value-decomposition-and-determinants}

The function `svd(M)` takes an arbitrary matrix argument, `M`, and
calculates the singular value decomposition of `M`. This consists of a
matrix of orthonormal columns `U` with the same column space as `M`, a
second matrix of orthonormal columns `V` whose column space is the row
space of `M` and a diagonal matrix of positive entries `D` such that
`M = U %*% D %*% t(V)`. `D` is actually returned as a vector of the
diagonal elements. The result of `svd(M)` is actually a list of three
components named `d`, `u` and `v`, with evident meanings.

If `M` is in fact square, then, it is not hard to see that

``` r
> absdetM <- prod(svd(M)$d)
```

calculates the absolute value of the determinant of `M`. If this
calculation were needed often with a variety of matrices it could be
defined as an R function

``` r
> absdet <- function(M) prod(svd(M)$d)
```

after which we could use `absdet()` as just another R function. As a
further trivial but potentially useful example, you might like to
consider writing a function, say `tr()`, to calculate the trace of a
square matrix. \[Hint: You will not need to use an explicit loop. Look
again at the `diag()` function.\]

R has a builtin function `det` to calculate a determinant, including the
sign, and another, `determinant`, to give the sign and modulus
(optionally on log scale),
 
### Least squares fitting and the QR decomposition {#sec-least-squares-fitting-and-the-qr-decomposition}

The function `lsfit()` returns a list giving results of a least squares
fitting procedure. An assignment such as

``` r
> ans <- lsfit(X, y)
```

gives the results of a least squares fit where `y` is the vector of
observations and `X` is the design matrix. See the help facility for
more details, and also for the follow-up function `ls.diag()` for, among
other things, regression diagnostics. Note that a grand mean term is
automatically included and need not be included explicitly as a column
of `X`. Further note that you almost always will prefer using `lm(.)`
(see [Linear models](Linear-models.html)) to `lsfit()` for regression
modelling.

Another closely related function is `qr()` and its allies. Consider the
following assignments

``` r
> Xplus <- qr(X)
> b <- qr.coef(Xplus, y)
> fit <- qr.fitted(Xplus, y)
> res <- qr.resid(Xplus, y)
```

These compute the orthogonal projection of `y` onto the range of `X` in
`fit`, the projection onto the orthogonal complement in `res` and the
coefficient vector for the projection in `b`, that is, `b` is
essentially the result of the [MATLAB]{.small} 'backslash' operator.

It is not assumed that `X` has full column rank. Redundancies will be
discovered and removed as they are found.

This alternative is the older, low-level way to perform least squares
calculations. Although still useful in some contexts, it would now
generally be replaced by the statistical models features, as will be
discussed in [Statistical models in R](Statistical-models-in-R.html).
 
## Forming partitioned matrices, `cbind()` and `rbind()` {#sec-forming-partitioned-matrices-cbind-and-rbind}

As we have already seen informally, matrices can be built up from other
vectors and matrices by the functions `cbind()` and `rbind()`. Roughly
`cbind()` forms matrices by binding together matrices horizontally, or
column-wise, and `rbind()` vertically, or row-wise.

In the assignment

``` r
> X <- cbind(arg_1, arg_2, arg_3, …)
```

the arguments to `cbind()` must be either vectors of any length, or
matrices with the same column size, that is the same number of rows. The
result is a matrix with the concatenated arguments `arg_1`{.variable},
`arg_2`{.variable}, ... forming the columns.

If some of the arguments to `cbind()` are vectors they may be shorter
than the column size of any matrices present, in which case they are
cyclically extended to match the matrix column size (or the length of
the longest vector if no matrices are given).

The function `rbind()` does the corresponding operation for rows. In
this case any vector argument, possibly cyclically extended, are of
course taken as row vectors.

Suppose `X1` and `X2` have the same number of rows. To combine these by
columns into a matrix `X`, together with an initial column of `1`s we
can use

``` r
> X <- cbind(1, X1, X2)
```

The result of `rbind()` or `cbind()` always has matrix status. Hence
`cbind(x)` and `rbind(x)` are possibly the simplest ways explicitly to
allow the vector `x` to be treated as a column or row matrix
respectively.
 
## The concatenation function, `c()`, with arrays {#sec-the-concatenation-function-c-with-arrays}

It should be noted that whereas `cbind()` and `rbind()` are
concatenation functions that respect `dim` attributes, the basic `c()`
function does not, but rather clears numeric objects of all `dim` and
`dimnames` attributes. This is occasionally useful in its own right.

The official way to coerce an array back to a simple vector object is to
use `as.vector()`

``` r
> vec <- as.vector(X)
```

However a similar result can be achieved by using `c()` with just one
argument, simply for this side-effect:

``` r
> vec <- c(X)
```

There are slight differences between the two, but ultimately the choice
between them is largely a matter of style (with the former being
preferable).
 
## Frequency tables from factors {#sec-frequency-tables-from-factors}

Recall that a factor defines a partition into groups. Similarly a pair
of factors defines a two way cross classification, and so on. The
function `table()` allows frequency tables to be calculated from equal
length factors. If there are *k* factor arguments, the result is a
*k*-way array of frequencies.

Suppose, for example, that `statef` is a factor giving the state code
for each entry in a data vector. The assignment

``` r
> statefr <- table(statef)
```

gives in `statefr` a table of frequencies of each state in the sample.
The frequencies are ordered and labelled by the `levels` attribute of
the factor. This simple case is equivalent to, but more convenient than,

``` r
> statefr <- tapply(statef, statef, length)
```

Further suppose that `incomef` is a factor giving a suitably defined
"income class" for each entry in the data vector, for example with the
`cut()` function:

``` R
> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef
```

Then to calculate a two-way table of frequencies:

``` R
> table(incomef,statef)
         statef
incomef   act nsw nt qld sa tas vic wa
  (35,45]   1   1  0   1  0   0   1  0
  (45,55]   1   1  1   1  2   0   1  3
  (55,65]   0   3  1   3  2   2   2  1
  (65,75]   0   1  0   0  0   0   1  0
```

Extension to higher-way frequency tables is immediate.
 
# Lists and data frames {#sec-lists-and-data-frames}

 
## Lists {#sec-lists}

An R *list* is an object consisting of an ordered collection of objects
known as its *components*.

There is no particular need for the components to be of the same mode or
type, and, for example, a list could consist of a numeric vector, a
logical value, a matrix, a complex vector, a character array, a
function, and so on. Here is a simple example of how to make a list:

``` r
> Lst <- list(name="Fred", wife="Mary", no.children=3,
              child.ages=c(4,7,9))
```

Components are always *numbered* and may always be referred to as such.
Thus if `Lst` is the name of a list with four components, these may be
individually referred to as `Lst[[1]]`, `Lst[[2]]`, `Lst[[3]]` and
`Lst[[4]]`. If, further, `Lst[[4]]` is a vector subscripted array then
`Lst[[4]][1]` is its first entry.

If `Lst` is a list, then the function `length(Lst)` gives the number of
(top level) components it has.

Components of lists may also be *named*, and in this case the component
may be referred to either by giving the component name as a character
string in place of the number in double square brackets, or, more
conveniently, by giving an expression of the form

``` R
> name$component_name
```

for the same thing.

This is a very useful convention as it makes it easier to get the right
component if you forget the number.

So in the simple example given above:

`Lst$name` is the same as `Lst[[1]]` and is the string `"Fred"`,

`Lst$wife` is the same as `Lst[[2]]` and is the string `"Mary"`,

`Lst$child.ages[1]` is the same as `Lst[[4]][1]` and is the number `4`.

Additionally, one can also use the names of the list components in
double square brackets, i.e., `Lst[["name"]]` is the same as `Lst$name`.
This is especially useful, when the name of the component to be
extracted is stored in another variable as in

``` r
> x <- "name"; Lst[[x]]
```

It is very important to distinguish `Lst[[1]]` from `Lst[1]`.
`[[``…`{.variable}`]]` is the operator used to select a single
element, whereas `[``…`{.variable}`]` is a general subscripting
operator. Thus the former is the *first object in the list* `Lst`, and
if it is a named list the name is *not* included. The latter is a
*sublist of the list `Lst` consisting of the first entry only. If it is
a named list, the names are transferred to the sublist.*

The names of components may be abbreviated down to the minimum number of
letters needed to identify them uniquely. Thus `Lst$coefficients` may be
minimally specified as `Lst$coe` and `Lst$covariance` as `Lst$cov`.

The vector of names is in fact simply an attribute of the list like any
other and may be handled as such. Other structures besides lists may, of
course, similarly be given a *names* attribute also.
 
## Constructing and modifying lists {#sec-constructing-and-modifying-lists}

New lists may be formed from existing objects by the function `list()`.
An assignment of the form

``` r
> Lst <- list(name_1=object_1, …, name_m=object_m)
```

sets up a list `Lst` of *m* components using `object_1`{.variable}, ...,
`object_m`{.variable} for the components and giving them names as
specified by the argument names, (which can be freely chosen). If these
names are omitted, the components are numbered only. The components used
to form the list are *copied* when forming the new list and the
originals are not affected.

Lists, like any subscripted object, can be extended by specifying
additional components. For example

``` r
> Lst[5] <- list(matrix=Mat)
```

 
### Concatenating lists {#sec-concatenating-lists}

When the concatenation function `c()` is given list arguments, the
result is an object of mode list also, whose components are those of the
argument lists joined together in sequence.

``` r
> list.ABC <- c(list.A, list.B, list.C)
```

Recall that with vector objects as arguments the concatenation function
similarly joined together all arguments into a single vector structure.
In this case all other attributes, such as `dim` attributes, are
discarded.
 
## Data frames {#sec-data-frames}

A *data frame* is a list with class `"data.frame"`. There are
restrictions on lists that may be made into data frames, namely

-   The components must be vectors (numeric, character, or logical),
    factors, numeric matrices, lists, or other data frames.
-   Matrices, lists, and data frames provide as many variables to the
    new data frame as they have columns, elements, or variables,
    respectively.
-   Vector structures appearing as variables of the data frame must all
    have the *same length*, and matrix structures must all have the
    *same number of rows*.

A data frame may for many purposes be regarded as a matrix with columns
possibly of differing modes and attributes. It may be displayed in
matrix form, and its rows and columns extracted using matrix indexing
conventions.

 
### Making data frames {#sec-making-data-frames}

Objects satisfying the restrictions placed on the columns (components)
of a data frame may be used to form one using the function `data.frame`:

``` r
> accountants <- data.frame(home=statef, loot=incomes, shot=incomef)
```

A list whose components conform to the restrictions of a data frame may
be *coerced* into a data frame using the function `as.data.frame()`

The simplest way to construct a data frame from scratch is to use the
`read.table()` function to read an entire data frame from an external
file. This is discussed further in [Reading data from
files](Reading-data-from-files.html).
 
### `attach()` and `detach()` {#sec-attach-and-detach}

The `$` notation, such as `accountants$home`, for list components is not
always very convenient. A useful facility would be somehow to make the
components of a list or data frame temporarily visible as variables
under their component name, without the need to quote the list name
explicitly each time.

The `attach()` function takes a 'database' such as a list or data frame
as its argument. Thus suppose `lentils` is a data frame with three
variables `lentils$u`, `lentils$v`, `lentils$w`. The attach

``` R
> attach(lentils)
```

places the data frame in the search path at position 2, and provided
there are no variables `u`, `v` or `w` in position 1, `u`, `v` and `w`
are available as variables from the data frame in their own right. At
this point an assignment such as

``` r
> u <- v+w
```

does not replace the component `u` of the data frame, but rather masks
it with another variable `u` in the working directory at position 1 on
the search path. To make a permanent change to the data frame itself,
the simplest way is to resort once again to the `$` notation:

``` r
> lentils$u <- v+w
```

However the new value of component `u` is not visible until the data
frame is detached and attached again.

To detach a data frame, use the function

``` R
> detach()
```

More precisely, this statement detaches from the search path the entity
currently at position 2. Thus in the present context the variables `u`,
`v` and `w` would be no longer visible, except under the list notation
as `lentils$u` and so on. Entities at positions greater than 2 on the
search path can be detached by giving their number to `detach`, but it
is much safer to always use a name, for example by `detach(lentils)` or
`detach("lentils")`

> **Note:** In R lists and data frames can only be attached at position
> 2 or above, and what is attached is a *copy* of the original object.
> You can alter the attached values *via* `assign`, but the original
> list or data frame is unchanged.
 
### Working with data frames {#sec-working-with-data-frames}

A useful convention that allows you to work with many different problems
comfortably together in the same working directory is

-   gather together all variables for any well defined and separate
    problem in a data frame under a suitably informative name;
-   when working with a problem attach the appropriate data frame at
    position 2, and use the working directory at level 1 for operational
    quantities and temporary variables;
-   before leaving a problem, add any variables you wish to keep for
    future reference to the data frame using the `$` form of assignment,
    and then `detach()`;
-   finally remove all unwanted variables from the working directory and
    keep it as clean of left-over temporary variables as possible.

In this way it is quite simple to work with many problems in the same
directory, all of which have variables named `x`, `y` and `z`, for
example.
 
### Attaching arbitrary lists {#sec-attaching-arbitrary-lists}

`attach()` is a generic function that allows not only directories and
data frames to be attached to the search path, but other classes of
object as well. In particular any object of mode `"list"` may be
attached in the same way:

``` R
> attach(any.old.list)
```

Anything that has been attached can be detached by `detach`, by position
number or, preferably, by name.
 
### Managing the search path {#sec-managing-the-search-path}

The function `search` shows the current search path and so is a very
useful way to keep track of which data frames and lists (and packages)
have been attached and detached. Initially it gives

``` R
> search()
[1] ".GlobalEnv"   "Autoloads"    "package:base"
```

where `.GlobalEnv` is the workspace.[^18]

After `lentils` is attached we have

``` R
> search()
[1] ".GlobalEnv"   "lentils"      "Autoloads"    "package:base"
> ls(2)
[1] "u" "v" "w"
```

and as we see `ls` (or `objects`) can be used to examine the contents of
any position on the search path.

Finally, we detach the data frame and confirm it has been removed from
the search path.

``` R
> detach("lentils")
> search()
[1] ".GlobalEnv"   "Autoloads"    "package:base"
```

[^18]: See the on-line help for `autoload` for the meaning of the second
term.
 
# Reading data from files {#sec-reading-data-from-files}

Large data objects will usually be read as values from external files
rather than entered during an R session at the keyboard. R input
facilities are simple and their requirements are fairly strict and even
rather inflexible. There is a clear presumption by the designers of R
that you will be able to modify your input files using other tools, such
as file editors or Perl[^19] to fit in with the
requirements of R. Generally this is very simple.

If variables are to be held mainly in data frames, as we strongly
suggest they should be, an entire data frame can be read directly with
the `read.table()` function. There is also a more primitive input
function, `scan()`, that can be called directly.

For more details on importing data into R and also exporting data, see
the *R Data Import/Export* manual.


[^19]: Under UNIX, the utilities `sed` or`awk` can be used.
 
## The `read.table()` function {#sec-the-read.table-function}

To read an entire data frame directly, the external file will normally
have a special form.

-   The first line of the file should have a *name* for each variable in
    the data frame.
-   Each additional line of the file has as its first item a *row label*
    and the values for each variable.

If the file has one fewer item in its first line than in its second,
this arrangement is presumed to be in force. So the first few lines of a
file to be read as a data frame might look as follows.

::: blockquotes-block
> +-----------------------------------------------------------------------+
> | ``` R                                                                 |
> | Input file form with names and row labels:                            |
> |                                                                       |
> |      Price    Floor     Area   Rooms     Age  Cent.heat               |
> | 01   52.00    111.0      830     5       6.2      no                  |
> | 02   54.75    128.0      710     5       7.5      no                  |
> | 03   57.50    101.0     1000     5       4.2      no                  |
> | 04   57.50    131.0      690     6       8.8      no                  |
> | 05   59.75     93.0      900     5       1.9     yes                  |
> | ...                                                                   |
> | ```                                                                   |
> +-----------------------------------------------------------------------+
:::

By default numeric items (except row labels) are read as numeric
variables and non-numeric variables, such as `Cent.heat` in the example,
as character variables. This can be changed if necessary.

The function `read.table()` can then be used to read the data frame
directly

``` r
> HousePrice <- read.table("houses.data")
```

Often you will want to omit including the row labels directly and use
the default labels. In this case the file may omit the row label column
as in the following.

::: blockquotes-block
> +-----------------------------------------------------------------------+
> | ``` R                                                                 |
> | Input file form without row labels:                                   |
> |                                                                       |
> | Price    Floor     Area   Rooms     Age  Cent.heat                    |
> | 52.00    111.0      830     5       6.2      no                       |
> | 54.75    128.0      710     5       7.5      no                       |
> | 57.50    101.0     1000     5       4.2      no                       |
> | 57.50    131.0      690     6       8.8      no                       |
> | 59.75     93.0      900     5       1.9     yes                       |
> | ...                                                                   |
> | ```                                                                   |
> +-----------------------------------------------------------------------+
:::

The data frame may then be read as

``` r
> HousePrice <- read.table("houses.data", header=TRUE)
```

where the `header=TRUE` option specifies that the first line is a line
of headings, and hence, by implication from the form of the file, that
no explicit row labels are given.

 
## The `scan()` function {#sec-the-scan-function}

Suppose the data vectors are of equal length and are to be read in
parallel. Further suppose that there are three vectors, the first of
mode character and the remaining two of mode numeric, and the file is
`input.dat`. The first step is to use `scan()` to read in the
three vectors as a list, as follows

``` r
> inp <- scan("input.dat", list("",0,0))
```

The second argument is a dummy list structure that establishes the mode
of the three vectors to be read. The result, held in `inp`, is a list
whose components are the three vectors read in. To separate the data
items into three separate vectors, use assignments like

``` r
> label <- inp[[1]]; x <- inp[[2]]; y <- inp[[3]]
```

More conveniently, the dummy list can have named components, in which
case the names can be used to access the vectors read in. For example

``` r
> inp <- scan("input.dat", list(id="", x=0, y=0))
```

If you wish to access the variables separately they may either be
re-assigned to variables in the working frame:

``` r
> label <- inp$id; x <- inp$x; y <- inp$y
```

or the list may be attached at position 2 of the search path (see
[Attaching arbitrary lists](Attaching-arbitrary-lists.html)).

If the second argument is a single value and not a list, a single vector
is read in, all components of which must be of the same mode as the
dummy value.

``` r
> X <- matrix(scan("light.dat", 0), ncol=5, byrow=TRUE)
```

There are more elaborate input facilities available and these are
detailed in the manuals.
 
## Accessing builtin datasets {#sec-accessing-builtin-datasets}

Around 100 datasets are supplied with R (in package **datasets**), and
others are available in packages (including the recommended packages
supplied with R). To see the list of datasets currently available use

``` R
data()
```

All the datasets supplied with R are available directly by name.
However, many packages still use the obsolete convention in which `data`
was also used to load datasets into R, for example

``` R
data(infert)
```

and this can still be used with the standard packages (as in this
example). In most cases this will load an R object of the same name.
However, in a few cases it loads several objects, so see the on-line
help for the object to see what to expect.

### 7.3.1 Loading data from other R packages {#sec-7.3.1-loading-data-from-other-r-packages}

To access data from a particular package, use the `package` argument,
for example

``` R
data(package="rpart")
data(Puromycin, package="datasets")
```

If a package has been attached by `library`, its datasets are
automatically included in the search.

User-contributed packages can be a rich source of datasets.
 
## Editing data {#sec-editing-data}

When invoked on a data frame or matrix, `edit` brings up a separate
spreadsheet-like environment for editing. This is useful for making
small changes once a data set has been read. The command

``` r
> xnew <- edit(xold)
```

will allow you to edit your data set `xold`, and on completion the
changed object is assigned to `xnew`. If you want to alter the original
dataset `xold`, the simplest way is to use `fix(xold)`, which is
equivalent to `xold <- edit(xold)`.

Use

``` r
> xnew <- edit(data.frame())
```

to enter new data via the spreadsheet interface.
 
# Probability distributions {#sec-probability-distributions}

 
## R as a set of statistical tables {#sec-r-as-a-set-of-statistical-tables}

One convenient use of R is to provide a comprehensive set of statistical
tables. Functions are provided to evaluate the cumulative distribution
function P(X \<= x), the probability density function and the quantile
function (given *q*, the smallest *x* such that P(X \<= x) \> q), and to
simulate from the distribution.

::: blockquotes-block
>   Distribution        R name       additional arguments
>   ------------------- ------------ -----------------------
>   beta                `beta`       `shape1, shape2, ncp`
>   binomial            `binom`      `size, prob`
>   Cauchy              `cauchy`     `location, scale`
>   chi-squared         `chisq`      `df, ncp`
>   exponential         `exp`        `rate`
>   F                   `f`          `df1, df2, ncp`
>   gamma               `gamma`      `shape, scale`
>   geometric           `geom`       `prob`
>   hypergeometric      `hyper`      `m, n, k`
>   log-normal          `lnorm`      `meanlog, sdlog`
>   logistic            `logis`      `location, scale`
>   negative binomial   `nbinom`     `size, prob`
>   normal              `norm`       `mean, sd`
>   Poisson             `pois`       `lambda`
>   signed rank         `signrank`   `n`
>   Student's t         `t`          `df, ncp`
>   uniform             `unif`       `min, max`
>   Weibull             `weibull`    `shape, scale`
>   Wilcoxon            `wilcox`     `m, n`
:::

Prefix the name given here by '`d`' for the density,
'`p`' for the CDF, '`q`' for the quantile function and
'`r`' for simulation (*r*andom deviates). The first argument is
`x` for `d``xxx`{.variable}, `q` for `p``xxx`{.variable}, `p` for
`q``xxx`{.variable} and `n` for `r``xxx`{.variable} (except for
`rhyper`, `rsignrank` and `rwilcox`, for which it is `nn`). In not quite
all cases is the non-centrality parameter `ncp` currently available: see
the on-line help for details.

The `p``xxx`{.variable} and `q``xxx`{.variable} functions all have
logical arguments `lower.tail` and `log.p` and the `d``xxx`{.variable}
ones have `log`. This allows, e.g., getting the cumulative (or
"integrated") *hazard* function, H(t) = - log(1 - F(t)), by

``` R
 - pxxx(t, ..., lower.tail = FALSE, log.p = TRUE)
```

or more accurate log-likelihoods (by
`d``xxx`{.variable}`(..., log = TRUE)`), directly.

In addition there are functions `ptukey` and `qtukey` for the
distribution of the studentized range of samples from a normal
distribution, and `dmultinom` and `rmultinom` for the multinomial
distribution. Further distributions are available in contributed
packages, notably
[**SuppDists**](https://CRAN.R-project.org/package=SuppDists).

Here are some examples

``` c
> ## 2-tailed p-value for t distribution
> 2*pt(-2.43, df = 13)
> ## upper 1% point for an F(2, 7) distribution
> qf(0.01, 2, 7, lower.tail = FALSE)
```

See the on-line help on `RNG` for how random-number generation is done
in R.
 
## Examining the distribution of a set of data {#sec-examining-the-distribution-of-a-set-of-data}

Given a (univariate) set of data we can examine its distribution in a
large number of ways. The simplest is to examine the numbers. Two
slightly different summaries are given by `summary` and `fivenum` and a
display of the numbers by `stem` (a "stem and leaf" plot).

``` c
> attach(faithful)
> summary(eruptions)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
  1.600   2.163   4.000   3.488   4.454   5.100
> fivenum(eruptions)
[1] 1.6000 2.1585 4.0000 4.4585 5.1000
> stem(eruptions)

  The decimal point is 1 digit(s) to the left of the |

  16 | 070355555588
  18 | 000022233333335577777777888822335777888
  20 | 00002223378800035778
  22 | 0002335578023578
  24 | 00228
  26 | 23
  28 | 080
  30 | 7
  32 | 2337
  34 | 250077
  36 | 0000823577
  38 | 2333335582225577
  40 | 0000003357788888002233555577778
  42 | 03335555778800233333555577778
  44 | 02222335557780000000023333357778888
  46 | 0000233357700000023578
  48 | 00000022335800333
  50 | 0370
```

A stem-and-leaf plot is like a histogram, and R has a function `hist` to
plot histograms.

``` c
> hist(eruptions)
## make the bins smaller, make a plot of density
> hist(eruptions, seq(1.6, 5.2, 0.2), prob=TRUE)
> lines(density(eruptions, bw=0.1))
> rug(eruptions) # show the actual data points
```

More elegant density plots can be made by `density`, and we added a line
produced by `density` in this example. The bandwidth `bw` was chosen by
trial-and-error as the default gives too much smoothing (it usually does
for "interesting" densities). (Better automated methods of bandwidth
choice are available, and in this example `bw = "SJ"` gives a good
result.)

![images/hist](images/hist.png)

We can plot the empirical cumulative distribution function by using the
function `ecdf`.

``` c
> plot(ecdf(eruptions), do.points=FALSE, verticals=TRUE)
```

This distribution is obviously far from any standard distribution. How
about the right-hand mode, say eruptions of longer than 3 minutes? Let
us fit a normal distribution and overlay the fitted CDF.

``` c
> long <- eruptions[eruptions > 3]
> plot(ecdf(long), do.points=FALSE, verticals=TRUE)
> x <- seq(3, 5.4, 0.01)
> lines(x, pnorm(x, mean=mean(long), sd=sqrt(var(long))), lty=3)
```

![images/ecdf](images/ecdf.png)

Quantile-quantile (Q-Q) plots can help us examine this more carefully.

``` R
par(pty="s")       # arrange for a square figure region
qqnorm(long); qqline(long)
```

which shows a reasonable fit but a shorter right tail than one would
expect from a normal distribution. Let us compare this with some
simulated data from a *t* distribution

![images/QQ](images/QQ.png)

``` r
x <- rt(250, df = 5)
qqnorm(x); qqline(x)
```

which will usually (if it is a random sample) show longer tails than
expected for a normal. We can make a Q-Q plot against the generating
distribution by

``` c
qqplot(qt(ppoints(250), df = 5), x, xlab = "Q-Q plot for t dsn")
qqline(x)
```

Finally, we might want a more formal test of agreement with normality
(or not). R provides the Shapiro-Wilk test

``` R
> shapiro.test(long)

         Shapiro-Wilk normality test

data:  long
W = 0.9793, p-value = 0.01052
```

and the Kolmogorov-Smirnov test

``` R
> ks.test(long, "pnorm", mean = mean(long), sd = sqrt(var(long)))

         One-sample Kolmogorov-Smirnov test

data:  long
D = 0.0661, p-value = 0.4284
alternative hypothesis: two.sided
```

(Note that the distribution theory is not valid here as we have
estimated the parameters of the normal distribution from the same
sample.)
 
## One- and two-sample tests {#sec-one--and-two-sample-tests}

So far we have compared a single sample to a normal distribution. A much
more common operation is to compare aspects of two samples. Note that in
R, all "classical" tests including the ones used below are in package
**stats** which is normally loaded.

Consider the following sets of data on the latent heat of the fusion of
ice (*cal/gm*) from Rice (1995, p.490)

``` R
Method A: 79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97
          80.05 80.03 80.02 80.00 80.02
Method B: 80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97
```

Boxplots provide a simple graphical comparison of the two samples.

``` r
A <- scan()
79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97
80.05 80.03 80.02 80.00 80.02

B <- scan()
80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97

boxplot(A, B)
```

which indicates that the first group tends to give higher results than
the second.

![images/ice](images/ice.png)

To test for the equality of the means of the two examples, we can use an
*unpaired* *t*-test by

``` c
> t.test(A, B)

         Welch Two Sample t-test

data:  A and B
t = 3.2499, df = 12.027, p-value = 0.00694
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.01385526 0.07018320
sample estimates:
mean of x mean of y
 80.02077  79.97875
```

which does indicate a significant difference, assuming normality. By
default the R function does not assume equality of variances in the two
samples. We can use the F test to test for equality in the variances,
provided that the two samples are from normal populations.

``` c
> var.test(A, B)

         F test to compare two variances

data:  A and B
F = 0.5837, num df = 12, denom df =  7, p-value = 0.3938
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
 0.1251097 2.1052687
sample estimates:
ratio of variances
         0.5837405
```

which shows no evidence of a significant difference, and so we can use
the classical *t*-test that assumes equality of the variances.

``` c
> t.test(A, B, var.equal=TRUE)

         Two Sample t-test

data:  A and B
t = 3.4722, df = 19, p-value = 0.002551
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.01669058 0.06734788
sample estimates:
mean of x mean of y
 80.02077  79.97875
```

All these tests assume normality of the two samples. The two-sample
Wilcoxon (or Mann-Whitney) test only assumes a common continuous
distribution under the null hypothesis.

``` R
> wilcox.test(A, B)

         Wilcoxon rank sum test with continuity correction

data:  A and B
W = 89, p-value = 0.007497
alternative hypothesis: true location shift is not equal to 0

Warning message:
Cannot compute exact p-value with ties in: wilcox.test(A, B)
```

Note the warning: there are several ties in each sample, which suggests
strongly that these data are from a discrete distribution (probably due
to rounding).

There are several ways to compare graphically the two samples. We have
already seen a pair of boxplots. The following

``` c
> plot(ecdf(A), do.points=FALSE, verticals=TRUE, xlim=range(A, B))
> plot(ecdf(B), do.points=FALSE, verticals=TRUE, add=TRUE)
```

will show the two empirical CDFs, and `qqplot` will perform a Q-Q plot
of the two samples. The Kolmogorov-Smirnov test is of the maximal
vertical distance between the two ecdf's, assuming a common continuous
distribution:

``` R
> ks.test(A, B)

         Two-sample Kolmogorov-Smirnov test

data:  A and B
D = 0.5962, p-value = 0.05919
alternative hypothesis: two-sided

Warning message:
cannot compute correct p-values with ties in: ks.test(A, B)
```
 
# Grouping, loops and conditional execution {#sec-grouping-loops-and-conditional-execution}

 
## Grouped expressions {#sec-grouped-expressions}

R is an expression language in the sense that its only command type is a
function or expression which returns a result. Even an assignment is an
expression whose result is the value assigned, and it may be used
wherever any expression may be used; in particular multiple assignments
are possible.

Commands may be grouped together in braces,
`{``expr_1`{.variable}`; ``…`{.variable}`; ``expr_m`{.variable}`}`, in
which case the value of the group is the result of the last expression
in the group evaluated. Since such a group is also an expression it may,
for example, be itself included in parentheses and used as part of an
even larger expression, and so on.
 
## Control statements {#sec-control-statements}

 
### Conditional execution: `if` statements {#sec-conditional-execution:-if-statements}

The language has available a conditional construction of the form

``` R
> if (expr_1) expr_2 else expr_3
```

where `expr_1`{.variable} must evaluate to a single logical value and
the result of the entire expression is then evident.

The "short-circuit" operators `&&` and `||` are often used as part of
the condition in an `if` statement. Whereas `&` and `|` apply
element-wise to vectors, `&&` and `||` apply to vectors of length one,
and only evaluate their second argument if necessary.

There is a vectorized version of the `if`/`else` construct, the `ifelse`
function. This has the form `ifelse(condition, a, b)` and returns a
vector of the same length as `condition`, with elements `a[i]` if
`condition[i]` is true, otherwise `b[i]` (where `a` and `b` are recycled
as necessary).
 
### Repetitive execution: `for` loops, `repeat` and `while` {#sec-repetitive-execution:-for-loops-repeat-and-while}

There is also a `for` loop construction which has the form

``` R
> for (name in expr_1) expr_2
```

where `name`{.variable} is the loop variable. `expr_1`{.variable} is a
vector expression, (often a sequence like `1:20`), and
`expr_2`{.variable} is often a grouped expression with its
sub-expressions written in terms of the dummy *name*.
`expr_2`{.variable} is repeatedly evaluated as `name`{.variable} ranges
through the values in the vector result of `expr_1`{.variable}.

As an example, suppose `ind` is a vector of class indicators and we wish
to produce separate plots of `y` versus `x` within classes. One
possibility here is to use `coplot()`,[^20] which
will produce an array of plots corresponding to each level of the
factor. Another way to do this, now putting all plots on the one
display, is as follows:

``` r
> xc <- split(x, ind)
> yc <- split(y, ind)
> for (i in 1:length(yc)) {
    plot(xc[[i]], yc[[i]])
    abline(lsfit(xc[[i]], yc[[i]]))
  }
```

(Note the function `split()` which produces a list of vectors obtained
by splitting a larger vector according to the classes specified by a
factor. This is a useful function, mostly used in connection with
boxplots. See the `help` facility for further details.)

> **Warning**: `for()` loops are used in R code much less often than in
> compiled languages. Code that takes a 'whole object' view is likely to
> be both clearer and faster in R.

Other looping facilities include the

``` R
> repeat expr
```

statement and the

``` R
> while (condition) expr
```

statement.

The `break` statement can be used to terminate any loop, possibly
abnormally. This is the only way to terminate `repeat` loops.

The `next` statement can be used to discontinue one particular cycle and
skip to the "next".

Control statements are most often used in connection with *functions*
which are discussed in @sec-writing-your-own-functions
, and where more examples
will emerge.

[^20]: to be discussed later, or use `xyplot` from package
[**lattice**](https://CRAN.R-project.org/package=lattice).
 
# Writing your own functions {#sec-writing-your-own-functions}

As we have seen informally along the way, the R language allows the user
to create objects of mode *function*. These are true R functions that
are stored in a special internal form and may be used in further
expressions and so on. In the process, the language gains enormously in
power, convenience and elegance, and learning to write useful functions
is one of the main ways to make your use of R comfortable and
productive.

It should be emphasized that most of the functions supplied as part of
the R system, such as `mean()`, `var()`, `postscript()` and so on, are
themselves written in R and thus do not differ materially from user
written functions.

A function is defined by an assignment of the form

``` r
> name <- function(arg_1, arg_2, …) expression
```

The `expression`{.variable} is an R expression, (usually a grouped
expression), that uses the arguments, `arg_i`{.variable}, to calculate a
value. The value of the expression is the value returned for the
function.

A call to the function then usually takes the form
`name`{.variable}`(``expr_1`{.variable}`, ``expr_2`{.variable}`, …)` and
may occur anywhere a function call is legitimate.

 
## Simple examples {#sec-simple-examples}

As a first example, consider a function to calculate the two sample
*t*-statistic, showing "all the steps". This is an artificial example,
of course, since there are other, simpler ways of achieving the same
end.

The function is defined as follows:

``` r
> twosam <- function(y1, y2) {
    n1  <- length(y1); n2  <- length(y2)
    yb1 <- mean(y1);   yb2 <- mean(y2)
    s1  <- var(y1);    s2  <- var(y2)
    s <- ((n1-1)*s1 + (n2-1)*s2)/(n1+n2-2)
    tst <- (yb1 - yb2)/sqrt(s*(1/n1 + 1/n2))
    tst
  }
```

With this function defined, you could perform two sample *t*-tests using
a call such as

``` r
> tstat <- twosam(data$male, data$female); tstat
```

As a second example, consider a function to emulate directly the
[MATLAB]{.small} backslash command, which returns the coefficients of
the orthogonal projection of the vector *y* onto the column space of the
matrix, *X*. (This is ordinarily called the least squares estimate of
the regression coefficients.) This would ordinarily be done with the
`qr()` function; however this is sometimes a bit tricky to use directly
and it pays to have a simple function such as the following to use it
safely.

Thus given a *n* by *1* vector *y* and an *n* by *p* matrix *X* then *X
\\ y* is defined as (X'X)\^{-}X'y, where (X'X)\^{-} is a generalized
inverse of *X\'X*.

``` r
> bslash <- function(X, y) {
  X <- qr(X)
  qr.coef(X, y)
}
```

After this object is created it may be used in statements such as

``` r
> regcoeff <- bslash(Xmat, yvar)
```

and so on.

The classical R function `lsfit()` does this job quite well, and
more[^21]. It in turn uses the functions `qr()` and
`qr.coef()` in the slightly counterintuitive way above to do this part
of the calculation. Hence there is probably some value in having just
this part isolated in a simple to use function if it is going to be in
frequent use. If so, we may wish to make it a matrix binary operator for
even more convenient use.

[^21]: See also the methods described in [Statistical models in
R](Statistical-models-in-R.html)
 
## Defining new binary operators {#sec-defining-new-binary-operators}

Had we given the `bslash()` function a different name, namely one of the
form

``` R
%anything%
```

it could have been used as a *binary operator* in expressions rather
than in function form. Suppose, for example, we choose `!` for the
internal character. The function definition would then start as

``` r
> "%!%" <- function(X, y) { … }
```

(Note the use of quote marks.) The function could then be used as
`X %!% y`. (The backslash symbol itself is not a convenient choice as it
presents special problems in this context.)

The matrix multiplication operator, `%*%`, and the outer product matrix
operator `%o%` are other examples of binary operators defined in this
way.
 
## Named arguments and defaults {#sec-named-arguments-and-defaults}

As first noted in [Generating regular
sequences](Generating-regular-sequences.html), if arguments to called
functions are given in the "`name`{.variable}`=``object`{.variable}"
form, they may be given in any order. Furthermore the argument sequence
may begin in the unnamed, positional form, and specify named arguments
after the positional arguments.

Thus if there is a function `fun1` defined by

``` r
> fun1 <- function(data, data.frame, graph, limit) {
    [function body omitted]
  }
```

then the function may be invoked in several ways, for example

``` r
> ans <- fun1(d, df, TRUE, 20)
> ans <- fun1(d, df, graph=TRUE, limit=20)
> ans <- fun1(data=d, limit=20, graph=TRUE, data.frame=df)
```

are all equivalent.

In many cases arguments can be given commonly appropriate default
values, in which case they may be omitted altogether from the call when
the defaults are appropriate. For example, if `fun1` were defined as

``` r
> fun1 <- function(data, data.frame, graph=TRUE, limit=20) { … }
```

it could be called as

``` r
> ans <- fun1(d, df)
```

which is now equivalent to the three cases above, or as

``` r
> ans <- fun1(d, df, limit=10)
```

which changes one of the defaults.

It is important to note that defaults may be arbitrary expressions, even
involving other arguments to the same function; they are not restricted
to be constants as in our simple example here.
 
## The '`…`{#sec-the-'…'-argument}

Another frequent requirement is to allow one function to pass on
argument settings to another. For example many graphics functions use
the function `par()` and functions like `plot()` allow the user to pass
on graphical parameters to `par()` to control the graphical output. (See
[The par() function](The-par_0028_0029-function.html), for more details
on the `par()` function.) This can be done by including an extra
argument, literally '`…`', of the function, which may then be
passed on. An outline example is given below.

``` r
fun1 <- function(data, data.frame, graph=TRUE, limit=20, ...) {
  [omitted statements]
  if (graph)
    par(pch="*", ...)
  [more omissions]
}
```

Less frequently, a function will need to refer to components of
'`…`'. The expression `list(...)` evaluates all such arguments
and returns them in a named list, while `..1`, `..2`, etc. evaluate them
one at a time, with '`..n`' returning the n'th unmatched
argument.
 
## Assignments within functions {#sec-assignments-within-functions}

Note that *any ordinary assignments done within the function are local
and temporary and are lost after exit from the function*. Thus the
assignment `X <- qr(X)` does not affect the value of the argument in the
calling program.

To understand completely the rules governing the scope of R assignments
the reader needs to be familiar with the notion of an evaluation
*frame*. This is a somewhat advanced, though hardly difficult, topic and
is not covered further here.

If global and permanent assignments are intended within a function, then
either the "superassignment" operator, `<<-` or the function `assign()`
can be used. See the `help` document for details.
 
## More advanced examples {#sec-more-advanced-examples}

 
### Efficiency factors in block designs {#sec-efficiency-factors-in-block-designs}

As a more complete, if a little pedestrian, example of a function,
consider finding the efficiency factors for a block design. (Some
aspects of this problem have already been discussed in [Index
matrices](Index-matrices.html).)

A block design is defined by two factors, say `blocks` (`b` levels) and
`varieties` (`v` levels). If *R* and *K* are the *v* by *v* and *b* by
*b* *replications* and *block size* matrices, respectively, and *N* is
the *b* by *v* incidence matrix, then the efficiency factors are defined
as the eigenvalues of the matrix E = I_v - R\^{-1/2}N'K\^{-1}NR\^{-1/2}
= I_v - A'A, where A = K\^{-1/2}NR\^{-1/2}. One way to write the
function is given below.

``` r
> bdeff <- function(blocks, varieties) {
    blocks <- as.factor(blocks)             # minor safety move
    b <- length(levels(blocks))
    varieties <- as.factor(varieties)       # minor safety move
    v <- length(levels(varieties))
    K <- as.vector(table(blocks))           # remove dim attr
    R <- as.vector(table(varieties))        # remove dim attr
    N <- table(blocks, varieties)
    A <- 1/sqrt(K) * N * rep(1/sqrt(R), rep(b, v))
    sv <- svd(A)
    list(eff=1 - sv$d^2, blockcv=sv$u, varietycv=sv$v)
}
```

It is numerically slightly better to work with the singular value
decomposition on this occasion rather than the eigenvalue routines.

The result of the function is a list giving not only the efficiency
factors as the first component, but also the block and variety canonical
contrasts, since sometimes these give additional useful qualitative
information.
 
### Dropping all names in a printed array {#sec-dropping-all-names-in-a-printed-array}

For printing purposes with large matrices or arrays, it is often useful
to print them in close block form without the array names or numbers.
Removing the `dimnames` attribute will not achieve this effect, but
rather the array must be given a `dimnames` attribute consisting of
empty strings. For example to print a matrix, `X`

``` r
> temp <- X
> dimnames(temp) <- list(rep("", nrow(X)), rep("", ncol(X)))
> temp; rm(temp)
```

This can be much more conveniently done using a function,
`no.dimnames()`, shown below, as a "wrap around" to achieve the same
result. It also illustrates how some effective and useful user functions
can be quite short.

``` c
no.dimnames <- function(a) {
  ## Remove all dimension names from an array for compact printing.
  d <- list()
  l <- 0
  for(i in dim(a)) {
    d[[l <- l + 1]] <- rep("", i)
  }
  dimnames(a) <- d
  a
}
```

With this function defined, an array may be printed in close format
using

``` R
> no.dimnames(X)
```

This is particularly useful for large integer arrays, where patterns are
the real interest rather than the values.
 
### Recursive numerical integration {#sec-recursive-numerical-integration}

Functions may be recursive, and may themselves define functions within
themselves. Note, however, that such functions, or indeed variables, are
not inherited by called functions in higher evaluation frames as they
would be if they were on the search path.

The example below shows a naive way of performing one-dimensional
numerical integration. The integrand is evaluated at the end points of
the range and in the middle. If the one-panel trapezium rule answer is
close enough to the two panel, then the latter is returned as the value.
Otherwise the same process is recursively applied to each panel. The
result is an adaptive integration process that concentrates function
evaluations in regions where the integrand is farthest from linear.
There is, however, a heavy overhead, and the function is only
competitive with other algorithms when the integrand is both smooth and
very difficult to evaluate.

The example is also given partly as a little puzzle in R programming.

``` r
area <- function(f, a, b, eps = 1.0e-06, lim = 10) {
  fun1 <- function(f, a, b, fa, fb, a0, eps, lim, fun) {
    ## function ‘fun1’ is only visible inside ‘area’
    d <- (a + b)/2
    h <- (b - a)/4
    fd <- f(d)
    a1 <- h * (fa + fd)
    a2 <- h * (fd + fb)
    if(abs(a0 - a1 - a2) < eps || lim == 0)
      return(a1 + a2)
    else {
      return(fun(f, a, d, fa, fd, a1, eps, lim - 1, fun) +
             fun(f, d, b, fd, fb, a2, eps, lim - 1, fun))
    }
  }
  fa <- f(a)
  fb <- f(b)
  a0 <- ((fa + fb) * (b - a))/2
  fun1(f, a, b, fa, fb, a0, eps, lim, fun1)
}
```

 
## Scope {#sec-scope}

The discussion in this section is somewhat more technical than in other
parts of this document. However, it details one of the major differences
between [S-PLUS]{.small} and R.

The symbols which occur in the body of a function can be divided into
three classes; formal parameters, local variables and free variables.
The formal parameters of a function are those occurring in the argument
list of the function. Their values are determined by the process of
*binding* the actual function arguments to the formal parameters. Local
variables are those whose values are determined by the evaluation of
expressions in the body of the functions. Variables which are not formal
parameters or local variables are called free variables. Free variables
become local variables if they are assigned to. Consider the following
function definition.

``` c
f <- function(x) {
  y <- 2*x
  print(x)
  print(y)
  print(z)
}
```

In this function, `x` is a formal parameter, `y` is a local variable and
`z` is a free variable.

In R the free variable bindings are resolved by first looking in the
environment in which the function was created. This is called *lexical
scope*. First we define a function called `cube`.

``` r
cube <- function(n) {
  sq <- function() n*n
  n*sq()
}
```

The variable `n` in the function `sq` is not an argument to that
function. Therefore it is a free variable and the scoping rules must be
used to ascertain the value that is to be associated with it. Under
static scope ([S-PLUS]{.small}) the value is that associated with a
global variable named `n`. Under lexical scope (R) it is the parameter
to the function `cube` since that is the active binding for the variable
`n` at the time the function `sq` was defined. The difference between
evaluation in R and evaluation in [S-PLUS]{.small} is that
[S-PLUS]{.small} looks for a global variable called `n` while R first
looks for a variable called `n` in the environment created when `cube`
was invoked.

``` r
## first evaluation in S
S> cube(2)
Error in sq(): Object "n" not found
Dumped
S> n <- 3
S> cube(2)
[1] 18
## then the same function evaluated in R
R> cube(2)
[1] 8
```

Lexical scope can also be used to give functions *mutable state*. In the
following example we show how R can be used to mimic a bank account. A
functioning bank account needs to have a balance or total, a function
for making withdrawals, a function for making deposits and a function
for stating the current balance. We achieve this by creating the three
functions within `account` and then returning a list containing them.
When `account` is invoked it takes a numerical argument `total` and
returns a list containing the three functions. Because these functions
are defined in an environment which contains `total`, they will have
access to its value.

The special assignment operator, `<<-`, is used to change the value
associated with `total`. This operator looks back in enclosing
environments for an environment that contains the symbol `total` and
when it finds such an environment it replaces the value, in that
environment, with the value of right hand side. If the global or
top-level environment is reached without finding the symbol `total` then
that variable is created and assigned to there. For most users `<<-`
creates a global variable and assigns the value of the right hand side
to it[^22]. Only when `<<-` has been used in a
function that was returned as the value of another function will the
special behavior described here occur.

``` r
open.account <- function(total) {
  list(
    deposit = function(amount) {
      if(amount <= 0)
        stop("Deposits must be positive!\n")
      total <<- total + amount
      cat(amount, "deposited.  Your balance is", total, "\n\n")
    },
    withdraw = function(amount) {
      if(amount > total)
        stop("You don't have that much money!\n")
      total <<- total - amount
      cat(amount, "withdrawn.  Your balance is", total, "\n\n")
    },
    balance = function() {
      cat("Your balance is", total, "\n\n")
    }
  )
}

ross <- open.account(100)
robert <- open.account(200)

ross$withdraw(30)
ross$balance()
robert$balance()

ross$deposit(50)
ross$balance()
ross$withdraw(500)
```

[^22]: In some sense this mimics the behavior in [S-PLUS]{.small} since in
[S-PLUS]{.small} this operator always creates or assigns to a global
variable.
 
## Customizing the environment {#sec-customizing-the-environment}

Users can customize their environment in several different ways. There
is a site initialization file and every directory can have its own
special initialization file. Finally, the special functions `.First` and
`.Last` can be used.

The location of the site initialization file is taken from the value of
the `R_PROFILE` environment variable. If that variable is unset, the
file `Rprofile.site` in the R home subdirectory `etc`
is used. This file should contain the commands that you want to execute
every time R is started under your system. A second, personal, profile
file named `.Rprofile`[^23] can be placed in
any directory. If R is invoked in that directory then that file will be
sourced. This file gives individual users control over their workspace
and allows for different startup procedures in different working
directories. If no `.Rprofile` file is found in the startup
directory, then R looks for a `.Rprofile` file in the user's
home directory and uses that (if it exists). If the environment variable
`R_PROFILE_USER` is set, the file it points to is used instead of the
`.Rprofile` files.

Any function named `.First()` in either of the two profile files or in
the `.RData` image has a special status. It is automatically
performed at the beginning of an R session and may be used to initialize
the environment. For example, the definition in the example below alters
the prompt to `$` and sets up various other useful things that can then
be taken for granted in the rest of the session.

Thus, the sequence in which files are executed is,
`Rprofile.site`, the user profile, `.RData` and then
`.First()`. A definition in later files will mask definitions in earlier
files.

``` c
> .First <- function() {
  options(prompt="$ ", continue="+\t")  # $ is the prompt
  options(digits=5, length=999)         # custom numbers and printout
  x11()                                 # for graphics
  par(pch = "+")                        # plotting character
  source(file.path(Sys.getenv("HOME"), "R", "mystuff.R"))
                                        # my personal functions
  library(MASS)                         # attach a package
}
```

Similarly a function `.Last()`, if defined, is (normally) executed at
the very end of the session. An example is given below.

``` r
> .Last <- function() {
  graphics.off()                        # a small safety measure.
  cat(paste(date(),"\nAdios\n"))        # Is it time for lunch?
}
```

[^23]: So it is hidden under UNIX.
 
## Classes, generic functions and object orientation {#sec-classes-generic-functions-and-object-orientation}

The class of an object determines how it will be treated by what are
known as *generic* functions. Put the other way round, a generic
function performs a task or action on its arguments *specific to the
class of the argument itself*. If the argument lacks any `class`
attribute, or has a class not catered for specifically by the generic
function in question, there is always a *default action* provided.

An example makes things clearer. The class mechanism offers the user the
facility of designing and writing generic functions for special
purposes. Among the other generic functions are `plot()` for displaying
objects graphically, `summary()` for summarizing analyses of various
types, and `anova()` for comparing statistical models.

The number of generic functions that can treat a class in a specific way
can be quite large. For example, the functions that can accommodate in
some fashion objects of class `"data.frame"` include

``` r
[     [[<-    any    as.matrix
[<-   mean    plot   summary
```

A currently complete list can be got by using the `methods()` function:

``` R
> methods(class="data.frame")
```

Conversely the number of classes a generic function can handle can also
be quite large. For example the `plot()` function has a default method
and variants for objects of classes `"data.frame"`, `"density"`,
`"factor"`, and more. A complete list can be got again by using the
`methods()` function:

``` R
> methods(plot)
```

For many generic functions the function body is quite short, for example

``` R
> coef
function (object, ...)
UseMethod("coef")
```

The presence of `UseMethod` indicates this is a generic function. To see
what methods are available we can use `methods()`

``` R
> methods(coef)
[1] coef.aov*         coef.Arima*       coef.default*     coef.listof*
[5] coef.nls*         coef.summary.nls*

   Non-visible functions are asterisked
```

In this example there are six methods, none of which can be seen by
typing its name. We can read these by either of

``` r
> getAnywhere("coef.aov")
A single object matching 'coef.aov' was found
It was found in the following places
  registered S3 method for coef from namespace stats
  namespace:stats
with value

function (object, ...)
{
    z <- object$coef
    z[!is.na(z)]
}

> getS3method("coef", "aov")
function (object, ...)
{
    z <- object$coef
    z[!is.na(z)]
}
```

A function named `gen`{.variable}`.``cl`{.variable} will be invoked by
the generic `gen`{.variable} for class `cl`{.variable}, so do not name
functions in this style unless they are intended to be methods.

The reader is referred to the *R Language Definition* for a more
complete discussion of this mechanism.
 
# Statistical models in R {#sec-statistical-models-in-r}

This section presumes the reader has some familiarity with statistical
methodology, in particular with regression analysis and the analysis of
variance. Later we make some rather more ambitious presumptions, namely
that something is known about generalized linear models and nonlinear
regression.

The requirements for fitting statistical models are sufficiently well
defined to make it possible to construct general tools that apply in a
broad spectrum of problems.

R provides an interlocking suite of facilities that make fitting
statistical models very simple. As we mention in the introduction, the
basic output is minimal, and one needs to ask for the details by calling
extractor functions.

 
## Defining statistical models; formulae {#sec-defining-statistical-models-formulae}

The template for a statistical model is a linear regression model with
independent, homoscedastic errors

::: display
``` R
y_i = sum_{j=0}^p beta_j x_{ij} + e_i,     i = 1, …, n,
```
:::

where the e_i are NID(0, sigma\^2). In matrix terms this would be
written

::: display
``` R
y = X  beta + e
```
:::

where the *y* is the response vector, *X* is the *model matrix* or
*design matrix* and has columns *x_0, x_1, ..., x_p*, the determining
variables. Very often *x_0* will be a column of ones defining an
*intercept* term.

### Examples {#sec-examples}

Before giving a formal specification, a few examples may usefully set
the picture.

Suppose `y`, `x`, `x0`, `x1`, `x2`, ... are numeric variables, `X` is a
matrix and `A`, `B`, `C`, ... are factors. The following formulae on the
left side below specify statistical models as described on the right.

`y ~ x`\
`y ~ 1 + x`

:   Both imply the same simple linear regression model of *y* on *x*.
    The first has an implicit intercept term, and the second an explicit
    one.

`y ~ 0 + x`\
`y ~ -1 + x`\
`y ~ x - 1`

:   Simple linear regression of *y* on *x* through the origin (that is,
    without an intercept term).

`log(y) ~ x1 + x2`

:   Multiple regression of the transformed variable, log(y), on *x1* and
    *x2* (with an implicit intercept term).

`y ~ poly(x,2)`\
`y ~ 1 + x + I(x^2)`

:   Polynomial regression of *y* on *x* of degree 2. The first form uses
    orthogonal polynomials, and the second uses explicit powers, as
    basis.

`y ~ X + poly(x,2)`

:   Multiple regression *y* with model matrix consisting of the matrix
    *X* as well as polynomial terms in *x* to degree 2.

`y ~ A`

:   Single classification analysis of variance model of *y*, with
    classes determined by *A*.

`y ~ A + x`

:   Single classification analysis of covariance model of *y*, with
    classes determined by *A*, and with covariate *x*.

`y ~ A*B`\
`y ~ A + B + A:B`\
`y ~ B %in% A`\
`y ~ A/B`

:   Two factor non-additive model of *y* on *A* and *B*. The first two
    specify the same crossed classification and the second two specify
    the same nested classification. In abstract terms all four specify
    the same model subspace.

`y ~ (A + B + C)^2`\
`y ~ A*B*C - A:B:C`

:   Three factor experiment but with a model containing main effects and
    two factor interactions only. Both formulae specify the same model.

`y ~ A * x`\
`y ~ A/x`\
`y ~ A/(1 + x) - 1`

:   Separate simple linear regression models of *y* on *x* within the
    levels of *A*, with different codings. The last form produces
    explicit estimates of as many different intercepts and slopes as
    there are levels in *A*.

`y ~ A*B + Error(C)`

:   An experiment with two treatment factors, *A* and *B*, and error
    strata determined by factor *C*. For example a split plot
    experiment, with whole plots (and hence also subplots), determined
    by factor *C*.

The operator `~` is used to define a *model formula* in R. The form, for
an ordinary linear model, is

``` R
response ~ op_1 term_1 op_2 term_2 op_3 term_3 …
```

where

`response`{.variable}

:   is a vector or matrix, (or expression evaluating to a vector or
    matrix) defining the response variable(s).

`op_i`{.variable}

:   is an operator, either `+` or `-`, implying the inclusion or
    exclusion of a term in the model, (the first is optional).

`term_i`{.variable}

:   is either

    -   a vector or matrix expression, or `1`,
    -   a factor, or
    -   a *formula expression* consisting of factors, vectors or
        matrices connected by *formula operators*.

    In all cases each term defines a collection of columns either to be
    added to or removed from the model matrix. A `1` stands for an
    intercept column and is by default included in the model matrix
    unless explicitly removed.

The *formula operators* are similar in effect to the Wilkinson and
Rogers notation used by such programs as Glim and Genstat. One
inevitable change is that the operator `.`' becomes '`:` since the
period is a valid name character in R.

The notation is summarized below (based on Chambers & Hastie, 1992,
p.29):

`Y`{.variable}` ~ ``M`{.variable}

:   `Y`{.variable} is modeled as `M`{.variable}.

`M_1`{.variable}` + ``M_2`{.variable}

:   Include `M_1`{.variable} and `M_2`{.variable}.

`M_1`{.variable}` - ``M_2`{.variable}

:   Include `M_1`{.variable} leaving out terms of `M_2`{.variable}.

`M_1`{.variable}` : ``M_2`{.variable}

:   The tensor product of `M_1`{.variable} and `M_2`{.variable}. If both
    terms are factors, then the "subclasses" factor.

`M_1`{.variable}` %in% ``M_2`{.variable}

:   Similar to `M_1`{.variable}`:``M_2`{.variable}, but with a different
    coding.

`M_1`{.variable}` * ``M_2`{.variable}

:   `M_1`{.variable}` + ``M_2`{.variable}` + ``M_1`{.variable}`:``M_2`{.variable}.

`M_1`{.variable}` / ``M_2`{.variable}

:   `M_1`{.variable}` + ``M_2`{.variable}` %in% ``M_1`{.variable}.

`M`{.variable}`^``n`{.variable}

:   All terms in `M`{.variable} together with "interactions" up to order
    `n`{.variable}

`I(``M`{.variable}`)`

:   Insulate `M`{.variable}. Inside `M`{.variable} all operators have
    their normal arithmetic meaning, and that term appears in the model
    matrix.

Note that inside the parentheses that usually enclose function arguments
all operators have their normal arithmetic meaning. The function `I()`
is an identity function used to allow terms in model formulae to be
defined using arithmetic operators.

Note particularly that the model formulae specify the *columns of the
model matrix*, the specification of the parameters being implicit. This
is not the case in other contexts, for example in specifying nonlinear
models.

 
### Contrasts {#sec-contrasts}

We need at least some idea how the model formulae specify the columns of
the model matrix. This is easy if we have continuous variables, as each
provides one column of the model matrix (and the intercept will provide
a column of ones if included in the model).

What about a *k*-level factor `A`? The answer differs for unordered and
ordered factors. For *unordered* factors *k - 1* columns are generated
for the indicators of the second, ..., *k*th levels of the factor. (Thus
the implicit parameterization is to contrast the response at each level
with that at the first.) For *ordered* factors the *k - 1* columns are
the orthogonal polynomials on *1, ..., k*, omitting the constant term.

Although the answer is already complicated, it is not the whole story.
First, if the intercept is omitted in a model that contains a factor
term, the first such term is encoded into *k* columns giving the
indicators for all the levels. Second, the whole behavior can be changed
by the `options` setting for `contrasts`. The default setting in R is

``` r
options(contrasts = c("contr.treatment", "contr.poly"))
```

The main reason for mentioning this is that R and S have different
defaults for unordered factors, S using Helmert contrasts. So if you
need to compare your results to those of a textbook or paper which used
[S-PLUS]{.small}, you will need to set

``` r
options(contrasts = c("contr.helmert", "contr.poly"))
```

This is a deliberate difference, as treatment contrasts (R's default)
are thought easier for newcomers to interpret.

We have still not finished, as the contrast scheme to be used can be set
for each term in the model using the functions `contrasts` and `C`.

We have not yet considered interaction terms: these generate the
products of the columns introduced for their component terms.

Although the details are complicated, model formulae in R will normally
generate the models that an expert statistician would expect, provided
that marginality is preserved. Fitting, for example, a model with an
interaction but not the corresponding main effects will in general lead
to surprising results, and is for experts only.
 
## Linear models {#sec-linear-models}

The basic function for fitting ordinary multiple models is `lm()`, and a
streamlined version of the call is as follows:

``` r
> fitted.model <- lm(formula, data = data.frame)
```

For example

``` r
> fm2 <- lm(y ~ x1 + x2, data = production)
```

would fit a multiple regression model of *y* on *x1* and *x2* (with
implicit intercept term).

The important (but technically optional) parameter `data = production`
specifies that any variables needed to construct the model should come
first from the `production` *data frame*. *This is the case regardless
of whether data frame `production` has been attached on the search path
or not*.
 
## Generic functions for extracting model information {#sec-generic-functions-for-extracting-model-information}

The value of `lm()` is a fitted model object; technically a list of
results of class `"lm"`. Information about the fitted model can then be
displayed, extracted, plotted and so on by using generic functions that
orient themselves to objects of class `"lm"`. These include

``` c
add1    deviance   formula      predict  step
alias   drop1      kappa        print    summary
anova   effects    labels       proj     vcov
coef    family     plot         residuals
```

A brief description of the most commonly used ones is given below.

[]{#index-anova}

`anova(``object_1`{.variable}`, ``object_2`{.variable}`)`

Compare a submodel with an outer model and produce an analysis of
variance table.

`coef(``object`{.variable}`)`

Extract the regression coefficient (matrix).

Long form: `coefficients(``object`{.variable}`)`.

`deviance(``object`{.variable}`)`

Residual sum of squares, weighted if appropriate.

`formula(``object`{.variable}`)`

Extract the model formula.

`plot(``object`{.variable}`)`

Produce four plots, showing residuals, fitted values and some
diagnostics.

`predict(``object`{.variable}`, newdata=``data.frame`{.variable}`)`

The data frame supplied must have variables specified with the same
labels as the original. The value is a vector or matrix of predicted
values corresponding to the determining variable values in
`data.frame`{.variable}.

`print(``object`{.variable}`)`

Print a concise version of the object. Most often used implicitly.

`residuals(``object`{.variable}`)`

Extract the (matrix of) residuals, weighted as appropriate.

Short form: `resid(``object`{.variable}`)`.

`step(``object`{.variable}`)`

Select a suitable model by adding or dropping terms and preserving
hierarchies. The model with the smallest value of AIC (Akaike's An
Information Criterion) discovered in the stepwise search is returned.

`summary(``object`{.variable}`)`

Print a comprehensive summary of the results of the regression analysis.

`vcov(``object`{.variable}`)`

Returns the variance-covariance matrix of the main parameters of a
fitted model object.
 
## Analysis of variance and model comparison {#sec-analysis-of-variance-and-model-comparison}

The model fitting function
`aov(``formula`{.variable}`, data=``data.frame`{.variable}`)` operates
at the simplest level in a very similar way to the function `lm()`, and
most of the generic functions listed in the table in [Generic functions
for extracting model
information](Generic-functions-for-extracting-model-information.html)
apply.

It should be noted that in addition `aov()` allows an analysis of models
with multiple error strata such as split plot experiments, or balanced
incomplete block designs with recovery of inter-block information. The
model formula

``` R
response ~ mean.formula + Error(strata.formula)
```

specifies a multi-stratum experiment with error strata defined by the
`strata.formula`{.variable}. In the simplest case,
`strata.formula`{.variable} is simply a factor, when it defines a two
strata experiment, namely between and within the levels of the factor.

For example, with all determining variables factors, a model formula
such as that in:

``` r
> fm <- aov(yield ~ v + n*p*k + Error(farms/blocks), data=farm.data)
```

would typically be used to describe an experiment with mean model
`v + n*p*k` and three error strata, namely "between farms", "within
farms, between blocks" and "within blocks".

 
### ANOVA tables {#sec-anova-tables}

Note also that the analysis of variance table (or tables) are for a
sequence of fitted models. The sums of squares shown are the decrease in
the residual sums of squares resulting from an inclusion of *that term*
in the model at *that place* in the sequence. Hence only for orthogonal
experiments will the order of inclusion be inconsequential.

For multistratum experiments the procedure is first to project the
response onto the error strata, again in sequence, and to fit the mean
model to each projection. For further details, see Chambers & Hastie
(1992).

A more flexible alternative to the default full ANOVA table is to
compare two or more models directly using the `anova()` function.

``` R
> anova(fitted.model.1, fitted.model.2, …)
```

The display is then an ANOVA table showing the differences between the
fitted models when fitted in sequence. The fitted models being compared
would usually be an hierarchical sequence, of course. This does not give
different information to the default, but rather makes it easier to
comprehend and control.
 
## Updating fitted models {#sec-updating-fitted-models}

The `update()` function is largely a convenience function that allows a
model to be fitted that differs from one previously fitted usually by
just a few additional or removed terms. Its form is

``` r
> new.model <- update(old.model, new.formula)
```

In the `new.formula`{.variable} the special name consisting of a period,
`.`, only, can be used to stand for "the corresponding part of the old
model formula". For example,

``` r
> fm05 <- lm(y ~ x1 + x2 + x3 + x4 + x5, data = production)
> fm6  <- update(fm05, . ~ . + x6)
> smf6 <- update(fm6, sqrt(.) ~ .)
```

would fit a five variate multiple regression with variables (presumably)
from the data frame `production`, fit an additional model including a
sixth regressor variable, and fit a variant on the model where the
response had a square root transform applied.

Note especially that if the `data=` argument is specified on the
original call to the model fitting function, this information is passed
on through the fitted model object to `update()` and its allies.

The name '`.`' can also be used in other contexts, but with
slightly different meaning. For example

``` r
> fmfull <- lm(y ~ . , data = production)
```

would fit a model with response `y` and regressor variables *all other
variables in the data frame `production`*.

Other functions for exploring incremental sequences of models are
`add1()`, `drop1()` and `step()`. The names of these give a good clue to
their purpose, but for full details see the on-line help.
 
## Generalized linear models {#sec-generalized-linear-models}

Generalized linear modeling is a development of linear models to
accommodate both non-normal response distributions and transformations
to linearity in a clean and straightforward way. A generalized linear
model may be described in terms of the following sequence of
assumptions:

-   There is a response, *y*, of interest and stimulus variables x_1,
    x_2, ..., whose values influence the distribution of the response.

-   The stimulus variables influence the distribution of *y* through *a
    single linear function, only*. This linear function is called the
    *linear predictor*, and is usually written

    ::: display
    ``` R
    eta = beta_1 x_1 + beta_2 x_2 + … + beta_p x_p,
    ```
    :::

    hence x_i has no influence on the distribution of *y* if and only if
    beta_i is zero.

-   The distribution of *y* is of the form

    ::: display
    ``` R
    f_Y(y; mu, phi)
      = exp((A/phi) * (y lambda(mu) - gamma(lambda(mu))) + tau(y, phi))
    ```
    :::

    where phi is a *scale parameter* (possibly known), and is constant
    for all observations, *A* represents a prior weight, assumed known
    but possibly varying with the observations, and \$\\mu\$ is the mean
    of *y*. So it is assumed that the distribution of *y* is determined
    by its mean and possibly a scale parameter as well.

-   The mean, mu, is a smooth invertible function of the linear
    predictor:

    ::: display
    ``` R
    mu = m(eta),    eta = m^{-1}(mu) = ell(mu)
    ```
    :::

    and this inverse function, ell(), is called the *link function*.

These assumptions are loose enough to encompass a wide class of models
useful in statistical practice, but tight enough to allow the
development of a unified methodology of estimation and inference, at
least approximately. The reader is referred to any of the current
reference works on the subject for full details, such as McCullagh &
Nelder (1989) or Dobson (1990).

 
### Families {#sec-families}

The class of generalized linear models handled by facilities supplied in
R includes *gaussian*, *binomial*, *poisson*, *inverse gaussian* and
*gamma* response distributions and also *quasi-likelihood* models where
the response distribution is not explicitly specified. In the latter
case the *variance function* must be specified as a function of the
mean, but in other cases this function is implied by the response
distribution.

Each response distribution admits a variety of link functions to connect
the mean with the linear predictor. Those automatically available are
shown in the following table:

::: blockquotes-block
>   Family name          Link functions
>   -------------------- ------------------------------------------------------------------------------
>   `binomial`           `logit`, `probit`, `log`, `cloglog`
>   `gaussian`           `identity`, `log`, `inverse`
>   `Gamma`              `identity`, `inverse`, `log`
>   `inverse.gaussian`   `1/mu^2`, `identity`, `inverse`, `log`
>   `poisson`            `identity`, `log`, `sqrt`
>   `quasi`              `logit`, `probit`, `cloglog`, `identity`, `inverse`, `log`, `1/mu^2`, `sqrt`
:::

The combination of a response distribution, a link function and various
other pieces of information that are needed to carry out the modeling
exercise is called the *family* of the generalized linear model.
 
### The `glm()` function {#sec-the-glm-function}

Since the distribution of the response depends on the stimulus variables
through a single linear function *only*, the same mechanism as was used
for linear models can still be used to specify the linear part of a
generalized model. The family has to be specified in a different way.

The R function to fit a generalized linear model is `glm()` which uses
the form

``` r
> fitted.model <- glm(formula, family=family.generator, data=data.frame)
```

The only new feature is the `family.generator`{.variable}, which is the
instrument by which the family is described. It is the name of a
function that generates a list of functions and expressions that
together define and control the model and estimation process. Although
this may seem a little complicated at first sight, its use is quite
simple.

The names of the standard, supplied family generators are given under
"Family Name" in the table in [Families](Families.html). Where there is
a choice of links, the name of the link may also be supplied with the
family name, in parentheses as a parameter. In the case of the `quasi`
family, the variance function may also be specified in this way.

Some examples make the process clear.

### The `gaussian` family {#sec-the-gaussian-family}

A call such as

``` r
> fm <- glm(y ~ x1 + x2, family = gaussian, data = sales)
```

achieves the same result as

``` r
> fm <- lm(y ~ x1+x2, data=sales)
```

but much less efficiently. Note how the gaussian family is not
automatically provided with a choice of links, so no parameter is
allowed. If a problem requires a gaussian family with a nonstandard
link, this can usually be achieved through the `quasi` family, as we
shall see later.

### The `binomial` family {#sec-the-binomial-family}

Consider a small, artificial example, from Silvey (1970).

On the Aegean island of Kalythos the male inhabitants suffer from a
congenital eye disease, the effects of which become more marked with
increasing age. Samples of islander males of various ages were tested
for blindness and the results recorded. The data is shown below:


The problem we consider is to fit both logistic and probit models to
this data, and to estimate for each model the LD50, that is the age at
which the chance of blindness for a male inhabitant is 50%.

If *y* is the number of blind at age *x* and *n* the number tested, both
models have the form y \~ B(n, F(beta_0 + beta_1 x)) where for the
probit case, F(z) = Phi(z) is the standard normal distribution function,
and in the logit case (the default), F(z) = e\^z/(1+e\^z). In both cases
the LD50 is LD50 = - beta_0/beta_1 that is, the point at which the
argument of the distribution function is zero.

The first step is to set the data up as a data frame

``` r
> kalythos <- data.frame(x = c(20,35,45,55,70), n = rep(50,5),
                         y = c(6,17,26,37,44))
```

To fit a binomial model using `glm()` there are three possibilities for
the response:

-   If the response is a *vector* it is assumed to hold *binary* data,
    and so must be a *0/1* vector.
-   If the response is a *two-column matrix* it is assumed that the
    first column holds the number of successes for the trial and the
    second holds the number of failures.
-   If the response is a *factor*, its first level is taken as
    failure (0) and all other levels as 'success' (1).

Here we need the second of these conventions, so we add a matrix to our
data frame:

``` r
> kalythos$Ymat <- cbind(kalythos$y, kalythos$n - kalythos$y)
```

To fit the models we use

``` r
> fmp <- glm(Ymat ~ x, family = binomial(link=probit), data = kalythos)
> fml <- glm(Ymat ~ x, family = binomial, data = kalythos)
```

Since the logit link is the default the parameter may be omitted on the
second call. To see the results of each fit we could use

``` R
> summary(fmp)
> summary(fml)
```

Both models fit (all too) well. To find the LD50 estimate we can use a
simple function:

``` r
> ld50 <- function(b) -b[1]/b[2]
> ldp <- ld50(coef(fmp)); ldl <- ld50(coef(fml)); c(ldp, ldl)
```

The actual estimates from this data are 43.663 years and 43.601 years
respectively.

### Poisson models {#sec-poisson-models}

With the Poisson family the default link is the `log`, and in practice
the major use of this family is to fit surrogate Poisson log-linear
models to frequency data, whose actual distribution is often
multinomial. This is a large and important subject we will not discuss
further here. It even forms a major part of the use of non-gaussian
generalized models overall.

Occasionally genuinely Poisson data arises in practice and in the past
it was often analyzed as gaussian data after either a log or a
square-root transformation. As a graceful alternative to the latter, a
Poisson generalized linear model may be fitted as in the following
example:

``` r
> fmod <- glm(y ~ A + B + x, family = poisson(link=sqrt),
              data = worm.counts)
```

### Quasi-likelihood models {#sec-quasi-likelihood-models}

For all families the variance of the response will depend on the mean
and will have the scale parameter as a multiplier. The form of
dependence of the variance on the mean is a characteristic of the
response distribution; for example for the poisson distribution Var(y) =
mu.

For quasi-likelihood estimation and inference the precise response
distribution is not specified, but rather only a link function and the
form of the variance function as it depends on the mean. Since
quasi-likelihood estimation uses formally identical techniques to those
for the gaussian distribution, this family provides a way of fitting
gaussian models with non-standard link functions or variance functions,
incidentally.

For example, consider fitting the non-linear regression y = theta_1 z_1
/ (z_2 - theta_2) + e which may be written alternatively as y = 1 /
(beta_1 x_1 + beta_2 x_2) + e where x_1 = z_2/z_1, x_2 = -1/z_1, beta_1
= 1/theta_1, and beta_2 = theta_2/theta_1. Supposing a suitable data
frame to be set up we could fit this non-linear regression as

``` r
> nlfit <- glm(y ~ x1 + x2 - 1,
               family = quasi(link=inverse, variance=constant),
               data = biochem)
```

The reader is referred to the manual and the help document for further
information, as needed.
 
## Nonlinear least squares and maximum likelihood models {#sec-nonlinear-least-squares-and-maximum-likelihood-models}

Certain forms of nonlinear model can be fitted by Generalized Linear
Models (`glm()`). But in the majority of cases we have to approach the
nonlinear curve fitting problem as one of nonlinear optimization. R's
nonlinear optimization routines are `optim()`, `nlm()` and `nlminb()`,
We seek the parameter values that minimize some index of lack-of-fit,
and they do this by trying out various parameter values iteratively.
Unlike linear regression for example, there is no guarantee that the
procedure will converge on satisfactory estimates. All the methods
require initial guesses about what parameter values to try, and
convergence may depend critically upon the quality of the starting
values.

 
### Least squares {#sec-least-squares}

One way to fit a nonlinear model is by minimizing the sum of the squared
errors (SSE) or residuals. This method makes sense if the observed
errors could have plausibly arisen from a normal distribution.

Here is an example from Bates & Watts (1988), page 51. The data are:

``` r
> x <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56,
         1.10, 1.10)
> y <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200)
```

The fit criterion to be minimized is:

``` r
> fn <- function(p) sum((y - (p[1] * x)/(p[2] + x))^2)
```

In order to do the fit we need initial estimates of the parameters. One
way to find sensible starting values is to plot the data, guess some
parameter values, and superimpose the model curve using those values.

``` r
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 200 * xfit/(0.1 + xfit)
> lines(spline(xfit, yfit))
```

We could do better, but these starting values of 200 and 0.1 seem
adequate. Now do the fit:

``` r
> out <- nlm(fn, p = c(200, 0.1), hessian = TRUE)
```

After the fitting, `out$minimum` is the SSE, and `out$estimate` are the
least squares estimates of the parameters. To obtain the approximate
standard errors (SE) of the estimates we do:

``` R
> sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))
```

The `2` which is subtracted in the line above represents the number of
parameters. A 95% confidence interval would be the parameter estimate
+/- 1.96 SE. We can superimpose the least squares fit on a new plot:

``` r
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 212.68384222 * xfit/(0.06412146 + xfit)
> lines(spline(xfit, yfit))
```

The standard package **stats** provides much more extensive facilities
for fitting non-linear models by least squares. The model we have just
fitted is the Michaelis-Menten model, so we can use

``` r
> df <- data.frame(x=x, y=y)
> fit <- nls(y ~ SSmicmen(x, Vm, K), df)
> fit
Nonlinear regression model
  model:  y ~ SSmicmen(x, Vm, K)
   data:  df
          Vm            K
212.68370711   0.06412123
 residual sum-of-squares:  1195.449
> summary(fit)

Formula: y ~ SSmicmen(x, Vm, K)

Parameters:
    Estimate Std. Error t value Pr(>|t|)
Vm 2.127e+02  6.947e+00  30.615 3.24e-11
K  6.412e-02  8.281e-03   7.743 1.57e-05

Residual standard error: 10.93 on 10 degrees of freedom

Correlation of Parameter Estimates:
      Vm
K 0.7651
```
 
### Maximum likelihood {#sec-maximum-likelihood}

Maximum likelihood is a method of nonlinear model fitting that applies
even if the errors are not normal. The method finds the parameter values
which maximize the log likelihood, or equivalently which minimize the
negative log-likelihood. Here is an example from Dobson (1990), pp.
108--111. This example fits a logistic model to dose-response data,
which clearly could also be fit by `glm()`. The data are:

``` r
> x <- c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113,
         1.8369, 1.8610, 1.8839)
> y <- c( 6, 13, 18, 28, 52, 53, 61, 60)
> n <- c(59, 60, 62, 56, 63, 59, 62, 60)
```

The negative log-likelihood to minimize is:

``` r
> fn <- function(p)
   sum( - (y*(p[1]+p[2]*x) - n*log(1+exp(p[1]+p[2]*x))
           + log(choose(n, y)) ))
```

We pick sensible starting values and do the fit:

``` r
> out <- nlm(fn, p = c(-50,20), hessian = TRUE)
```

After the fitting, `out$minimum` is the negative log-likelihood, and
`out$estimate` are the maximum likelihood estimates of the parameters.
To obtain the approximate SEs of the estimates we do:

``` R
> sqrt(diag(solve(out$hessian)))
```

A 95% confidence interval would be the parameter estimate +/- 1.96 SE.
 
## Some non-standard models {#sec-some-non-standard-models}

We conclude this chapter with just a brief mention of some of the other
facilities available in R for special regression and data analysis
problems.

-   []{#index-Mixed-models} **Mixed models.** The recommended
    [**nlme**](https://CRAN.R-project.org/package=nlme) package provides
    functions `lme()` and `nlme()` for linear and non-linear
    mixed-effects models, that is linear and non-linear regressions in
    which some of the coefficients correspond to random effects. These
    functions make heavy use of formulae to specify the models.

-   []{#index-Local-approximating-regressions} **Local approximating
    regressions.** The `loess()` function fits a nonparametric
    regression by using a locally weighted regression. Such regressions
    are useful for highlighting a trend in messy data or for data
    reduction to give some insight into a large data set.

    Function `loess` is in the standard package **stats**, together with
    code for projection pursuit regression.

-   []{#index-Robust-regression} **Robust regression.** There are
    several functions available for fitting regression models in a way
    resistant to the influence of extreme outliers in the data. Function
    `lqs` in the recommended package
    [**MASS**](https://CRAN.R-project.org/package=MASS) provides
    state-of-art algorithms for highly-resistant fits. Less resistant
    but statistically more efficient methods are available in packages,
    for example function `rlm` in package
    [**MASS**](https://CRAN.R-project.org/package=MASS).

-   []{#index-Additive-models} **Additive models.** This technique aims
    to construct a regression function from smooth additive functions of
    the determining variables, usually one for each determining
    variable. Functions `avas` and `ace` in package
    [**acepack**](https://CRAN.R-project.org/package=acepack) and
    functions `bruto` and `mars` in package
    [**mda**](https://CRAN.R-project.org/package=mda) provide some
    examples of these techniques in user-contributed packages to R. An
    extension is **Generalized Additive Models**, implemented in
    user-contributed packages
    [**gam**](https://CRAN.R-project.org/package=gam) and
    [**mgcv**](https://CRAN.R-project.org/package=mgcv).

-   []{#index-Tree_002dbased-models} **Tree-based models.** Rather than
    seek an explicit global linear model for prediction or
    interpretation, tree-based models seek to bifurcate the data,
    recursively, at critical points of the determining variables in
    order to partition the data ultimately into groups that are as
    homogeneous as possible within, and as heterogeneous as possible
    between. The results often lead to insights that other data analysis
    methods tend not to yield.

    Models are again specified in the ordinary linear model form. The
    model fitting function is `tree()`, but many other generic functions
    such as `plot()` and `text()` are well adapted to displaying the
    results of a tree-based model fit in a graphical way.

    Tree models are available in R *via* the user-contributed packages
    [**rpart**](https://CRAN.R-project.org/package=rpart) and
    [**tree**](https://CRAN.R-project.org/package=tree).
 
# Graphical procedures {#sec-graphical-procedures}

Graphical facilities are an important and extremely versatile component
of the R environment. It is possible to use the facilities to display a
wide variety of statistical graphs and also to build entirely new types
of graph.

The graphics facilities can be used in both interactive and batch modes,
but in most cases, interactive use is more productive. Interactive use
is also easy because at startup time R initiates a graphics *device
driver* which opens a special *graphics window* for the display of
interactive graphics. Although this is done automatically, it may useful
to know that the command used is `X11()` under UNIX, `windows()` under
Windows and `quartz()` under macOS. A new device can always be opened by
`dev.new()`.

Once the device driver is running, R plotting commands can be used to
produce a variety of graphical displays and to create entirely new kinds
of display.

Plotting commands are divided into three basic groups:

-   **High-level** plotting functions create a new plot on the graphics
    device, possibly with axes, labels, titles and so on.
-   **Low-level** plotting functions add more information to an existing
    plot, such as extra points, lines and labels.
-   **Interactive** graphics functions allow you interactively add
    information to, or extract information from, an existing plot, using
    a pointing device such as a mouse.

In addition, R maintains a list of *graphical parameters* which can be
manipulated to customize your plots.

This manual only describes what are known as 'base' graphics. A separate
graphics sub-system in package **grid** coexists with base -- it is more
powerful but harder to use. There is a recommended package
[**lattice**](https://CRAN.R-project.org/package=lattice) which builds
on **grid** and provides ways to produce multi-panel plots akin to those
in the *Trellis* system in S.

 
## High-level plotting commands {#sec-high-level-plotting-commands}

High-level plotting functions are designed to generate a complete plot
of the data passed as arguments to the function. Where appropriate,
axes, labels and titles are automatically generated (unless you request
otherwise.) High-level plotting commands always start a new plot,
erasing the current plot if necessary.

 
### The `plot()` function {#sec-the-plot-function}

One of the most frequently used plotting functions in R is the `plot()`
function. This is a *generic* function: the type of plot produced is
dependent on the type or *class* of the first argument.

`plot(``x`{.variable}`, ``y`{.variable}`)`\
`plot(``xy`{.variable}`)`

:   If `x`{.variable} and `y`{.variable} are vectors,
    `plot(``x`{.variable}`, ``y`{.variable}`)` produces a scatterplot of
    `y`{.variable} against `x`{.variable}. The same effect can be
    produced by supplying one argument (second form) as either a list
    containing two elements `x`{.variable} and `y`{.variable} or a
    two-column matrix.

`plot(``x`{.variable}`)`

:   If `x`{.variable} is a time series, this produces a time-series
    plot. If `x`{.variable} is a numeric vector, it produces a plot of
    the values in the vector against their index in the vector. If
    `x`{.variable} is a complex vector, it produces a plot of imaginary
    versus real parts of the vector elements.

`plot(``f`{.variable}`)`\
`plot(``f`{.variable}`, ``y`{.variable}`)`

:   `f`{.variable} is a factor object, `y`{.variable} is a numeric
    vector. The first form generates a bar plot of `f`{.variable}; the
    second form produces boxplots of `y`{.variable} for each level of
    `f`{.variable}.

`plot(``df`{.variable}`)`\
`plot(~ ``expr`{.variable}`)`\
`plot(``y`{.variable}` ~ ``expr`{.variable}`)`

:   `df`{.variable} is a data frame, `y`{.variable} is any object,
    `expr`{.variable} is a list of object names separated by `+`
    (e.g., `a + b + c`). The first two forms produce distributional
    plots of the variables in a data frame (first form) or of a number
    of named objects (second form). The third form plots `y`{.variable}
    against every object named in `expr`{.variable}.
 
### Displaying multivariate data {#sec-displaying-multivariate-data}

R provides two very useful functions for representing multivariate data.
If `X` is a numeric matrix or data frame, the command

``` R
> pairs(X)
```

produces a pairwise scatterplot matrix of the variables defined by the
columns of `X`, that is, every column of `X` is plotted against every
other column of `X` and the resulting *n(n-1)* plots are arranged in a
matrix with plot scales constant over the rows and columns of the
matrix.

When three or four variables are involved a *coplot* may be more
enlightening. If `a` and `b` are numeric vectors and `c` is a numeric
vector or factor object (all of the same length), then the command

``` R
> coplot(a ~ b | c)
```

produces a number of scatterplots of `a` against `b` for given values of
`c`. If `c` is a factor, this simply means that `a` is plotted against
`b` for every level of `c`. When `c` is numeric, it is divided into a
number of *conditioning intervals* and for each interval `a` is plotted
against `b` for values of `c` within the interval. The number and
position of intervals can be controlled with `given.values=` argument to
`coplot()`---the function `co.intervals()` is useful for selecting
intervals. You can also use two *given* variables with a command like

``` R
> coplot(a ~ b | c + d)
```

which produces scatterplots of `a` against `b` for every joint
conditioning interval of `c` and `d`.

The `coplot()` and `pairs()` function both take an argument `panel=`
which can be used to customize the type of plot which appears in each
panel. The default is `points()` to produce a scatterplot but by
supplying some other low-level graphics function of two vectors `x` and
`y` as the value of `panel=` you can produce any type of plot you wish.
An example panel function useful for coplots is `panel.smooth()`.
 
### Display graphics {#sec-display-graphics}

Other high-level graphics functions produce different types of plots.
Some examples are:

`qqnorm(x)`\
`qqline(x)`\
`qqplot(x, y)`

:   []{#index-qqnorm-1}

    Distribution-comparison plots. The first form plots the numeric
    vector `x` against the expected Normal order scores (a normal scores
    plot) and the second adds a straight line to such a plot by drawing
    a line through the distribution and data quartiles. The third form
    plots the quantiles of `x` against those of `y` to compare their
    respective distributions.

`hist(x)`\
`hist(x, nclass=``n`{.variable}`)`\
`hist(x, breaks=``b`{.variable}`, …)`

:   []{#index-hist-1}

    Produces a histogram of the numeric vector `x`. A sensible number of
    classes is usually chosen, but a recommendation can be given with
    the `nclass=` argument. Alternatively, the breakpoints can be
    specified exactly with the `breaks=` argument. If the
    `probability=TRUE` argument is given, the bars represent relative
    frequencies divided by bin width instead of counts.

`dotchart(x, …)`

:   []{#index-dotchart}

    Constructs a dotchart of the data in `x`. In a dotchart the *y*-axis
    gives a labelling of the data in `x` and the *x*-axis gives its
    value. For example it allows easy visual selection of all data
    entries with values lying in specified ranges.

`image(x, y, z, …)`\
`contour(x, y, z, …)`\
`persp(x, y, z, …)`

:   []{#index-image}

    Plots of three variables. The `image` plot draws a grid of
    rectangles using different colours to represent the value of `z`,
    the `contour` plot draws contour lines to represent the value of
    `z`, and the `persp` plot draws a 3D surface.
 
### Arguments to high-level plotting functions {#sec-arguments-to-high-level-plotting-functions}

There are a number of arguments which may be passed to high-level
graphics functions, as follows:

`add=TRUE`

:   Forces the function to act as a low-level graphics function,
    superimposing the plot on the current plot (some functions only).

`axes=FALSE`

:   Suppresses generation of axes---useful for adding your own custom
    axes with the `axis()` function. The default, `axes=TRUE`, means
    include axes.

`log="x"`\
`log="y"`\
`log="xy"`

:   Causes the *x*, *y* or both axes to be logarithmic. This will work
    for many, but not all, types of plot.

`type=`

:   The `type=` argument controls the type of plot produced, as follows:

    `type="p"`

    :   Plot individual points (the default)

    `type="l"`

    :   Plot lines

    `type="b"`

    :   Plot points connected by lines (*both*)

    `type="o"`

    :   Plot points overlaid by lines

    `type="h"`

    :   Plot vertical lines from points to the zero axis
        (*high-density*)

    `type="s"`\
    `type="S"`

    :   Step-function plots. In the first form, the top of the vertical
        defines the point; in the second, the bottom.

    `type="n"`

    :   No plotting at all. However axes are still drawn (by default)
        and the coordinate system is set up according to the data. Ideal
        for creating plots with subsequent low-level graphics functions.

`xlab=``string`{.variable}\
`ylab=``string`{.variable}

:   Axis labels for the *x* and *y* axes. Use these arguments to change
    the default labels, usually the names of the objects used in the
    call to the high-level plotting function.

`main=``string`{.variable}

:   Figure title, placed at the top of the plot in a large font.

`sub=``string`{.variable}

:   Sub-title, placed just below the *x*-axis in a smaller font.
 
## Low-level plotting commands {#sec-low-level-plotting-commands}

Sometimes the high-level plotting functions don't produce exactly the
kind of plot you desire. In this case, low-level plotting commands can
be used to add extra information (such as points, lines or text) to the
current plot.

Some of the more useful low-level plotting functions are:

`points(x, y)`\
`lines(x, y)`

:   []{#index-points}

    Adds points or connected lines to the current plot. `plot()`'s
    `type=` argument can also be passed to these functions (and defaults
    to `"p"` for `points()` and `"l"` for `lines()`.)

`text(x, y, labels, …)`

:   []{#index-text}

    Add text to a plot at points given by `x, y`. Normally `labels` is
    an integer or character vector in which case `labels[i]` is plotted
    at point `(x[i], y[i])`. The default is `1:length(x)`.

    **Note**: This function is often used in the sequence

    ``` R
    > plot(x, y, type="n"); text(x, y, names)
    ```

    The graphics parameter `type="n"` suppresses the points but sets up
    the axes, and the `text()` function supplies special characters, as
    specified by the character vector `names` for the points.

`abline(a, b)`\
`abline(h=``y`{.variable}`)`\
`abline(v=``x`{.variable}`)`\
`abline(``lm.obj`{.variable}`)`

:   []{#index-abline}

    Adds a line of slope `b` and intercept `a` to the current plot.
    `h=``y`{.variable} may be used to specify *y*-coordinates for the
    heights of horizontal lines to go across a plot, and
    `v=``x`{.variable} similarly for the *x*-coordinates for vertical
    lines. Also `lm.obj`{.variable} may be list with a `coefficients`
    component of length 2 (such as the result of model-fitting
    functions,) which are taken as an intercept and slope, in that
    order.

`polygon(x, y, …)`

:   []{#index-polygon}

    Draws a polygon defined by the ordered vertices in (`x`, `y`) and
    (optionally) shade it in with hatch lines, or fill it if the
    graphics device allows the filling of figures.

`legend(x, y, legend, …)`

:   []{#index-legend}

    Adds a legend to the current plot at the specified position.
    Plotting characters, line styles, colors etc., are identified with
    the labels in the character vector `legend`. At least one other
    argument `v`{.variable} (a vector the same length as `legend`) with
    the corresponding values of the plotting unit must also be given, as
    follows:

    `legend( , fill=``v`{.variable}`)`

    :   Colors for filled boxes

    `legend( , col=``v`{.variable}`)`

    :   Colors in which points or lines will be drawn

    `legend( , lty=``v`{.variable}`)`

    :   Line styles

    `legend( , lwd=``v`{.variable}`)`

    :   Line widths

    `legend( , pch=``v`{.variable}`)`

    :   Plotting characters (character vector)

`title(main, sub)`

:   []{#index-title}

    Adds a title `main` to the top of the current plot in a large font
    and (optionally) a sub-title `sub` at the bottom in a smaller font.

`axis(side, …)`

:   []{#index-axis}

    Adds an axis to the current plot on the side given by the first
    argument (1 to 4, counting clockwise from the bottom.) Other
    arguments control the positioning of the axis within or beside the
    plot, and tick positions and labels. Useful for adding custom axes
    after calling `plot()` with the `axes=FALSE` argument.

Low-level plotting functions usually require some positioning
information (e.g., *x* and *y* coordinates) to determine where to place
the new plot elements. Coordinates are given in terms of *user
coordinates* which are defined by the previous high-level graphics
command and are chosen based on the supplied data.

Where `x` and `y` arguments are required, it is also sufficient to
supply a single argument being a list with elements named `x` and `y`.
Similarly a matrix with two columns is also valid input. In this way
functions such as `locator()` (see below) may be used to specify
positions on a plot interactively.

 
### Mathematical annotation {#sec-mathematical-annotation}

In some cases, it is useful to add mathematical symbols and formulae to
a plot. This can be achieved in R by specifying an *expression* rather
than a character string in any one of `text`, `mtext`, `axis`, or
`title`. For example, the following code draws the formula for the
Binomial probability function:

``` R
> text(x, y, expression(paste(bgroup("(", atop(n, x), ")"), p^x, q^{n-x})))
```

More information, including a full listing of the features available can
obtained from within R using the commands:

``` R
> help(plotmath)
> example(plotmath)
> demo(plotmath)
```
 
### Hershey vector fonts {#sec-hershey-vector-fonts}

It is possible to specify Hershey vector fonts for rendering text when
using the `text` and `contour` functions. There are three reasons for
using the Hershey fonts:

-   Hershey fonts can produce better output, especially on a computer
    screen, for rotated and/or small text.
-   Hershey fonts provide certain symbols that may not be available in
    the standard fonts. In particular, there are zodiac signs,
    cartographic symbols and astronomical symbols.
-   Hershey fonts provide Cyrillic and Japanese (Kana and Kanji)
    characters.

More information, including tables of Hershey characters can be obtained
from within R using the commands:

``` R
> help(Hershey)
> demo(Hershey)
> help(Japanese)
> demo(Japanese)
```
 
## Interacting with graphics {#sec-interacting-with-graphics}

R also provides functions which allow users to extract or add
information to a plot using a mouse. The simplest of these is the
`locator()` function:

`locator(n, type)`

:   []{#index-locator}

    Waits for the user to select locations on the current plot using the
    left mouse button. This continues until `n` (default 512) points
    have been selected, or another mouse button is pressed. The `type`
    argument allows for plotting at the selected points and has the same
    effect as for high-level graphics commands; the default is no
    plotting. `locator()` returns the locations of the points selected
    as a list with two components `x` and `y`.

`locator()` is usually called with no arguments. It is particularly
useful for interactively selecting positions for graphic elements such
as legends or labels when it is difficult to calculate in advance where
the graphic should be placed. For example, to place some informative
text near an outlying point, the command

``` R
> text(locator(1), "Outlier", adj=0)
```

may be useful. (`locator()` will be ignored if the current device, such
as `postscript` does not support interactive pointing.)

`identify(x, y, labels)`

:   []{#index-identify}

    Allow the user to highlight any of the points defined by `x` and `y`
    (using the left mouse button) by plotting the corresponding
    component of `labels` nearby (or the index number of the point if
    `labels` is absent). Returns the indices of the selected points when
    another button is pressed.

Sometimes we want to identify particular *points* on a plot, rather than
their positions. For example, we may wish the user to select some
observation of interest from a graphical display and then manipulate
that observation in some way. Given a number of *(x, y)* coordinates in
two numeric vectors `x` and `y`, we could use the `identify()` function
as follows:

``` R
> plot(x, y)
> identify(x, y)
```

The `identify()` functions performs no plotting itself, but simply
allows the user to move the mouse pointer and click the left mouse
button near a point. If there is a point near the mouse pointer it will
be marked with its index number (that is, its position in the `x`/`y`
vectors) plotted nearby. Alternatively, you could use some informative
string (such as a case name) as a highlight by using the `labels`
argument to `identify()`, or disable marking altogether with the
`plot = FALSE` argument. When the process is terminated (see above),
`identify()` returns the indices of the selected points; you can use
these indices to extract the selected points from the original vectors
`x` and `y`.
 
## Using graphics parameters {#sec-using-graphics-parameters}

When creating graphics, particularly for presentation or publication
purposes, R's defaults do not always produce exactly that which is
required. You can, however, customize almost every aspect of the display
using *graphics parameters*. R maintains a list of a large number of
graphics parameters which control things such as line style, colors,
figure arrangement and text justification among many others. Every
graphics parameter has a name (such as `col`, which controls colors,)
and a value (a color number, for example.)

A separate list of graphics parameters is maintained for each active
device, and each device has a default set of parameters when
initialized. Graphics parameters can be set in two ways: either
permanently, affecting all graphics functions which access the current
device; or temporarily, affecting only a single graphics function call.

 
### Permanent changes: The `par()` function {#sec-permanent-changes:-the-par-function}

The `par()` function is used to access and modify the list of graphics
parameters for the current graphics device.

`par()`

:   Without arguments, returns a list of all graphics parameters and
    their values for the current device.

`par(c("col", "lty"))`

:   With a character vector argument, returns only the named graphics
    parameters (again, as a list.)

`par(col=4, lty=2)`

:   With named arguments (or a single list argument), sets the values of
    the named graphics parameters, and returns the original values of
    the parameters as a list.

Setting graphics parameters with the `par()` function changes the value
of the parameters *permanently*, in the sense that all future calls to
graphics functions (on the current device) will be affected by the new
value. You can think of setting graphics parameters in this way as
setting "default" values for the parameters, which will be used by all
graphics functions unless an alternative value is given.

Note that calls to `par()` *always* affect the global values of graphics
parameters, even when `par()` is called from within a function. This is
often undesirable behavior---usually we want to set some graphics
parameters, do some plotting, and then restore the original values so as
not to affect the user's R session. You can restore the initial values
by saving the result of `par()` when making changes, and restoring the
initial values when plotting is complete.

``` r
> oldpar <- par(col=4, lty=2)
  … plotting commands …
> par(oldpar)
```

To save and restore *all* settable[^24] graphical
parameters use

``` r
> oldpar <- par(no.readonly=TRUE)
  … plotting commands …
> par(oldpar)
```

[^24]: Some graphics parameters such as the size of the current device are
for information only.
 
### Temporary changes: Arguments to graphics functions {#sec-temporary-changes:-arguments-to-graphics-functions}

Graphics parameters may also be passed to (almost) any graphics function
as named arguments. This has the same effect as passing the arguments to
the `par()` function, except that the changes only last for the duration
of the function call. For example:

``` R
> plot(x, y, pch="+")
```

produces a scatterplot using a plus sign as the plotting character,
without changing the default plotting character for future plots.

Unfortunately, this is not implemented entirely consistently and it is
sometimes necessary to set and reset graphics parameters using `par()`.
 
## Graphics parameters list {#sec-graphics-parameters-list}

The following sections detail many of the commonly-used graphical
parameters. The R help documentation for the `par()` function provides a
more concise summary; this is provided as a somewhat more detailed
alternative.

Graphics parameters will be presented in the following form:

`name`{.variable}`=``value`{.variable}

:   A description of the parameter's effect. `name`{.variable} is the
    name of the parameter, that is, the argument name to use in calls to
    `par()` or a graphics function. `value`{.variable} is a typical
    value you might use when setting the parameter.

Note that `axes` is **not** a graphics parameter but an argument to a
few `plot` methods: see `xaxt` and `yaxt`.

 
### Graphical elements {#sec-graphical-elements}

R plots are made up of points, lines, text and polygons (filled
regions.) Graphical parameters exist which control how these *graphical
elements* are drawn, as follows:

`pch="+"`

:   Character to be used for plotting points. The default varies with
    graphics drivers, but it is usually a circle. Plotted points tend to
    appear slightly above or below the appropriate position unless you
    use `"."` as the plotting character, which produces centered points.

`pch=4`

:   When `pch` is given as an integer between 0 and 25 inclusive, a
    specialized plotting symbol is produced. To see what the symbols
    are, use the command

    ``` R
    > legend(locator(1), as.character(0:25), pch = 0:25)
    ```

    Those from 21 to 25 may appear to duplicate earlier symbols, but can
    be coloured in different ways: see the help on `points` and its
    examples.

    In addition, `pch` can be a character or a number in the range
    `32:255` representing a character in the current font.

`lty=2`

:   Line types. Alternative line styles are not supported on all
    graphics devices (and vary on those that do) but line type 1 is
    always a solid line, line type 0 is always invisible, and line types
    2 and onwards are dotted or dashed lines, or some combination of
    both.

`lwd=2`

:   Line widths. Desired width of lines, in multiples of the "standard"
    line width. Affects axis lines as well as lines drawn with
    `lines()`, etc. Not all devices support this, and some have
    restrictions on the widths that can be used.

`col=2`

:   Colors to be used for points, lines, text, filled regions and
    images. A number from the current palette (see `?palette`) or a
    named colour.

`col.axis`\
`col.lab`\
`col.main`\
`col.sub`

:   The color to be used for axis annotation, *x* and *y* labels, main
    and sub-titles, respectively.

`font=2`

:   An integer which specifies which font to use for text. If possible,
    device drivers arrange so that `1` corresponds to plain text, `2` to
    bold face, `3` to italic, `4` to bold italic and `5` to a symbol
    font (which include Greek letters).

`font.axis`\
`font.lab`\
`font.main`\
`font.sub`

:   The font to be used for axis annotation, *x* and *y* labels, main
    and sub-titles, respectively.

`adj=-0.1`

:   Justification of text relative to the plotting position. `0` means
    left justify, `1` means right justify and `0.5` means to center
    horizontally about the plotting position. The actual value is the
    proportion of text that appears to the left of the plotting
    position, so a value of `-0.1` leaves a gap of 10% of the text width
    between the text and the plotting position.

`cex=1.5`

:   Character expansion. The value is the desired size of text
    characters (including plotting characters) relative to the default
    text size.

`cex.axis`\
`cex.lab`\
`cex.main`\
`cex.sub`

:   The character expansion to be used for axis annotation, *x* and *y*
    labels, main and sub-titles, respectively.
 
### Axes and tick marks {#sec-axes-and-tick-marks}

Many of R's high-level plots have axes, and you can construct axes
yourself with the low-level `axis()` graphics function. Axes have three
main components: the *axis line* (line style controlled by the `lty`
graphics parameter), the *tick marks* (which mark off unit divisions
along the axis line) and the *tick labels* (which mark the units.) These
components can be customized with the following graphics parameters.

`lab=c(5, 7, 12)`

:   The first two numbers are the desired number of tick intervals on
    the *x* and *y* axes respectively. The third number is the desired
    length of axis labels, in characters (including the decimal point.)
    Choosing a too-small value for this parameter may result in all tick
    labels being rounded to the same number!

`las=1`

:   Orientation of axis labels. `0` means always parallel to axis, `1`
    means always horizontal, and `2` means always perpendicular to the
    axis.

`mgp=c(3, 1, 0)`

:   Positions of axis components. The first component is the distance
    from the axis label to the axis position, in text lines. The second
    component is the distance to the tick labels, and the final
    component is the distance from the axis position to the axis line
    (usually zero). Positive numbers measure outside the plot region,
    negative numbers inside.

`tck=0.01`

:   Length of tick marks, as a fraction of the size of the plotting
    region. When `tck` is small (less than 0.5) the tick marks on the
    *x* and *y* axes are forced to be the same size. A value of 1 gives
    grid lines. Negative values give tick marks outside the plotting
    region. Use `tck=0.01` and `mgp=c(1,-1.5,0)` for internal tick
    marks.

`xaxs="r"`\
`yaxs="i"`

:   Axis styles for the *x* and *y* axes, respectively. With styles
    `"i"` (internal) and `"r"` (the default) tick marks always fall
    within the range of the data, however style `"r"` leaves a small
    amount of space at the edges.
 
### Figure margins {#sec-figure-margins}

A single plot in R is known as a `figure` and comprises a *plot region*
surrounded by margins (possibly containing axis labels, titles, etc.)
and (usually) bounded by the axes themselves.

A typical figure is

![images/fig11](images/fig11.png)

Graphics parameters controlling figure layout include:

`mai=c(1, 0.5, 0.5, 0)`

:   Widths of the bottom, left, top and right margins, respectively,
    measured in inches.

`mar=c(4, 2, 2, 1)`

:   Similar to `mai`, except the measurement unit is text lines.

`mar` and `mai` are equivalent in the sense that setting one changes the
value of the other. The default values chosen for this parameter are
often too large; the right-hand margin is rarely needed, and neither is
the top margin if no title is being used. The bottom and left margins
must be large enough to accommodate the axis and tick labels.
Furthermore, the default is chosen without regard to the size of the
device surface: for example, using the `postscript()` driver with the
`height=4` argument will result in a plot which is about 50% margin
unless `mar` or `mai` are set explicitly. When multiple figures are in
use (see below) the margins are reduced, however this may not be enough
when many figures share the same page.
 
### Multiple figure environment {#sec-multiple-figure-environment}

R allows you to create an *n* by *m* array of figures on a single page.
Each figure has its own margins, and the array of figures is optionally
surrounded by an *outer margin*, as shown in the following figure.

![images/fig12](images/fig12.png)

The graphical parameters relating to multiple figures are as follows:

`mfcol=c(3, 2)`\
`mfrow=c(2, 4)`

:   Set the size of a multiple figure array. The first value is the
    number of rows; the second is the number of columns. The only
    difference between these two parameters is that setting `mfcol`
    causes figures to be filled by column; `mfrow` fills by rows.

    The layout in the Figure could have been created by setting
    `mfrow=c(3,2)`; the figure shows the page after four plots have been
    drawn.

    Setting either of these can reduce the base size of symbols and text
    (controlled by `par("cex")` and the pointsize of the device). In a
    layout with exactly two rows and columns the base size is reduced by
    a factor of 0.83: if there are three or more of either rows or
    columns, the reduction factor is 0.66.

`mfg=c(2, 2, 3, 2)`

:   Position of the current figure in a multiple figure environment. The
    first two numbers are the row and column of the current figure; the
    last two are the number of rows and columns in the multiple figure
    array. Set this parameter to jump between figures in the array. You
    can even use different values for the last two numbers than the
    *true* values for unequally-sized figures on the same page.

`fig=c(4, 9, 1, 4)/10`

:   Position of the current figure on the page. Values are the positions
    of the left, right, bottom and top edges respectively, as a
    percentage of the page measured from the bottom left corner. The
    example value would be for a figure in the bottom right of the page.
    Set this parameter for arbitrary positioning of figures within a
    page. If you want to add a figure to a current page, use `new=TRUE`
    as well (unlike S).

`oma=c(2, 0, 3, 0)`\
`omi=c(0, 0, 0.8, 0)`

:   Size of outer margins. Like `mar` and `mai`, the first measures in
    text lines and the second in inches, starting with the bottom margin
    and working clockwise.

Outer margins are particularly useful for page-wise titles, etc. Text
can be added to the outer margins with the `mtext()` function with
argument `outer=TRUE`. There are no outer margins by default, however,
so you must create them explicitly using `oma` or `omi`.

More complicated arrangements of multiple figures can be produced by the
`split.screen()` and `layout()` functions, as well as by the **grid**
and [**lattice**](https://CRAN.R-project.org/package=lattice) packages.
 
## Device drivers {#sec-device-drivers}

R can generate graphics (of varying levels of quality) on almost any
type of display or printing device. Before this can begin, however, R
needs to be informed what type of device it is dealing with. This is
done by starting a *device driver*. The purpose of a device driver is to
convert graphical instructions from R ("draw a line," for example) into
a form that the particular device can understand.

Device drivers are started by calling a device driver function. There is
one such function for every device driver: type `help(Devices)` for a
list of them all. For example, issuing the command

``` R
> postscript()
```

causes all future graphics output to be sent to the printer in
PostScript format. Some commonly-used device drivers are:

`X11()`

:   []{#index-X11}

    For use with the X11 window system on Unix-alikes

`windows()`

:   []{#index-windows}

    For use on Windows

`quartz()`

:   []{#index-quartz}

    For use on macOS

`postscript()`

:   []{#index-postscript}

    For printing on PostScript printers, or creating PostScript graphics
    files.

`pdf()`

:   []{#index-pdf}

    Produces a PDF file, which can also be included into PDF files.

`png()`

:   []{#index-png}

    Produces a bitmap PNG file. (Not always available: see its help
    page.)

`jpeg()`

:   []{#index-jpeg}

    Produces a bitmap JPEG file, best used for `image` plots. (Not
    always available: see its help page.)

When you have finished with a device, be sure to terminate the device
driver by issuing the command

``` R
> dev.off()
```

This ensures that the device finishes cleanly; for example in the case
of hardcopy devices this ensures that every page is completed and has
been sent to the printer. (This will happen automatically at the normal
end of a session.)

 
### PostScript diagrams for typeset documents {#sec-postscript-diagrams-for-typeset-documents}

By passing the `file` argument to the `postscript()` device driver
function, you may store the graphics in PostScript format in a file of
your choice. The plot will be in landscape orientation unless the
`horizontal=FALSE` argument is given, and you can control the size of
the graphic with the `width` and `height` arguments (the plot will be
scaled as appropriate to fit these dimensions.) For example, the command

``` c
> postscript("file.ps", horizontal=FALSE, height=5, pointsize=10)
```

will produce a file containing PostScript code for a figure five inches
high, perhaps for inclusion in a document. It is important to note that
if the file named in the command already exists, it will be overwritten.
This is the case even if the file was only created earlier in the same R
session.

Many usages of PostScript output will be to incorporate the figure in
another document. This works best when *encapsulated* PostScript is
produced: R always produces conformant output, but only marks the output
as such when the `onefile=FALSE` argument is supplied. This unusual
notation stems from S-compatibility: it really means that the output
will be a single page (which is part of the EPSF specification). Thus to
produce a plot for inclusion use something like

``` c
> postscript("plot1.eps", horizontal=FALSE, onefile=FALSE,
             height=8, width=6, pointsize=10)
```
 
### Multiple graphics devices {#sec-multiple-graphics-devices}

In advanced use of R it is often useful to have several graphics devices
in use at the same time. Of course only one graphics device can accept
graphics commands at any one time, and this is known as the *current
device*. When multiple devices are open, they form a numbered sequence
with names giving the kind of device at any position.

The main commands used for operating with multiple devices, and their
meanings are as follows:

`X11()`

:   \[UNIX\]

`windows()`\
`win.printer()`\
`win.metafile()`

:   \[Windows\]

`quartz()`

:   \[macOS\]

`postscript()`\
`pdf()`\
`png()`\
`jpeg()`\
`tiff()`\
`bitmap()`\
`…`

:   Each new call to a device driver function opens a new graphics
    device, thus extending by one the device list. This device becomes
    the current device, to which graphics output will be sent.

`dev.list()`

:   []{#index-dev_002elist}

    Returns the number and name of all active devices. The device at
    position 1 on the list is always the *null device* which does not
    accept graphics commands at all.

`dev.next()`\
`dev.prev()`

:   []{#index-dev_002enext}

    Returns the number and name of the graphics device next to, or
    previous to the current device, respectively.

`dev.set(which=``k`{.variable}`)`

:   []{#index-dev_002eset}

    Can be used to change the current graphics device to the one at
    position `k`{.variable} of the device list. Returns the number and
    label of the device.

`dev.off(``k`{.variable}`)`

:   []{#index-dev_002eoff}

    Terminate the graphics device at point `k`{.variable} of the device
    list. For some devices, such as `postscript` devices, this will
    either print the file immediately or correctly complete the file for
    later printing, depending on how the device was initiated.

`dev.copy(device, …, which=``k`{.variable}`)`\
`dev.print(device, …, which=``k`{.variable}`)`

:   Make a copy of the device `k`{.variable}. Here `device` is a device
    function, such as `postscript`, with extra arguments, if needed,
    specified by '`…`'. `dev.print` is similar, but the copied
    device is immediately closed, so that end actions, such as printing
    hardcopies, are immediately performed.

`graphics.off()`

:   Terminate all graphics devices on the list, except the null device.
 
## Dynamic graphics {#sec-dynamic-graphics}

R does not have builtin capabilities for dynamic or interactive
graphics, e.g. rotating point clouds or to "brushing" (interactively
highlighting) points. However, extensive dynamic graphics facilities are
available in the system GGobi by Swayne, Cook and Buja available from

> <http://ggobi.org/>

and these can be accessed from R via the package
[**rggobi**](https://CRAN.R-project.org/package=rggobi), described at
<http://ggobi.org/rggobi.html>.

Also, package [**rgl**](https://CRAN.R-project.org/package=rgl) provides
ways to interact with 3D plots, for example of surfaces.
 
# Packages {#sec-packages}

All R functions and datasets are stored in *packages*. Only when a
package is loaded are its contents available. This is done both for
efficiency (the full list would take more memory and would take longer
to search than a subset), and to aid package developers, who are
protected from name clashes with other code. The process of developing
packages is described in [Creating R
packages](https://cran.r-project.org/doc/manuals/R-exts.html#Creating-R-packages)
in Writing R Extensions. Here, we will describe them from a user's point
of view.

To see which packages are installed at your site, issue the command

``` r
> library()
```

with no arguments. To load a particular package (e.g., the
[**boot**](https://CRAN.R-project.org/package=boot) package containing
functions from Davison & Hinkley (1997)), use a command like

``` r
> library(boot)
```

Users connected to the Internet can use the `install.packages()` and
`update.packages()` functions (available through the `Packages` menu in
the Windows and macOS GUIs, see [Installing
packages](https://cran.r-project.org/doc/manuals/R-admin.html#Installing-packages)
in R Installation and Administration) to install and update packages.

To see which packages are currently loaded, use

``` R
> search()
```

to display the search list. Some packages may be loaded but not
available on the search list (see [Namespaces](Namespaces.html)): these
will be included in the list given by

``` R
> loadedNamespaces()
```

To see a list of all available help topics in an installed package, use

``` R
> help.start()
```

to start the HTML help system, and then navigate to the package listing
in the `Reference` section.

 
## Standard packages {#sec-standard-packages}

The standard (or *base*) packages are considered part of the R source
code. They contain the basic functions that allow R to work, and the
datasets and standard statistical and graphical functions that are
described in this manual. They should be automatically available in any
R installation. See [R
packages](../R-FAQ/Which-add-on-packages-exist-for-R_003f.html#which-add-on-packages-exist-for-r_003f)
in R FAQ, for a complete list.
 
## Contributed packages and CRAN {#sec-contributed-packages-and-cran}

There are thousands of contributed packages for R, written by many
different authors. Some of these packages implement specialized
statistical methods, others give access to data or hardware, and others
are designed to complement textbooks. Some (the *recommended* packages)
are distributed with every binary distribution of R. Most are available
for download from CRAN (<https://CRAN.R-project.org/> and its mirrors)
and other repositories such as Bioconductor
(<https://www.bioconductor.org/>). The *R FAQ* contains a list of CRAN
packages current at the time of release, but the collection of available
packages changes very frequently.
 
## Namespaces {#sec-namespaces}

Packages have *namespaces*, which do three things: they allow the
package writer to hide functions and data that are meant only for
internal use, they prevent functions from breaking when a user (or other
package writer) picks a name that clashes with one in the package, and
they provide a way to refer to an object within a particular package.

For example, `t()` is the transpose function in R, but users might
define their own function named `t`. Namespaces prevent the user's
definition from taking precedence, and breaking every function that
tries to transpose a matrix.

There are two operators that work with namespaces. The double-colon
operator `::` selects definitions from a particular namespace. In the
example above, the transpose function will always be available as
`base::t`, because it is defined in the `base` package. Only functions
that are exported from the package can be retrieved in this way.

The triple-colon operator `:::` may be seen in a few places in R code:
it acts like the double-colon operator but also allows access to hidden
objects. Users are more likely to use the `getAnywhere()` function,
which searches multiple packages.

Packages are often inter-dependent, and loading one may cause others to
be automatically loaded. The colon operators described above will also
cause automatic loading of the associated package. When packages with
namespaces are loaded automatically they are not added to the search
list.
 
# OS facilities {#sec-os-facilities}

R has quite extensive facilities to access the OS under which it is
running: this allows it to be used as a scripting language and that
ability is much used by R itself, for example to install packages.

Because R's own scripts need to work across all platforms, considerable
effort has gone into make the scripting facilities as
platform-independent as is feasible.

 
## Files and directories {#sec-files-and-directories}

There are many functions to manipulate files and directories. Here are
pointers to some of the more commonly used ones.

To create an (empty) file or directory, use `file.create` or
`dir.create`. (These are the analogues of the POSIX utilities `touch`
and `mkdir`.) For temporary files and directories in the R session
directory see `tempfile`.

Files can be removed by either `file.remove` or `unlink`: the latter can
remove directory trees.

For directory listings use `list.files` (also available as `dir`) or
`list.dirs`. These can select files using a regular expression: to
select by wildcards use `Sys.glob`.

Many types of information on a filepath (including for example if it is
a file or directory) can be found by `file.info`.

There are several ways to find out if a file 'exists' (a file can exist
on the filesystem and not be visible to the current user). There are
functions `file.exists`, `file.access` and `file_test` with various
versions of this test: `file_test` is a version of the POSIX `test`
command for those familiar with shell scripting.

Function `file.copy` is the R analogue of the POSIX command `cp`.

Choosing files can be done interactively by `file.choose`: the Windows
port has the more versatile functions `choose.files` and `choose.dir`
and there are similar functions in the **tcltk** package:
`tk_choose.files` and `tk_choose.dir`.

Functions `file.show` and `file.edit` will display and edit one or more
files in a way appropriate to the R port, using the facilities of a
console (such as RGui on Windows or R.app on macOS) if one is in use.

There is some support for *links* in the filesystem: see functions
`file.link` and `Sys.readlink`.
 
## Filepaths {#sec-filepaths}

With a few exceptions, R relies on the underlying OS functions to
manipulate filepaths. Some aspects of this are allowed to depend on the
OS, and do, even down to the version of the OS. There are POSIX
standards for how OSes should interpret filepaths and many R users
assume POSIX compliance: but Windows does not claim to be compliant and
other OSes may be less than completely compliant.

The following are some issues which have been encountered with
filepaths.

-   POSIX filesystems are case-sensitive, so `foo.png` and
    `Foo.PNG` are different files. However, the defaults on
    Windows and macOS are to be case-insensitive, and FAT filesystems
    (commonly used on removable storage) are not normally case-sensitive
    (and all filepaths may be mapped to lower case).
-   Almost all the Windows' OS services support the use of slash or
    backslash as the filepath separator, and R converts the known
    exceptions to the form required by Windows.
-   The behaviour of filepaths with a trailing slash is OS-dependent.
    Such paths are not valid on Windows and should not be expected to
    work. POSIX-2008 requires such paths to match only directories, but
    earlier versions allowed them to also match files. So they are best
    avoided.
-   Multiple slashes in filepaths such as `/abc//def` are valid
    on POSIX filesystems and treated as if there was only one slash.
    They are *usually* accepted by Windows' OS functions. However,
    leading double slashes may have a different meaning.
-   Windows' UNC filepaths (such as `\\server\dir1\dir2\file`
    and `\\?\UNC\server\dir1\dir2\file`) are not supported, but
    they may work in some R functions. POSIX filesystems are allowed to
    treat a leading double slash specially.
-   Windows allows filepaths containing drives and relative to the
    current directory on a drive, e.g. `d:foo/bar` refers to
    `d:/a/b/c/foo/bar` if the current directory *on drive
    `d:`* is `/a/b/c`. It is intended that these work,
    but the use of absolute paths is safer.

Functions `basename` and `dirname` select parts of a file path: the
recommended way to assemble a file path from components is `file.path`.
Function `pathexpand` does 'tilde expansion', substituting values for
home directories (the current user's, and perhaps those of other users).

On filesystems with links, a single file can be referred to by many
filepaths. Function `normalizePath` will find a canonical filepath.

Windows has the concepts of short ('8.3') and long file names:
`normalizePath` will return an absolute path using long file names and
`shortPathName` will return a version using short names. The latter does
not contain spaces and uses backslash as the separator, so is sometimes
useful for exporting names from R.

File *permissions* are a related topic. R has support for the POSIX
concepts of read/write/execute permission for owner/group/all but this
may be only partially supported on the filesystem, so for example on
Windows only read-only files (for the account running the R session) are
recognized. Access Control Lists (ACLs) are employed on several
filesystems, but do not have an agreed standard and R has no facilities
to control them. Use `Sys.chmod` to change permissions.
 
## System commands {#sec-system-commands}

Functions `system` and `system2` are used to invoke a system command and
optionally collect its output. `system2` is a little more general but
its main advantage is that it is easier to write cross-platform code
using it.

`system` behaves differently on Windows from other OSes (because the API
C call of that name does). Elsewhere it invokes a shell to run the
command: the Windows port of R has a function `shell` to do that.

To find out if the OS includes a command, use `Sys.which`, which
attempts to do this in a cross-platform way (unfortunately it is not a
standard OS service).

Function `shQuote` will quote filepaths as needed for commands in the
current OS.
 
## Compression and Archives {#sec-compression-and-archives}

Recent versions of R have extensive facilities to read and write
compressed files, often transparently. Reading of files in R is to a
very large extent done by *connections*, and the `file` function which
is used to open a connection to a file (or a URL) and is able to
identify the compression used from the 'magic' header of the file.

The type of compression which has been supported for longest is `gzip`
compression, and that remains a good general compromise. Files
compressed by the earlier Unix `compress` utility can also be read, but
these are becoming rare. Two other forms of compression, those of the
`bzip2` and `xz` utilities are also available. These generally achieve
higher rates of compression (depending on the file, much higher) at the
expense of slower decompression and much slower compression.

There is some confusion between `xz` and `lzma` compression (see
<https://en.wikipedia.org/wiki/Xz> and
<https://en.wikipedia.org/wiki/LZMA>): R can read files compressed by
most versions of either.

File archives are single files which contain a collection of files, the
most common ones being 'tarballs' and zip files as used to distribute R
packages. R can list and unpack both (see functions `untar` and `unzip`)
and create both (for `zip` with the help of an external program).
 
## B.1 Invoking R from the command line {#sec-b.1-invoking-r-from-the-command-line}

When working at a command line on UNIX or Windows, the command
'`R`' can be used both for starting the main R program in the
form

::: display
``` R
R [options] [<infile] [>outfile],
```
:::

or, via the `R CMD` interface, as a wrapper to various R tools (e.g.,
for processing files in R documentation format or manipulating add-on
packages) which are not intended to be called "directly".

At the Windows command-line, `Rterm.exe` is preferred to `R`.

You need to ensure that either the environment variable `TMPDIR` is
unset or it points to a valid place to create temporary files and
directories.

Most options control what happens at the beginning and at the end of an
R session. The startup mechanism is as follows (see also the on-line
help for topic '`Startup`' for more information, and the
section below for some Windows-specific details).

-   Unless `--no-environ` was given, R searches for user and
    site files to process for setting environment variables. The name of
    the site file is the one pointed to by the environment variable
    `R_ENVIRON`; if this is unset,
    `R_HOME`{.variable}`/etc/Renviron.site` is used (if it
    exists). The user file is the one pointed to by the environment
    variable `R_ENVIRON_USER` if this is set; otherwise, files
    `.Renviron` in the current or in the user's home directory
    (in that order) are searched for. These files should contain lines
    of the form '`name`{.variable}`=``value`{.variable}'. (See
    `help("Startup")` for a precise description.) Variables you might
    want to set include `R_PAPERSIZE` (the default paper size),
    `R_PRINTCMD` (the default print command) and `R_LIBS` (specifies the
    list of R library trees searched for add-on packages).
-   Then R searches for the site-wide startup profile unless the command
    line option `--no-site-file` was given. The name of this
    file is taken from the value of the `R_PROFILE` environment
    variable. If that variable is unset, the default
    `R_HOME`{.variable}`/etc/Rprofile.site` is used if this
    exists.
-   Then, unless `--no-init-file` was given, R searches for a
    user profile and sources it. The name of this file is taken from the
    environment variable `R_PROFILE_USER`; if unset, a file called
    `.Rprofile` in the current directory or in the user's home
    directory (in that order) is searched for.
-   It also loads a saved workspace from file `.RData` in the
    current directory if there is one (unless `--no-restore` or
    `--no-restore-data` was specified).
-   Finally, if a function `.First()` exists, it is executed. This
    function (as well as `.Last()` which is executed at the end of the R
    session) can be defined in the appropriate startup profiles, or
    reside in `.RData`.

In addition, there are options for controlling the memory available to
the R process (see the on-line help for topic '`Memory`' for
more information). Users will not normally need to use these unless they
are trying to limit the amount of memory used by R.

R accepts the following command-line options.

`--help`\
`-h`

:   Print short help message to standard output and exit successfully.

`--version`

:   Print version information to standard output and exit successfully.

`--encoding=``enc`{.variable}

:   Specify the encoding to be assumed for input from the console or
    `stdin`. This needs to be an encoding known to `iconv`: see its help
    page. (`--encoding ``enc`{.variable} is also accepted.) The input is
    re-encoded to the locale R is running in and needs to be
    representable in the latter's encoding (so e.g. you cannot re-encode
    Greek text in a French locale unless that locale uses the UTF-8
    encoding).

`RHOME`

:   Print the path to the R "home directory" to standard output and exit
    successfully. Apart from the front-end shell script and the man
    page, R installation puts everything (executables, packages, etc.)
    into this directory.

`--save`\
`--no-save`

:   Control whether data sets should be saved or not at the end of the R
    session. If neither is given in an interactive session, the user is
    asked for the desired behavior when ending the session with
    [q()]{.kbd}; in non-interactive use one of these must be specified
    or implied by some other option (see below).

`--no-environ`

:   Do not read any user file to set environment variables.

`--no-site-file`

:   Do not read the site-wide profile at startup.

`--no-init-file`

:   Do not read the user's profile at startup.

`--restore`\
`--no-restore`\
`--no-restore-data`

:   Control whether saved images (file `.RData` in the
    directory where R was started) should be restored at startup or not.
    The default is to restore. (`--no-restore` implies all the
    specific `--no-restore-*` options.)

`--no-restore-history`

:   Control whether the history file (normally file `.Rhistory`
    in the directory where R was started, but can be set by the
    environment variable `R_HISTFILE`) should be restored at startup or
    not. The default is to restore.

`--no-Rconsole`

:   (Windows only) Prevent loading the `Rconsole` file at
    startup.

`--vanilla`

:   Combine `--no-save`, `--no-environ`,
    `--no-site-file`, `--no-init-file` and
    `--no-restore`. Under Windows, this also includes
    `--no-Rconsole`.

`-f ``file`{.variable}\
`--file=``file`{.variable}

:   (not `Rgui.exe`) Take input from `file`{.variable}: '`-`'
    means `stdin`. Implies `--no-save` unless `--save`
    has been set. On a Unix-alike, shell metacharacters should be
    avoided in `file`{.variable} (but spaces are allowed).

`-e ``expression`{.variable}

:   (not `Rgui.exe`) Use `expression`{.variable} as an input line. One
    or more `-e` options can be used, but not together with
    `-f` or `--file`. Implies `--no-save`
    unless `--save` has been set. (There is a limit of 10,000
    bytes on the total length of expressions used in this way.
    Expressions containing spaces or shell metacharacters will need to
    be quoted.)

`--no-readline`

:   (UNIX only) Turn off command-line editing via **readline**. This is
    useful when running R from within Emacs using the ESS ("Emacs Speaks
    Statistics") package. See @sec-appendix-c-the-command-line-editor
    , for more information.
    Command-line editing is enabled for default interactive use (see
    `--interactive`). This option also affects tilde-expansion:
    see the help for `path.expand`.

`--min-vsize=``N`{.variable}\
`--min-nsize=``N`{.variable}

:   For expert use only: set the initial trigger sizes for garbage
    collection of vector heap (in bytes) and *cons cells* (number)
    respectively. Suffix '`M`' specifies megabytes or millions
    of cells respectively. The defaults are 6Mb and 350k respectively
    and can also be set by environment variables `R_NSIZE` and
    `R_VSIZE`.

`--max-ppsize=``N`{.variable}

:   Specify the maximum size of the pointer protection stack as
    `N`{.variable} locations. This defaults to 10000, but can be
    increased to allow large and complicated calculations to be done.
    Currently the maximum value accepted is 100000.

`--quiet`\
`--silent`\
`-q`

:   Do not print out the initial copyright and welcome messages.

`--no-echo`

:   Make R run as quietly as possible. This option is intended to
    support programs which use R to compute results for them. It implies
    `--quiet` and `--no-save`.

`--interactive`

:   (UNIX only) Assert that R really is being run interactively even if
    input has been redirected: use if input is from a FIFO or pipe and
    fed from an interactive program. (The default is to deduce that R is
    being run interactively if and only if `stdin` is connected
    to a terminal or `pty`.) Using `-e`, `-f` or
    `--file` asserts non-interactive use even if
    `--interactive` is given.

    Note that this does not turn on command-line editing.

`--ess`

:   (Windows only) Set `Rterm` up for use by `R-inferior-mode` in ESS,
    including asserting interactive use (without the command-line
    editor) and no buffering of `stdout`.

`--verbose`

:   Print more information about progress, and in particular set R's
    option `verbose` to `TRUE`. R code uses this option to control the
    printing of diagnostic messages.

`--debugger=``name`{.variable}\
`-d ``name`{.variable}

:   (UNIX only) Run R through debugger `name`{.variable}. For most
    debuggers (the exceptions are `valgrind` and recent versions of
    `gdb`), further command line options are disregarded, and should
    instead be given when starting the R executable from inside the
    debugger.

`--gui=``type`{.variable}\
`-g ``type`{.variable}

:   (UNIX only) Use `type`{.variable} as graphical user interface (note
    that this also includes interactive graphics). Currently, possible
    values for `type`{.variable} are '`X11`' (the default) and,
    provided that '`Tcl/Tk`' support is available,
    '`Tk`'. (For back-compatibility, '`x11`' and
    '`tk`' are accepted.)

`--arch=``name`{.variable}

:   (UNIX only) Run the specified sub-architecture.

`--args`

:   This flag does nothing except cause the rest of the command line to
    be skipped: this can be useful to retrieve values from it with
    `commandArgs(TRUE)`.

Note that input and output can be redirected in the usual way (using
'`<`' and '`>`'), but the line length limit of 4095
bytes still applies. Warning and error messages are sent to the error
channel (`stderr`).

The command `R CMD` allows the invocation of various tools which are
useful in conjunction with R, but not intended to be called "directly".
The general form is

``` bash
R CMD command args
```

where `command`{.variable} is the name of the tool and `args`{.variable}
the arguments passed on to it.

Currently, the following tools are available.

`BATCH`

:   Run R in batch mode. Runs `R --restore --save` with possibly further
    options (see `?BATCH`).

`COMPILE`

:   (UNIX only) Compile C, C++, Fortran ... files for use with R.

`SHLIB`

:   Build shared library for dynamic loading.

`INSTALL`

:   Install add-on packages.

`REMOVE`

:   Remove add-on packages.

`build`

:   Build (that is, package) add-on packages.

`check`

:   Check add-on packages.

`LINK`

:   (UNIX only) Front-end for creating executable programs.

`Rprof`

:   Post-process R profiling files.

`Rdconv`\
`Rd2txt`

:   Convert Rd format to various other formats, including HTML, LaTeX,
    plain text, and extracting the examples. `Rd2txt` can be used as
    shorthand for `Rd2conv -t txt`.

`Rd2pdf`

:   Convert Rd format to PDF.

`Stangle`

:   Extract S/R code from Sweave or other vignette documentation

`Sweave`

:   Process Sweave or other vignette documentation

`Rdiff`

:   Diff R output ignoring headers etc

`config`

:   Obtain configuration information

`javareconf`

:   (Unix only) Update the Java configuration variables

`rtags`

:   (Unix only) Create Emacs-style tag files from C, R, and Rd files

`open`

:   (Windows only) Open a file via Windows' file associations

`texify`

:   (Windows only) Process (La)TeX files with R's style files

Use

``` bash
R CMD command --help
```

to obtain usage information for each of the tools accessible via the
`R CMD` interface.

In addition, you can use options `--arch=`,
`--no-environ`, `--no-init-file`,
`--no-site-file` and `--vanilla` between `R` and
`CMD`: these affect any R processes run by the tools. (Here
`--vanilla` is equivalent to
`--no-environ --no-site-file --no-init-file`.) However, note
that `R CMD` does not of itself use any R startup files (in particular,
neither user nor site `Renviron` files), and all of the R
processes run by these tools (except `BATCH`) use
`--no-restore`. Most use `--vanilla` and so invoke no
R startup files: the current exceptions are `INSTALL`, `REMOVE`,
`Sweave` and `SHLIB` (which uses
`--no-site-file --no-init-file`).

``` bash
R CMD cmd args
```

for any other executable `cmd`{.variable} on the path or given by an
absolute filepath: this is useful to have the same environment as R or
the specific commands run under, for example to run `ldd` or `pdflatex`.
Under Windows `cmd`{.variable} can be an executable or a batch file, or
if it has extension `.sh` or `.pl` the appropriate interpreter (if
available) is called to run it.
 
## B.2 Invoking R under Windows {#sec-b.2-invoking-r-under-windows}

There are two ways to run R under Windows. Within a terminal window
(e.g. `cmd.exe` or a more capable shell), the methods described in the
previous section may be used, invoking by `R.exe` or more directly by
`Rterm.exe`. For interactive use, there is a console-based GUI
(`Rgui.exe`).

The startup procedure under Windows is very similar to that under UNIX,
but references to the 'home directory' need to be clarified, as this is
not always defined on Windows. If the environment variable `R_USER` is
defined, that gives the home directory. Next, if the environment
variable `HOME` is defined, that gives the home directory. After those
two user-controllable settings, R tries to find system defined home
directories. It first tries to use the Windows \"personal\" directory
(typically `My Documents` in recent versions of Windows). If that fails,
and environment variables `HOMEDRIVE` and `HOMEPATH` are defined (and
they normally are) these define the home directory. Failing all those,
the home directory is taken to be the starting directory.

You need to ensure that either the environment variables `TMPDIR`, `TMP`
and `TEMP` are either unset or one of them points to a valid place to
create temporary files and directories.

Environment variables can be supplied as
'`name`{.variable}`=``value`{.variable}' pairs on the command
line.

If there is an argument ending `.RData` (in any case) it is
interpreted as the path to the workspace to be restored: it implies
`--restore` and sets the working directory to the parent of the
named file. (This mechanism is used for drag-and-drop and file
association with `RGui.exe`, but also works for `Rterm.exe`. If the
named file does not exist it sets the working directory if the parent
directory exists.)

The following additional command-line options are available when
invoking `RGui.exe`.

`--mdi`\
`--sdi`\
`--no-mdi`

:   Control whether `Rgui` will operate as an MDI program (with multiple
    child windows within one main window) or an SDI application (with
    multiple top-level windows for the console, graphics and pager). The
    command-line setting overrides the setting in the user's
    `Rconsole` file.

`--debug`

:   Enable the "Break to debugger" menu item in `Rgui`, and trigger a
    break to the debugger during command line processing.

Under Windows with `R CMD` you may also specify your own
`.bat`, `.exe`, `.sh` or `.pl` file.
It will be run under the appropriate interpreter (Perl for
`.pl`) with several environment variables set appropriately,
including `R_HOME`, `R_OSTYPE`, `PATH`, `BSTINPUTS` and `TEXINPUTS`. For
example, if you already have `latex.exe` on your path, then

``` bash
R CMD latex.exe mydoc
```

will run LaTeX on `mydoc.tex`, with the path to R's
`share/texmf` macros appended to `TEXINPUTS`. (Unfortunately,
this does not help with the MiKTeX build of LaTeX, but
`R CMD texify mydoc` will work in that case.)
 
## B.3 Invoking R under macOS {#sec-b.3-invoking-r-under-macos}

There are two ways to run R under macOS. Within a `Terminal.app` window
by invoking `R`, the methods described in the first subsection apply.
There is also console-based GUI (`R.app`) that by default is installed
in the `Applications` folder on your system. It is a standard
double-clickable macOS application.

The startup procedure under macOS is very similar to that under UNIX,
but `R.app` does not make use of command-line arguments. The 'home
directory' is the one inside the R.framework, but the startup and
current working directory are set as the user's home directory unless a
different startup directory is given in the Preferences window
accessible from within the GUI.
 
## B.4 Scripting with R {#sec-b.4-scripting-with-r}

If you just want to run a file `foo.R` of R commands, the
recommended way is to use `R CMD BATCH foo.R`. If you want to run this
in the background or as a batch job use OS-specific facilities to do so:
for example in most shells on Unix-alike OSes `R CMD BATCH foo.R &` runs
a background job.

You can pass parameters to scripts via additional arguments on the
command line: for example (where the exact quoting needed will depend on
the shell in use)

``` bash
R CMD BATCH "--args arg1 arg2" foo.R &
```

will pass arguments to a script which can be retrieved as a character
vector by

``` r
args <- commandArgs(TRUE)
```

This is made simpler by the alternative front-end `Rscript`, which can
be invoked by

``` R
Rscript foo.R arg1 arg2
```

and this can also be used to write executable script files like (at
least on Unix-alikes, and in some Windows shells)

``` r
#! /path/to/Rscript
args <- commandArgs(TRUE)
...
q(status=<exit status code>)
```

If this is entered into a text file `runfoo` and this is made
executable (by `chmod 755 runfoo`), it can be invoked for different
arguments by

``` R
runfoo arg1 arg2
```

For further options see `help("Rscript")`. This writes R output to
`stdout` and `stderr`, and this can be redirected in
the usual way for the shell running the command.

If you do not wish to hardcode the path to `Rscript` but have it in your
path (which is normally the case for an installed R except on Windows,
but e.g. macOS users may need to add `/usr/local/bin` to their
path), use

``` R
#! /usr/bin/env Rscript
...
```

At least in Bourne and bash shells, the `#!` mechanism does **not**
allow extra arguments like `#! /usr/bin/env Rscript --vanilla`.

One thing to consider is what `stdin()` refers to. It is commonplace to
write R scripts with segments like

``` r
chem <- scan(n=24)
2.90 3.10 3.40 3.40 3.70 3.70 2.80 2.50 2.40 2.40 2.70 2.20
5.28 3.37 3.03 3.03 28.95 3.77 3.40 2.20 3.50 3.60 3.70 3.70
```

and `stdin()` refers to the script file to allow such traditional usage.
If you want to refer to the process's `stdin`, use `"stdin"` as
a `file` connection, e.g. `scan("stdin", ...)`.

Another way to write executable script files (suggested by François
Pinard) is to use a *here document* like

``` bash
#!/bin/sh
[environment variables can be set here]
R --no-echo [other options] <<EOF

   R program goes here...

EOF
```

but here `stdin()` refers to the program source and `"stdin"` will not
be usable.

Short scripts can be passed to `Rscript` on the command-line *via* the
`-e` flag. (Empty scripts are not accepted.)

Note that on a Unix-alike the input filename (such as `foo.R`)
should not contain spaces nor shell metacharacters.
 
# A sample session {#sec-appendix-a-a-sample-session}

The following session is intended to introduce to you some features of
the R environment by using them. Many features of the system will be
unfamiliar and puzzling at first, but this puzzlement will soon
disappear.

`Start R appropriately for your platform (see `[`Invoking R`](Invoking-R.html)`).`

:   The R program begins, with a banner.

    (Within R code, the prompt on the left hand side will not be shown
    to avoid confusion.)

`help.start()`

:   Start the HTML interface to on-line help (using a web browser
    available at your machine). You should briefly explore the features
    of this facility with the mouse.

    Iconify the help window and move on to the next part.

`x <- rnorm(50)`\
`y <- rnorm(x)`

:   Generate two pseudo-random normal vectors of *x*- and
    *y*-coordinates.

`plot(x, y)`

:   Plot the points in the plane. A graphics window will appear
    automatically.

`ls()`

:   See which R objects are now in the R workspace.

`rm(x, y)`

:   Remove objects no longer needed. (Clean up).

`x <- 1:20`

:   Make *x = (1, 2, ..., 20)*.

`w <- 1 + sqrt(x)/2`

:   A 'weight' vector of standard deviations.

`dummy <- data.frame(x=x, y= x + rnorm(x)*w)`\
`dummy`

:   Make a *data frame* of two columns, *x* and *y*, and look at it.

`fm <- lm(y ~ x, data=dummy)`\
`summary(fm)`

:   Fit a simple linear regression and look at the analysis. With `y` to
    the left of the tilde, we are modelling *y* dependent on *x*.

`fm1 <- lm(y ~ x, data=dummy, weight=1/w^2)`\
`summary(fm1)`

:   Since we know the standard deviations, we can do a weighted
    regression.

`attach(dummy)`

:   Make the columns in the data frame visible as variables.

`lrf <- lowess(x, y)`

:   Make a nonparametric local regression function.

`plot(x, y)`

:   Standard point plot.

`lines(x, lrf$y)`

:   Add in the local regression.

`abline(0, 1, lty=3)`

:   The true regression line: (intercept 0, slope 1).

`abline(coef(fm))`

:   Unweighted regression line.

`abline(coef(fm1), col = "red")`

:   Weighted regression line.

`detach()`

:   Remove data frame from the search path.

`plot(fitted(fm), resid(fm),`\
`     xlab="Fitted values",`\
`     ylab="Residuals",`\
`     main="Residuals vs Fitted")`

:   A standard regression diagnostic plot to check for
    heteroscedasticity. Can you see it?

`qqnorm(resid(fm), main="Residuals Rankit Plot")`

:   A normal scores plot to check for skewness, kurtosis and outliers.
    (Not very useful here.)

`rm(fm, fm1, lrf, x, dummy)`

:   Clean up again.

The next section will look at data from the classical experiment of
Michelson to measure the speed of light. This dataset is available in
the `morley` object, but we will read it to illustrate the `read.table`
function.

`filepath <- system.file("data", "morley.tab" , package="datasets")`\
`filepath`

:   Get the path to the data file.

`file.show(filepath)`

:   Optional. Look at the file.

`mm <- read.table(filepath)`\
`mm`

:   Read in the Michelson data as a data frame, and look at it. There
    are five experiments (column `Expt`) and each has 20 runs (column
    `Run`) and `sl` is the recorded speed of light, suitably coded.

`mm$Expt <- factor(mm$Expt)`\
`mm$Run <- factor(mm$Run)`

:   Change `Expt` and `Run` into factors.

`attach(mm)`

:   Make the data frame visible at position 2 (the default).

`plot(Expt, Speed, main="Speed of Light Data", xlab="Experiment No.")`

:   Compare the five experiments with simple boxplots.

`fm <- aov(Speed ~ Run + Expt, data=mm)`\
`summary(fm)`

:   Analyze as a randomized block, with 'runs' and 'experiments' as
    factors.

`fm0 <- update(fm, . ~ . - Run)`\
`anova(fm0, fm)`

:   Fit the sub-model omitting 'runs', and compare using a formal
    analysis of variance.

`detach()`\
`rm(fm, fm0)`

:   Clean up before moving on.

We now look at some more graphical features: contour and image plots.

`x <- seq(-pi, pi, len=50)`\
`y <- x`

:   *x* is a vector of 50 equally spaced values in the interval \[-pi\\,
    pi\]. *y* is the same.

`f <- outer(x, y, function(x, y) cos(y)/(1 + x^2))`

:   *f* is a square matrix, with rows and columns indexed by *x* and *y*
    respectively, of values of the function cos(y)/(1 + x\^2).

`oldpar <- par(no.readonly = TRUE)`\
`par(pty="s")`

:   Save the plotting parameters and set the plotting region to
    "square".

`contour(x, y, f)`\
`contour(x, y, f, nlevels=15, add=TRUE)`

:   Make a contour map of *f*; add in more lines for more detail.

`fa <- (f-t(f))/2`

:   `fa` is the "asymmetric part" of *f*. (`t()` is transpose).

`contour(x, y, fa, nlevels=15)`

:   Make a contour plot, ...

`par(oldpar)`

:   ... and restore the old graphics parameters.

`image(x, y, f)`\
`image(x, y, fa)`

:   Make some high density image plots, (of which you can get hardcopies
    if you wish), ...

`objects(); rm(x, y, f, fa)`

:   ... and clean up before moving on.

R can do complex arithmetic, also.

`th <- seq(-pi, pi, len=100)`\
`z <- exp(1i*th)`

:   `1i` is used for the complex number *i*.

`par(pty="s")`\
`plot(z, type="l")`

:   Plotting complex arguments means plot imaginary versus real parts.
    This should be a circle.

`w <- rnorm(100) + rnorm(100)*1i`

:   Suppose we want to sample points within the unit circle. One method
    would be to take complex numbers with standard normal real and
    imaginary parts ...

`w <- ifelse(Mod(w) > 1, 1/w, w)`

:   ... and to map any outside the circle onto their reciprocal.

`plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+",xlab="x", ylab="y")`\
`lines(z)`

:   All points are inside the unit circle, but the distribution is not
    uniform.

`w <- sqrt(runif(100))*exp(2*pi*runif(100)*1i)`\
`plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+", xlab="x", ylab="y")`\
`lines(z)`

:   The second method uses the uniform distribution. The points should
    now look more evenly spaced over the disc.

`rm(th, w, z)`

:   Clean up again.

`q()`

:   Quit the R program. You will be asked if you want to save the R
    workspace, and for an exploratory session like this, you probably do
    not want to save it.
 
# Invoking R {#sec-appendix-b-invoking-r}

Users of R on Windows or macOS should read the OS-specific section
first, but command-line use is also supported.

 
# The command-line editor {#sec-appendix-c-the-command-line-editor}

## Preliminaries {#sec-c.1-preliminaries}

When the GNU **readline** library is available at the time R is
configured for compilation under UNIX, an inbuilt command line editor
allowing recall, editing and re-submission of prior commands is used.
Note that other versions of **readline** exist and may be used by the
inbuilt command line editor: this is most common on macOS. You can find
out which version (if any) is available by running `extSoftVersion()` in
an R session.

It can be disabled (useful for usage with ESS [^25])
using the startup option `--no-readline`.

Windows versions of R have somewhat simpler command-line editing: see
'`Console`' under the '`Help`' menu of the GUI, and
the file `README.Rterm` for command-line editing under
`Rterm.exe`.

When using R with GNU[^26] **readline** capabilities,
the functions described below are available, as well as others
(probably) documented in `man readline` or `info readline` on your
system.

Many of these use either Control or Meta characters. Control characters,
such as [Control-m]{.kbd}, are obtained by holding the CTRL down while
you press the m key, and are written as [C-m]{.kbd} below. Meta
characters, such as [Meta-b]{.kbd}, are typed by holding down
META[^27] and pressing b, and written as [M-b]{.kbd}
in the following. If your terminal does not have a META key enabled, you
can still type Meta characters using two-character sequences starting
with [ESC]{.kbd}. Thus, to enter [M-b]{.kbd}, you could type ESCb. The
[ESC]{.kbd} character sequences are also allowed on terminals with real
Meta keys. Note that case is significant for Meta characters.

Some but not all versions[^28] of **readline** will
recognize resizing of the terminal window so this is best avoided.

## Editing actions {#sec-c.2-editing-actions}

The R program keeps a history of the command lines you type, including
the erroneous lines, and commands in your history may be recalled,
changed if necessary, and re-submitted as new commands. In Emacs-style
command-line editing any straight typing you do while in this editing
phase causes the characters to be inserted in the command you are
editing, displacing any characters to the right of the cursor. In *vi*
mode character insertion mode is started by [M-i]{.kbd} or [M-a]{.kbd},
characters are typed and insertion mode is finished by typing a further
ESC. (The default is Emacs-style, and only that is described here: for
*vi* mode see the **readline** documentation.)

Pressing the RET command at any time causes the command to be
re-submitted.

Other editing actions are summarized in the following table.

## Command-line editor summary {#sec-c.3-command-line-editor-summary}

### Command recall and vertical motion {#sec-command-recall-and-vertical-motion}

[C-p]{.kbd}

:   Go to the previous command (backwards in the history).

[C-n]{.kbd}

:   Go to the next command (forwards in the history).

[C-r `text`{.variable}]{.kbd}

:   Find the last command with the `text`{.variable} string in it. This
    can be cancelled by `C-g` (and on some versions of R by `C-c`).

On most terminals, you can also use the up and down arrow keys instead
of [C-p]{.kbd} and [C-n]{.kbd}, respectively.

### Horizontal motion of the cursor {#sec-horizontal-motion-of-the-cursor}

[C-a]{.kbd}

:   Go to the beginning of the command.

[C-e]{.kbd}

:   Go to the end of the line.

[M-b]{.kbd}

:   Go back one word.

[M-f]{.kbd}

:   Go forward one word.

[C-b]{.kbd}

:   Go back one character.

[C-f]{.kbd}

:   Go forward one character.

On most terminals, you can also use the left and right arrow keys
instead of [C-b]{.kbd} and [C-f]{.kbd}, respectively.

### Editing and re-submission {#sec-editing-and-re-submission}

[`text`{.variable}]{.kbd}

:   Insert `text`{.variable} at the cursor.

[C-f `text`{.variable}]{.kbd}

:   Append `text`{.variable} after the cursor.

[[DEL]{.key}]{.kbd}

:   Delete the previous character (left of the cursor).

[C-d]{.kbd}

:   Delete the character under the cursor.

[M-d]{.kbd}

:   Delete the rest of the word under the cursor, and "save" it.

[C-k]{.kbd}

:   Delete from cursor to end of command, and "save" it.

[C-y]{.kbd}

:   Insert (yank) the last "saved" text here.

[C-t]{.kbd}

:   Transpose the character under the cursor with the next.

[M-l]{.kbd}

:   Change the rest of the word to lower case.

[M-c]{.kbd}

:   Change the rest of the word to upper case.

[[RET]{.key}]{.kbd}

:   Re-submit the command to R.

The final RET terminates the command line editing sequence.

The **readline** key bindings can be customized in the usual way *via* a
`~/.inputrc` file. These customizations can be conditioned on
application `R`, that is by including a section like

``` R
$if R
  "\C-xd": "q('no')\n"
$endif
```

[^25]: The 'Emacs Speaks Statistics' package; see the URL
<https://ESS.R-project.org/>

[^26]: It is possible to build R using an emulation of GNU **readline**,
such as one based on NetBSD's **editline** (also known as **libedit**),
in which case only a subset of the capabilities may be provided.

[^27]: On a PC keyboard this is usually the Alt key, occasionally the
'Windows' key. On a Mac keyboard normally no meta key is available.

[^28]: In particular, not versions 6.3 or later: this is worked around as
from R 3.4.0.
 
# Function and variable index {#sec-appendix-d-function-and-variable-index}

  ----------------------------------- ----------------------------------------------------------------------
  Jump to:                            [**!**](#function-and-variable-index_vr_symbol-1){.summary-letter}  
                                      [**%**](#function-and-variable-index_vr_symbol-2){.summary-letter}  
                                      [**&**](#function-and-variable-index_vr_symbol-3){.summary-letter}  
                                      [**\***](#function-and-variable-index_vr_symbol-4){.summary-letter}  
                                      [**+**](#function-and-variable-index_vr_symbol-5){.summary-letter}  
                                      [**-**](#function-and-variable-index_vr_symbol-6){.summary-letter}  
                                      [**.**](#function-and-variable-index_vr_symbol-7){.summary-letter}  
                                      [**/**](#function-and-variable-index_vr_symbol-8){.summary-letter}  
                                      [**:**](#function-and-variable-index_vr_symbol-9){.summary-letter}  
                                      [**\<**](#function-and-variable-index_vr_symbol-10){.summary-letter}  
                                      [**=**](#function-and-variable-index_vr_symbol-11){.summary-letter}  
                                      [**\>**](#function-and-variable-index_vr_symbol-12){.summary-letter}  
                                      [**?**](#function-and-variable-index_vr_symbol-13){.summary-letter}  
                                      [**\^**](#function-and-variable-index_vr_symbol-14){.summary-letter}  
                                      [**\|**](#function-and-variable-index_vr_symbol-15){.summary-letter}  
                                      [**\~**](#function-and-variable-index_vr_symbol-16){.summary-letter}
                                       \
                                      [**A**](#function-and-variable-index_vr_letter-a){.summary-letter}  
                                      [**B**](#function-and-variable-index_vr_letter-b){.summary-letter}  
                                      [**C**](#function-and-variable-index_vr_letter-c){.summary-letter}  
                                      [**D**](#function-and-variable-index_vr_letter-d){.summary-letter}  
                                      [**E**](#function-and-variable-index_vr_letter-e){.summary-letter}  
                                      [**F**](#function-and-variable-index_vr_letter-f){.summary-letter}  
                                      [**G**](#function-and-variable-index_vr_letter-g){.summary-letter}  
                                      [**H**](#function-and-variable-index_vr_letter-h){.summary-letter}  
                                      [**I**](#function-and-variable-index_vr_letter-i){.summary-letter}  
                                      [**J**](#function-and-variable-index_vr_letter-j){.summary-letter}  
                                      [**K**](#function-and-variable-index_vr_letter-k){.summary-letter}  
                                      [**L**](#function-and-variable-index_vr_letter-l){.summary-letter}  
                                      [**M**](#function-and-variable-index_vr_letter-m){.summary-letter}  
                                      [**N**](#function-and-variable-index_vr_letter-n){.summary-letter}  
                                      [**O**](#function-and-variable-index_vr_letter-o){.summary-letter}  
                                      [**P**](#function-and-variable-index_vr_letter-p){.summary-letter}  
                                      [**Q**](#function-and-variable-index_vr_letter-q){.summary-letter}  
                                      [**R**](#function-and-variable-index_vr_letter-r){.summary-letter}  
                                      [**S**](#function-and-variable-index_vr_letter-s){.summary-letter}  
                                      [**T**](#function-and-variable-index_vr_letter-t){.summary-letter}  
                                      [**U**](#function-and-variable-index_vr_letter-u){.summary-letter}  
                                      [**V**](#function-and-variable-index_vr_letter-v){.summary-letter}  
                                      [**W**](#function-and-variable-index_vr_letter-w){.summary-letter}  
                                      [**X**](#function-and-variable-index_vr_letter-x){.summary-letter}  

  ----------------------------------- ----------------------------------------------------------------------

  ----------------- ----------------------------------------------------------------------------------------------- ----------------- -----------------------------------------------------------------------
                    Index Entry                                                                                                       Section

                                                                                                                                      

  !                                                                                                                                   

                    [`!`](Logical-vectors.html#index-_0021):                                                                          [Logical vectors](Logical-vectors.html)

                    [`!=`](Logical-vectors.html#index-_0021_003d):                                                                    [Logical vectors](Logical-vectors.html)

                                                                                                                                      

  \%                                                                                                                                  

                    [`%*%`](Multiplication.html#index-_0025_002a_0025):                                                               [Multiplication](Multiplication.html)

                    [`%o%`](The-outer-product-of-two-arrays.html#index-_0025o_0025):                                                  [The outer product of two arrays](The-outer-product-of-two-arrays.html)

                                                                                                                                      

  &                                                                                                                                   

                    [`&`](Logical-vectors.html#index-_0026):                                                                          [Logical vectors](Logical-vectors.html)

                    [`&&`](Conditional-execution.html#index-_0026_0026):                                                              [Conditional execution](Conditional-execution.html)

                                                                                                                                      

  \*                                                                                                                                  

                    [`*`](Vector-arithmetic.html#index-_002a):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  \+                                                                                                                                  

                    [`+`](Vector-arithmetic.html#index-):                                                                             [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  \-                                                                                                                                  

                    [`-`](Vector-arithmetic.html#index--):                                                                            [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  .                                                                                                                                   

                    [`.`](Updating-fitted-models.html#index-.):                                                                       [Updating fitted models](Updating-fitted-models.html)

                    [`.First`](Customizing-the-environment.html#index-.first):                                                        [Customizing the environment](Customizing-the-environment.html)

                    [`.Last`](Customizing-the-environment.html#index-.last):                                                          [Customizing the environment](Customizing-the-environment.html)

                                                                                                                                      

  /                                                                                                                                   

                    [`/`](Vector-arithmetic.html#index-_002f):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  :                                                                                                                                   

                    [`:`](Generating-regular-sequences.html#index-_003a):                                                             [Generating regular sequences](Generating-regular-sequences.html)

                    [`::`](Namespaces.html#index-_003a_003a):                                                                         [Namespaces](Namespaces.html)

                    [`:::`](Namespaces.html#index-_003a_003a_003a):                                                                   [Namespaces](Namespaces.html)

                                                                                                                                      

  \<                                                                                                                                  

                    [`<`](Logical-vectors.html#index-_003c):                                                                          [Logical vectors](Logical-vectors.html)

                    [`<<-`](Scope.html#index-_003c_003c-):                                                                            [Scope](Scope.html)

                    [`<=`](Logical-vectors.html#index-_003c_003d):                                                                    [Logical vectors](Logical-vectors.html)

                                                                                                                                      

  =                                                                                                                                   

                    [`==`](Logical-vectors.html#index-_003d_003d):                                                                    [Logical vectors](Logical-vectors.html)

                                                                                                                                      

  \>                                                                                                                                  

                    [`>`](Logical-vectors.html#index-_003e):                                                                          [Logical vectors](Logical-vectors.html)

                    [`>=`](Logical-vectors.html#index-_003e_003d):                                                                    [Logical vectors](Logical-vectors.html)

                                                                                                                                      

  ?                                                                                                                                   

                    [`?`](Getting-help.html#index-_003f):                                                                             [Getting help](Getting-help.html)

                    [`??`](Getting-help.html#index-_003f_003f):                                                                       [Getting help](Getting-help.html)

                                                                                                                                      

  \^                                                                                                                                  

                    [`^`](Vector-arithmetic.html#index-_005e):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  \|                                                                                                                                  

                    [`|`](Logical-vectors.html#index-_007c):                                                                          [Logical vectors](Logical-vectors.html)

                    [`||`](Conditional-execution.html#index-_007c_007c):                                                              [Conditional execution](Conditional-execution.html)

                                                                                                                                      

  \~                                                                                                                                  

                    [`~`](Formulae-for-statistical-models.html#index-_007e):                                                          [Formulae for statistical models](Formulae-for-statistical-models.html)

                                                                                                                                      

  A                                                                                                                                   

                    [`abline`](Low-level-plotting-commands.html#index-abline):                                                        [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`ace`](Some-non-standard-models.html#index-ace):                                                                 [Some non-standard models](Some-non_002dstandard-models.html)

                    [`add1`](Updating-fitted-models.html#index-add1):                                                                 [Updating fitted models](Updating-fitted-models.html)

                    [`anova`](Generic-functions-for-extracting-model-information.html#index-anova):                                   [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`anova`](ANOVA-tables.html#index-anova-1):                                                                       [ANOVA tables](ANOVA-tables.html)

                    [`aov`](Analysis-of-variance-and-model-comparison.html#index-aov):                                                [Analysis of variance and model
                                                                                                                                      comparison](Analysis-of-variance-and-model-comparison.html)

                    [`aperm`](Generalized-transpose-of-an-array.html#index-aperm):                                                    [Generalized transpose of an
                                                                                                                                      array](Generalized-transpose-of-an-array.html)

                    [`array`](The-array_0028_0029-function.html#index-array):                                                         [The array() function](The-array_0028_0029-function.html)

                    [`as.data.frame`](Making-data-frames.html#index-as.data.frame):                                                   [Making data frames](Making-data-frames.html)

                    [`as.vector`](The-concatenation-function-c_0028_0029-with-arrays.html#index-as.vector):                           [The concatenation function c() with
                                                                                                                                      arrays](The-concatenation-function-c_0028_0029-with-arrays.html)

                    [`attach`](attach_0028_0029-and-detach_0028_0029.html#index-attach):                                              [attach() and detach()](attach_0028_0029-and-detach_0028_0029.html)

                    [`attr`](Getting-and-setting-attributes.html#index-attr):                                                          @sec-getting-and-setting-attributes

                    [`attr`](Getting-and-setting-attributes.html#index-attr-1):                                                        @sec-getting-and-setting-attributes

                    [`attributes`](Getting-and-setting-attributes.html#index-attributes):                                              @sec-getting-and-setting-attributes

                    [`attributes`](Getting-and-setting-attributes.html#index-attributes-1):                                            @sec-getting-and-setting-attributes

                    [`avas`](Some-non-standard-models.html#index-avas):                                                               [Some non-standard models](Some-non_002dstandard-models.html)

                    [`axis`](Low-level-plotting-commands.html#index-axis):                                                            [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                                                                                                                                      

  B                                                                                                                                   

                    [`boxplot`](One--and-two-sample-tests.html#index-boxplot):                                                        [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                    [`break`](Repetitive-execution.html#index-break):                                                                 [Repetitive execution](Repetitive-execution.html)

                    [`bruto`](Some-non-standard-models.html#index-bruto):                                                             [Some non-standard models](Some-non_002dstandard-models.html)

                                                                                                                                      

  C                                                                                                                                   

                    [`c`](Vectors-and-assignment.html#index-c):                                                                       [Vectors and assignment](Vectors-and-assignment.html)

                    [`c`](Character-vectors.html#index-c-1):                                                                          [Character vectors](Character-vectors.html)

                    [`c`](The-concatenation-function-c_0028_0029-with-arrays.html#index-c-2):                                         [The concatenation function c() with
                                                                                                                                      arrays](The-concatenation-function-c_0028_0029-with-arrays.html)

                    [`c`](Concatenating-lists.html#index-c-3):                                                                        [Concatenating lists](Concatenating-lists.html)

                    [`C`](Contrasts.html#index-c):                                                                                    [Contrasts](Contrasts.html)

                    [`cbind`](Forming-partitioned-matrices.html#index-cbind):                                                         [Forming partitioned matrices](Forming-partitioned-matrices.html)

                    [`coef`](Generic-functions-for-extracting-model-information.html#index-coef):                                     [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`coefficients`](Generic-functions-for-extracting-model-information.html#index-coefficients):                     [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`contour`](Display-graphics.html#index-contour):                                                                 [Display graphics](Display-graphics.html)

                    [`contrasts`](Contrasts.html#index-contrasts):                                                                    [Contrasts](Contrasts.html)

                    [`coplot`](Displaying-multivariate-data.html#index-coplot):                                                       [Displaying multivariate data](Displaying-multivariate-data.html)

                    [`cos`](Vector-arithmetic.html#index-cos):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`crossprod`](Index-matrices.html#index-crossprod):                                                               [Index matrices](Index-matrices.html)

                    [`crossprod`](Multiplication.html#index-crossprod-1):                                                             [Multiplication](Multiplication.html)

                    [`cut`](Frequency-tables-from-factors.html#index-cut):                                                            [Frequency tables from factors](Frequency-tables-from-factors.html)

                                                                                                                                      

  D                                                                                                                                   

                    [`data`](Accessing-builtin-datasets.html#index-data):                                                             [Accessing builtin datasets](Accessing-builtin-datasets.html)

                    [`data.frame`](Making-data-frames.html#index-data.frame):                                                         [Making data frames](Making-data-frames.html)

                    [`density`](Examining-the-distribution-of-a-set-of-data.html#index-density):                                      [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`det`](Singular-value-decomposition-and-determinants.html#index-det):                                            [Singular value decomposition and
                                                                                                                                      determinants](Singular-value-decomposition-and-determinants.html)

                    [`detach`](attach_0028_0029-and-detach_0028_0029.html#index-detach):                                              [attach() and detach()](attach_0028_0029-and-detach_0028_0029.html)

                    [`determinant`](Singular-value-decomposition-and-determinants.html#index-determinant):                            [Singular value decomposition and
                                                                                                                                      determinants](Singular-value-decomposition-and-determinants.html)

                    [`dev.list`](Multiple-graphics-devices.html#index-dev.list):                                                      [Multiple graphics devices](Multiple-graphics-devices.html)

                    [`dev.next`](Multiple-graphics-devices.html#index-dev.next):                                                      [Multiple graphics devices](Multiple-graphics-devices.html)

                    [`dev.off`](Multiple-graphics-devices.html#index-dev.off):                                                        [Multiple graphics devices](Multiple-graphics-devices.html)

                    [`dev.prev`](Multiple-graphics-devices.html#index-dev.prev):                                                      [Multiple graphics devices](Multiple-graphics-devices.html)

                    [`dev.set`](Multiple-graphics-devices.html#index-dev.set):                                                        [Multiple graphics devices](Multiple-graphics-devices.html)

                    [`deviance`](Generic-functions-for-extracting-model-information.html#index-deviance):                             [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`diag`](Multiplication.html#index-diag):                                                                         [Multiplication](Multiplication.html)

                    [`dim`](Arrays.html#index-dim):                                                                                   [Arrays](Arrays.html)

                    [`dotchart`](Display-graphics.html#index-dotchart):                                                               [Display graphics](Display-graphics.html)

                    [`drop1`](Updating-fitted-models.html#index-drop1):                                                               [Updating fitted models](Updating-fitted-models.html)

                                                                                                                                      

  E                                                                                                                                   

                    [`ecdf`](Examining-the-distribution-of-a-set-of-data.html#index-ecdf):                                            [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`edit`](Editing-data.html#index-edit):                                                                           [Editing data](Editing-data.html)

                    [`eigen`](Eigenvalues-and-eigenvectors.html#index-eigen):                                                         [Eigenvalues and eigenvectors](Eigenvalues-and-eigenvectors.html)

                    [`else`](Conditional-execution.html#index-else):                                                                  [Conditional execution](Conditional-execution.html)

                    [`Error`](Analysis-of-variance-and-model-comparison.html#index-error):                                            [Analysis of variance and model
                                                                                                                                      comparison](Analysis-of-variance-and-model-comparison.html)

                    [`example`](Getting-help.html#index-example):                                                                     [Getting help](Getting-help.html)

                    [`exp`](Vector-arithmetic.html#index-exp):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  F                                                                                                                                   

                    [`F`](Logical-vectors.html#index-f):                                                                              [Logical vectors](Logical-vectors.html)

                    [`factor`](Factors.html#index-factor):                                                                            [Factors](Factors.html)

                    [`FALSE`](Logical-vectors.html#index-false):                                                                      [Logical vectors](Logical-vectors.html)

                    [`fivenum`](Examining-the-distribution-of-a-set-of-data.html#index-fivenum):                                      [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`for`](Repetitive-execution.html#index-for):                                                                     [Repetitive execution](Repetitive-execution.html)

                    [`formula`](Generic-functions-for-extracting-model-information.html#index-formula):                               [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`function`](Writing-your-own-functions.html#index-function):                                                     @sec-writing-your-own-functions 

                                                                                                                                      

  G                                                                                                                                   

                    [`getAnywhere`](Object-orientation.html#index-getanywhere):                                                       [Object orientation](Object-orientation.html)

                    [`getS3method`](Object-orientation.html#index-gets3method):                                                       [Object orientation](Object-orientation.html)

                    [`glm`](The-glm_0028_0029-function.html#index-glm):                                                               [The glm() function](The-glm_0028_0029-function.html)

                                                                                                                                      

  H                                                                                                                                   

                    [`help`](Getting-help.html#index-help):                                                                           [Getting help](Getting-help.html)

                    [`help`](Getting-help.html#index-help-1):                                                                         [Getting help](Getting-help.html)

                    [`help.search`](Getting-help.html#index-help.search):                                                             [Getting help](Getting-help.html)

                    [`help.start`](Getting-help.html#index-help.start):                                                               [Getting help](Getting-help.html)

                    [`hist`](Examining-the-distribution-of-a-set-of-data.html#index-hist):                                            [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`hist`](Display-graphics.html#index-hist-1):                                                                     [Display graphics](Display-graphics.html)

                                                                                                                                      

  I                                                                                                                                   

                    [`identify`](Interacting-with-graphics.html#index-identify):                                                      [Interacting with graphics](Interacting-with-graphics.html)

                    [`if`](Conditional-execution.html#index-if):                                                                      [Conditional execution](Conditional-execution.html)

                    [`if`](Conditional-execution.html#index-if-1):                                                                    [Conditional execution](Conditional-execution.html)

                    [`ifelse`](Conditional-execution.html#index-ifelse):                                                              [Conditional execution](Conditional-execution.html)

                    [`image`](Display-graphics.html#index-image):                                                                     [Display graphics](Display-graphics.html)

                    [`is.na`](Missing-values.html#index-is.na):                                                                       [Missing values](Missing-values.html)

                    [`is.nan`](Missing-values.html#index-is.nan):                                                                     [Missing values](Missing-values.html)

                                                                                                                                      

  J                                                                                                                                   

                    [`jpeg`](Device-drivers.html#index-jpeg):                                                                         [Device drivers](Device-drivers.html)

                                                                                                                                      

  K                                                                                                                                   

                    [`ks.test`](Examining-the-distribution-of-a-set-of-data.html#index-ks.test):                                      [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                                                                                                                                      

  L                                                                                                                                   

                    [`legend`](Low-level-plotting-commands.html#index-legend):                                                        [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`length`](Vector-arithmetic.html#index-length):                                                                  [Vector arithmetic](Vector-arithmetic.html)

                    [`length`](The-intrinsic-attributes-mode-and-length.html#index-length-1):                                         [The intrinsic attributes mode and
                                                                                                                                      length](The-intrinsic-attributes-mode-and-length.html)

                    [`levels`](Factors.html#index-levels):                                                                            [Factors](Factors.html)

                    [`lines`](Low-level-plotting-commands.html#index-lines):                                                          [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`list`](Lists.html#index-list):                                                                                  @sec-lists

                    [`lm`](Linear-models.html#index-lm):                                                                              [Linear models](Linear-models.html)

                    [`lme`](Some-non-standard-models.html#index-lme):                                                                 [Some non-standard models](Some-non_002dstandard-models.html)

                    [`locator`](Interacting-with-graphics.html#index-locator):                                                        [Interacting with graphics](Interacting-with-graphics.html)

                    [`loess`](Some-non-standard-models.html#index-loess):                                                             [Some non-standard models](Some-non_002dstandard-models.html)

                    [`loess`](Some-non-standard-models.html#index-loess-1):                                                           [Some non-standard models](Some-non_002dstandard-models.html)

                    [`log`](Vector-arithmetic.html#index-log):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`lqs`](Some-non-standard-models.html#index-lqs):                                                                 [Some non-standard models](Some-non_002dstandard-models.html)

                    [`lsfit`](Least-squares-fitting-and-the-QR-decomposition.html#index-lsfit):                                       [Least squares fitting and the QR
                                                                                                                                      decomposition](Least-squares-fitting-and-the-QR-decomposition.html)

                                                                                                                                      

  M                                                                                                                                   

                    [`mars`](Some-non-standard-models.html#index-mars):                                                               [Some non-standard models](Some-non_002dstandard-models.html)

                    [`max`](Vector-arithmetic.html#index-max):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`mean`](Vector-arithmetic.html#index-mean):                                                                      [Vector arithmetic](Vector-arithmetic.html)

                    [`methods`](Object-orientation.html#index-methods):                                                               [Object orientation](Object-orientation.html)

                    [`min`](Vector-arithmetic.html#index-min):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`mode`](The-intrinsic-attributes-mode-and-length.html#index-mode):                                               [The intrinsic attributes mode and
                                                                                                                                      length](The-intrinsic-attributes-mode-and-length.html)

                                                                                                                                      

  N                                                                                                                                   

                    [`NA`](Missing-values.html#index-na):                                                                             [Missing values](Missing-values.html)

                    [`NaN`](Missing-values.html#index-nan):                                                                           [Missing values](Missing-values.html)

                    [`ncol`](Matrix-facilities.html#index-ncol):                                                                      [Matrix facilities](Matrix-facilities.html)

                    [`next`](Repetitive-execution.html#index-next):                                                                   [Repetitive execution](Repetitive-execution.html)

                    [`nlm`](Nonlinear-least-squares-and-maximum-likelihood-models.html#index-nlm):                                    [Nonlinear least squares and maximum likelihood
                                                                                                                                      models](Nonlinear-least-squares-and-maximum-likelihood-models.html)

                    [`nlm`](Least-squares.html#index-nlm-1):                                                                          [Least squares](Least-squares.html)

                    [`nlm`](Maximum-likelihood.html#index-nlm-2):                                                                     [Maximum likelihood](Maximum-likelihood.html)

                    [`nlme`](Some-non-standard-models.html#index-nlme):                                                               [Some non-standard models](Some-non_002dstandard-models.html)

                    [`nlminb`](Nonlinear-least-squares-and-maximum-likelihood-models.html#index-nlminb):                              [Nonlinear least squares and maximum likelihood
                                                                                                                                      models](Nonlinear-least-squares-and-maximum-likelihood-models.html)

                    [`nrow`](Matrix-facilities.html#index-nrow):                                                                      [Matrix facilities](Matrix-facilities.html)

                                                                                                                                      

  O                                                                                                                                   

                    [`optim`](Nonlinear-least-squares-and-maximum-likelihood-models.html#index-optim):                                [Nonlinear least squares and maximum likelihood
                                                                                                                                      models](Nonlinear-least-squares-and-maximum-likelihood-models.html)

                    [`order`](Vector-arithmetic.html#index-order):                                                                    [Vector arithmetic](Vector-arithmetic.html)

                    [`ordered`](Ordered-factors.html#index-ordered):                                                                  [Ordered factors](Ordered-factors.html)

                    [`ordered`](Ordered-factors.html#index-ordered-1):                                                                [Ordered factors](Ordered-factors.html)

                    [`outer`](The-outer-product-of-two-arrays.html#index-outer):                                                      [The outer product of two arrays](The-outer-product-of-two-arrays.html)

                                                                                                                                      

  P                                                                                                                                   

                    [`pairs`](Displaying-multivariate-data.html#index-pairs):                                                         [Displaying multivariate data](Displaying-multivariate-data.html)

                    [`par`](The-par_0028_0029-function.html#index-par):                                                               [The par() function](The-par_0028_0029-function.html)

                    [`paste`](Character-vectors.html#index-paste):                                                                    [Character vectors](Character-vectors.html)

                    [`pdf`](Device-drivers.html#index-pdf):                                                                           [Device drivers](Device-drivers.html)

                    [`persp`](Display-graphics.html#index-persp):                                                                     [Display graphics](Display-graphics.html)

                    [`plot`](Generic-functions-for-extracting-model-information.html#index-plot):                                     [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`plot`](The-plot_0028_0029-function.html#index-plot-1):                                                          [The plot() function](The-plot_0028_0029-function.html)

                    [`pmax`](Vector-arithmetic.html#index-pmax):                                                                      [Vector arithmetic](Vector-arithmetic.html)

                    [`pmin`](Vector-arithmetic.html#index-pmin):                                                                      [Vector arithmetic](Vector-arithmetic.html)

                    [`png`](Device-drivers.html#index-png):                                                                           [Device drivers](Device-drivers.html)

                    [`points`](Low-level-plotting-commands.html#index-points):                                                        [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`polygon`](Low-level-plotting-commands.html#index-polygon):                                                      [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`postscript`](Device-drivers.html#index-postscript):                                                             [Device drivers](Device-drivers.html)

                    [`predict`](Generic-functions-for-extracting-model-information.html#index-predict):                               [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`print`](Generic-functions-for-extracting-model-information.html#index-print):                                   [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`prod`](Vector-arithmetic.html#index-prod):                                                                      [Vector arithmetic](Vector-arithmetic.html)

                                                                                                                                      

  Q                                                                                                                                   

                    [`qqline`](Examining-the-distribution-of-a-set-of-data.html#index-qqline):                                        [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`qqline`](Display-graphics.html#index-qqline-1):                                                                 [Display graphics](Display-graphics.html)

                    [`qqnorm`](Examining-the-distribution-of-a-set-of-data.html#index-qqnorm):                                        [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`qqnorm`](Display-graphics.html#index-qqnorm-1):                                                                 [Display graphics](Display-graphics.html)

                    [`qqplot`](Display-graphics.html#index-qqplot):                                                                   [Display graphics](Display-graphics.html)

                    [`qr`](Least-squares-fitting-and-the-QR-decomposition.html#index-qr):                                             [Least squares fitting and the QR
                                                                                                                                      decomposition](Least-squares-fitting-and-the-QR-decomposition.html)

                    [`quartz`](Device-drivers.html#index-quartz):                                                                     [Device drivers](Device-drivers.html)

                                                                                                                                      

  R                                                                                                                                   

                    [`range`](Vector-arithmetic.html#index-range):                                                                    [Vector arithmetic](Vector-arithmetic.html)

                    [`rbind`](Forming-partitioned-matrices.html#index-rbind):                                                         [Forming partitioned matrices](Forming-partitioned-matrices.html)

                    [`read.table`](The-read.table_0028_0029-function.html#index-read.table):                                          [The read.table() function](The-read_002etable_0028_0029-function.html)

                    [`rep`](Generating-regular-sequences.html#index-rep):                                                             [Generating regular sequences](Generating-regular-sequences.html)

                    [`repeat`](Repetitive-execution.html#index-repeat):                                                               [Repetitive execution](Repetitive-execution.html)

                    [`resid`](Generic-functions-for-extracting-model-information.html#index-resid):                                   [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`residuals`](Generic-functions-for-extracting-model-information.html#index-residuals):                           [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`rlm`](Some-non-standard-models.html#index-rlm):                                                                 [Some non-standard models](Some-non_002dstandard-models.html)

                    [`rm`](Data-permanency-and-removing-objects.html#index-rm):                                                       [Data permanency and removing
                                                                                                                                      objects](Data-permanency-and-removing-objects.html)

                                                                                                                                      

  S                                                                                                                                   

                    [`scan`](The-scan_0028_0029-function.html#index-scan):                                                            [The scan() function](The-scan_0028_0029-function.html)

                    [`sd`](The-function-tapply_0028_0029-and-ragged-arrays.html#index-sd):                                            [The function tapply() and ragged
                                                                                                                                      arrays](The-function-tapply_0028_0029-and-ragged-arrays.html)

                    [`search`](Managing-the-search-path.html#index-search):                                                           [Managing the search path](Managing-the-search-path.html)

                    [`seq`](Generating-regular-sequences.html#index-seq):                                                             [Generating regular sequences](Generating-regular-sequences.html)

                    [`shapiro.test`](Examining-the-distribution-of-a-set-of-data.html#index-shapiro.test):                            [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`sin`](Vector-arithmetic.html#index-sin):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`sink`](Executing-commands-from-or-diverting-output-to-a-file.html#index-sink):                                  [Executing commands from or diverting output to a
                                                                                                                                      file](Executing-commands-from-or-diverting-output-to-a-file.html)

                    [`solve`](Linear-equations-and-inversion.html#index-solve):                                                       [Linear equations and inversion](Linear-equations-and-inversion.html)

                    [`sort`](Vector-arithmetic.html#index-sort):                                                                      [Vector arithmetic](Vector-arithmetic.html)

                    [`source`](Executing-commands-from-or-diverting-output-to-a-file.html#index-source):                              [Executing commands from or diverting output to a
                                                                                                                                      file](Executing-commands-from-or-diverting-output-to-a-file.html)

                    [`split`](Repetitive-execution.html#index-split):                                                                 [Repetitive execution](Repetitive-execution.html)

                    [`sqrt`](Vector-arithmetic.html#index-sqrt):                                                                      [Vector arithmetic](Vector-arithmetic.html)

                    [`stem`](Examining-the-distribution-of-a-set-of-data.html#index-stem):                                            [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`step`](Generic-functions-for-extracting-model-information.html#index-step):                                     [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`step`](Updating-fitted-models.html#index-step-1):                                                               [Updating fitted models](Updating-fitted-models.html)

                    [`sum`](Vector-arithmetic.html#index-sum):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`summary`](Examining-the-distribution-of-a-set-of-data.html#index-summary):                                      [Examining the distribution of a set of
                                                                                                                                      data](Examining-the-distribution-of-a-set-of-data.html)

                    [`summary`](Generic-functions-for-extracting-model-information.html#index-summary-1):                             [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`svd`](Singular-value-decomposition-and-determinants.html#index-svd):                                            [Singular value decomposition and
                                                                                                                                      determinants](Singular-value-decomposition-and-determinants.html)

                                                                                                                                      

  T                                                                                                                                   

                    [`T`](Logical-vectors.html#index-t):                                                                              [Logical vectors](Logical-vectors.html)

                    [`t`](Generalized-transpose-of-an-array.html#index-t):                                                            [Generalized transpose of an
                                                                                                                                      array](Generalized-transpose-of-an-array.html)

                    [`t.test`](One--and-two-sample-tests.html#index-t.test):                                                          [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                    [`table`](Index-matrices.html#index-table):                                                                       [Index matrices](Index-matrices.html)

                    [`table`](Frequency-tables-from-factors.html#index-table-1):                                                      [Frequency tables from factors](Frequency-tables-from-factors.html)

                    [`tan`](Vector-arithmetic.html#index-tan):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`tapply`](The-function-tapply_0028_0029-and-ragged-arrays.html#index-tapply):                                    [The function tapply() and ragged
                                                                                                                                      arrays](The-function-tapply_0028_0029-and-ragged-arrays.html)

                    [`text`](Low-level-plotting-commands.html#index-text):                                                            [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`title`](Low-level-plotting-commands.html#index-title):                                                          [Low-level plotting commands](Low_002dlevel-plotting-commands.html)

                    [`tree`](Some-non-standard-models.html#index-tree):                                                               [Some non-standard models](Some-non_002dstandard-models.html)

                    [`TRUE`](Logical-vectors.html#index-true):                                                                        [Logical vectors](Logical-vectors.html)

                                                                                                                                      

  U                                                                                                                                   

                    [`unclass`](The-class-of-an-object.html#index-unclass):                                                           [The class of an object](The-class-of-an-object.html)

                    [`update`](Updating-fitted-models.html#index-update):                                                             [Updating fitted models](Updating-fitted-models.html)

                                                                                                                                      

  V                                                                                                                                   

                    [`var`](Vector-arithmetic.html#index-var):                                                                        [Vector arithmetic](Vector-arithmetic.html)

                    [`var`](The-function-tapply_0028_0029-and-ragged-arrays.html#index-var-1):                                        [The function tapply() and ragged
                                                                                                                                      arrays](The-function-tapply_0028_0029-and-ragged-arrays.html)

                    [`var.test`](One--and-two-sample-tests.html#index-var.test):                                                      [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                    [`vcov`](Generic-functions-for-extracting-model-information.html#index-vcov):                                     [Generic functions for extracting model
                                                                                                                                      information](Generic-functions-for-extracting-model-information.html)

                    [`vector`](Vectors-and-assignment.html#index-vector):                                                             [Vectors and assignment](Vectors-and-assignment.html)

                                                                                                                                      

  W                                                                                                                                   

                    [`while`](Repetitive-execution.html#index-while):                                                                 [Repetitive execution](Repetitive-execution.html)

                    [`wilcox.test`](One--and-two-sample-tests.html#index-wilcox.test):                                                [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                    [`windows`](Device-drivers.html#index-windows):                                                                   [Device drivers](Device-drivers.html)

                                                                                                                                      

  X                                                                                                                                   

                    [`X11`](Device-drivers.html#index-x11):                                                                           [Device drivers](Device-drivers.html)

                                                                                                                                      
  ----------------- ----------------------------------------------------------------------------------------------- ----------------- -----------------------------------------------------------------------

  ----------------------------------- ----------------------------------------------------------------------
  Jump to:                            [**!**](#function-and-variable-index_vr_symbol-1){.summary-letter}  
                                      [**%**](#function-and-variable-index_vr_symbol-2){.summary-letter}  
                                      [**&**](#function-and-variable-index_vr_symbol-3){.summary-letter}  
                                      [**\***](#function-and-variable-index_vr_symbol-4){.summary-letter}  
                                      [**+**](#function-and-variable-index_vr_symbol-5){.summary-letter}  
                                      [**-**](#function-and-variable-index_vr_symbol-6){.summary-letter}  
                                      [**.**](#function-and-variable-index_vr_symbol-7){.summary-letter}  
                                      [**/**](#function-and-variable-index_vr_symbol-8){.summary-letter}  
                                      [**:**](#function-and-variable-index_vr_symbol-9){.summary-letter}  
                                      [**\<**](#function-and-variable-index_vr_symbol-10){.summary-letter}  
                                      [**=**](#function-and-variable-index_vr_symbol-11){.summary-letter}  
                                      [**\>**](#function-and-variable-index_vr_symbol-12){.summary-letter}  
                                      [**?**](#function-and-variable-index_vr_symbol-13){.summary-letter}  
                                      [**\^**](#function-and-variable-index_vr_symbol-14){.summary-letter}  
                                      [**\|**](#function-and-variable-index_vr_symbol-15){.summary-letter}  
                                      [**\~**](#function-and-variable-index_vr_symbol-16){.summary-letter}
                                       \
                                      [**A**](#function-and-variable-index_vr_letter-a){.summary-letter}  
                                      [**B**](#function-and-variable-index_vr_letter-b){.summary-letter}  
                                      [**C**](#function-and-variable-index_vr_letter-c){.summary-letter}  
                                      [**D**](#function-and-variable-index_vr_letter-d){.summary-letter}  
                                      [**E**](#function-and-variable-index_vr_letter-e){.summary-letter}  
                                      [**F**](#function-and-variable-index_vr_letter-f){.summary-letter}  
                                      [**G**](#function-and-variable-index_vr_letter-g){.summary-letter}  
                                      [**H**](#function-and-variable-index_vr_letter-h){.summary-letter}  
                                      [**I**](#function-and-variable-index_vr_letter-i){.summary-letter}  
                                      [**J**](#function-and-variable-index_vr_letter-j){.summary-letter}  
                                      [**K**](#function-and-variable-index_vr_letter-k){.summary-letter}  
                                      [**L**](#function-and-variable-index_vr_letter-l){.summary-letter}  
                                      [**M**](#function-and-variable-index_vr_letter-m){.summary-letter}  
                                      [**N**](#function-and-variable-index_vr_letter-n){.summary-letter}  
                                      [**O**](#function-and-variable-index_vr_letter-o){.summary-letter}  
                                      [**P**](#function-and-variable-index_vr_letter-p){.summary-letter}  
                                      [**Q**](#function-and-variable-index_vr_letter-q){.summary-letter}  
                                      [**R**](#function-and-variable-index_vr_letter-r){.summary-letter}  
                                      [**S**](#function-and-variable-index_vr_letter-s){.summary-letter}  
                                      [**T**](#function-and-variable-index_vr_letter-t){.summary-letter}  
                                      [**U**](#function-and-variable-index_vr_letter-u){.summary-letter}  
                                      [**V**](#function-and-variable-index_vr_letter-v){.summary-letter}  
                                      [**W**](#function-and-variable-index_vr_letter-w){.summary-letter}  
                                      [**X**](#function-and-variable-index_vr_letter-x){.summary-letter}  

  ----------------------------------- ----------------------------------------------------------------------
 
# Concept index {#sec-appendix-e-concept-index}

  ------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  Jump to:     [**A**](#concept-index_cp_letter-a){.summary-letter}   [**B**](#concept-index_cp_letter-b){.summary-letter}   [**C**](#concept-index_cp_letter-c){.summary-letter}   [**D**](#concept-index_cp_letter-d){.summary-letter}   [**E**](#concept-index_cp_letter-e){.summary-letter}   [**F**](#concept-index_cp_letter-f){.summary-letter}   [**G**](#concept-index_cp_letter-g){.summary-letter}   [**I**](#concept-index_cp_letter-i){.summary-letter}   [**K**](#concept-index_cp_letter-k){.summary-letter}   [**L**](#concept-index_cp_letter-l){.summary-letter}   [**M**](#concept-index_cp_letter-m){.summary-letter}   [**N**](#concept-index_cp_letter-n){.summary-letter}   [**O**](#concept-index_cp_letter-o){.summary-letter}   [**P**](#concept-index_cp_letter-p){.summary-letter}   [**Q**](#concept-index_cp_letter-q){.summary-letter}   [**R**](#concept-index_cp_letter-r){.summary-letter}   [**S**](#concept-index_cp_letter-s){.summary-letter}   [**T**](#concept-index_cp_letter-t){.summary-letter}   [**U**](#concept-index_cp_letter-u){.summary-letter}   [**V**](#concept-index_cp_letter-v){.summary-letter}   [**W**](#concept-index_cp_letter-w){.summary-letter}  
  ------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

  ----------------- -------------------------------------------------------------------------------------------------------- ----------------- ---------------------------------------------------------------------
                    Index Entry                                                                                                                Section

                                                                                                                                               

  A                                                                                                                                            

                    [Accessing builtin datasets](Accessing-builtin-datasets.html#index-accessing-builtin-datasets):                            [Accessing builtin datasets](Accessing-builtin-datasets.html)

                    [Additive models](Some-non-standard-models.html#index-additive-models):                                                    [Some non-standard models](Some-non_002dstandard-models.html)

                    [Analysis of variance](Analysis-of-variance-and-model-comparison.html#index-analysis-of-variance):                         [Analysis of variance and model
                                                                                                                                               comparison](Analysis-of-variance-and-model-comparison.html)

                    [Arithmetic functions and operators](Vector-arithmetic.html#index-arithmetic-functions-and-operators):                     [Vector arithmetic](Vector-arithmetic.html)

                    [Arrays](Arrays.html#index-arrays):                                                                                        [Arrays](Arrays.html)

                    [Assignment](Vectors-and-assignment.html#index-assignment):                                                                [Vectors and assignment](Vectors-and-assignment.html)

                    [Attributes](Objects.html#index-attributes):                                                                               [Objects](Objects.html)

                                                                                                                                               

  B                                                                                                                                            

                    [Binary operators](Defining-new-binary-operators.html#index-binary-operators):                                             [Defining new binary operators](Defining-new-binary-operators.html)

                    [Box plots](One--and-two-sample-tests.html#index-box-plots):                                                               [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                                                                                                                                               

  C                                                                                                                                            

                    [Character vectors](Character-vectors.html#index-character-vectors):                                                       [Character vectors](Character-vectors.html)

                    [Classes](The-class-of-an-object.html#index-classes):                                                                      [The class of an object](The-class-of-an-object.html)

                    [Classes](Object-orientation.html#index-classes-1):                                                                        [Object orientation](Object-orientation.html)

                    [Concatenating lists](Concatenating-lists.html#index-concatenating-lists):                                                 [Concatenating lists](Concatenating-lists.html)

                    [Contrasts](Contrasts.html#index-contrasts):                                                                               [Contrasts](Contrasts.html)

                    [Control statements](Control-statements.html#index-control-statements):                                                    [Control statements](Control-statements.html)

                    [CRAN](Contributed-packages-and-CRAN.html#index-cran):                                                                     [Contributed packages and CRAN](Contributed-packages-and-CRAN.html)

                    [Customizing the environment](Customizing-the-environment.html#index-customizing-the-environment):                         [Customizing the environment](Customizing-the-environment.html)

                                                                                                                                               

  D                                                                                                                                            

                    [Data frames](Data-frames.html#index-data-frames):                                                                         @sec-data-frames

                    [Default values](Named-arguments-and-defaults.html#index-default-values):                                                  [Named arguments and defaults](Named-arguments-and-defaults.html)

                    [Density estimation](Examining-the-distribution-of-a-set-of-data.html#index-density-estimation):                           [Examining the distribution of a set of
                                                                                                                                               data](Examining-the-distribution-of-a-set-of-data.html)

                    [Determinants](Singular-value-decomposition-and-determinants.html#index-determinants):                                     [Singular value decomposition and
                                                                                                                                               determinants](Singular-value-decomposition-and-determinants.html)

                    [Diverting input and                                                                                                       [Executing commands from or diverting output to a
                    output](Executing-commands-from-or-diverting-output-to-a-file.html#index-diverting-input-and-output):                      file](Executing-commands-from-or-diverting-output-to-a-file.html)

                    [Dynamic graphics](Dynamic-graphics.html#index-dynamic-graphics):                                                          [Dynamic graphics](Dynamic-graphics.html)

                                                                                                                                               

  E                                                                                                                                            

                    [Eigenvalues and eigenvectors](Eigenvalues-and-eigenvectors.html#index-eigenvalues-and-eigenvectors):                      [Eigenvalues and eigenvectors](Eigenvalues-and-eigenvectors.html)

                    [Empirical CDFs](Examining-the-distribution-of-a-set-of-data.html#index-empirical-cdfs):                                   [Examining the distribution of a set of
                                                                                                                                               data](Examining-the-distribution-of-a-set-of-data.html)

                                                                                                                                               

  F                                                                                                                                            

                    [Factors](Factors.html#index-factors):                                                                                     [Factors](Factors.html)

                    [Factors](Contrasts.html#index-factors-1):                                                                                 [Contrasts](Contrasts.html)

                    [Families](Families.html#index-families):                                                                                  [Families](Families.html)

                    [Formulae](Formulae-for-statistical-models.html#index-formulae):                                                           [Formulae for statistical
                                                                                                                                               models](Formulae-for-statistical-models.html)

                                                                                                                                               

  G                                                                                                                                            

                    [Generalized linear models](Generalized-linear-models.html#index-generalized-linear-models):                               [Generalized linear models](Generalized-linear-models.html)

                    [Generalized transpose of an                                                                                               [Generalized transpose of an
                    array](Generalized-transpose-of-an-array.html#index-generalized-transpose-of-an-array):                                    array](Generalized-transpose-of-an-array.html)

                    [Generic functions](Object-orientation.html#index-generic-functions):                                                      [Object orientation](Object-orientation.html)

                    [Graphics device drivers](Device-drivers.html#index-graphics-device-drivers):                                              [Device drivers](Device-drivers.html)

                    [Graphics parameters](The-par_0028_0029-function.html#index-graphics-parameters):                                          [The par() function](The-par_0028_0029-function.html)

                    [Grouped expressions](Grouped-expressions.html#index-grouped-expressions):                                                 [Grouped expressions](Grouped-expressions.html)

                                                                                                                                               

  I                                                                                                                                            

                    [Indexing of and by arrays](Array-indexing.html#index-indexing-of-and-by-arrays):                                          [Array indexing](Array-indexing.html)

                    [Indexing vectors](Index-vectors.html#index-indexing-vectors):                                                             [Index vectors](Index-vectors.html)

                                                                                                                                               

  K                                                                                                                                            

                    [Kolmogorov-Smirnov                                                                                                        [Examining the distribution of a set of
                    test](Examining-the-distribution-of-a-set-of-data.html#index-kolmogorov-smirnov-test):                                     data](Examining-the-distribution-of-a-set-of-data.html)

                                                                                                                                               

  L                                                                                                                                            

                    [Least squares                                                                                                             [Least squares fitting and the QR
                    fitting](Least-squares-fitting-and-the-QR-decomposition.html#index-least-squares-fitting):                                 decomposition](Least-squares-fitting-and-the-QR-decomposition.html)

                    [Linear equations](Linear-equations-and-inversion.html#index-linear-equations):                                            [Linear equations and inversion](Linear-equations-and-inversion.html)

                    [Linear models](Linear-models.html#index-linear-models):                                                                   [Linear models](Linear-models.html)

                    [Lists](Lists.html#index-lists):                                                                                           @sec-lists

                    [Local approximating regressions](Some-non-standard-models.html#index-local-approximating-regressions):                    [Some non-standard models](Some-non_002dstandard-models.html)

                    [Loops and conditional                                                                                                     [Loops and conditional
                    execution](Loops-and-conditional-execution.html#index-loops-and-conditional-execution):                                    execution](Loops-and-conditional-execution.html)

                                                                                                                                               

  M                                                                                                                                            

                    [Matrices](Arrays.html#index-matrices):                                                                                    [Arrays](Arrays.html)

                    [Matrix multiplication](Multiplication.html#index-matrix-multiplication):                                                  [Multiplication](Multiplication.html)

                    [Maximum likelihood](Maximum-likelihood.html#index-maximum-likelihood):                                                    [Maximum likelihood](Maximum-likelihood.html)

                    [Missing values](Missing-values.html#index-missing-values):                                                                [Missing values](Missing-values.html)

                    [Mixed models](Some-non-standard-models.html#index-mixed-models):                                                          [Some non-standard models](Some-non_002dstandard-models.html)

                                                                                                                                               

  N                                                                                                                                            

                    [Named arguments](Named-arguments-and-defaults.html#index-named-arguments):                                                [Named arguments and defaults](Named-arguments-and-defaults.html)

                    [Namespace](Namespaces.html#index-namespace):                                                                              [Namespaces](Namespaces.html)

                    [Nonlinear least                                                                                                           [Nonlinear least squares and maximum likelihood
                    squares](Nonlinear-least-squares-and-maximum-likelihood-models.html#index-nonlinear-least-squares):                        models](Nonlinear-least-squares-and-maximum-likelihood-models.html)

                                                                                                                                               

  O                                                                                                                                            

                    [Object orientation](Object-orientation.html#index-object-orientation):                                                    [Object orientation](Object-orientation.html)

                    [Objects](Objects.html#index-objects):                                                                                     [Objects](Objects.html)

                    [One- and two-sample tests](One--and-two-sample-tests.html#index-one--and-two-sample-tests):                               [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                    [Ordered factors](Factors.html#index-ordered-factors):                                                                     [Factors](Factors.html)

                    [Ordered factors](Contrasts.html#index-ordered-factors-1):                                                                 [Contrasts](Contrasts.html)

                    [Outer products of arrays](The-outer-product-of-two-arrays.html#index-outer-products-of-arrays):                           [The outer product of two
                                                                                                                                               arrays](The-outer-product-of-two-arrays.html)

                                                                                                                                               

  P                                                                                                                                            

                    [Packages](R-and-statistics.html#index-packages):                                                                          [R and statistics](R-and-statistics.html)

                    [Packages](Packages.html#index-packages-1):                                                                                @sec-packages

                    [Probability distributions](Probability-distributions.html#index-probability-distributions):                               [Probability distributions](Probability-distributions.html)

                                                                                                                                               

  Q                                                                                                                                            

                    [QR decomposition](Least-squares-fitting-and-the-QR-decomposition.html#index-qr-decomposition):                            [Least squares fitting and the QR
                                                                                                                                               decomposition](Least-squares-fitting-and-the-QR-decomposition.html)

                    [Quantile-quantile                                                                                                         [Examining the distribution of a set of
                    plots](Examining-the-distribution-of-a-set-of-data.html#index-quantile-quantile-plots):                                    data](Examining-the-distribution-of-a-set-of-data.html)

                                                                                                                                               

  R                                                                                                                                            

                    [Reading data from files](Reading-data-from-files.html#index-reading-data-from-files):                                     [Reading data from files](Reading-data-from-files.html)

                    [Recycling rule](Vector-arithmetic.html#index-recycling-rule):                                                             [Vector arithmetic](Vector-arithmetic.html)

                    [Recycling rule](The-recycling-rule.html#index-recycling-rule-1):                                                          [The recycling rule](The-recycling-rule.html)

                    [Regular sequences](Generating-regular-sequences.html#index-regular-sequences):                                            [Generating regular sequences](Generating-regular-sequences.html)

                    [Removing objects](Data-permanency-and-removing-objects.html#index-removing-objects):                                      [Data permanency and removing
                                                                                                                                               objects](Data-permanency-and-removing-objects.html)

                    [Robust regression](Some-non-standard-models.html#index-robust-regression):                                                [Some non-standard models](Some-non_002dstandard-models.html)

                                                                                                                                               

  S                                                                                                                                            

                    [Scope](Scope.html#index-scope):                                                                                           [Scope](Scope.html)

                    [Search path](Managing-the-search-path.html#index-search-path):                                                            [Managing the search path](Managing-the-search-path.html)

                    [Shapiro-Wilk test](Examining-the-distribution-of-a-set-of-data.html#index-shapiro-wilk-test):                             [Examining the distribution of a set of
                                                                                                                                               data](Examining-the-distribution-of-a-set-of-data.html)

                    [Singular value                                                                                                            [Singular value decomposition and
                    decomposition](Singular-value-decomposition-and-determinants.html#index-singular-value-decomposition):                     determinants](Singular-value-decomposition-and-determinants.html)

                    [Statistical models](Statistical-models-in-R.html#index-statistical-models):                                               [Statistical models in R](Statistical-models-in-R.html)

                    [Student's *t* test](One--and-two-sample-tests.html#index-student_0027s-t-test):                                           [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                                                                                                                                               

  T                                                                                                                                            

                    [Tabulation](Frequency-tables-from-factors.html#index-tabulation):                                                         [Frequency tables from factors](Frequency-tables-from-factors.html)

                    [Tree-based models](Some-non-standard-models.html#index-tree-based-models):                                                [Some non-standard models](Some-non_002dstandard-models.html)

                                                                                                                                               

  U                                                                                                                                            

                    [Updating fitted models](Updating-fitted-models.html#index-updating-fitted-models):                                        [Updating fitted models](Updating-fitted-models.html)

                                                                                                                                               

  V                                                                                                                                            

                    [Vectors](Simple-manipulations-numbers-and-vectors.html#index-vectors):                                                    [Simple manipulations numbers and
                                                                                                                                               vectors](Simple-manipulations-numbers-and-vectors.html)

                                                                                                                                               

  W                                                                                                                                            

                    [Wilcoxon test](One--and-two-sample-tests.html#index-wilcoxon-test):                                                       [One- and two-sample tests](One_002d-and-two_002dsample-tests.html)

                    [Workspace](Data-permanency-and-removing-objects.html#index-workspace):                                                    [Data permanency and removing
                                                                                                                                               objects](Data-permanency-and-removing-objects.html)

                    [Writing functions](Writing-your-own-functions.html#index-writing-functions):                                              @sec-writing-your-own-functions 

                                                                                                                                               
  ----------------- -------------------------------------------------------------------------------------------------------- ----------------- ---------------------------------------------------------------------

  ------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  Jump to:     [**A**](#concept-index_cp_letter-a){.summary-letter}   [**B**](#concept-index_cp_letter-b){.summary-letter}   [**C**](#concept-index_cp_letter-c){.summary-letter}   [**D**](#concept-index_cp_letter-d){.summary-letter}   [**E**](#concept-index_cp_letter-e){.summary-letter}   [**F**](#concept-index_cp_letter-f){.summary-letter}   [**G**](#concept-index_cp_letter-g){.summary-letter}   [**I**](#concept-index_cp_letter-i){.summary-letter}   [**K**](#concept-index_cp_letter-k){.summary-letter}   [**L**](#concept-index_cp_letter-l){.summary-letter}   [**M**](#concept-index_cp_letter-m){.summary-letter}   [**N**](#concept-index_cp_letter-n){.summary-letter}   [**O**](#concept-index_cp_letter-o){.summary-letter}   [**P**](#concept-index_cp_letter-p){.summary-letter}   [**Q**](#concept-index_cp_letter-q){.summary-letter}   [**R**](#concept-index_cp_letter-r){.summary-letter}   [**S**](#concept-index_cp_letter-s){.summary-letter}   [**T**](#concept-index_cp_letter-t){.summary-letter}   [**U**](#concept-index_cp_letter-u){.summary-letter}   [**V**](#concept-index_cp_letter-v){.summary-letter}   [**W**](#concept-index_cp_letter-w){.summary-letter}  
  ------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 
# References {#sec-appendix-f-references}

D. M. Bates and D. G. Watts (1988), *Nonlinear Regression Analysis and
Its Applications.* John Wiley & Sons, New York.

Richard A. Becker, John M. Chambers and Allan R. Wilks (1988), *The New
S Language.* Chapman & Hall, New York. This book is often called the
"*Blue Book*".

John M. Chambers and Trevor J. Hastie eds. (1992), *Statistical Models
in S.* Chapman & Hall, New York. This is also called the "*White Book*".

John M. Chambers (1998) *Programming with Data*. Springer, New York.
This is also called the "*Green Book*".

A. C. Davison and D. V. Hinkley (1997), *Bootstrap Methods and Their
Applications*, Cambridge University Press.

Annette J. Dobson (1990), *An Introduction to Generalized Linear
Models*, Chapman and Hall, London.

Peter McCullagh and John A. Nelder (1989), *Generalized Linear Models.*
Second edition, Chapman and Hall, London.

John A. Rice (1995), *Mathematical Statistics and Data Analysis.* Second
edition. Duxbury Press, Belmont, CA.

S. D. Silvey (1970), *Statistical Inference.* Penguin, London.
