\input texinfo
@c %**start of header
@setfilename R-intro.info
@settitle An Introduction to R
@setchapternewpage on
@c %**end of header

@syncodeindex fn vr

@dircategory Programming
@direntry
* R Introduction: (R-intro).    An introduction to R.
@end direntry

@finalout

@include R-defs.texi
@include version.texi

@ifnottex
@macro RIcopyright{}
Copyright @copyright{} 1990 W.@: N.@: Venables@*
Copyright @copyright{} 1992 W.@: N.@: Venables & D.@: M.@: Smith@*
Copyright @copyright{} 1997 R.@: Gentleman & R.@: Ihaka@*
Copyright @copyright{} 1997, 1998 M.@: Maechler@*
@Rcopyright{1999}
@end macro
@end ifnottex

@iftex
@macro RIcopyright{}
@noindent
Copyright @copyright{} 1990 W.@: N.@: Venables
Copyright @copyright{} 1992 W.@: N.@: Venables & D.@: M.@: Smith
Copyright @copyright{} 1997 R.@: Gentleman & R.@: Ihaka
Copyright @copyright{} 1997, 1998 M.@: Maechler
@Rcopyright{1999}
@end macro
@end iftex

@c NOTE:
@c Conversion to PDF fails if sectioning titles contain (user-defined)
@c macros such as @R{}.  Hence in section titles we changed @R{} to R.
@c Revert when this is fixed.

@ifinfo
This is an introduction to R.

@RIcopyright{}

@permission{}

@end ifinfo

@titlepage
@title An Introduction to R
@subtitle Notes on @R{}: A Programming Environment for Data Analysis and Graphics
@subtitle Version @value{VERSION}
@author W. N. Venables, D. M. Smith 
@author and the R Development Core Team

@page
@vskip 0pt plus 1filll

@flushleft
@RIcopyright{}
@end flushleft

@permission{}

@end titlepage

@ifnothtml
@contents
@end ifnothtml

@ifnottex
@node Top, Preface, (dir), (dir)
@top An Introduction to R

This is an introduction to R (``GNU S''), a language and environment for
statistical computing and graphics.  R is similar to the award-winning 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 modeling and graphics.

The current version of this document is @value{VERSION}.
@end ifnottex

@menu
* Preface::                     
* Introduction and preliminaries::  
* Simple manipulations numbers and vectors::  
* Objects::                     
* Factors::                     
* Arrays and matrices::         
* Lists and data frames::       
* Reading data from files::     
* Probability distributions::   
* Loops and conditional execution::  
* Writing your own functions::  
* Statistical models in R::     
* Graphics::                    
* A sample session::            
* Invoking R::                  
* The command line editor::     
* Function and variable index::  
* Concept index::               
* References::                  
@end menu

@node Preface, Introduction and preliminaries, Top, Top
@unnumbered Preface

This introduction to @R{} is derived from an original set of notes
describing the @Sl{} and @SPLUS{} environments written by Bill Venables
and David M. Smith (Insightful Corporation).  We have made a number of
small changes to reflect differences between the @R{} and @Sl{}
programs, and expanded some of the material.

@c FIXME:
@c Would something like this still be needed?  There are certainly no
@c such features ...
@c  @R{} is an ongoing project and its capabilities do not presently
@c  match those of @Sl{}.  In this guide we have adopted the convention
@c  that any feature that we intend to implement is noted as such at the
@c  beginning of the section where the feature is described.  Users can
@c  contribute to the project by implementing any of these that remain.
@c </FIXME>

We would like to extend warm thanks to Bill Venables 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 @email{R-core@@r-project.org}.

@subheading Suggestions to the reader

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.  In
this case, @ref{Graphics} on the graphics facilities can be read at
almost any time and need not wait until all the preceding sections have
been digested.

@menu
* Introduction and preliminaries::
@end menu

@node Introduction and preliminaries, Simple manipulations numbers and vectors, Preface, Top
@chapter Introduction and preliminaries

@menu
* The R environment::           
* Related software and documentation::  
* R and statistics::            
* R and the window system::     
* Using R interactively::       
* Getting help::                
* R commands; case sensitivity etc::  
* Recall and correction of previous commands::  
* Executing commands from or diverting output to a file::  
* Data permanency and removing objects::  
@end menu

@node The R environment, Related software and documentation, Introduction and preliminaries, Introduction and preliminaries
@section The R environment

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

@itemize @bullet
@item
an effective data handling and storage facility,
@item
a suite of operators for calculations on arrays, in particular matrices,
@item
a large, coherent, integrated collection of intermediate tools for data
analysis,
@item
graphical facilities for data analysis and display either directly at
the computer or on hardcopy, and
@item
a well developed, simple and effective programming language 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 @Sl{} language.)
@end itemize

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.  As such it is very dynamic, and new releases have not
always been fully backwards compatible with previous releases.  Some
users welcome the changes because of the bonus of new technology and new
methods that come with new releases; others seem to be more worried by
the fact that old code no longer works.  Although @R{} is intended as a
programming language, one should regard most programs written in @R{} as
essentially ephemeral.

@node Related software and documentation, R and statistics, The R environment, Introduction and preliminaries
@section Related software and documentation

@R{} can be regarded as an implementation of the @Sl{} language which
was developed at Bell Laboratories by Rick Becker, John Chambers and
Allan Wilks, and also forms the basis of the @SPLUS{} systems.

The evolution of the @Sl{} language is characterized by four books by
John Chambers and coauthors.  For @R{}, the basic reference is @emph{The
New @Sl{} 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 @Sl{} (@Sl{} version 3)
are covered in @emph{Statistical Models in @Sl{}} edited by John M.@:
Chambers and Trevor J.@: Hastie.  @xref{References}, for precise
references.

In addition, documentation for @Sl{}/@SPLUS{} can typically be used with
@R{}, keeping the differences between the @Sl{} implementations in mind.
@xref{What documentation exists for R?, , , R-FAQ, The R statistical
system FAQ}.

@node R and statistics, R and the window system, Related software and documentation, Introduction and preliminaries
@section R and statistics
@cindex Packages

Our introduction to the @R{} environment did not mention
@emph{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.  Some of these are
built into the base @R{} environment, but many are supplied as
@emph{packages}. (Currently the distinction is largely a matter of
historical accident.)  There are about 8 packages supplied with @R{}
(called ``standard'' packages) and many more are available through the
@acronym{CRAN} family of Internet sites (via
@uref{http://cran.r-project.org}).

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

There is an important difference in philosophy between @Sl{} (and hence
@R{}) and the other main statistical systems.  In @Sl{} 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.

@node R and the window system, Using R interactively, R and statistics, Introduction and preliminaries
@section 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 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.

@node Using R interactively, Getting help, R and the window system, Introduction and preliminaries
@section Using R interactively

When you use the @R{} program it issues a prompt when it expects input
commands.  The default prompt is @samp{@code{>}}, 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 @samp{@code{$}}.

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

@enumerate
@item
Create a separate sub-directory, say @file{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.

@example
$ mkdir work
$ cd work
@end example

@item
Start the @R{} program with the command

@example
$ R
@end example

@item
At this point @R{} commands may be issued (see later).

@item
To quit the @R{} program the command is

@example
> q()
@end example

At this point you will be asked whether you want to save the data from
your @R{} session.  You can respond @kbd{yes}, @kbd{no} or @kbd{cancel}
(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.

@end enumerate

Further @R{} sessions are simple.

@enumerate

@item
Make @file{work} the working directory and start the program as before:

@example
$ cd work
$ R
@end example

@item
Use the @R{} program, terminating with the @code{q()} command at the end
of the session.

@end enumerate

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 @file{Start In} field in your @R{} shortcut.
Then launch @R{} by double clicking on the icon.

@section 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 @ref{A sample session}.

@node Getting help, R commands; case sensitivity etc, Using R interactively, Introduction and preliminaries
@section Getting help with functions and features
@findex help

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

@example
> help(solve)
@end example
@findex help

An alternative is

@example
> ?solve
@end example
@findex ?

For a feature specified by special characters, the argument must be
enclosed in double or single quotes, making it a ``character string'':

@example
> help("[[")
@end example

Either form of quote mark may be used to escape the other, as in the
string @code{"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
@findex help.start
@example
> help.start()
@end example

@noindent
which will launch a Web browser (@code{netscape} on UNIX) 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 @code{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 you
bearings quickly and to understand the breadth of what @R{} has to
offer.

@findex help.search
The @code{help.search} command allows searching for help in various
ways: try @code{?help.search} for details and examples.

The examples on a help topic can normally be run by 
@findex example
@example
> example(@var{topic})
@end example


Windows versions of @R{} have other optional help systems: use

@example
> ?help
@end example

@noindent
for further details.

@node R commands; case sensitivity etc, Recall and correction of previous commands, Getting help, Introduction and preliminaries
@section R commands, case sensitivity, etc.

Technically @R{} is an @emph{expression language} with a very simple
syntax.  It is @emph{case sensitive} as are most UNIX based packages, so
@code{A} and @code{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 @emph{locale} in use).  Normally all alphanumeric
symbols are allowed (and in some countries this includes accented
letters) plus @samp{@code{.}}@footnote{C programmers should note that
@samp{@code{_}} is not available, but @samp{@code{.}} is and is often
used to separate words in @R{} names.}, with the restriction that a name
cannot start with a digit.

Elementary commands consist of either @emph{expressions} or
@emph{assignments}.  If an expression is given as a command, it is
evaluated, printed, 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 (@samp{@code{;}}), or by a
newline.  Elementary commands can be grouped together into one compound
expression by braces (@samp{@code{@{}} and @samp{@code{@}}}).
@emph{Comments} can be put almost@footnote{@strong{not} inside strings,
nor within the argument list of a function definition} anywhere,
starting with a hashmark (@samp{@code{#}}), 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

@smallexample
+
@end smallexample

@noindent
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.

@node Recall and correction of previous commands, Executing commands from or diverting output to a file, R commands; case sensitivity etc, Introduction and preliminaries
@section 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 @emph{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 @key{DEL} key or added with the
other keys.  More details are provided later: @pxref{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
@strong{readline} library.

Alternatively, the Emacs text editor provides more general support
mechanisms (via @acronym{ESS}, @emph{Emacs Speaks Statistics}) for
working interactively with @R{}.  @xref{R and Emacs, , , R-FAQ, The R
statistical system FAQ}.

@node Executing commands from or diverting output to a file, Data permanency and removing objects, Recall and correction of previous commands, Introduction and preliminaries
@section Executing commands from or diverting output to a file
@cindex Diverting input and output

If commands are stored on an external file, say @file{commands.R} in the
working directory @file{work}, they may be executed at any time in an
@R{} session with the command

@example
> source("commands.R")
@end example
@findex source

For Windows @strong{Source} is also available on the
@strong{File} menu.  The function @code{sink},

@example
> sink("record.lis")
@end example
@findex sink

@noindent
will divert all subsequent output from the console to an external file,
@file{record.lis}.  The command

@example
> sink()
@end example

@noindent
restores it to the console once again.

@node Data permanency and removing objects,  , Executing commands from or diverting output to a file, Introduction and preliminaries
@section Data permanency and removing objects

The entities that @R{} creates and manipulates are known as
@emph{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 session).  The @R{} command

@example
> objects()
@end example

@noindent
(alternatively, @code{ls()} can be used to display the names of the
objects which are currently stored within @R{}.  The collection of
objects currently stored is called the @emph{workspace}.
@cindex Workspace

To remove objects the function @code{rm} is available:

@example
> rm(x, y, z, ink, junk, temp, foo, bar)
@end example
@findex rm
@cindex Removing objects

All objects created during an @R{} sessions 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 @file{.RData}@footnote{The leading ``dot'' in
this file name makes it @emph{invisible} in normal file listings in
UNIX.} in the current directory.

When @R{} is started at later time it reloads the workspace from this
file.  At the same time the associated command 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
@code{x} and @code{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.

@node Simple manipulations numbers and vectors, Objects, Introduction and preliminaries, Top
@chapter Simple manipulations; numbers and vectors
@cindex Vectors

@menu
* Vectors and assignment::      
* Vector arithmetic::           
* Generating regular sequences::  
* Logical vectors::             
* Missing values::              
* Character vectors::           
* Index vectors::               
* Other types of objects::      
@end menu

@node Vectors and assignment, Vector arithmetic, Simple manipulations numbers and vectors, Simple manipulations numbers and vectors
@section Vectors and assignment

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

@example
> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)
@end example
@findex c
@findex vector

This is an @emph{assignment} statement using the @emph{function}
@code{c()} which in this context can take an arbitrary number of vector
@emph{arguments} and whose value is a vector got by concatenating its
arguments end to end.@footnote{With other than vector types of argument,
such as @code{list} mode arguments, the action of @code{c()} is rather
different.  See @ref{Concatenating lists}.}

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

Notice that the assignment operator (@samp{@code{<-}}) is @strong{not}
the usual @samp{@code{=}} operator, which is reserved for another
purpose.  It consists of the two characters @samp{@code{<}} (``less
than'') and @samp{@code{-}} (``minus'') occurring strictly side-by-side
and it `points' to the object receiving the value of the expression.
@c In this text, the assignment operator is printed as @samp{<-}, rather
@c than ``@code{<-}''.
@footnote{The underscore character, @samp{@code{_}} is an allowable
synonym for the left pointing assignment operator @samp{@code{<-}},
however we discourage this option, as it can easily lead to much less
readable code.}
@cindex Assignment

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

@example
> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))
@end example

@noindent
The usual operator, @code{<-}, 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

@example
> c(10.4, 5.6, 3.1, 6.4, 21.7) -> x
@end example

If an expression is used as a complete command, the value is printed
@emph{and lost}@footnote{Actually, it is still available as
@code{.Last.value} before any other statements are executed}.  So now if we
were to use the command

@example
> 1/x
@end example

@noindent
the reciprocals of the five values would be printed at the terminal (and
the value of @code{x}, of course, unchanged).

The further assignment

@example
> y <- c(x, 0, x)
@end example

@noindent
would create a vector @code{y} with 11 entries consisting of two copies
of @code{x} with a zero in the middle place.

@node Vector arithmetic, Generating regular sequences, Vectors and assignment, Simple manipulations numbers and vectors
@section 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 @emph{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
@cindex Recycling rule

@example
> v <- 2*x + y + 1
@end example

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

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

Two statistical functions are @code{mean(x)} which calculates the sample
mean, which is the same as @code{sum(x)/length(x)},
@findex mean
and @code{var(x)} which gives

@example
sum((x-mean(x))^2)/(length(x)-1)
@end example
@findex var

@noindent
or sample variance.  If the argument to @code{var()} is an
@math{n}-by-@math{p} matrix the value is a @math{p}-by-@math{p} sample
covariance matrix got by regarding the rows as independent
@math{p}-variate sample vectors.

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

Note that @code{max} and @code{min} select the largest and smallest
values in their arguments, even if they are given several vectors.  The
@emph{parallel} maximum and minimum functions @code{pmax} and
@code{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.
@findex pmax
@findex pmin

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

@example
sqrt(-17)
@end example

@noindent
will give @code{NaN} and a warning, but

@example
sqrt(-17+0i)
@end example

@noindent
will do the computations as complex numbers.

@menu
* Generating regular sequences::
@end menu

@node Generating regular sequences, Logical vectors, Vector arithmetic, Simple manipulations numbers and vectors
@section Generating regular sequences
@cindex Regular sequences

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

The construction @code{30:1} may be used to generate a sequence
backwards.

@findex seq
The function @code{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
@code{seq(2,10)} is the same vector as @code{2:10}.

Parameters to @code{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 parameters may be named
@code{from=@var{value}} and @code{to=@var{value}}; thus
@code{seq(1,30)}, @code{seq(from=1, to=30)} and @code{seq(to=30,
from=1)} are all the same as @code{1:30}.  The next two parameters to
@code{seq()} may be named @code{by=@var{value}} and
@code{length=@var{value}}, which specify a step size and a length for
the sequence respectively.  If neither of these is given, the default
@code{by=1} is assumed.

For example

@example
> seq(-5, 5, by=.2) -> s3
@end example

@noindent
generates in @code{s3} the vector @code{c(-5.0, -4.8, -4.6, @dots{},
4.6, 4.8, 5.0)}.  Similarly

@example
> s4 <- seq(length=51, from=-5, by=.2)
@end example

@noindent
generates the same vector in @code{s4}.

The fifth parameter may be named @code{along=@var{vector}}, which if
used must be the only parameter, and creates a sequence @code{1, 2,
@dots{}, length(@var{vector})}, or the empty sequence if the vector is
empty (as it can be).

A related function is @code{rep()}
@findex rep
which can be used for replicating an object in various complicated ways.

The simplest form is

@example
> s5 <- rep(x, times=5)
@end example

@noindent
which will put five copies of @code{x} end-to-end in @code{s5}.

@node Logical vectors, Missing values, Generating regular sequences, Simple manipulations numbers and vectors
@section Logical vectors

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

Logical vectors are generated by @emph{conditions}.  For example

@example
> temp <- x > 13
@end example

@noindent
sets @code{temp} as a vector of the same length as @code{x} with values
@code{FALSE} corresponding to elements of @code{x} where the condition
is @emph{not} met and @code{TRUE} where it is.

The logical operators are @code{<}, @code{<=}, @code{>}, @code{>=},
@code{==} for exact equality and @code{!=} for inequality.
@findex <
@findex <=
@findex >
@findex >=
@findex ==
@findex !=
In addition if @code{c1} and @code{c2} are logical expressions, then
@w{@code{c1 & c2}} is their intersection (@emph{``and''}), @w{@code{c1 | c2}}
is their union (@emph{``or''}), and @code{!c1} is the negation of
@code{c1}.
@findex !
@findex |
@findex &

Logical vectors may be used in ordinary arithmetic, in which case they
are @emph{coerced} into numeric vectors, @code{FALSE} becoming @code{0}
and @code{TRUE} becoming @code{1}.  However there are situations where
logical vectors and their coerced numeric counterparts are not
equivalent, for example see the next subsection.

@node Missing values, Character vectors, Logical vectors, Simple manipulations numbers and vectors
@section Missing values
@cindex 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 @code{NA}.
@findex NA
In general any operation on an @code{NA} becomes an @code{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.

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

@example
> z <- c(1:3,NA);  ind <- is.na(z)
@end example

Notice that the logical expression @code{x == NA} is quite different
from @code{is.na(x)} since @code{NA} is not really a value but a marker
for a quantity that is not available.  Thus @code{x == NA} is a vector
of the same length as @code{x} @emph{all} of whose values are @code{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 @emph{Not a Number},
@code{NaN},
@findex NaN
values.  Examples are

@example
> 0/0
@end example

@noindent
or

@example
> Inf - Inf
@end example

@noindent
which both give @code{NaN} since the result cannot be defined sensibly.

In summary, @code{is.na(xx)} is @code{TRUE} @emph{both} for @code{NA}
and @code{NaN} values.  To differentiate these, @code{is.nan(xx)} is only
@code{TRUE} for @code{NaN}s.
@findex is.nan

@node Character vectors, Index vectors, Missing values, Simple manipulations numbers and vectors
@section Character vectors
@cindex 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.,
@code{"x-values"}, @code{"New iteration results"}.

Character vectors may be concatenated into a vector by the @code{c()}
function; examples of their use will emerge frequently.
@findex c

@findex paste
The @code{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 parameter,
@code{sep=@var{string}}, which changes it to @code{@var{string}},
possibly empty.

For example

@example
> labs <- paste(c("X","Y"), 1:10, sep="")
@end example

@noindent
makes @code{labs} into the character vector

@example
c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10")
@end example

Note particularly that recycling of short lists takes place here too;
thus @code{c("X", "Y")} is repeated 5 times to match the sequence
@code{1:10}.
@footnote{@code{paste(..., collapse=@var{ss})} allows to collapse the
arguments into a single character string putting @var{ss} in between, and
there are more tools for character manipulation, see the help
for @code{sub} and @code{substring}.}

@node Index vectors, Other types of objects, Character vectors, Simple manipulations numbers and vectors
@section Index vectors; selecting and modifying subsets of a data set
@cindex Indexing vectors

Subsets of the elements of a vector may be selected by appending to the
name of the vector an @emph{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.

@enumerate

@item
@strong{A logical vector}.  In this case the index vector must be of the
same length as the vector from which elements are to be selected.
Values corresponding to @code{TRUE} in the index vector are selected and
those corresponding to @code{FALSE} omitted.  For example

@example
> y <- x[!is.na(x)]
@end example

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

@example
> (x+1)[(!is.na(x)) & x>0] -> z
@end example

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

@item
@strong{A vector of positive integral quantities}.  In this case the
values in the index vector must lie in the the set @{1, 2, @dots{},
@code{length(x)}@}.  The corresponding elements of the vector are
selected and concatenated, @emph{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 @code{x[6]} is the sixth component of
@code{x} and

@example
> x[1:10]
@end example

@noindent
selects the first 10 elements of @code{x} (assuming @code{length(x)} is
no less than 10).  Also

@example
> c("x","y")[rep(c(1,2,2,1), times=4)]
@end example

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

@item
@strong{A vector of negative integral quantities}.  Such an index vector
specifies the values to be @emph{excluded} rather than included.  Thus

@example
> y <- x[-(1:5)]
@end example

@noindent
gives @code{y} all but the first five elements of @code{x}.

@item
@strong{A vector of character strings}.  This possibility only applies
where an object has a @code{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.

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

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

@end enumerate

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

The vector assigned must match the length of the index vector, and in
the case of a logical index vector it must again be the same length as
the vector it is indexing.

For example

@example
> x[is.na(x)] <- 0
@end example

@noindent
replaces any missing values in @code{x} by zeros and

@example
> y[y < 0] <- -y[y < 0]
@end example

@noindent
has the same effect as

@example
> y <- abs(y)
@end example

@node Other types of objects,  , Index vectors, Simple manipulations numbers and vectors
@section 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.

@itemize @bullet
@item
@emph{matrices} or more generally @emph{arrays} are multi-dimensional
generalizations of vectors.  In fact, they @emph{are} vectors that can
be indexed by two or more indices and will be printed in special ways.
@xref{Arrays and matrices}.

@item
@emph{factors} provide compact ways to handle categorical data.
@xref{Factors}.

@item
@emph{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.  @xref{Lists}.

@item
@emph{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.
@xref{Data frames}.

@item
@emph{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{}.  @xref{Writing your own functions}.

@end itemize

@node Objects, Factors, Simple manipulations numbers and vectors, Top
@chapter Objects, their modes and attributes
@cindex Objects
@cindex Attributes

@c FIXME:
@c This needs to be re-written for R.  We really have data types (as
@c returned by typeof()) and that functions mode() and storage.mode()
@c are for S compatibility mostly.  Hence in particular, there is no
@c intrinsic attribute `mode'.

@menu
* The intrinsic attributes mode and length::  
* Changing the length of an object::  
* Getting and setting attributes::  
* The class of an object::      
@end menu

@node The intrinsic attributes mode and length, Changing the length of an object, Objects, Objects
@section Intrinsic attributes: mode and length

The entities @R{} operates on are technically known as @emph{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 @emph{mode}, namely @emph{numeric}@footnote{@emph{numeric} mode is
actually an amalgam of two distinct modes, namely @emph{integer} and
@emph{double} precision, as explained in the manual.}, @emph{complex},
@emph{logical} and @emph{character} respectively.

Vectors must have their values @emph{all of the same mode}.  Thus any
given vector must be unambiguously either @emph{logical},
@emph{numeric}, @emph{complex} or @emph{character}.  The only mild
exception to this rule is the special ``value'' listed as @code{NA} for
quantities not available.  Note that a vector can be empty and still
have a mode.  For example the empty character string vector is listed as
@code{character(0)} and the empty numeric vector as @code{numeric(0)}.

@R{} also operates on objects called @emph{lists}, which are of mode
@emph{list}.  These are ordered sequences of objects which individually
can be of any mode.  @emph{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 @emph{function} and
@emph{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 @emph{formulae} used with modeling in @R{}.

By the @emph{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 @emph{length}.  The
functions @code{mode(@var{object})} and @code{length(@var{object})} can be
used to find out the mode and length of any defined structure
@footnote{Note however that @code{length(@var{object})} does not always
contain intrinsic useful information, e.g., when @code{@var{object}} is a
function.}.

Further properties of an object are usually provided by
@code{attributes(@var{object})}, see @ref{Getting and setting attributes}.
Because of this, @emph{mode} and @emph{length} are also called ``intrinsic
attributes'' of an object.
@findex mode
@findex length

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

@example
> z <- 0:9
@end example

@noindent
we could put

@example
> digits <- as.character(z)
@end example

@noindent
after which @code{digits} is the character vector @code{c("0", "1", "2",
@dots{}, "9")}.  A further @emph{coercion}, or change of mode,
reconstructs the numerical vector again:

@example
> d <- as.integer(digits)
@end example

@noindent
Now @code{d} and @code{z} are the same.@footnote{In general, coercion
from numeric to character and back again will not be exactly reversible,
because of roundoff errors in the character representation.}  There is a
large collection of functions of the form @code{as.@var{something}()}
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.

@c </FIXME>

@node Changing the length of an object, Getting and setting attributes, The intrinsic attributes mode and length, Objects
@section Changing the length of an object

An ``empty'' object may still have a mode.  For example

@example
> e <- numeric()
@end example

@noindent
makes @code{e} an empty vector structure of mode numeric.  Similarly
@code{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

@example
> e[3] <- 17
@end example

@noindent
now makes @code{e} a vector of length 3, (the first two components of
which are at this point both @code{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 @code{scan()} function for input.  (See @ref{The scan()
function}.)

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

@example
> alpha <- alpha[2 * 1:5]
@end example

@noindent
makes it an object of length 5 consisting of just the former components
with even index.  The old indices are not retained, of course.

@node Getting and setting attributes, The class of an object, Changing the length of an object, Objects
@section Getting and setting attributes
@findex attr
@findex attributes

The function @code{attributes(@var{object})}
@findex attributes
gives a list of all the non-intrinsic attributes currently defined for
that object.  The function @code{attr(@var{object}, @var{name})}
@findex attr
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 @code{@var{object}} or to
change an existing one.  For example

@example
> attr(z,"dim") <- c(10,10)
@end example

@noindent
allows @R{} to treat @code{z} as if it were a 10-by-10 matrix.

@node The class of an object,  , Getting and setting attributes, Objects
@section The class of an object
@cindex Classes

A special attribute known as the @emph{class} of the object is used to
allow for an object oriented style of programming in @R{}.

For example if an object has class @code{"data.frame"}, it will be
printed in a certain way, the @code{plot()} function will display it
graphically in a certain way, and other so-called generic functions such
as @code{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
@code{unclass()}.
@findex unclass
For example if @code{winter} has the class @code{"data.frame"} then

@example
> winter
@end example

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

@example
> unclass(winter)
@end example

@noindent
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 @ref{Object
orientation}, but only briefly.

@node Factors, Arrays and matrices, Objects, Top
@chapter Ordered and unordered factors
@cindex Factors
@cindex Ordered factors

A @emph{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 @emph{ordered} and @emph{unordered} factors.
While the ``real'' application of factors is with model formulae
(@pxref{Contrasts}), we here look at

@section A specific example

Suppose, for example, we have a sample of 30 tax accountants from all
the states and territories of Australia@footnote{Foreign 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.}  and their individual state of origin is specified by a
character vector of state mnemonics as

@example
> 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")
@end example

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

A @emph{factor} is similarly created using the @code{factor()} function:
@findex factor

@example
> statef <- factor(state)
@end example

The @code{print()} function handles factors slightly differently from
other objects:

@example
> 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
@end example

To find out the levels of a factor the function @code{levels()} can be
used.
@findex levels

@example
> levels(statef)
[1] "act" "nsw" "nt"  "qld" "sa"  "tas" "vic" "wa"
@end example

@menu
* The function tapply() and ragged arrays::  
* Ordered factors::             
@end menu

@node The function tapply() and ragged arrays, Ordered factors, Factors, Factors
@section The function @code{tapply()} and ragged arrays
@findex tapply

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

@example
> 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)
@end example

To calculate the sample mean income for each state we can now use the
special function @code{tapply()}:

@example
> incmeans <- tapply(incomes, statef, mean)
@end example

@noindent
giving a means vector with the components labelled by the levels

@example
   act    nsw     nt    qld     sa    tas    vic     wa
44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250
@end example

The function @code{tapply()} is used to apply a function, here
@code{mean()}, to each group of components of the first argument, here
@code{incomes}, defined by the levels of the second component, here
@code{statef}@footnote{Note that @code{tapply()} also works in this case
when its second argument is not a factor, e.g.,
@samp{@code{tapply(incomes, state)}}, and this is true for quite a few
other functions, since arguments are @emph{coerced} to factors when
necessary (using @code{as.factor()}).}, 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 @code{var()} to calculate the sample variance, such a function
is a very simple one liner, specified by the assignment:

@example
> stderr <- function(x) sqrt(var(x)/length(x))
@end example

@noindent
(Writing functions will be considered later in @ref{Writing your own
functions}.)  After this assignment, the standard errors are calculated
by

@example
> incster <- tapply(incomes, statef, stderr)
@end example

@noindent
and the values calculated are then

@example
> 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
@end example

As an exercise you may care to find the usual 95% confidence limits for
the state mean incomes.  To do this you could use @code{tapply()} once
more with the @code{length()} function to find the sample sizes, and the
@code{qt()} function to find the percentage points of the appropriate
@math{t}-distributions.

The function @code{tapply()} can 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 category) what happens can be thought of as follows.  The
values in the vector are collected into groups corresponding to the
distinct entries in the category.  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 category.

The combination of a vector and a labelling factor is an example of what
is sometimes called a @emph{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.


@node Ordered factors,  , The function tapply() and ragged arrays, Factors
@section Ordered factors
@findex ordered

The levels of factors are stored in alphabetical order, or in the order
they were specified to @code{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 @code{ordered()}
@findex ordered
function creates such ordered factors but is otherwise identical to
@code{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.


@node Arrays and matrices, Lists and data frames, Factors, Top
@chapter Arrays and matrices

@menu
* Arrays::                      
* Array indexing::              
* Index arrays::                
* The array() function::        
* The outer product of two arrays::  
* Generalized transpose of an array::  
* Matrix facilities::           
* Forming partitioned matrices::  
* The concatenation function c() with arrays::  
* Frequency tables from factors::  
@end menu

@node Arrays, Array indexing, Arrays and matrices, Arrays and matrices
@section Arrays
@cindex Arrays
@cindex Matrices

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 positive integers.  If its length is
@math{k} then the array is @math{k}-dimensional, i.e., a matrix is a
@math{2}-dimensional array.  The values in the
dimension vector give the upper limits for each of the @math{k}
subscripts.  The lower limits are always 1.

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

@example
> dim(z) <- c(3,5,100)
@end example
@findex dim

@noindent
gives it the @emph{dim} attribute that allows it to be treated as a
@math{3} by @math{5} by @math{100} array.

Other functions such as @code{matrix()} and @code{array()} are available
for simpler and more natural looking assignments, as we shall see in
@ref{The array() function}.

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 @code{a} is
@code{c(3,4,2)} then there are @math{3 * 4 * 2 = 24} entries in @code{a}
and the data vector holds them in the order @code{a[1,1,1], a[2,1,1],
@dots{}, a[2,4,2], a[3,4,2]}.

@node Array indexing, Index arrays, Arrays, Arrays and matrices
@section Array indexing.  Subsections of an array
@cindex Indexing of and by arrays

Individual elements of an array may be referenced, as above, 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 @emph{index vectors} in place of subscripts; however
@emph{if any index position is given an empty index vector, then the
full range of that subscript is taken}.

Continuing the previous example, @code{a[2,,]} is a @math{4 * 2} array
with dimension vector @code{c(4,2)} and data vector containing the
values

@example
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])
@end example

@noindent
in that order.  @code{a[,,]} stands for the entire array, which is the
same as omitting the subscripts entirely and using @code{a} alone.

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

Also, if an array name is given with just @emph{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.

@menu
* Index arrays::
* The array() function::
@end menu

@node Index arrays, The array() function, Array indexing, Arrays and matrices
@section Index arrays

As well as an index vector in any subscript position, an array may be
used with a single @emph{index array} 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 @math{4} by @math{5} array @code{X} and we wish to do
the following:

@itemize @bullet
@item
Extract elements @code{X[1,3]}, @code{X[2,2]} and @code{X[3,1]} as a
vector structure, and
@item
Replace these entries in the array @code{X} by 0s.
@end itemize
In this case we need a @math{3} by @math{2} subscript array, as in the
following example.

@example
> x <- array(1:20,dim=c(4,5))   # @r{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                             # @r{@code{i} is a 3 by 2 index array.}
     [,1] [,2]
[1,]    1    3
[2,]    2    2
[3,]    3    1
> x[i]                          # @r{Extract those elements}
[1] 9 6 3
> x[i] <- 0                     # @r{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
>
@end example

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

@example
> 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)
@end example

To construct the incidence matrix, @code{N} say, we could use

@example
> N <- crossprod(Xb, Xv)
@end example
@findex crossprod

However a simpler direct way of producing this matrix is to use
@code{table()}:
@findex table

@example
> N <- table(blocks, varieties)
@end example

@node The array() function, The outer product of two arrays, Index arrays, Arrays and matrices
@section The @code{array()} function
@findex array

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

@example
> Z <- array(@var{data_vector}, @var{dim_vector})
@end example

For example, if the vector @code{h} contains 24, or fewer, numbers then
the command

@example
> Z <- array(h, dim=c(3,4,2))
@end example

@noindent
would use @code{h} to set up @math{3} by @math{4} by @math{2} array in
@code{Z}.  If the size of @code{h} is exactly 24 the result is the same
as

@example
> dim(Z) <- c(3,4,2)
@end example

However if @code{h} is shorter than 24, its values recycled from the
beginning again to make it up to size 24 (@pxref{The recycling rule}).
As an extreme but common example

@example
> Z <- array(0, c(3,4,2))
@end example

@noindent
makes @code{Z} an array of all zeros.

At this point @code{dim(Z)} stands for the dimension vector
@code{c(3,4,2)}, and @code{Z[1:24]} stands for the data vector as it was
in @code{h}, and @code{Z[]} with an empty subscript or @code{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
@code{dim} attributes of operands generally need to be the same, and
this becomes the dimension vector of the result.  So if @code{A},
@code{B} and @code{C} are all similar arrays, then

@example
> D <- 2*A*B + C + 1
@end example

@noindent
makes @code{D} a similar array with data vector the result of the
evident element by element operations.  However the precise rule
concerning mixed array and vector calculations has to be considered a
little more carefully.

@menu
* The recycling rule::          
@end menu

@node The recycling rule,  , The array() function, The array() function
@subsection Mixed vector and array arithmetic.  The recycling rule
@cindex 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.

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

@node The outer product of two arrays, Generalized transpose of an array, The array() function, Arrays and matrices
@section The outer product of two arrays
@cindex Outer products of arrays

An important operation on arrays is the @emph{outer product}.  If
@code{a} and @code{b} are two numeric arrays, their outer product is an
array whose dimension vector is got 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 @code{a} with
those of @code{b}.  The outer product is formed by the special operator
@code{%o%}:
@findex %o%

@example
> ab <- a %o% b
@end example

An alternative is

@example
> ab <- outer(a, b, "*")
@end example
@findex outer

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

@example
> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)
@end example

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 @ref{Writing your own
functions}.

@unnumberedsubsec An example: Determinants of 2 by 2 digit matrices

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

The problem is to find the determinants, @math{ad - bc}, of all possible
matrices of this form and represent the frequency with which each value
occurs as a @emph{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 @code{outer()} function twice:

@example
> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(as.numeric(names(fr)), fr, type="h",
       xlab="Determinant", ylab="Frequency")
@end example

Notice the coercion of the @code{names} attribute of the frequency table
to numeric in order to recover the range of the determinant values.  The
``obvious'' way of doing this problem with @code{for} loops, to be
discussed in @ref{Loops and conditional execution}, is so inefficient as
to be impractical.

It is also perhaps surprising that about 1 in 20 such matrices is
singular.

@node Generalized transpose of an array, Matrix facilities, The outer product of two arrays, Arrays and matrices
@section Generalized transpose of an array
@cindex Generalized transpose of an array

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

@example
> B <- aperm(A, c(2,1))
@end example

@noindent
is just the transpose of @code{A}.  For this special case a simpler
function @code{t()}
@findex t
is available, so we could have used @code{B <- t(A)}.

@node Matrix facilities, Forming partitioned matrices, Generalized transpose of an array, Arrays and matrices
@section Matrix facilities

@macro xTx{}
 @eqn{@strong{x}'@strong{x}, x'x}
@end macro
@macro xxT{}
 @eqn{@strong{x}@strong{x}', x x'}
@end macro

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 @code{t(X)} is the matrix transpose function, as
noted above.  The functions @code{nrow(A)} and @code{ncol(A)} give the
number of rows and columns in the matrix @code{A} respectively.
@findex nrow
@findex ncol

@menu
* Multiplication::              
* Linear equations and inversion::  
* Eigenvalues and eigenvectors::  
* Singular value decomposition and determinants::  
* Least squares fitting and the QR decomposition::  
@end menu

@node  Multiplication, Linear equations and inversion, Matrix facilities, Matrix facilities
@subsection Matrix multiplication

@cindex Matrix multiplication
The operator @code{%*%} is used for matrix multiplication.
@findex %*%
An @math{n} by @math{1} or @math{1} by @math{n} matrix may of course be
used as an @math{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, @code{A} and @code{B} are square matrices of the same
size, then

@example
> A * B
@end example

@noindent
is the matrix of element by element products and

@example
> A %*% B
@end example

@noindent
is the matrix product.  If @code{x} is a vector, then

@example
> x %*% A %*% x
@end example

@noindent
is a quadratic form.@footnote{Note that @code{x %*% x} is ambiguous, as
it could mean either @xTx{} or @xxT{}, where @eqn{@strong{x},x} is
the column form.  In such cases the smaller matrix seems implicitly to be the
interpretation adopted, so the scalar @xTx{} is in
this case the result.  The matrix @xxT{} may be
calculated either by @code{cbind(x) %*% x} or @code{x %*% rbind(x)}
since the result of @code{rbind()} or @code{cbind()} is always a
matrix.}

@findex crossprod
The function @code{crossprod()} forms ``crossproducts'', meaning that
@code{crossprod(X, y)} is the same as @code{t(X) %*% y} but the
operation is more efficient.  If the second argument to
@code{crossprod()} is omitted it is taken to be the same as the first.

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

@node Linear equations and inversion, Eigenvalues and eigenvectors, Multiplication, Matrix facilities
@subsection Linear equations and inversion

@cindex Linear equations
@findex solve
Solving linear equations is the inverse of matrix multiplication.
When after

@example
> b <- A %*% x
@end example

@noindent
only @code{A} and @code{b} are given, the vector @code{x} is the
solution of that linear equation system.  In @R{},

@example
> solve(A,b)
@end example

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

@example
solve(A)
@end example

@noindent
but rarely is needed.  Numerically, it is both inefficient and unstable
to compute @code{x <- solve(A) %*% b} instead of @code{solve(A,b)}.

The quadratic form
@eqn{@strong{x' A^{-1} x},@ @code{x %*% A^@{-1@} %*% x} @ }
which is used in multivariate computations, should be computed by something like
@code{x %*% solve(A,x)}, rather than computing the inverse of @code{A}.

@node Eigenvalues and eigenvectors, Singular value decomposition and determinants, Linear equations and inversion, Matrix facilities
@subsection Eigenvalues and eigenvectors
@cindex Eigenvalues and eigenvectors

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

@example
> ev <- eigen(Sm)
@end example

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

@example
> evals <- eigen(Sm)$values
@end example

@noindent
@code{evals} now holds the vector of eigenvalues and the second
component is discarded.  If the expression

@example
> eigen(Sm)
@end example

@noindent
is used by itself as a command the two components are printed, with
their names.

@node Singular value decomposition and determinants, Least squares fitting and the QR decomposition, Eigenvalues and eigenvectors, Matrix facilities
@subsection Singular value decomposition and determinants
@cindex Singular value decomposition

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

If @code{M} is in fact square, then, it is not hard to see that

@example
> absdetM <- prod(svd(M)$d)
@end example

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

@example
> absdet <- function(M) prod(svd(M)$d)
@end example

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

@c Functions will be discussed formally later in these notes.

@node Least squares fitting and the QR decomposition,  , Singular value decomposition and determinants, Matrix facilities
@subsection Least squares fitting and the QR decomposition
@cindex Least squares fitting
@cindex QR decomposition

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

@example
> ans <- lsfit(X, y)
@end example
@findex lsfit

@noindent
gives the results of a least squares fit where @code{y} is the vector of
observations and @code{X} is the design matrix.  See the help facility
for more details, and also for the follow-up function @code{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 @code{X}.

@findex qr
Another closely related function is @code{qr()} and its allies.
Consider the following assignments

@example
> Xplus <- qr(X)
> b <- qr.coef(Xplus, y)
> fit <- qr.fitted(Xplus, y)
> res <- qr.resid(Xplus, y)
@end example

@noindent
These compute the orthogonal projection of @code{y} onto the range of
@code{X} in @code{fit}, the projection onto the orthogonal complement in
@code{res} and the coefficient vector for the projection in @code{b},
that is, @code{b} is essentially the result of the @sc{Matlab}
`backslash' operator.

It is not assumed that @code{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 @ref{Statistical models in R}.


@node Forming partitioned matrices, The concatenation function c() with arrays, Matrix facilities, Arrays and matrices
@section Forming partitioned matrices, @code{cbind()} and @code{rbind()}
@findex cbind
@findex rbind

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

In the assignment

@example
> X <- cbind(@var{arg_1}, @var{arg_2}, @var{arg_3}, @dots{})
@end example

@noindent
the arguments to @code{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 @var{arg_1},
@var{arg_2}, @dots{} forming the columns.

If some of the arguments to @code{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 @code{rbind()} does the corresponding operation for rows.
In this case any vector argument, possibly cyclically extended, are of
course taken as row vectors.

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

@example
> X <- cbind(1, X1, X2)
@end example

The result of @code{rbind()} or @code{cbind()} always has matrix status.
Hence @code{cbind(x)} and @code{rbind(x)} are possibly the simplest ways
explicitly to allow the vector @code{x} to be treated as a column or row
matrix respectively.

@node The concatenation function c() with arrays, Frequency tables from factors, Forming partitioned matrices, Arrays and matrices
@section The concatenation function, @code{c()}, with arrays

It should be noted that whereas @code{cbind()} and @code{rbind()} are
concatenation functions that respect @code{dim} attributes, the basic
@code{c()} function does not, but rather clears numeric objects of all
@code{dim} and @code{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 @code{as.vector()}

@example
> vec <- as.vector(X)
@end example
@findex as.vector

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

@example
> vec <- c(X)
@end example
@findex c

There are slight differences between the two, but ultimately the choice
between them is largely a matter of style (with the former being
preferable).

@node Frequency tables from factors,  , The concatenation function c() with arrays, Arrays and matrices
@section Frequency tables from factors
@cindex Tabulation

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

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

@example
> statefr <- table(statef)
@end example

@noindent
gives in @code{statefr} a table of frequencies of each state in the
sample.  The frequencies are ordered and labelled by the levels
attribute of the category.  This simple case is equivalent to, but more
convenient than,

@example
> statefr <- tapply(statef, statef, length)
@end example

Further suppose that @code{incomef} is a category giving a suitably
defined ``income class'' for each entry in the data vector, for example
with the @code{cut()} function:

@example
> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef
@end example
@findex cut

Then to calculate a two-way table of frequencies:

@example
> 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
@end example

Extension to higher way frequency tables is immediate.

@node Lists and data frames, Reading data from files, Arrays and matrices, Top
@chapter Lists and data frames

@menu
* Lists::                       
* Constructing and modifying lists::  
* Data frames::                 
@end menu

@node Lists, Constructing and modifying lists, Lists and data frames, Lists and data frames
@section Lists
@cindex Lists

An @R{} @emph{list} is an object consisting of an ordered collection of
objects known as its @emph{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:

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

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

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

Components of lists may also be @emph{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

@example
> @var{name}$@var{component_name}
@end example

@noindent
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:

@code{Lst$name} is the same as @code{Lst[[1]]} and is the string
@code{"Fred"},

@code{Lst$wife} is the same as @code{Lst[[2]]} and is the string
@code{"Mary"},

@code{Lst$child.ages[1]} is the same as @code{Lst[[4]][1]} and is the
number @code{4}.

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

@example
> x <- "name"; Lst[[x]]
@end example

It is very important to distinguish @code{Lst[[1]]} from @code{Lst[1]}.
@samp{@code{[[@var{@dots{}}]]}} is the operator used to select a single
element, whereas @samp{@code{[@var{@dots{}}]}} is a general
subscripting operator.  Thus the former is the @emph{first object in the
list} @code{Lst}, and if it is a named list the name is @emph{not}
included.  The latter is a @emph{sublist of the list @code{Lst}
consisting of the first entry only.  If it is a named list, the name is
transferred to the sublist.}

The names of components may be abbreviated down to the minimum number of
letters needed to identify them uniquely.  Thus @code{Lst$coefficients}
may be minimally specified as @code{Lst$coe} and @code{Lst$covariance}
as @code{Lst$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 @emph{names} attribute also.

@node Constructing and modifying lists, Data frames, Lists, Lists and data frames
@section Constructing and modifying lists

New lists may be formed from existing objects by the function
@code{list()}.  An assignment of the form

@example
> Lst <- list(@var{name_1}=@var{object_1}, @var{@dots{}}, @var{name_m}=@var{object_m})
@end example

@noindent
sets up a list @code{Lst} of @math{m} components using @var{object_1},
@dots{}, @var{object_m} 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 @emph{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

@example
> Lst[5] <- list(matrix=Mat)
@end example

@menu
* Concatenating lists::         
@end menu

@node Concatenating lists,  , Constructing and modifying lists, Constructing and modifying lists
@subsection Concatenating lists
@cindex Concatenating lists

@findex c
When the concatenation function @code{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.

@example
> list.ABC <- c(list.A, list.B, list.C)
@end example

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 @code{dim} attributes, are
discarded.


@node Data frames,  , Constructing and modifying lists, Lists and data frames
@section Data frames
@cindex Data frames

A @emph{data frame} is a list with class @code{"data.frame"}.  There are
restrictions on lists that may be made into data frames, namely

@itemize @bullet
@item
The components must be vectors (numeric, character, or logical),
factors, numeric matrices, lists, or other data frames.
@item
Matrices, lists, and data frames provide as many variables to the new
data frame as they have columns, elements, or variables, respectively.
@item
Numeric vectors and factors are included as is, and non-numeric
(character and logical) vectors are coerced to be factors, whose levels
are the unique values appearing in the vector.
@item
Vector structures appearing as variables of the data frame must all have
the @emph{same length}, and matrix structures must all have the same
@emph{row size}.
@end itemize

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.

@menu
* Making data frames::          
* attach() and detach()::       
* Working with data frames::    
* Attaching arbitrary lists::   
* Managing the search path::    
@end menu

@node Making data frames, attach() and detach(), Data frames, Data frames
@subsection 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
@code{data.frame}:
@findex data.frame

@example
> accountants <- data.frame(home=statef, loot=income, shot=incomef)
@end example

A list whose components conform to the restrictions of a data frame may
be @emph{coerced} into a data frame using the function
@code{as.data.frame()}
@findex as.data.frame

The simplest way to construct a data frame from scratch is to use the
@code{read.table()} function to read an entire data frame from an
external file.  This is discussed further in @ref{Reading data from
files}.

@node attach() and detach(), Working with data frames, Making data frames, Data frames
@subsection @code{attach() and @code{detach()}}
@findex attach
@findex detach

The @code{$} notation, such as @code{accountants$statef}, 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 @code{attach()} function, as well as having a directory name as its
argument, may also have a data frame.  Thus suppose @code{lentils} is a
data frame with three variables @code{lentils$u}, @code{lentils$v},
@code{lentils$w}.  The attach

@example
> attach(lentils)
@end example

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

@example
> u <- v+w
@end example

@noindent
does not replace the component @code{u} of the data frame, but rather
masks it with another variable @code{u} in the working directory at
@w{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
@code{$} notation:

@example
> lentils$u <- v+w
@end example

However the new value of component @code{u} is not visible until the
data frame is detached and attached again.

To detach a data frame, use the function

@example
> detach()
@end example

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

@quotation
@strong{NOTE}: With the current release of @R{} the search path can
contain at most 20 items.  Avoid attaching the same data frame more than
once.  Always detach the data frame as soon as you have finished using
its variables.
@end quotation

@quotation
@strong{NOTE}: With the current release of @R{} lists and data frames
can only be attached at position 2 or above.  It is not possible to
directly assign into an attached list or data frame (thus, to some
extent they are static).
@c This is likely to change in the next year or so.
@end quotation

@node Working with data frames, Attaching arbitrary lists, attach() and detach(), Data frames
@subsection Working with data frames

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

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

In this way it is quite simple to work with many problems in the same
directory, all of which have variables named @code{x}, @code{y} and
@code{z}, for example.

@node Attaching arbitrary lists, Managing the search path, Working with data frames, Data frames
@subsection Attaching arbitrary lists

@code{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 @code{"list"} may be
attached in the same way:

@example
> attach(any.old.list)
@end example

Anything that has been attached can be detached by @code{detach}, by
position number or, preferably, by name.

@node Managing the search path,  , Attaching arbitrary lists, Data frames
@subsection Managing the search path
@findex search
@cindex Search path

The function @code{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

@example
> search()
[1] ".GlobalEnv"   "Autoloads"    "package:base"
@end example
@noindent
where @code{.GlobalEnv} is the workspace.@footnote{See the on-line help
for @code{autoload} for the meaning of the second term.}

After @code{lentils} is attached we have

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

@noindent
and as we see @code{ls} (or @code{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.

@example
> detach("lentils")
> search()
[1] ".GlobalEnv"   "Autoloads"    "package:base"
@end example

@node Reading data from files, Probability distributions, Lists and data frames, Top
@chapter Reading data from files
@cindex 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@footnote{Under UNIX, the utilities Sed or
Awk can be used.} 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 @code{read.table()} function.  There is also a more primitive input
function, @code{scan()}, that can be called directly.

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

@menu
* The read.table() function::   
* The scan() function::         
* Accessing builtin datasets::  
* Editing data::                
@end menu

@node The read.table() function, The scan() function, Reading data from files, Reading data from files
@section The @code{read.table()} function
@findex read.table

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

@itemize @bullet
@item
The first line of the file should have a @emph{name} for each variable
in the data frame.

@item
Each additional line of the file has its first item a @emph{row label}
and the values for each variable.
@end itemize

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.

@quotation
@cartouche
@smallexample
@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
...
@end smallexample
@end cartouche
@end quotation

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

The function @code{read.table()} can then be used to read the data frame
directly

@example
> HousePrice <- read.table("houses.data")
@end example

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.

@quotation
@cartouche
@smallexample
@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
...
@end smallexample
@end cartouche
@end quotation

The data frame may then be read as

@example
> HousePrice <- read.table("houses.data", header=TRUE)
@end example

@noindent
where the @code{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.

@menu
* The scan() function::
@end menu

@node The scan() function, Accessing builtin datasets, The read.table() function, Reading data from files
@section The @code{scan()} function
@findex scan

Suppose the data vectors are of equal length and are to be read in 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
@file{input.dat}.  The first step is to use @code{scan()} to read in the
three vectors as a list, as follows

@example
> inp <- scan("input.dat", list("",0,0))
@end example

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

@example
> label <- inp[[1]]; x <- inp[[2]]; y <- inp[[3]]
@end example

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

@example
> inp <- scan("input.dat", list(id="", x=0, y=0))
@end example

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

@example
> label <- in$id; x <- in$x; y <- in$y
@end example

@noindent
or the list may be attached at @w{position 2} of the search path
(@pxref{Attaching arbitrary lists}).

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.

@example
> X <- matrix(scan("light.dat", 0), ncol=5, byrow=TRUE)
@end example

There are more elaborate input facilities available and these are
detailed in the manual.

@node Accessing builtin datasets, Editing data, The scan() function, Reading data from files
@section Accessing builtin datasets
@cindex Accessing builtin datasets
@findex data

Over fifty datasets are supplied with @R{}, and others are available in
packages (including the standard packages supplied with @R{}).  Unlike
@SPLUS{}, these datasets have to be loaded explicitly, using the function
@code{data}.  To see the list of datasets in the base system use

@example
data()
@end example

@noindent
and to load one of these use, for example,

@example
data(infert)
@end example

@noindent
In most cases this will load an @R{} object of the same name, usually a
data frame.  However, in a few cases it loads several objects, so see the
on-line help for the object to see what to expect.

@subsection Loading data from other R packages

To access data from another package, use the @code{package} argument,
for example

@example
data(package="nls")
data(Puromycin, package="nls")
@end example

If a package has been attached by @code{library}, its datasets are
automatically included in the search, so that

@example
library(nls)
data()
data(Puromycin)
@end example

@noindent
will list all the datasets in all the currently attached packages (at
least @pkg{base} and @pkg{nls}) and then load @code{Puromycin} from the
first package in which such a dataset is found.

User-contributed packages can be a rich source of datasets.

@node Editing data,  , Accessing builtin datasets, Reading data from files
@section Editing data

@findex edit
When invoked on a data frame or matrix, @code{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

@smallexample
> xnew <- edit(xold)
@end smallexample

@noindent
will allow you to edit your data set @code{xold}, and on completion the
changed object is assigned to @code{xnew}.

Use

@example
> xnew <- edit(data.frame())
@end example

@noindent
to enter new data via the spreadsheet interface.


@node Probability distributions, Loops and conditional execution, Reading data from files, Top
@chapter Probability distributions
@cindex Probability distributions

@menu
* R as a set of statistical tables::  
* Examining the distribution of a set of data::  
* One- and two-sample tests::   
@end menu

@node R as a set of statistical tables, Examining the distribution of a set of data, Probability distributions, Probability distributions
@section 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 @eqn{P(X \le x), P(X <= x)},
the probability density function and the quantile function (given
@math{q}, the smallest @math{x} such that @eqn{P(X \le x) > q, P(X <= x) > q}),
and to simulate from the distribution.

@quotation
@multitable{Distribution namessss}{names, names}{arguments, arguments}

@item Distribution @tab @R{} name @tab additional arguments

@item beta @tab @code{beta} @tab @code{shape1, shape2, ncp}
@item binomial @tab  @code{binom} @tab @code{size, prob}
@item Cauchy @tab @code{cauchy} @tab @code{location, scale}
@item chi-squared @tab @code{chisq} @tab @code{df, ncp}
@item exponential @tab @code{exp} @tab @code{rate}
@item F @tab @code{f} @tab @code{df1, df1, ncp}
@item gamma @tab @code{gamma} @tab @code{shape, scale}
@item geometric @tab @code{geom} @tab @code{prob}
@item hypergeometric @tab @code{hyper} @tab @code{m, n, k}
@item log-normal @tab @code{lnorm} @tab @code{meanlog, sdlog}
@item logistic @tab @code{logis} @tab @code{location, scale}
@item negative binomial @tab @code{nbinom} @tab @code{size, prob}
@item normal @tab @code{norm} @tab @code{mean, sd}
@item Poisson @tab @code{pois} @tab @code{lambda}
@item Student's t @tab @code{t} @tab @code{df, ncp}
@item uniform @tab @code{unif} @tab @code{min, max}
@item Weibull @tab @code{weibull} @tab @code{shape, scale}
@item Wilcoxon @tab @code{wilcox} @tab @code{m, n}

@end multitable
@end quotation

@noindent
Prefix the name given here by @samp{d} for the density, @samp{p} for the
CDF, @samp{q} for the quantile function and @samp{r} for simulation
(@emph{r}andom deviates).  The first argument is @code{x} for
@code{d@var{xxx}}, @code{q} for @code{p@var{xxx}}, @code{p} for
@code{q@var{xxx}} and @code{n} for @code{r@var{xxx}} (except for
@code{rhyper} and @code{rwilcox}, for which it is @code{nn}).  The
non-centrality parameter @code{ncp} is currently only available for the
CDFs and a few other functions: see the on-line help for current
details.

The @code{p@var{xxx}} and @code{q@var{xxx}} functions all have logical
arguments @code{lower.tail} and @code{log.p} and the @code{d@var{xxx}}
ones have @code{log}.  This allows, e.g., getting the cumulative (or
``integrated'') @emph{hazard} function, @eqn{H(t) = - \log(1 - F(t)),
H(t) = - log(1 - F(t))}, by

@smallexample
 - p@var{xxx}(t, ..., lower.tail = FALSE, log.p = TRUE)
@end smallexample

@noindent
or more accurate log-likelihoods (by @code{d@var{xxx}(..., log =
TRUE)}), directly.

In addition there are functions @code{ptukey} and @code{qtukey} for the
distribution of the studentized range of samples from a normal
distribution.

Here are some examples

@example
> ## @r{2-tailed p-value for t distribution}
> 2*pt(-2.43, df = 13)
> ## @r{upper 1% point for an F(2, 7) distribution}
> qf(0.99, 2, 7)
@end example

@node  Examining the distribution of a set of data, One- and two-sample tests, R as a set of statistical tables, Probability distributions
@section 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 @code{summary} and
@code{fivenum}
@findex summary
@findex fivenum
and a display of the numbers by @code{stem} (a ``stem and leaf'' plot).
@findex stem

@example
> data(faithful)
> 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
@end example

A stem-and-leaf plot is like a histogram, and @R{} has a function
@code{hist} to plot histograms.
@findex hist

@example
> hist(eruptions)
## @r{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) # @r{show the actual data points}
@end example

@findex density
@cindex Density estimation
More elegant density plots can be made by @code{density}, and we added a
line produced by @code{density} in this example.  The bandwidth
@code{bw} was chosen by trial-and-error as the default gives too much
smoothing (it usually does for ``interesting'' densities).  (Automated
methods of bandwidth choice are implemented in packages @pkg{MASS} and
@pkg{KernSmooth}.)

@iftex
@image{images/hist,9cm}
@end iftex

We can plot the empirical cumulative distribution function by using
function @code{ecdf} in the standard package @pkg{stepfun}.
@findex ecdf
@cindex Empirical CDFs

@example
> library(stepfun)
> plot(ecdf(eruptions), do.points=FALSE, verticals=TRUE)
@end example

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.

@example
> 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)
@end example

@iftex
@image{images/ecdf,9cm}
@end iftex

Quantile-quantile (Q-Q) plots can help us examine this more carefully.
@cindex Quantile-quantile plots
@findex qqnorm
@findex qqline

@example
par(pty="s")
qqnorm(long); qqline(long)
@end example

@noindent
which shows a reasonable fit but a shorter right tail that one would
expect from a normal distribution.  Let us compare this with some
simulated data from a @math{t} distribution

@iftex
@image{images/QQ,7cm}
@end iftex

@example
x <- rt(250, df = 5)
qqnorm(x); qqline(x)
@end example

@noindent
which will usually (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

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

Finally, we might want a more formal test of agreement with normality
(or not).  Package @pkg{ctest} provides the Shapiro-Wilk test
@cindex Shapiro-Wilk test
@findex shapiro.test

@example
> library(ctest)
> shapiro.test(long)

         Shapiro-Wilk normality test

data:  long
W = 0.9793, p-value = 0.01052
@end example

@noindent
and the Kolmogorov-Smirnov test
@cindex Kolmogorov-Smirnov test
@findex ks.test

@example
> 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
@end example

@noindent
(Note that the distribution theory is not valid here as we
have estimated the parameters of the normal distribution from the same
sample.)

@node One- and two-sample tests,  , Examining the distribution of a set of data, Probability distributions
@section One- and two-sample tests
@cindex 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 @pkg{ctest}, which may need to be loaded explicitly using
@code{library(ctest)} to make the tests available.

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

@example
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
@end example

@noindent
Boxplots provide a simple graphical comparison of the two samples.

@example
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)
@end example
@findex boxplot
@cindex Box plots

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

@iftex
@image{images/ice,7cm}
@end iftex

To test for the equality of the means of the two examples, we can use
an @emph{unpaired} @math{t}-test by
@cindex Student's @math{t} test
@findex t.test

@example
> library(ctest)
> 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
@end example

@noindent
which does indicate a significant difference, assuming normality.  By
default the @R{} function does not assume equality of variances in the
two samples (in contrast to the similar @SPLUS{} @code{t.test}
function).  We can use the F test to test for equality in the variances,
provided that the two samples are from normal populations.

@example
> 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
@end example
@findex var.test

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

@example
> 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
@end example

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.

@cindex Wilcoxon test
@findex wilcox.test
@example
> wilcox.test(A, B)

         Wilcoxon rank sum test with continuity correction

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

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

@noindent
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

@example
> library(stepfun)
> plot(ecdf(A), do.points=FALSE, verticals=TRUE, xlim=range(A, B))
> plot(ecdf(B), do.points=FALSE, verticals=TRUE, add=TRUE)
@end example

@noindent
will show the two empirical CDFs, and @code{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:

@example
> 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)
@end example

@node Loops and conditional execution, Writing your own functions, Probability distributions, Top
@chapter More language features.  Loops and conditional execution
@cindex Loops and conditional execution

@menu
* Grouped expressions::         
* Control statements::          
@end menu

@node Grouped expressions, Control statements, Loops and conditional execution, Loops and conditional execution
@section Grouped expressions
@cindex 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, @code{@{@var{expr_1};
@var{@dots{}}; @var{expr_m}@}}, 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 a part of an even larger expression, and so on.

@node Control statements,  , Grouped expressions, Loops and conditional execution
@section Control statements
@cindex Control statements

@menu
* Conditional execution::       
* Repetitive execution::        
@end menu

@node Conditional execution, Repetitive execution, Control statements, Control statements
@subsection Conditional execution: @code{if} statements
@findex if

The language has available a conditional construction of the form

@example
> if (@var{expr_1}) @var{expr_2} else @var{expr_3}
@end example
@findex if
@findex else

@noindent
where @var{expr_1} must evaluate to a logical value and the result of
the entire expression is then evident.

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

@findex ifelse
There is a vectorized version of the @code{if}/@code{else} construct,
the @code{ifelse} function.  This has the form @code{ifelse(condition, a,
b)} and returns a vector of the length of its longest argument, with
elements @code{a[i]} if @code{condition[i]} is true, otherwise
@code{b[i]}.


@node Repetitive execution,  , Conditional execution, Control statements
@subsection Repetitive execution: @code{for} loops, @code{repeat} and @code{while}
@findex for

There is also a @code{for} loop construction which has the form

@example
> for (@code{@var{name}} in @var{expr_1}) @var{expr_2}
@end example

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

As an example, suppose @code{ind} is a vector of class indicators and we
wish to produce separate plots of @code{y} versus @code{x} within
classes.  One possibility here is to use @code{coplot()} to be discussed
later, 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:

@example
> 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]]))
  @}
@end example

@findex split

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

@quotation
@strong{WARNING}: @code{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{}.
@end quotation

Other looping facilities include the

@example
> repeat @var{expr}
@end example
@findex repeat

@noindent
statement and the

@example
> while (@var{condition}) @var{expr}
@end example
@findex while

@noindent
statement.

The @code{break} statement can be used to terminate any loop, possibly
abnormally.  This is the only way to terminate @code{repeat} loops.
@findex break

The @code{next} statement can be used to discontinue one particular
cycle and skip to the ``next''.
@findex next

Control statements are most often used in connection with
@emph{functions} which are discussed in @ref{Writing your own
functions}, and where more examples will emerge.


@node Writing your own functions, Statistical models in R, Loops and conditional execution, Top
@chapter Writing your own functions
@cindex Writing functions

As we have seen informally along the way, the @R{} language allows the
user to create objects of mode @emph{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 @code{mean()}, @code{var()},
@code{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

@example
> @var{name} <- function(@var{arg_1}, @var{arg_2}, @dots{}) @var{expression}
@end example
@findex function

@noindent
The @var{expression} is an @R{} expression, (usually a grouped
expression), that uses the arguments, @var{arg_i}, 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
@code{@var{name}(@var{expr_1}, @var{expr_2}, @dots{})} and may occur
anywhere a function call is legitimate.

@menu
* Simple examples::             
* Defining new binary operators::  
* Named arguments and defaults::  
* The ellipsis argument (...)::  
* Assignment within functions::  
* More advanced examples::      
* Scope::                       
* Customizing the environment::  
* Object orientation::          
@end menu

@node Simple examples, Defining new binary operators, Writing your own functions, Writing your own functions
@section Simple examples

As a first example, consider a function to calculate the two sample
@math{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:

@example
> 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(s2*(1/n1 + 1/n2))
    tst
  @}
@end example

With this function defined, you could perform two sample @math{t}-tests
using a call such as

@example
> tstat <- twosam(data$male, data$female); tstat
@end example

As a second example, consider a function to emulate directly the
@sc{Matlab} backslash command, which returns the coefficients of the
orthogonal projection of the vector @math{y} onto the column space of
the matrix, @math{X}.  (This is ordinarily called the least squares
estimates of the regression coefficients.)  This would ordinarily be
done with the @code{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 @math{n} by @math{1} vector @math{y} and an @math{n} by
@math{p} matrix @math{X} then @math{X \ y} is defined as
@ifnottex
(X'X)^@{-@}X'y, where (X'X)^@{-@}
@end ifnottex
@tex
$(X'X)^{-}X'y$, where $(X'X)^{-}$
@end tex
is a generalized inverse of @math{X'X}.

@example
> bslash <- function(X, y) @{
  X <- qr(X)
  qr.coef(X, y)
@}
@end example

After this object is created it is permanent, like all objects, and may
be used in statements such as

@example
> regcoeff <- bslash(Xmat, yvar)
@end example

@noindent
and so on.

The classical @R{} function @code{lsfit()} does this job quite well, and
more@footnote{See also the methods described in @ref{Statistical models
in R}}.  It in turn uses the functions @code{qr()} and @code{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.

@node Defining new binary operators, Named arguments and defaults, Simple examples, Writing your own functions
@section Defining new binary operators
@cindex Binary operators

Had we given the @code{bslash()} function a different name, namely one of
the form

@example
%@var{anything}%
@end example

@noindent
it could have been used as a @emph{binary operator} in expressions
rather than in function form.  Suppose, for example, we choose @code{!}
for the internal character.  The function definition would then start as

@example
> "%!%" <- function(X, y) @{ @dots{} @}
@end example

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

The matrix multiplication operator, @code{%*%}, and the outer product
matrix operator @code{%o%} are other examples of binary operators
defined in this way.

@node Named arguments and defaults, The ellipsis argument (...), Defining new binary operators, Writing your own functions
@section Named arguments and defaults
@cindex Named arguments
@cindex Default values

As first noted in @ref{Generating regular sequences}, if arguments to
called functions are given in the ``@code{@var{name}=@var{object}}''
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 @code{fun1} defined by

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

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

@example
> 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)
@end example

@noindent
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 @code{fun1} were defined
as

@example
> fun1 <- function(data, data.frame, graph=TRUE, limit=20) @{ @dots{} @}
@end example

@noindent
it could be called as

@example
> ans <- fun1(d, df)
@end example

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

@example
> ans <- fun1(d, df, limit=10)
@end example

@noindent
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.

@node The ellipsis argument (...), Assignment within functions, Named arguments and defaults, Writing your own functions
@section The @samp{@dots{}} argument

Another frequent requirement is to allow one function to pass on
argument settings to another.  For example many graphics functions use
the function @code{par()} and functions like @code{plot()} allow the
user to pass on graphical parameters to @code{par()} to control the
graphical output.  (@xref{The par() function}, for more details on the
@code{par()} function.)  This can be done by including an extra
argument, literally @samp{@dots{}}, of the function, which may then be
passed on.  An outline example is given in the example below.

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

@node Assignment within functions, More advanced examples, The ellipsis argument (...), Writing your own functions
@section Assignments within functions

Note that @emph{any ordinary assignments done within the function are
local and temporary and are lost after exit from the function}.  Thus
the assignment @code{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
@emph{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, @code{<<-} or the function
@code{assign()} can be used.  See the @code{help} document for details.
@SPLUS{} users should be aware that @code{<<-} has different semantics
in @R{}.  These are discussed further in @ref{Scope}.

@node More advanced examples, Scope, Assignment within functions, Writing your own functions
@section More advanced examples

@menu
* Efficiency factors in block designs::  
* Dropping all names in a printed array::  
* Recursive numerical integration::  
@end menu

@node Efficiency factors in block designs, Dropping all names in a printed array, More advanced examples, More advanced examples
@subsection 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 @ref{Index
arrays}.)

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

@example
> bdeff <- function(blocks, varieties) @{
    blocks <- as.factor(blocks)             # @r{minor safety move}
    b <- length(levels(blocks))
    varieties <- as.factor(varieties)       # @r{minor safety move}
    v <- length(levels(varieties))
    K <- as.vector(table(blocks))           # @r{remove dim attr}
    R <- as.vector(table(varieties))        # @r{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)
@}
@end example

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.

@node Dropping all names in a printed array, Recursive numerical integration, Efficiency factors in block designs, More advanced examples
@subsection 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 @code{dimnames} attribute will not achieve this effect, but
rather the array must be given a @code{dimnames} attribute consisting of
empty strings.  For example to print a matrix, @code{X}

@example
> temp <- X
> dimnames(temp) <- list(rep("", nrow(X)), rep("", ncol(X))
> temp; rm(temp)
@end example

This can be much more conveniently done using a function,
@code{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.

@example
no.dimnames <- function(a) @{
  ## @r{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
@}
@end example

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

@example
> no.dimnames(X)
@end example

This is particularly useful for large integer arrays, where patterns are
the real interest rather than the values.

@node Recursive numerical integration,  , Dropping all names in a printed array, More advanced examples
@subsection 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.

@example
area <- function(f, a, b, eps = 1.0e-06, lim = 10) @{
  fun1 <- function(f, a, b, fa, fb, a0, eps, lim, fun) @{
    ## @r{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)
@}
@end example

@menu
* Scope::
* Object orientation::
@end menu

@node Scope, Customizing the environment, More advanced examples, Writing your own functions
@section Scope
@cindex 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 @SPLUS{} 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
@emph{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.

@example
f <- function(x) @{
  y <- 2*x
  print(x)
  print(y)
  print(z)
@}
@end example

In this function, @code{x} is a formal parameter, @code{y} is a local
variable and @code{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
@emph{lexical scope}.  First we define a function called @code{cube}.

@example
cube <- function(n) @{
  sq <- function() n*n
  n*sq()
@}
@end example

The variable @code{n} in the function @code{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 (@SPLUS{}) the value is that associated with a global variable named
@code{n}.  Under lexical scope (@R{}) it is the parameter to the function
@code{cube} since that is the active binding for the variable @code{n} at
the time the function @code{sq} was defined.  The difference between
evaluation in @R{} and evaluation in @SPLUS{} is that @SPLUS{} looks for a
global variable called @code{n} while @R{} first looks for a variable
called @code{n} in the environment created when @code{cube} was invoked.

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

Lexical scope can also be used to give functions @emph{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 @code{account} and then returning a list
containing them.  When @code{account} is invoked it takes a numerical
argument @code{total} and returns a list containing the three functions.
Because these functions are defined in an environment which contains
@code{total}, they will have access to its value.

The special assignment operator, @code{<<-},
@findex <<-
is used to change the value associated with @code{total}.  This operator
looks back in enclosing environments for an environment that contains
the symbol @code{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 @code{total} then that variable is created and assigned to
there.  For most users @code{<<-} creates a global variable and assigns
the value of the right hand side to it@footnote{In some sense this
mimics the behavior in @SPLUS{} since in @SPLUS{} this operator always
creates or assigns to a global variable.}.  Only when @code{<<-} has
been used in a function that was returned as the value of another
function will the special behavior described here occur.

@example
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)
@end example

@node Customizing the environment, Object orientation, Scope, Writing your own functions
@section Customizing the environment
@cindex 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
@code{.First} and @code{.Last} can be used.

The location of the site initialization file is taken from the value of
the @env{R_PROFILE} environment variable.  If that variable is unset,
file @file{Rprofile} in the @R{} home subdirectory @file{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 @file{.Rprofile}@footnote{So it is hidden under UNIX.} 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 @file{.Rprofile} file is found in the
startup directory, than @R{} looks for a @file{.Rprofile} file in the
user's home directory and uses that (if it exists).

Any function named @code{.First()} in either of the two profile files or
in the @file{.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 @code{$} 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, @file{Rprofile},
@file{.Rprofile}, @file{.RData} and then @code{.First()}.  A definition
in later files will mask definitions in earlier files.

@example
> .First <- function() @{
  options(prompt="$ ", continue="+\t")  # @r{@code{$} is the prompt}
  options(digits=5, length=999)         # @r{custom numbers and printout}
  x11()                                 # @r{for graphics}
  par(pch = "+")                        # @r{plotting character}
  source(file.path(Sys.getenv("HOME"), "R", "mystuff.R"))
                                        # @r{my personal package}
  library(stepfun)                      # @r{attach the step function tools}
@}
@end example
@findex .First

Similarly a function @code{.Last()}, if defined, is executed at the very end
of the session.  An example is given below.

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

@node Object orientation,  , Customizing the environment, Writing your own functions
@section Classes, generic functions and object orientation
@cindex Classes
@cindex Generic functions
@cindex Object orientation

The class of an object determines how it will be treated by what are
known as @emph{generic} functions.  Put the other way round, a generic
function performs a task or action on its arguments @emph{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 @emph{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 @code{plot()} for
displaying objects graphically, @code{summary()} for summarizing
analyses of various types, and @code{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 @code{"data.frame"} include

@example
[     [[<-    any    as.matrix
[<-   model   plot   summary
@end example

A currently complete list can be got by using the @code{methods()}
function:

@example
> methods(class="data.frame")
@end example

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

@example
> methods(plot)
@end example

The reader is referred to the official references for a complete
discussion of this mechanism.


@node Statistical models in R, Graphics, Writing your own functions, Top
@chapter Statistical models in R
@cindex Statistical models

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.

@menu
* Formulae for statistical models::  
* Linear models::               
* Generic functions for extracting model information::  
* Analysis of variance and model comparison::  
* Updating fitted models::      
* Generalized linear models::   
* Nonlinear least squares and maximum likelihood models::  
* Some non-standard models::    
@end menu

@node Formulae for statistical models, Linear models, Statistical models in R, Statistical models in R
@section Defining statistical models; formulae
@cindex Formulae

The template for a statistical model is a linear regression model with
independent, homoscedastic errors
@ifnottex
@display
y_i = sum_@{j=0@}^p beta_j x_@{ij@} + e_i, @ @ @ @ i = 1, @dots{} n,
@end display
where the e_i are NID(0, sigma^2)
@end ifnottex
@tex
$$ y_i = \sum_{j=0}^p \beta_j x_{ij} + e_i,
   \qquad e_i \sim {\rm NID}(0,\sigma^2),
   \qquad i = 1, @dots{} ,n
$$
@end tex
In matrix terms this would be written
@ifnottex
    y = X @ beta + e @*
@end ifnottex
@tex
 $$ y = X \beta + e $$
@end tex
where the @math{y} is the response vector, @math{X} is the @emph{model
matrix} or @emph{design matrix} and has columns
@math{x_0, x_1, @dots{}, x_p},
the determining variables.  Very often @math{x_0}
will be a column of ones defining an @emph{intercept} term.

@subsubheading Examples

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

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

@table @code
@item y ~ x
@itemx y ~ 1 + x
Both imply the same simple linear regression model of @math{y} on
@math{x}.  The first has an implicit intercept term, and the second an
explicit one.

@item y ~ 0 + x
@itemx y ~ -1 + x
@itemx y ~ x - 1
Simple linear regression of @math{y} on @math{x} through the origin
(that is, without an intercept term).

@item log(y) ~ x1 + x2
Multiple regression of the transformed variable,
@ifnottex
log(y),
@end ifnottex
@tex
$\log(y)$,
@end tex
on @math{x1} and @math{x2} (with an implicit intercept term).

@item y ~ poly(x,2)
@itemx y ~ 1 + x + I(x^2)
Polynomial regression of @math{y} on @math{x} of degree 2.  The first
form uses orthogonal polynomials, and the second uses explicit powers,
as basis.

@item y ~ X + poly(x,2)
Multiple regression @math{y} with model matrix consisting of the matrix
@math{X} as well as polynomial terms in @math{x} to degree 2.

@item y ~ A
Single classification analysis of variance model of @math{y}, with
classes determined by @math{A}.

@item y ~ A + x
Single classification analysis of covariance model of @math{y}, with
classes determined by @math{A}, and with covariate @math{x}.

@item y ~ A*B
@itemx y ~ A + B + A:B
@itemx y ~ B %in% A
@itemx y ~ A/B
Two factor non-additive model of @math{y} on @math{A} and @math{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.

@item y ~ (A + B + C)^2
@itemx 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.

@item y ~ A * x
@itemx y ~ A/x
@itemx y ~ A/(1 + x) - 1
Separate simple linear regression models of @math{y} on @math{x} within
the levels of @math{A}, with different codings.  The last form produces
explicit estimates of as many different intercepts and slopes as there
are levels in @math{A}.

@item y ~ A*B + Error(C)
An experiment with two treatment factors, @math{A} and @math{B}, and
error strata determined by factor @math{C}.  For example a split plot
experiment, with whole plots (and hence also subplots), determined by
factor @math{C}.
@end table

@findex ~
The operator @code{~} is used to define a @emph{model formula} in @R{}.
The form, for an ordinary linear model, is

@example
@var{response} ~ @var{op_1} @var{term_1} @var{op_2} @var{term_2} @var{op_3} @var{term_3} @var{@dots{}}
@end example

@noindent
where

@table @var
@item response
is a vector or matrix, (or expression evaluating to a vector or matrix)
defining the response variable(s).
@item op_i
is an operator, either @code{+} or @code{-}, implying the inclusion or
exclusion of a term in the model, (the first is optional).
@item term_i
is either
@itemize @bullet
@item
a vector or matrix expression, or @code{1},
@item
a factor, or
@item
a @emph{formula expression} consisting of factors, vectors or matrices
connected by @emph{formula operators}.
@end itemize
In all cases each term defines a collection of columns either to be
added to or removed from the model matrix.  A @code{1} stands for an
intercept column and is by default included in the model matrix unless
explicitly removed.

@end table

The @emph{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 @samp{@code{.}} becomes
@samp{@code{:}} since the period is a valid name character in @R{}.

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

@table @code
@item @var{Y} ~ @var{M}
@var{Y} is modeled as @var{M}.

@item @var{M_1} + @var{M_2}
Include @var{M_1} and @var{M_2}.

@item @var{M_1} - @var{M_2}
Include @var{M_1} leaving out terms of @var{M_2}.

@item @var{M_1} : @var{M_2}
The tensor product of @var{M_1} and @var{M_2}.  If both terms are
factors, then the ``subclasses'' factor.

@item @var{M_1} %in% @var{M_2}
Similar to @code{@var{M_1}:@var{M_2}}, but with a different coding.

@item @var{M_1} * @var{M_2}
@code{@var{M_1} + @var{M_2} + @var{M_1}:@var{M_2}}.

@item @var{M_1} / @var{M_2}
@code{@var{M_1} + @var{M_2} %in% @var{M_1}}.

@item @var{M}^@var{n}
All terms in @var{M} together with ``interactions'' up to order @var{n}

@item I(@var{M})
Insulate @var{M}.  Inside @var{M} all operators have their normal
arithmetic meaning, and that term appears in the model matrix.
@end table

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

Note particularly that the model formulae specify the @emph{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.

@menu
* Contrasts::                   
@end menu

@node Contrasts,  , Formulae for statistical models, Formulae for statistical models
@subsection Contrasts
@cindex 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).

@cindex Factors
@cindex Ordered factors
What about a @math{k}-level factor @code{A}?  The answer differs for
unordered and ordered factors.  For @emph{unordered} factors @math{k -
1} columns are generated for the indicators of the second, @dots{},
@math{k}th levels of the factor. (Thus the implicit parameterization is
to contrast the response at each level with that at the first.)  For
@emph{ordered} factors the @math{k - 1} columns are the orthogonal
polynomials on @math{1, @dots{}, 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 @math{k} columns giving the
indicators for all the levels.  Second, the whole behavior can be
changed by the @code{options} setting for @code{contrasts}.  The default
setting in @R{} is

@example
options(contrasts = c("contr.treatment", "contr.poly"))
@end example

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

@example
options(contrasts = c("contr.helmert", "contr.poly"))
@end example

@noindent
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 @code{contrasts} and
@code{C}.
@findex contrasts
@findex 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, model with
interaction but not the corresponding main effects will in general lead
to surprising results, and is for experts only.


@node Linear models, Generic functions for extracting model information, Formulae for statistical models, Statistical models in R
@section Linear models
@cindex Linear models

The basic function for fitting ordinary multiple models is @code{lm()},
and a streamlined version of the call is as follows:
@findex lm

@example
> @var{fitted.model} <- lm(@var{formula}, data = @var{data.frame})
@end example

For example

@example
> fm2 <- lm(y ~ x1 + x2, data = production)
@end example

@noindent
would fit a multiple regression model of @math{y} on @math{x1} and
@math{x2} (with implicit intercept term).

The important but technically optional parameter @code{data =
production} specifies that any variables needed to construct the model
should come first from the @code{production} @emph{data frame}.
@emph{This is the case regardless of whether data frame
@code{production} has been attached on the search path or not}.

@node Generic functions for extracting model information, Analysis of variance and model comparison, Linear models, Statistical models in R
@section Generic functions for extracting model information

The value of @code{lm()} is a fitted model object; technically a list of
results of class @code{"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 @code{"lm"}.  These
include

@example
add1    coef       effects   kappa   predict  residuals
alias   deviance   family    labels  print    step
anova   drop1      formula   plot    proj     summary
@end example

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

@table @code
@findex anova
@item anova(@var{object_1}, @var{object_2})
Compare a submodel with an outer model and produce an analysis of
variance table.

@findex coefficients
@findex coef
@item coefficients(@var{object})
Extract the regression coefficient (matrix).

Short form: @code{coef(@var{object})}.

@findex deviance
@item deviance(@var{object})
Residual sum of squares, weighted if appropriate.

@findex formula
@item formula(@var{object})
Extract the model formula.

@findex plot
@item plot(@var{object})
Produce four plots, showing residuals, fitted values and some
diagnostics.

@findex predict
@item predict(@var{object}, newdata=@var{data.frame})
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
@var{data.frame}.

@c  @item @code{predict.gam(@var{object},}
@c  @item @w{@ @ @ @code{newdata=@var{data.frame})}}
@c  @tab @code{predict.gam()} is a safe alternative to @code{predict()} that
@c  can be used for @code{lm}, @code{glm} and @code{gam} fitted objects.  It
@c  must be used, for example, in cases where orthogonal polynomials are
@c  used as the original basis functions, and the addition of new data
@c  implies different basis functions to the original.

@findex print
@item print(@var{object})
Print a concise version of the object.  Most often used implicitly.

@findex residuals
@findex resid
@item residuals(@var{object})
Extract the (matrix of) residuals, weighted as appropriate.

Short form: @code{resid(@var{object})}.

@findex step
@item step(@var{object})
Select a suitable model by adding or dropping terms and preserving
hierarchies.  The model with the largest value of AIC (Akaike's An
Information Criterion) discovered in the stepwise search is returned.

@findex summary
@item summary(@var{object})
Print a comprehensive summary of the results of the regression analysis.
@end table

@node Analysis of variance and model comparison, Updating fitted models, Generic functions for extracting model information, Statistical models in R
@section Analysis of variance and model comparison
@cindex Analysis of variance

The model fitting function @code{aov(@code{formula},
data=@var{data.frame})}
@findex aov
operates at the simplest level in a very similar way to the function
@code{lm()}, and most of the generic functions listed in the table in
@ref{Generic functions for extracting model information} apply.

It should be noted that in addition @code{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

@example
@var{response} ~ @var{mean.formula} + Error(@var{strata.formula})
@end example
@findex Error

@noindent
specifies a multi-stratum experiment with error strata defined by the
@var{strata.formula}.  In the simplest case, @var{strata.formula} 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:

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

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

@menu
* ANOVA tables::                
@end menu

@node ANOVA tables,  , Analysis of variance and model comparison, Analysis of variance and model comparison
@subsection 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
@emph{that term} in the model at @emph{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 @code{anova()} function.
@findex anova

@example
> anova(@var{fitted.model.1}, @var{fitted.model.2}, @dots{})
@end example

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.

@node Updating fitted models, Generalized linear models, Analysis of variance and model comparison, Statistical models in R
@section Updating fitted models
@cindex Updating fitted models

The @code{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
@findex update

@example
> @var{new.model} <- update(@var{old.model}, @var{new.formula})
@end example

In the @var{new.formula} the special name consisting of a period,
@samp{@code{.}},
@findex .
only, can be used to stand for ``the corresponding part of the old model
formula''.  For example,

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

@noindent
would fit a five variate multiple regression with variables (presumably)
from the data frame @code{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 @code{data=} argument is specified on the
original call to the model fitting function, this information is passed on
through the fitted model object to @code{update()} and its allies.

The name @samp{.} can also be used in other contexts, but with slightly
different meaning.  For example

@example
> fmfull <- lm(y ~ . , data = production)
@end example

@noindent
would fit a model with response @code{y} and regressor variables
@emph{all other variables in the data frame @code{production}}.

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

@node Generalized linear models, Nonlinear least squares and maximum likelihood models, Updating fitted models, Statistical models in R
@section Generalized linear models
@cindex 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:

@itemize @bullet
@item
There is a response, @math{y}, of interest and stimulus variables
@ifnottex
x_1, x_2, @dots{},
@end ifnottex
@tex
$x_1$, $x_2$, @dots{},
@end tex
whose values influence the distribution of the response.

@item
The stimulus variables influence the distribution of @math{y} through
@emph{a single linear function, only}.  This linear function is called
the @emph{linear predictor}, and is usually written
@ifnottex
@display
eta = beta_1 x_1 + beta_2 x_2 + @dots{} + beta_p x_p,
@end display
hence x_i has no influence on the distribution of @math{y} if and only if
beta_i is zero.
@end ifnottex
@tex
$$ \eta = \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_p x_p, $$
hence $x_i$ has no influence on the distribution of @math{y} if and only
if $\beta_i=0$.
@end tex

@item
The distribution of @math{y} is of the form
@ifnottex
@display
f_Y(y; mu, phi)
  = exp((A/phi) * (y lambda(mu) - gamma(lambda(mu))) + tau(y, phi))
@end display
where phi is a @emph{scale parameter} (possibly known), and is constant
for all observations, @math{A} represents a prior weight, assumed known
but possibly varying with the observations, and $\mu$ is the mean of
@math{y}.
@end ifnottex
@tex
$$
f_Y(y;\mu,\varphi)
= \exp\left[{A \over \varphi}\left\{y\lambda(\mu) -
\gamma\left(\lambda(\mu)\right)\right\} + \tau(y,\varphi)\right]
$$
where $\varphi$ is a @emph{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 $y$.
@end tex
So it is assumed that the distribution of @math{y} is determined by its
mean and possibly a scale parameter as well.

@item
@ifnottex
The mean, mu, is a smooth invertible function of the linear predictor:
@display
mu = m(eta),    eta = m^@{-1@}(mu) = ell(mu)
@end display
and this inverse function, ell(), is called the @emph{link function}.
@end ifnottex
@tex
The mean, $\mu$, is a smooth invertible function of the linear predictor:
$$ \mu = m(\eta),\qquad \eta = m^{-1}(\mu) = \ell(\mu) $$
and this inverse function, $\ell()$, is called the @emph{link function}.
@end tex

@end itemize

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).

@menu
* Families::                    
* The glm() function::          
@end menu

@node Families, The glm() function, Generalized linear models, Generalized linear models
@subsection Families
@cindex Families

The class of generalized linear models handled by facilities supplied in
@R{} includes @emph{gaussian}, @emph{binomial}, @emph{poisson},
@emph{inverse gaussian} and @emph{gamma} response distributions and also
@emph{quasi-likelihood} models where the response distribution is not
explicitly specified.  In the latter case the @emph{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
as in the table below.

@quotation
@multitable @columnfractions 0.25 0.55

@item Family name @tab Link functions

@item @code{binomial} @tab @code{logit}, @code{probit}, @code{cloglog}
@item @code{gaussian} @tab @code{identity}
@item @code{Gamma} @tab @code{identity}, @code{inverse}, @code{log}
@item @code{inverse.gaussian} @tab @code{1/mu^2}
@item @code{poisson} @tab @code{identity}, @code{log}, @code{sqrt}
@item @code{quasi} @tab @code{logit}, @code{probit}, @code{cloglog},
@code{identity}, @code{inverse}, @code{log}, @code{1/mu^2}, @code{sqrt}

@end multitable
@end quotation

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 @emph{family} of the generalized linear model.

@node The glm() function,  , Families, Generalized linear models
@subsection The @code{glm()} function
@findex glm

Since the distribution of the response depends on the stimulus variables
through a single linear function @emph{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 @code{glm()}
which uses the form

@example
> @var{fitted.model} <- glm(@var{formula}, family=@var{family.generator}, data=@var{data.frame})
@end example

The only new feature is the @var{family.generator}, 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 @ref{Families}.  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 @code{quasi}
family, the variance function may also be specified in this way.

Some examples make the process clear.

@subsubheading The @code{gaussian} family

A call such as

@example
> fm <- glm(y ~ x1 + x2, family = gaussian, data = sales)
@end example

@noindent
achieves the same result as

@example
> fm <- lm(y ~ x1+x2, data=sales)
@end example

@noindent
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 @code{quasi} family, as
we shall see later.

@subsubheading The @code{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:

@iftex
@quotation
@end iftex
@multitable {No.@: tested::} {50} {50} {50} {50} {50}
@item Age:          @tab  20 @tab  35 @tab  45 @tab  55 @tab  70
@item No.@: tested: @tab  50 @tab  50 @tab  50 @tab  50 @tab  50
@item No.@: blind:  @tab  @w{ 6} @tab  17 @tab  26 @tab  37 @tab  44
@end multitable
@iftex
@end quotation
@end iftex

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 @math{y} is the number of blind at age @math{x} and @math{n} the
number tested, both models have the form
@ifnottex
y ~ B(n, F(beta_0 + beta_1 x))
@end ifnottex
@tex
$$ y \sim {\rm B}(n, F(\beta_0 + \beta_1 x)) $$
@end tex
where for the probit case,
@eqn{F(z) = \Phi(z), F(z) = Phi(z)}
is the standard normal distribution function, and in the logit case
(the default),
@eqn{F(z) = e^z/(1+e^z),F(z) = e^z/(1+e^z)}.
In both cases the LD50 is
@ifnottex
LD50 = - beta_0/beta_1
@end ifnottex
@tex
$$ \hbox{LD50} = -\beta_0/\beta_1 $$
@end tex
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

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

To fit a binomial model using @code{glm()} there are two possibilities
for the response:

@itemize @bullet
@item
If the response is a @emph{vector} it is assumed to hold @emph{binary}
data, and so must be a @math{0/1} vector.

@item
If the response is a @emph{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.
@end itemize

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

@example
> kalythos$Ymat <- cbind(kalythos$y, kalythos$n - kalythos$y)
@end example

To fit the models we use

@example
> fmp <- glm(Ymat ~ x, family = binomial(link=probit), data = kalythos)
> fml <- glm(Ymat ~ x, family = binomial, data = kalythos)
@end example

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

@example
> summary(fmp)
> summary(fml)
@end example

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

@example
> ld50 <- function(b) -b[1]/b[2]
> ldp <- ld50(coef(fmp)); ldl <- ld50(coef(fmp)); c(ldp, ldl)
@end example

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

@subsubheading Poisson models

With the Poisson family the default link is the @code{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:

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

@subsubheading 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
@eqn{\hbox{Var}[y] = \mu,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
@ifnottex
y = theta_1 z_1 / (z_2 - theta_2) + e
@end ifnottex
@tex
$$ y = {\theta_1z_1 \over z_2 - \theta_2} + e $$
@end tex
which may be written alternatively as
@ifnottex
y = 1 / (beta_1 x_1 + beta_2 x_2) + e
@end ifnottex
@tex
$$ y = {1 \over \beta_1x_1 + \beta_2x_2} + e $$
@end tex
where
@ifnottex
x_1 = z_2/z_1, x_2 = -1/x_1, beta_1 = 1/theta_1, and beta_2 =
theta_2/theta_1.
@end ifnottex
@tex
$x_1 = z_2/z_1$, $x_2=-1/x_1$, $\beta_1=1/\theta_1$ and
$\beta_2=\theta_2/\theta_1$.
@end tex
Supposing a suitable data frame to be set up we could fit this
non-linear regression as

@example
> nlfit <- glm(y ~ x1 + x2 - 1,
               family = quasi(link=inverse, variance=constant),
               data = biochem)
@end example

The reader is referred to the manual and the help document for further
information, as needed.

@node Nonlinear least squares and maximum likelihood models, Some non-standard models, Generalized linear models, Statistical models in R
@section Nonlinear least squares and maximum likelihood models
@cindex Nonlinear least squares

Certain forms of nonlinear model can be fitted by Generalized Linear
Models (@code{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 routine is @code{nlm()},
@findex nlm
which takes the place of @SPLUS{}'s @code{ms()} and @code{nlmin()}.  We
seek the parameter values that minimize some index of lack-of-fit, and
@code{nlm()} does that by trying out various parameter values
iteratively.  Unlike linear regression for example, there is no
guarantee that the procedure will converge on satisfactory estimates.
@code{nlm()} requires initial guesses about what parameter values to
try, and convergence depends critically upon the quality of the initial
guesses.

@menu
* Least squares::               
* Maximum likelihood::          
@end menu

@node Least squares, Maximum likelihood, Nonlinear least squares and maximum likelihood models, Nonlinear least squares and maximum likelihood models
@subsection 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:

@example
> 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)
@end example

The model to be fitted is:

@example
> fn <- function(p) sum((y - (p[1] * x)/(p[2] + x))^2)
@end example

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.

@example
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 200 * xfit/(0.1 + xfit)
> lines(spline(xfit, yfit))
@end example

We could do better, but these starting values of 200 and .1 seem
adequate.  Now do the fit:

@example
> out <- nlm(fn, p = c(200, 0.1), hessian = TRUE)
@end example
@findex nlm

After the fitting, @code{out$minimum} is the SSE, and
@code{out$estimates} are the least squares estimates of the parameters.
To obtain the approximate standard errors (SE) of the estimates we do:

@example
> sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))
@end example

The 2 in the line above represents the number of parameters.  A 95%
confidence interval would be the parameter estimate @eqn{\pm, +/-} 1.96
SE.  We can superimpose the least squares fit on a new plot:

@example
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 212.68384222 * xfit/(0.06412146 + xfit)
> lines(spline(xfit, yfit))
@end example

The standard package @pkg{nls} 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

@example
> 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
@end example

@node Maximum likelihood,  , Least squares, Nonlinear least squares and maximum likelihood models
@subsection Maximum likelihood
@cindex 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 @code{glm()}.  The data are:

@example
> 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)
@end example

The negative log-likelihood to minimize is:

@example
> fn <- function(p)
   sum( - (y*(p[1]+p[2]*x) - n*log(1+exp(p[1]+p[2]*x))
           + log(choose(n, y)) ))
@end example

@noindent
We pick sensible starting values and do the fit:

@example
> out <- nlm(fn, p = c(-50,20), hessian = TRUE)
@end example
@findex nlm

@noindent
After the fitting, @code{out$minimum} is the negative log-likelihood,
and @code{out$estimates} are the maximum likelihood estimates of the
parameters.  To obtain the approximate SEs of the estimates we do:

@example
> sqrt(diag(solve(out$hessian)))
@end example

A 95% confidence interval would be the parameter estimate @eqn{\pm, +/-}
1.96 SE.

@node Some non-standard models,  , Nonlinear least squares and maximum likelihood models, Statistical models in R
@section 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.

@itemize @bullet
@item
@cindex Mixed models
@strong{Mixed models.}  The user-contributed @pkg{nlme} package provides
functions @code{lme()} and @code{nlme()}
@findex lme
@findex 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.

@item
@cindex Local approximating regressions
@strong{Local approximating regressions.}  The @code{loess()}
@findex 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 @code{loess} is the standard package @pkg{modreg}, together
with code for projection pursuit regression.
@findex loess

@item
@cindex Robust regression
@strong{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 @code{lqs}
@findex lqs
in the standard package of the same name provides state-of-art
algorithms for highly-resistant fits.  Less resistant but statistically
more efficient methods are available in user-contributed packages, for
example function @code{rlm}
@findex rlm
in package @pkg{MASS}.

@item
@cindex Additive models
@strong{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 @code{avas} and
@code{ace}
@findex avas
@findex ace
in package @pkg{acepack} and functions @code{bruto} and @code{mars}
@findex bruto
@findex mars
in package @pkg{mda} provide some examples of these techniques in
user-contributed packages to @R{}.

@c  @item
@c  @cindex Generalized additive models
@c  @strong{Generalized additive models.}  This technique aims to construct
@c  a regression function from smooth additive functions of the determining
@c  variables, usually one for each determining variable.  The function
@c  @code{gam()} is in many ways similar to the other model fitting
@c  functions outlined above.  In addition there are other model fitting
@c  functions that do a similar job.  These include @code{avas()} and
@c  @code{ace()}.  On the other hand @code{ppreg()} is available for
@c  projection pursuit regression, but this technique is still very much in
@c  need of a complete theoretical treatment and further practical
@c  experience.  These latter functions are again conforming to an older
@c  protocol for model fitting functions and lack the convenience of the
@c  newer functions.

@item
@cindex Tree based models
@strong{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 @code{tree()},
@findex tree
but many other generic functions such as @code{plot()} and @code{text()}
are well adapted to displaying the results of a tree-based model fit in
a graphical way.

Tree models are available in @R{} @emph{via} the user-contributed
packages @pkg{rpart} and @pkg{tree}.

@end itemize

@node Graphics, A sample session, Statistical models in R, Top
@chapter 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 @emph{device
driver} which opens a special @emph{graphics window} for the display of
interactive graphics.  Although this is done automatically, it is useful
to know that the command used is @code{X11()} under UNIX,
@code{windows()} under Windows and @code{macintosh()} on MacOS 8/9.

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:

@itemize @bullet
@item
@strong{High-level} plotting functions create a new plot on the graphics
device, possibly with axes, labels, titles and so on.
@item
@strong{Low-level} plotting functions add more information to an
existing plot, such as extra points, lines and labels.
@item
@strong{Interactive} graphics functions allow you interactively add
information to, or extract information from, an existing plot, using a
pointing device such as a mouse.
@end itemize

In addition, @R{} maintains a list of @emph{graphical parameters} which
can be manipulated to customize your plots.

@menu
* High-level plotting commands::  
* Low-level plotting commands::  
* Interacting with graphics::   
* Using graphics parameters::   
* Graphics parameters::         
* Device drivers::              
* Dynamic graphics::            
@end menu

@node High-level plotting commands, Low-level plotting commands, Graphics, Graphics
@section 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.

@menu
* The plot() function::         
* Displaying multivariate data::  
* Display graphics::            
* Arguments to high-level plotting functions::  
@end menu

@node The plot() function, Displaying multivariate data, High-level plotting commands, High-level plotting commands
@subsection The @code{plot()} function
@findex plot

One of the most frequently used plotting functions in @R{} is the
@code{plot()} function.  This is a @emph{generic} function: the type of
plot produced is dependent on the type or @emph{class} of the first
argument.

@table @code

@item plot(@var{x}, @var{y})
@itemx plot(@var{xy})
If @var{x} and @var{y} are vectors, @code{plot(@var{x}, @var{y})}
produces a scatterplot of @var{y} against @var{x}.  The same effect can
be produced by supplying one argument (second form) as either a list
containing two elements @var{x} and @var{y} or a two-column matrix.

@item plot(@var{x})
If @var{x} is a time series, this produces a time-series plot, if
@var{x} is a numeric vector, it produce a plot of the values in the
vector against their index in the vector, and if @var{x} is a complex
vector, it produces a plot of imaginary versus real parts of the vector
elements.

@item plot(@var{f})
@itemx plot(@var{f}, @var{y})
@var{f} is a factor object, @var{y} is a numeric vector.  The first form
generates a bar plot of @var{f}; the second form produces boxplots of
@var{y} for each level of @var{f}.

@item plot(@var{df})
@itemx plot(~ @var{expr})
@itemx plot(@var{y} ~ @var{expr})
@var{df} is a data frame, @var{y} is any object, @var{expr} is a list
of object names separated by `@code{+}' (e.g., @code{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 @var{y} against every object named in @var{expr}.
@end table

@node Displaying multivariate data, Display graphics, The plot() function, High-level plotting commands
@subsection Displaying multivariate data

@R{} provides two very useful functions for representing multivariate
data.  If @code{X} is a numeric matrix or data frame, the command

@example
> pairs(X)
@end example
@findex pairs

@noindent
produces a pairwise scatterplot matrix of the variables defined by the
columns of @code{X}, that is, every column of @code{X} is plotted
against every other column of @code{X} and the resulting @math{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 @emph{coplot} may be more
enlightening.  If @code{a} and @code{b} are numeric vectors and @code{c}
is a numeric vector or factor object (all of the same length), then
the command

@example
> coplot(a ~ b | c)
@end example
@findex coplot

@noindent
produces a number of scatterplots of @code{a} against @code{b} for given
values of @code{c}.  If @code{c} is a factor, this simply means that
@code{a} is plotted against @code{b} for every level of @code{c}.  When
@code{c} is numeric, it is divided into a number of @emph{conditioning
intervals} and for each interval @code{a} is plotted against @code{b}
for values of @code{c} within the interval.  The number and position of
intervals can be controlled with @code{given.values=} argument to
@code{coplot()}---the function @code{co.intervals()} is useful for
selecting intervals.  You can also use two @emph{given} variables with a
command like

@example
> coplot(a ~ b | c + d)
@end example

@noindent
which produces scatterplots of @code{a} against @code{b} for every joint
conditioning interval of @code{c} and @code{d}.

The @code{coplot()} and @code{pairs()} function both take an argument
@code{panel=} which can be used to customize the type of plot which
appears in each panel.  The default is @code{points()} to produce a
scatterplot but by supplying some other low-level graphics function of
two vectors @code{x} and @code{y} as the value of @code{panel=} you can
produce any type of plot you wish.  An example panel function useful for
coplots is @code{panel.smooth()}.

@node Display graphics, Arguments to high-level plotting functions, Displaying multivariate data, High-level plotting commands
@subsection Display graphics

Other high-level graphics functions produce different types of plots.
Some examples are:

@table @code
@c  @item tsplot(x_1, x_2, @dots{})
@c  @findex tsplot
@c  Plots any number of time series on the same scale.  This automatic
@c  simultaneous scaling feature is also useful when the @code{x@var{_i}}'s
@c  are ordinary numeric vectors, in which case they are plotted against the
@c  numbers @math{1, 2, 3, @dots{}}.

@item qqnorm(x)
@itemx qqline(x)
@itemx qqplot(x, y)
@findex qqnorm
@findex qqline
@findex qqplot
Distribution-comparison plots.  The first form plots the numeric vector
@code{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 @code{x} against those of @code{y} to compare their
respective distributions.

@item hist(x)
@itemx hist(x, nclass=@var{n})
@itemx hist(x, breaks=@var{b}, @dots{})
@findex hist
Produces a histogram of the numeric vector @code{x}.  A sensible number
of classes is usually chosen, but a recommendation can be given with the
@code{nclass=} argument.  Alternatively, the breakpoints can be
specified exactly with the @code{breaks=} argument.  If the
@code{probability=TRUE} argument is given, the bars represent relative
frequencies instead of counts.

@item dotchart(x, @dots{})
@findex dotchart
Constructs a dotchart of the data in @code{x}.  In a dotchart the
@math{y}-axis gives a labelling of the data in @code{x} and the
@math{x}-axis gives its value.  For example it allows easy visual
selection of all data entries with values lying in specified ranges.

@item image(x, y, z, @dots{})
@itemx contour(x, y, z, @dots{})
@itemx persp(x, y, z, @dots{})
@findex image
@findex contour
@findex persp
Plots of three variables.  The @code{image} plot draws a grid of rectangles
using different colours to represent the value of @code{z}, the @code{contour}
plot draws contour lines to represent the value of @code{z}, and the
@code{persp} plot draws a 3D surface.
@end table

@node Arguments to high-level plotting functions,  , Display graphics, High-level plotting commands
@subsection Arguments to high-level plotting functions

There are a number of arguments which may be passed to high-level
graphics functions, as follows:

@table @code
@item add=TRUE
Forces the function to act as a low-level graphics function,
superimposing the plot on the current plot (some functions only).

@item axes=FALSE
Suppresses generation of axes---useful for adding your own custom axes
with the @code{axis()} function.  The default, @code{axes=TRUE}, means
include axes.

@item log="x"
@itemx log="y"
@itemx log="xy"
Causes the @math{x}, @math{y} or both axes to be logarithmic.  This will
work for many, but not all, types of plot.

@item type=
The @code{type=} argument controls the type of plot produced, as
follows:

@table @code
@item type="p"
Plot individual points (the default)
@item type="l"
Plot lines
@item type="b"
Plot points connected by lines (@emph{both})
@item type="o"
Plot points overlaid by lines
@item type="h"
Plot vertical lines from points to the zero axis (@emph{high-density})
@item type="s"
@itemx type="S"
Step-function plots.  In the first form, the top of the vertical defines
the point; in the second, the bottom.
@item 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.
@end table

@item xlab=@var{string}
@itemx ylab=@var{string}
Axis labels for the @math{x} and @math{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.

@item main=@var{string}
Figure title, placed at the top of the plot in a large font.

@item sub=@var{string}
Sub-title, placed just below the @math{x}-axis in a smaller font.
@end table

@node Low-level plotting commands, Interacting with graphics, High-level plotting commands, Graphics
@section 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:

@table @code
@item points(x, y)
@itemx lines(x, y)
@findex points
@findex lines
Adds points or connected lines to the current plot.  @code{plot()}'s
@code{type=} argument can also be passed to these functions (and
defaults to @code{"p"} for @code{points()} and @code{"l"} for
@code{lines()}.)

@item text(x, y, labels, @dots{})
@findex text
Add text to a plot at points given by @code{x, y}.  Normally
@code{labels} is an integer or character vector in which case
@code{labels[i]} is plotted at point @code{(x[i], y[i])}.  The default
is @code{1:length(x)}.

@strong{Note}: This function is often used in the sequence

@smallexample
> plot(x, y, type="n"); text(x, y, names)
@end smallexample

@noindent
The graphics parameter @code{type="n"} suppresses the points but sets up
the axes, and the @code{text()} function supplies special characters, as
specified by the character vector @code{names} for the points.

@item abline(a, b)
@itemx abline(h=@var{y})
@itemx abline(v=@var{x})
@itemx abline(@var{lm.obj})
@findex abline
Adds a line of slope @code{b} and intercept @code{a} to the current
plot.  @code{h=@var{y}} may be used to specify @math{y}-coordinates for
the heights of horizontal lines to go across a plot, and
@code{v=@var{x}} similarly for the @math{x}-coordinates for vertical
lines.  Also @var{lm.obj} may be list with a @code{coefficients}
component of length 2 (such as the result of model-fitting functions,)
which are taken as an intercept and slope, in that order.

@item polygon(x, y, @dots{})
@findex polygon
Draws a polygon defined by the ordered vertices in (@code{x}, @code{y})
and (optionally) shade it in with hatch lines, or fill it if the
graphics device allows the filling of figures.

@item legend(x, y, legend, @dots{})
@findex 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 @code{legend}.  At least one other argument @var{v}
(a vector the same length as @code{legend}) with the corresponding
values of the plotting unit must also be given, as follows:

@table @code
@item legend( , fill=@var{v})
Colors for filled boxes
@item legend( , col=@var{v})
Colors in which points or lines will be drawn
@item legend( , lty=@var{v})
Line styles
@item legend( , lwd=@var{v})
Line widths
@item legend( , pch=@var{v})
Plotting characters (character vector)
@end table

@item title(main, sub)
@findex title
Adds a title @code{main} to the top of the current plot in a large font
and (optionally) a sub-title @code{sub} at the bottom in a smaller font.

@item axis(side, @dots{})
@findex 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
@code{plot()} with the @code{axes=FALSE} argument.
@end table

Low-level plotting functions usually require some positioning
information (e.g., @math{x} and @math{y} coordinates) to determine where
to place the new plot elements.  Coordinates are given in terms of
@emph{user coordinates} which are defined by the previous high-level
graphics command and are chosen based on the supplied data.

Where @code{x} and @code{y} arguments are required, it is also
sufficient to supply a single argument being a list with elements named
@code{x} and @code{y}.  Similarly a matrix with two columns is also
valid input.  In this way functions such as @code{locator()} (see below)
may be used to specify positions on a plot interactively.

@menu
* Mathematical annotation::     
* Hershey vector fonts::        
@end menu

@node Mathematical annotation, Hershey vector fonts, Low-level plotting commands, Low-level plotting commands
@subsection 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 @emph{expression} rather
than a character string in any one of @code{text}, @code{mtext}, @code{axis},
or @code{title}.   For example, the following code draws the formula for
the Binomial probability function:

@smallexample
> text(x, y, expression(paste(bgroup("(", atop(n, x), ")"),
                              p^x, q^@{n-x@})))
@end smallexample

More information, including a full listing of the features available can
obtained from within @R{} using the commands:

@smallexample
> help(plotmath)
> example(plotmath)
@end smallexample

@node Hershey vector fonts,  , Mathematical annotation, Low-level plotting commands
@subsection Hershey vector fonts

It is possible to specify Hershey vector fonts for rendering text when using
the @code{text} and @code{contour} functions.  There are three reasons for
using the Hershey fonts:
@itemize @bullet
@item
Hershey fonts can produce better
output, especially on a computer screen, for rotated and/or small text.
@item
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.
@item
Hershey fonts provide cyrillic and japanese (Kana and Kanji) characters.
@end itemize

More information, including tables of Hershey characters can be obtained from
within @R{} using the commands:

@smallexample
> help(Hershey)
> example(Hershey)
> help(Japanese)
> example(Japanese)
@end smallexample

@node Interacting with graphics, Using graphics parameters, Low-level plotting commands, Graphics
@section 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
@code{locator()} function:

@table @code
@item locator(n, type)
@findex locator
Waits for the user to select locations on the current plot using the
left mouse button.  This continues until @code{n} (default 512) points
have been selected, or another mouse button is pressed (Unix, Windows)
or the mouse is clicked outside the graphics window (Mac).  The
@code{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.  @code{locator()} returns the locations of the points selected
as a list with two components @code{x} and @code{y}.
@end table

@code{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

@example
> text(locator(1), "Outlier", adj=0)
@end example

@noindent
may be useful.  @code{locator()} will still work if the current device
does not support a mouse; in this case the user will be prompted for
@math{x} and @math{y} coordinates.

@table @code
@item identify(x, y, labels)
@findex identify
Allow the user to highlight any of the points defined by @code{x} and
@code{y} (using the left mouse button) by plotting the corresponding
component of @code{labels} nearby (or the index number of the point if
@code{labels} is absent).  Returns the indices of the selected points
when another button is pressed (unix, Windows) or the mouse is clicked 
outside the graphics window (Mac).
@end table

Sometimes we want to identify particular @emph{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 @math{(x, y)}
coordinates in two numeric vectors @code{x} and @code{y}, we could use
the @code{identify()} function as follows:

@example
> plot(x, y)
> identify(x, y)
@end example

The @code{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.  The point nearest the mouse pointer will be
highlighted with its index number (that is, its position in the
@code{x}/@code{y} vectors) plotted nearby.  Alternatively, you could use
some informative string (such as a case name) as a highlight by using
the @code{labels} argument to @code{identify()}, or disable highlighting
altogether with the @code{plot=FALSE} argument.  When the process is
terminated (see above), @code{identify()} returns the indices of the
selected points; you can use these indices to extract the selected
points from the original vectors @code{x} and @code{y}.

@node Using graphics parameters, Graphics parameters, Interacting with graphics, Graphics
@section Using graphics parameters

When creating graphics, particularly for presentation or publication
purposes, @R{} does not always produce exactly that which is required.
You can, however, customize almost every aspect of the display using
@emph{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 `@code{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.

@menu
* The par() function::          
* Arguments to graphics functions::  
@end menu

@node The par() function, Arguments to graphics functions, Using graphics parameters, Using graphics parameters
@subsection Permanent changes: The @code{par()} function
@findex par
@cindex Graphics parameters

The @code{par()} function is used to access and modify the list of
graphics parameters for the current graphics device.

@table @code
@item par()
Without arguments, returns a list of all graphics parameters and their
values for the current device.
@item par(c("col", "lty"))
With a character vector argument, returns only the named graphics
parameters (again, as a list.)
@item 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.
@end table

Setting graphics parameters with the @code{par()} function changes the
value of the parameters @emph{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 @code{par()} @emph{always} affect the global values
of graphics parameters, even when @code{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 @code{par()} when
making changes, and restoring the initial values when plotting is
complete.

@example
> oldpar <- par(col=4, lty=2)
  @r{@dots{} plotting commands @dots{}}
> par(oldpar)
@end example

@node Arguments to graphics functions,  , The par() function, Using graphics parameters
@subsection 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 @code{par()} function, except that the changes only last for the
duration of the function call.  For example:

@example
> plot(x, y, pch="+")
@end example

@noindent
produces a scatterplot using a plus sign as the plotting character,
without changing the default plotting character for future plots.

@node Graphics parameters, Device drivers, Using graphics parameters, Graphics
@section Graphics parameters list

The following sections detail many of the commonly-used graphical
parameters.  The @R{} help documentation for the @code{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:

@table @code
@item @var{name}=@var{value}
A description of the parameter's effect.  @var{name} is the name of the
parameter, that is, the argument name to use in calls to @code{par()} or
a graphics function.  @var{value} is a typical value you might use when
setting the parameter.
@end table

@menu
* Graphical elements::          
* Axes and tick marks::         
* Figure margins::              
* Multiple figure environment::  
@end menu

@node Graphical elements, Axes and tick marks, Graphics parameters, Graphics parameters
@subsection Graphical elements

@R{} plots are made up of points, lines, text and polygons (filled
regions.) Graphical parameters exist which control how these
@emph{graphical elements} are drawn, as follows:

@table @code
@item pch="+"
Character to be used for plotting points.  The default varies with
graphics drivers, but it is usually
@ifnottex
a circle.
@end ifnottex
@tex
`$\circ$'.
@end tex
Plotted points tend to appear slightly above or below the appropriate
position unless you use @code{"."} as the plotting character, which
produces centered points.

@item pch=4
When @code{pch} is given as an integer between 0 and 18 inclusive, a
specialized plotting symbol is produced.  To see what the symbols are,
use the command

@example
> legend(locator(1), as.character(0:18), pch = 0:18)
@end example

@item 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, and line types 2 and onwards are dotted or dashed lines, or some
combination of both.

@item lwd=2
Line widths.  Desired width of lines, in multiples of the ``standard''
line width.  Affects axis lines as well as lines drawn with
@code{lines()}, etc.

@item col=2
Colors to be used for points, lines, text, filled regions and images.
Each of these graphic elements has a list of possible colors, and the
value of this parameter is an index to that list.  Obviously, this
parameter applies only to a limited range of devices.

@item font=2
An integer which specifies which font to use for text.  If possible,
device drivers arrange so that @code{1} corresponds to plain text,
@code{2} to bold face, @code{3} to italic and @code{4} to bold italic.

@item font.axis
@itemx font.lab
@itemx font.main
@itemx font.sub
The font to be used for axis annotation, @math{x} and @math{y} labels,
main and sub-titles, respectively.

@item adj=-0.1
Justification of text relative to the plotting position.  @code{0} means
left justify, @code{1} means right justify and @code{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 @code{-0.1} leaves a gap of 10% of the text width
between the text and the plotting position.

@item cex=1.5
Character expansion.  The value is the desired size of text characters
(including plotting characters) relative to the default text size.
@end table

@node Axes and tick marks, Figure margins, Graphical elements, Graphics parameters
@subsection Axes and tick marks

Many of @R{}'s high-level plots have axes, and you can construct axes
yourself with the low-level @code{axis()} graphics function.  Axes have
three main components: the @emph{axis line} (line style controlled by the
@code{lty} graphics parameter), the @emph{tick marks} (which mark off unit
divisions along the axis line) and the @emph{tick labels} (which mark the
units.) These components can be customized with the following graphics
parameters.

@table @code
@item lab=c(5, 7, 12)
The first two numbers are the desired number of tick intervals on the
@math{x} and @math{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!

@item las=1
Orientation of axis labels.  @code{0} means always parallel to axis,
@code{1} means always horizontal, and @code{2} means always
perpendicular to the axis.

@item 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.

@item tck=0.01
Length of tick marks, as a fraction of the size of the plotting region.
When @code{tck} is small (less than 0.5) the tick marks on the @math{x}
and @math{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 @code{tck=0.01} and @code{mgp=c(1,-1.5,0)} for internal
tick marks.

@item xaxs="s"
@itemx yaxs="d"
Axis styles for the @math{x} and @math{y} axes, respectively.  With
styles @code{"s"} (standard) and @code{"e"} (extended) the smallest and
largest tick marks always lie outside the range of the data.  Extended
axes may be widened slightly if any points are very near the edge.  This
style of axis can sometimes leave large blank gaps near the edges.  With
styles @code{"i"} (internal) and @code{"r"} (the default) tick marks
always fall within the range of the data, however style @code{"r"}
leaves a small amount of space at the edges.

Setting this parameter to @code{"d"} (direct axis) @emph{locks in} the
current axis and uses it for all future plots (or until the parameter is
set to one of the other values above, at least.) Useful for generating
series of fixed-scale plots.
@end table

@node Figure margins, Multiple figure environment, Axes and tick marks, Graphics parameters
@subsection Figure margins


A single plot in @R{} is known as a @code{figure} and comprises a
@emph{plot region} surrounded by margins (possibly containing axis
labels, titles, etc.) and (usually) bounded by the axes themselves.

@iftex
A typical figure is

@image{images/fig11,7cm}
@end iftex

Graphics parameters controlling figure layout include:

@table @code
@item mai=c(1, 0.5, 0.5, 0)
Widths of the bottom, left, top and right margins, respectively,
measured in inches.

@item mar=c(4, 2, 2, 1)
Similar to @code{mai}, except the measurement unit is text lines.
@end table

@code{mar} and @code{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 @code{postscript()} driver
with the @code{height=4} argument will result in a plot which is about
50% margin unless @code{mar} or @code{mai} are set explicitly.  When
multiple figures are in use (see below) the margins are reduced by half,
however this may not be enough when many figures share the same page.

@node Multiple figure environment,  , Figure margins, Graphics parameters
@subsection Multiple figure environment

@R{} allows you to create an @math{n} by @math{m} array of figures on a
single page.  Each figure has its own margins, and the array of figures
is optionally surrounded by an @emph{outer margin}, as shown in the
following figure.

@iftex
@image{images/fig12,6cm}
@end iftex

The graphical parameters relating to multiple figures are as follows:

@table @code
@item mfcol=c(3, 2)
@itemx mfrow=c(2, 4)
Set size of 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 @code{mfcol} causes figures to be
filled by column; @code{mfrow} fills by rows.

The layout in the Figure could have been created by setting
@code{mfrow=c(3,2)}; the figure shows the page after four plots have
been drawn.

@item mfg=c(2, 2, 3, 2)
Position of 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 @emph{true} values
for unequally-sized figures on the same page.

@item 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.

@item oma=c(2, 0, 3, 0)
@itemx omi=c(0, 0, 0.8, 0)
Size of outer margins.  Like @code{mar} and @code{mai}, the first
measures in text lines and the second in inches, starting with the
bottom margin and working clockwise.

@end table

Outer margins are particularly useful for page-wise titles, etc.  Text
can be added to the outer margins with the @code{mtext()} function with
argument @code{outer=TRUE}.  There are no outer margins by default,
however, so you must create them explicitly using @code{oma} or
@code{omi}.

More complicated arrangements of multiple figures can be produced by the
@code{split.screen()} and @code{layout()} functions.

@node Device drivers, Dynamic graphics, Graphics parameters, Graphics
@section Device drivers
@cindex Graphics 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 @emph{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 @code{help(Devices)}
for a list of them all.  For example, issuing the command

@example
> postscript()
@end example

@noindent
causes all future graphics output to be sent to the printer in
PostScript format.  Some commonly-used device drivers are:

@table @code
@item X11()
@findex X11
For use with the X11 window system
@item postscript()
@findex postscript
For printing on PostScript printers, or creating PostScript graphics
files.
@item pictex()
@findex pictex
Produces a La@TeX{} file
@end table

When you have finished with a device, be sure to terminate the device
driver by issuing the command

@example
> dev.off()
@end example

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.

@menu
* PostScript diagrams for typeset documents::  
* Multiple graphics devices::   
@end menu

@node PostScript diagrams for typeset documents, Multiple graphics devices, Device drivers, Device drivers
@subsection PostScript diagrams for typeset documents

By passing the @code{file} argument to the @code{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 @code{horizontal=FALSE} argument is given, and you can control the
size of the graphic with the @code{width} and @code{height} arguments
(the plot will be scaled as appropriate to fit these dimensions.) For
example, the command

@example
> postscript("file.ps", horizontal=FALSE, height=5, pointsize=10)
@end example

@noindent
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 @emph{encapsulated} PostScript
is produced: @R{} always produces conformant output, but only marks the
output as such when the @code{onefile=FALSE} argument is supplied.  This
unusual notation stems from @Sl{}-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

@example
> postscript("plot1.eps", horizontal=FALSE, onefile=FALSE,
             height=8, width=6, pointsize=10)
@end example


@node Multiple graphics devices,  , PostScript diagrams for typeset documents, Device drivers
@subsection 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
@emph{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:

@table @code
@item X11()
[Unix]
@item windows()
[Windows]
@item Macintosh()
[MaxOS 8/9]
@item postscript()
@itemx pictex()
@itemx @dots{}
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. (Some platforms may have
further devices available.)

@item dev.list()
@findex dev.list
Returns the number and name of all active devices.  The device at
position 1 on the list is always the @emph{null device} which does not
accept graphics commands at all.

@item dev.next()
@itemx dev.prev()
@findex dev.next
@findex dev.prev
Returns the number and name of the graphics device next to, or previous
to the current device, respectively.

@item dev.set(which=@var{k})
@findex dev.set
Can be used to change the current graphics device to the one at position
@var{k} of the device list.  Returns the number and label of the device.

@item dev.off(@var{k})
@findex dev.off
Terminate the graphics device at point @var{k} of the device list.  For
some devices, such as @code{postscript} devices, this will either print
the file immediately or correctly complete the file for later printing,
depending on how the device was initiated.

@item dev.copy(device, @dots{}, which=@var{k})
@itemx dev.print(device, @dots{}, which=@var{k})
Make a copy of the device @var{k}.  Here @code{device} is a device
function, such as @code{postscript}, with extra arguments, if needed,
specified by @samp{@dots{}}.  @code{dev.print} is similar, but the
copied device is immediately closed, so that end actions, such as
printing hardcopies, are immediately performed.

@item graphics.off()
Terminate all graphics devices on the list, except the null device.
@end table

@node Dynamic graphics,  , Device drivers, Graphics
@section Dynamic graphics
@cindex Dynamic graphics

@R{} does not (currently) have a builtin function for dynamic graphics,
for example to rotate point clouds or to ``brush'' (interactively
highlight) plots.  However, extensive dynamic graphics facilities are
available in the system XGobi by Swayne, Cook and Buja available from

@quotation
@uref{http://www.research.att.com/areas/stat/xgobi/}
@end quotation

@noindent
and these can be accessed from @R{} via the package @pkg{xgobi}.

XGobi currently runs under the X Windows system on either Unix or
Windows, and @R{} interfaces are available for each.


@node A sample session, Invoking R, Graphics, Top
@appendix 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.  This is written for the UNIX user.  Those using Windows or
Macintosh will need to adapt the discussion slightly.

@table @code
@item
Login, start your windowing system.  You should also have the file
@file{morley.tab} in your working directory.  If not, seek the local
expert (or get it yourself from the @file{base/data} subdirectory of the
default @R{} library tree).  If you have, proceed.
@item $ R
Start @R{} as appropriate for your platform.

The @R{} program begins, with a banner.

(Within @R{}, the prompt on the left hand side will not be shown to
avoid confusion.)

@item 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.

@item x <- rnorm(50)
@itemx y <- rnorm(x)
Generate two pseudo-random normal vectors of @math{x}- and
@math{y}-coordinates.

@item plot(x, y)
Plot the points in the plane.  A graphics window will appear automatically.

@item ls()
See which @R{} objects are now in the @R{} workspace.

@item rm(x, y)
Remove objects no longer needed. (Clean up).

@item x <- 1:20
Make @math{x = (1, 2, @dots{}, 20)}.

@item w <- 1 + sqrt(x)/2
A `weight' vector of standard deviations.

@item dummy <- data.frame(x=x, y= x + rnorm(x)*w)
@itemx dummy
Make a @emph{data frame} of two columns, @math{x} and @math{y}, and look
at it.

@item fm <- lm(y ~ x, data=dummy)
@itemx summary(fm)
Fit a simple linear regression of @math{y} on @math{x} and look at the
analysis.

@item fm1 <- lm(y ~ x, data=dummy, weight=1/w^2)
@itemx summary(fm1)
Since we know the standard deviations, we can do a weighted regression.

@item attach(dummy)
Make the columns in the data frame visible as variables.

@item lrf <- lowess(x, y)
Make a nonparametric local regression function.

@item plot(x, y)
Standard point plot.

@item lines(x, lrf$y)
Add in the local regression.

@item abline(0, 1, lty=3)
The true regression line: (intercept 0, slope 1).

@item abline(coef(fm))
Unweighted regression line.

@item abline(coef(fm1), col = "red")
Weighted regression line.

@item detach()
Remove data frame from the search path.

@item plot(fitted(fm), resid(fm),
@itemx @w{@ @ @ @ @ xlab="Fitted values"},
@itemx @w{@ @ @ @ @ ylab="Residuals"},
@itemx @w{@ @ @ @ @ main="Residuals vs Fitted")}
A standard regression diagnostic plot to check for heteroscedasticity.
Can you see it?

@item qqnorm(resid(fm), main="Residuals Rankit Plot")
A normal scores plot to check for skewness, kurtosis and outliers.  (Not
very useful here.)

@item rm(fm, fm1, lrf, x, dummy)
Clean up again.
@end table

The next section will look at data from the classical experiment of
Michaelson and Morley to measure the speed of light.

@table @code
@item file.show("morley.tab")
Optional.  Look at the file.

@item mm <- read.table("morley.tab")
@itemx mm
Read in the Michaelson and Morley data as a data frame, and look at it.
There are five experiments (column @code{Expt}) and each has 20 runs
(column @code{Run}) and @code{sl} is the recorded speed of light,
suitably coded.

@item mm$Expt <- factor(mm$Expt)
@itemx mm$Run <- factor(mm$Run)
Change @code{Expt} and @code{Run} into factors.

@item attach(mm)
Make the data frame visible at position 2 (the default).

@item plot(Expt, Speed, main="Speed of Light Data", xlab="Experiment No.")
Compare the five experiments with simple boxplots.

@item fm <- aov(Speed ~ Run + Expt, data=mm)
@itemx summary(fm)
Analyze as a randomized block, with `runs' and `experiments' as factors.

@item fm0 <- update(fm, . ~ . - Run)
@itemx anova(fm0, fm)
Fit the sub-model omitting `runs', and compare using a formal analysis
of variance.

@item detach()
@itemx rm(fm, fm0)
Clean up before moving on.

@end table

We now look at some more graphical features: contour and image plots.

@table @code
@item x <- seq(-pi, pi, len=50)
@itemx y <- x
@math{x} is a vector of 50 equally spaced values in
@ifnottex
the interval [-pi\, pi].
@end ifnottex
@iftex
@tex
$-\pi\leq x \leq \pi$.
@end tex
@end iftex
@math{y} is the same.

@item f <- outer(x, y, function(x, y) cos(y)/(1 + x^2))
@math{f} is a square matrix, with rows and columns indexed by @math{x}
and @math{y} respectively, of values of the function
@eqn{\cos(y)/(1 + x^2),cos(y)/(1 + x^2)}.

@item oldpar <- par(no.readonly = TRUE)
@itemx par(pty="s")
Save the plotting parameters and set the plotting region to ``square''.

@item contour(x, y, f)
@itemx contour(x, y, f, nlevels=15, add=TRUE)
Make a contour map of @math{f}; add in more lines for more detail.

@item fa <- (f-t(f))/2
@code{fa} is the ``asymmetric part'' of @math{f}.  (@code{t()} is
transpose).

@item contour(x, y, fa, nint=15)
Make a contour plot, @dots{}

@item par(oldpar)
@dots{} and restore the old graphics parameters.

@item image(x, y, f)
@itemx image(x, y, fa)
Make some high density image plots, (of which you can get
hardcopies if you wish), @dots{}

@item objects(); rm(x, y, f, fa)
@dots{} and clean up before moving on.
@end table

@R{} can do complex arithmetic, also.

@table @code
@item th <- seq(-pi, pi, len=100)
@itemx z <- exp(1i*th)
@code{1i} is used for the complex number @math{i}.

@item par(pty="s")
@itemx plot(z, type="l")
Plotting complex arguments means plot imaginary versus real parts.  This
should be a circle.

@item 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 @dots{}

@item w <- ifelse(Mod(w) > 1, 1/w, w)
@dots{} and to map any outside the circle onto their reciprocal.

@item plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+",xlab="x", ylab="y")
@itemx lines(z)
@itemx
All points are inside the unit circle, but the distribution is not
uniform.

@item w <- sqrt(runif(100))*exp(2*pi*runif(100)*1i)
@itemx plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+", xlab="x", ylab="y")
@itemx lines(z)
@itemx
The second method uses the uniform distribution.  The points should now
look more evenly spaced over the disc.

@item rm(th, w, z)
Clean up again.

@item 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.
@end table

@node Invoking R, The command line editor, A sample session, Top
@appendix Invoking R

@menu
* Invoking R under UNIX::       
* Invoking R under Windows::    
* Invoking R on a Macintosh::   
@end menu

@node Invoking R under UNIX, Invoking R under Windows, Invoking R, Invoking R
@section Invoking R under UNIX

The command @samp{R} can be used both for starting the main @R{} program
with a variety of options in the form

@display
@code{R} [@var{options}] [@code{<}@var{infile}] [@code{>}@var{outfile}],
@end display

@noindent
or, via the @code{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''.

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 @samp{Startup} for more information).

@itemize @bullet
@item
Unless @option{--no-environ} was given, @R{} searches for the file
@file{.Renviron} in the current directory, or failing that the file
pointed to by the environment variable @env{R_ENVIRON}, or if that is
not set, the file @file{.Renviron} in the user's home directory.  The
first of these files found is processed to set environment variables. It
should contain lines of the form @samp{@var{name}=@var{value}}.
(See @code{help(Startup)} for a precise description.)
Variables you might want to set include @env{R_PAPERSIZE} (the default
paper size), @env{R_PRINTCMD} (the default print command) and
@env{R_LIBS} (specifies the list of @R{} library trees searched for
add-on packages).

@item
Then @R{} searches for the site-wide startup profile unless the command
line option @option{--no-site-file} was given.  The name of this file is
taken from the value of the @env{R_PROFILE} environment variable.  If
that variable is unset, the default is @file{$R_HOME/etc/Rprofile}.

@item
Then, unless @option{--no-init-file} was given, @R{} searches for a file
called @file{.Rprofile} in the current directory or in the user's home
directory (in that order) and sources it.

@item
It also loads a saved image from @file{.RData} if there is one (unless
@option{--no-restore} or @option{--no-restore-data} was specified).

@item
Finally, if a function @code{.First} exists, it is executed.  This
function (as well as @code{.Last} which is executed at the end of the
@R{} session) can be defined in the appropriate startup profiles, or
reside in @file{.RData}.
@end itemize

In addition, there are options for controlling the memory available to
the @R{} process (see the on-line help for topic @samp{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.

@table @option
@item --help
@itemx -h
Print short help message to standard output and exit successfully.

@item --version
Print version information to standard output and exit successfully.

@item 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.

@item --save
@itemx --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 @kbd{q()};
in batch mode, one of these must be specified.

@item --no-environ
Do not read any user file to set environment variables.

@item --no-site-file
Do not read the site-wide profile at startup.

@item --no-init-file
Do not read the user's profile at startup.

@item --restore
@itemx --no-restore
@itemx --no-restore-data
Control whether saved images (file @file{.RData} in the directory where
@R{} was started) should be restored at startup or not.  The default is
to restore. (@option{--no-restore} implies all the specific
@option{--no-restore-*} options.)

@item --no-restore-history
Control whether the history file (normally file @file{.Rhistory} in the
directory where @R{} was started, but can be set by the environment
variable @env{R_HISTFILE}) should be restored at startup or not.  The
default is to restore.

@item --vanilla
Combine @option{--no-save}, @option{--no-environ}
@option{--no-site-file}, @option{--no-init-file} and @option{--no-restore}.

@item --no-readline
Turn off command-line editing via @strong{readline}.  This is useful
when running @R{} from within Emacs using the @acronym{ESS} (``Emacs
Speaks Statistics'') package.  @xref{The command line editor}, for more
information.

@item --min-vsize=@var{N}
@itemx --max-vsize=@var{N}
Specify the minimum or maximum amount of memory used for variable size
objects by setting the ``vector heap'' size to @var{N} bytes.  Here,
@var{N} must either be an integer or an integer ending with @samp{M},
@samp{K}, or @samp{k}, meaning `Mega' (2^20), (computer) `Kilo' (2^10),
or regular `kilo' (1000).

@item --min-nsize=@var{N}
@itemx --max-nsize=@var{N}
Specify the amount of memory used for fixed size objects by setting the
number of ``cons cells'' to @var{N}.  See the previous option for
details on @var{N}.  A cons cell takes 28 bytes on a 32-bit machine, and
usually 56 bytes on a 64-bit machine.

@item --quiet
@itemx --silent
@itemx -q
Do not print out the initial copyright and welcome messages.

@item --slave
Make @R{} run as quietly as possible.  This option is intended to
support programs which use @R{} to compute results for them.

@item --verbose
Print more information about progress, and in particular set @R{}'s
option @code{verbose} to @code{TRUE}.  @R{} code uses this option to
control the printing of diagnostic messages.

@item --debugger=@var{name}
@itemx -d @var{name}
Run @R{} through debugger @var{name}.  Note that in this case, further
command line options are disregarded, and should instead be given when
starting the @R{} executable from inside the debugger.

@item --gui=@var{type}
@itemx -g @var{type}
Use @var{type} as graphical user interface (note that this also includes
interactive graphics).  Currently, possible values for @var{type} are
@samp{X11} (the default), @samp{gnome} provided that @acronym{GNOME}
support is available, and @samp{none}.
@end table

Note that input and output can be redirected in the usual way (using
@samp{<} and @samp{>}).

The command @code{R CMD} allows the invokation of various tools which
are useful in conjunction with @R{}, but not intended to be called
``directly''.  The general form is

@example
R CMD @var{command} @var{args}
@end example

@noindent
where @var{command} is the name of the tool and @var{args} the arguments
passed on to it.

Currently, the following tools are available.

@table @code
@item BATCH
Run @R{} in batch mode.
@item COMPILE
Compile files for use with @R{}.
@item SHLIB
Build shared library for dynamic loading.
@item INSTALL
Install add-on packages.
@item REMOVE
Remove add-on packages.
@item build
Build add-on packages.
@item check
Check add-on packages.
@item LINK
Front-end for creating executable programs.
@item Rprof
Post-process @R{} profiling files.
@item Rdconv
Convert Rd format to various other formats, including @HTML{}, Nroff,
La@TeX{}, plain text, and S documentation format.
@item Rd2dvi
Convert Rd format to DVI/PDF.
@item Rd2txt
Convert Rd format to text.
@item Rdindex
Extract index information from Rd files.
@item Sd2Rd
Convert S documentation to Rd format.
@end table

The first five tools (i.e., @code{BATCH}, @code{COMPILE}, @code{SHLIB},
@code{INSTALL}, and @code{REMOVE}) can also be invoked ``directly''
without the @code{CMD} option, i.e., in the form @code{R @var{command}
@var{args}}.

Use

@example
R CMD @var{command} --help
@end example

@noindent
to obtain usage information for each of the tools accessible via the
@code{R CMD} interface.

@node Invoking R under Windows, Invoking R on a Macintosh, Invoking R under UNIX, Invoking R
@section Invoking R under Windows

The startup procedure under Windows is very similar to that under UNIX,
but necessarily not identical. There are two versions of @R{} for
Windows, the console-based version (@code{RGui.exe}) and a terminal
version (@code{Rterm.exe}, principally intended for batch use).

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 @samp{Startup} for more information).  References to the
`home directory' need to be clarified, as this is not always defined on
Windows.  If the environment variable @env{R_USER} is defined, that
gives the home directory. Next, if the environment variable @env{HOME}
is defined, that gives the home directory. If not, and environment
variables @env{HOMEDRIVE} and @env{HOMEPATH} are defined (and they
normally are under Windows NT/2000) these define the home
directory. Failing all those, the home directory is taken to be the
starting directory.

@itemize @bullet
@item
If a file @file{.Renviron} in the current directory, or if that is not
set, the file @file{.Renviron} in the user's home directory, exists, it
is read to set environment variables.  Reading is suppressed by the flag
@option{--no-environ}.  The variables are set by separate lines of the
form @samp{@var{name}=@var{value}}.  (See @code{help(Startup)} for a
precise description.)  Variables you might want to set include
@env{R_PAPERSIZE} (the default paper size) and @env{R_LIBS} (controls
the list of @R{} library trees searched for add-on packages).

Environment variables can also be supplied as
@samp{@var{name}=@var{value}} pairs at the end of  the command line.

@item
Then @R{} searches for the site-wide startup profile unless the command
line option @option{--no-site-file} was given.  The name of this file is
taken from the value of the @env{R_PROFILE} environment variable.  If
that variable is unset, the default is @file{$R_HOME/etc/Rprofile}.

@item
Then, unless @option{--no-init-file} was given, @R{} searches for a file
called @file{.Rprofile} in the current directory or in the user's home
directory (in that order) and sources it.

@item
It also loads a saved image from @file{.RData} if there is one (unless
@option{--no-restore} or @option{--no-restore-data} was specified).

@item
Finally, if a function @code{.First} exists, it is executed.  This
function (as well as @code{.Last} which is executed at the end of the
@R{} session) can be defined in the appropriate startup profiles, or
reside in @file{.RData}.
@end itemize

In addition, there are options for controlling the memory available to
the @R{} process (see the on-line help for topic @samp{Memory} for more
information).  Users will not normally need to use these.  @R{} for
Windows has an additional flag @code{--max-mem-size} described below,
which by default does limit the memory usage of an @R{} process.

@R{} for Windows accepts the following command line options.

@table @option
@item --version
Print version info and exit.

@item --mdi
@itemx --sdi
@itemx --no-mdi
Control whether @code{Rgui} will operate as an MDI program (the default,
with multiple child windows within one main window) or an SDI application
(with multiple top-level windows for the console, graphics and pager).

@item --save
@itemx --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 @kbd{q()};
in batch mode, one of these must be specified.

@item --no-site-file
Suppress reading of the site-wide startup profile.

@item --no-init-file
Suppress reading of the directory or user's @file{.Rprofile} file.

@item --no-environ
Suppress reading of @file{.Renviron}.

@item --restore
@itemx --no-restore
@itemx --no-restore-data
Control whether saved images (file @file{.RData} in the directory where
@R{} was started) should be restored at startup or not.
The default is to restore. (@option{--no-restore} implies all the specific
@option{--no-restore-*} options.)

@item --no-restore-history
Control whether the history file (normally file @file{.Rhistory} in the
directory where @R{} was started, but can be set by the environment
variable @env{R_HISTFILE}) should be restored at startup or not.  The
default is to restore.

@item --vanilla
Combine @option{--no-save}, @option{--no-restore},
@option{--no-site-file}, @option{--no-init-file} and
@option{--no-environ}.

@item --min-vsize=@var{N}
@itemx --max-vsize=@var{N}
Specify the minimum or maximum amount of memory used for variable size
objects by setting the ``vector heap'' size to @var{N} bytes.  Here,
@var{N} must either be an integer or an integer ending with @samp{M},
@samp{K}, or @samp{k}, meaning `Mega' (2^20), (computer) `Kilo' (2^10),
or regular `kilo' (1000).

@item --min-nsize=@var{N}
@itemx --max-nsize=@var{N}
Specify the amount of memory used for fixed size objects by setting the
number of ``cons cells'' to @var{N}.  See the previous option for
details on @var{N}.  A cons cell takes 28 bytes.

@item --max-mem-size=@var{N}
Specify a limit for the amount of memory to be used both for @R{}
objects and working areas (`@code{malloc}-ed' memory).  This is set by
default to the smaller of 256Mb and the amount of physical RAM in the machine.


@item --quiet
@itemx --silent
@itemx -q
Don't print startup message.

@item --slave
Make @R{} run as quietly as possible.

@item --verbose
Print more information about progress, and in particular set @R{}'s
option @code{verbose} to @code{TRUE}.  @R{} code uses this option to
control the printing of diagnostic messages.

@item --ess
Set @code{Rterm} up for use by @code{R-inferior-mode} in @acronym{ESS}.
@end table

The command @code{Rcmd} allows the invokation of various tools which
are useful in conjunction with @R{}, but not intended to be called
``directly''.  The general form is

@example
Rcmd @var{command} @var{args}
@end example

@noindent
where @var{command} is the name of the tool and @var{args} the arguments
passed on to it.

Currently, the following tools are available.

@table @code
@item INSTALL
Install add-on packages.
@item REMOVE
Remove add-on packages.
@item SHLIB
Make a DLL for use with @code{dyn.load}.
@item BATCH
Run @R{} in batch mode.
@item build
Build add-on packages.
@item check
Check add-on packages.
@item Rdconv
Convert Rd format to various other formats, including @HTML{}, Nroff,
La@TeX{}, plain text, and S documentation format.
@item Rd2dvi.sh
Convert Rd format to DVI/PDF.
@item Rd2txt
Convert Rd format to text.
@item Rdindex
Extract index information from Rd files.
@item Sd2Rd
Convert S documentation to Rd format.
@end table

Use

@example
Rcmd @var{command} --help
@end example

@noindent
to obtain usage information for each of the tools accessible via the
@code{Rcmd} interface.

@node Invoking R on a Macintosh,  , Invoking R under Windows, Invoking R
@section Invoking R on a Macintosh

This section applies to the classic MacOS port, and not to ports of the
Unix version to MacOS X.

The @R{} application startup is similar to that of UNIX and Windows
systems.  Only a console-based version of @R{} for Macintosh exists, and
there is no equivalent to the @code{R CMD} engine.
 
When you double-click on the @R{} application icon a small window will
appear.  This is equivalent to the command line under other ports.  You
can enter the startup options, or simply press @key{RET} on the keyboard
to start @R{} running with default options.  The main @R{} program can be
started with a variety of options in the form

@display
@code{R} [@var{options}]
@end display

@noindent
as listed later in this section.  Using [@code{<}@var{infile}]
[@code{>}@var{outfile}] equivalent to UNIX can be selected at startup
using the standard Macintosh interface but they are not currently fully
implemented, so you can ignore these.

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 @samp{Startup} for more information).

@itemize @bullet
@item
Unless @option{--no-environ} was given, @R{} looks for the file
@file{.Renviron} in the @file{etc} subdirectory of @R{} application
folder. The file found is processed to set environment variables. It
should contain lines of the form @samp{@var{name}=@var{value}}.  (See
@code{help(Startup)} for a precise description.)

@item
Then, unless @option{--no-init-file} was given, @R{} looks for a file
called @file{.Rprofile} in the @file{etc} subdirectory of @R{}
application folder and processes it (as @R{} code).

@item
It then loads a saved image from file @file{.RData} if there is one (unless
@option{--no-restore} or @option{--no-restore-data} was specified).
This file looked for in the @file{etc} subdirectory of @R{}
application folder.

@item
Finally, if a function @code{.First} exists, it is executed.  This
function (as well as @code{.Last} which is executed at the end of the
@R{} session) can be defined in the appropriate startup profiles, or
reside in @file{.RData}.
@end itemize

In addition, there are options for controlling the memory available to
the @R{} process (see the on-line help for topic @samp{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{}.  Please note that
under MacOS, you have to reserve in advance the amount of memory
assigned to the @R{} application from the `Finder/Information' menu as
it is usually done for any Macintosh software.  An amount of 32000k is
sufficient to run all the demos and examples in the base distribution.

@R{} accepts the following command-line options.

@table @option
@item --version
Print version information to standard output and exit successfully.

@item --save
@itemx --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 @kbd{q()};
in batch mode, one of these must be specified.

@item --no-init-file
Do not read the user's profile at startup.

@item --restore
@itemx --no-restore
@itemx --no-restore-data
Control whether saved images (file @file{:etc:.RData} in the directory
where @R{} was started) should be restored at startup or not.  The
default is to restore. (@option{--no-restore} implies all the specific
@option{--no-restore-*} options.)

@item --no-restore-history
Control whether the history file (normally file @file{.Rhistory} in the
directory where @R{} was started, but can be set by the environment
variable @env{R_HISTFILE}) should be restored at startup or not.  The
default is to restore.

@item --vanilla
Combine @option{--no-save}, @option{--no-environ},
@option{--no-site-file}, @option{--no-init-file} and
@option{--no-restore}.

@item --min-vsize=@var{N}
@itemx --max-vsize=@var{N}
Specify the minimum or maximum amount of memory used for variable size
objects by setting the ``vector heap'' size to @var{N} bytes.  Here,
@var{N} must either be an integer or an integer ending with @samp{M},
@samp{K}, or @samp{k}, meaning `Mega' (2^20), (computer) `Kilo' (2^10),
or regular `kilo' (1000).

@item --min-nsize=@var{N}
@itemx --max-nsize=@var{N}
Specify the amount of memory used for fixed size objects by setting the
number of ``cons cells'' to @var{N}.  See the previous option for
details on @var{N}.  A cons cell takes 28 bytes on a 32-bit machine, and
usually 56 bytes on a 64-bit machine.

@item --quiet
@itemx --silent
@itemx -q
Do not print out the initial copyright and welcome messages.

@item --slave
Make @R{} run as quietly as possible.

@item --verbose
Print more information about progress, and in particular set @R{}'s
option @code{verbose} to @code{TRUE}.  @R{} code uses this option to
control the printing of diagnostic messages.

@end table


@node The command line editor, Function and variable index, Invoking R, Top
@appendix The command line editor

@section Preliminaries

When the @acronym{GNU} @strong{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: this appendix does @strong{not} apply to @acronym{GNOME}
interface under UNIX, only to the standard command-line interface.

It can be disabled (useful for usage with @acronym{ESS}@footnote{The
`Emacs Speaks Statistics' package; see the @acronym{URL}
@uref{http://ess.stat.wisc.edu/}}) using the startup option
@option{--no-readline}.

Windows versions of @R{} have somewhat simpler command-line editing: see
@samp{Console} under the @samp{Help} menu of the @acronym{GUI}, and the
file @file{README.Rterm} for command-line editing under
@code{Rterm.exe}.

When using @R{} with @strong{readline} capabilities, the functions
described below are available.

Many of these use either Control or Meta characters.  Control
characters, such as @kbd{Control-m}, are obtained by holding the
@key{CTRL} down while you press the @key{m} key, and are written as
@kbd{C-m} below.  Meta characters, such as @kbd{Meta-b}, are typed by
holding down @key{META} and pressing @key{b}, and written as @kbd{M-b}
in the following.  If your terminal does not have a @key{META} key, you
can still type Meta characters using two-character sequences starting
with @kbd{ESC}.  Thus, to enter @kbd{M-b}, you could type
@key{ESC}@key{b}.  The @kbd{ESC} character sequences are also allowed on
terminals with real Meta keys.  Note that case is significant for Meta
characters.

@section Editing actions

The @R{} program keeps a history of the commands you type, including the
error 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 @emph{vi} mode
character insertion mode is started by @kbd{M-i} or @kbd{M-a},
characters are typed and insertion mode is finished by typing a further
@key{ESC}.

Pressing the @key{RET} command at any time causes the command to be
re-submitted.

Other editing actions are summarized in the following table.

Unfortunately it does not seem to be possible to bind the motion keys,
for example, to the arrow keys, which is something of a nuisance.

@section Command line editor summary

@subheading Command recall and vertical motion

@table @kbd
@item C-p
Go to the previous command (backwards in the history).
@item C-n
Go to the next command (forwards in the history).
@item C-r @var{text}
Find the last command with the @var{text} string in it.
@end table

On most terminals, you can also use the up and down arrow keys instead
of @kbd{C-p} and @kbd{C-n}, respectively.

@subheading Horizontal motion of the cursor

@table @kbd
@item C-a
Go to the beginning of the command.
@item C-e
Go to the end of the line.
@item M-b
Go back one word.
@item M-f
Go forward one word.
@item C-b
Go back one character.
@item C-f
Go forward one character.
@end table

On most terminals, you can also use the left and right arrow keys
instead of @kbd{C-b} and @kbd{C-f}, respectively.

@subheading Editing and re-submission

@table @kbd
@item @var{text}
Insert @var{text} at the cursor.
@item C-f @var{text}
Append @var{text} after cursor.
@item @key{DEL}
Delete the previous character (left of the cursor).
@item C-d
Delete the character under the cursor.
@item M-d
Delete rest of the word under the cursor, and ``save'' it.
@item C-k
Delete from cursor to end of command, and ``save'' it.
@item C-y
Insert (yank) the last ``saved'' text here.
@item C-t
Transpose the character under the cursor with the next.
@item M-l
Change the rest of the word to lower case.
@item M-c
Change the rest of the word to upper case.
@item @key{RET}
Re-submit the command to @R{}.
@end table

The final @key{RET} terminates the command line editing sequence.

@node Function and variable index, Concept index, The command line editor, Top
@appendix Function and variable index

@printindex vr

@node Concept index, References, Function and variable index, Top
@appendix Concept index

@printindex cp

@node References,  , Concept index, Top
@appendix References

D.@: M.@: Bates and  D.@: G.@: Watts (1988), @emph{Nonlinear Regression
Analysis and Its Applications.} John Wiley & Sons, New York.

Richard A.@: Becker, John M.@: Chambers and Allan R.@: Wilks (1988),
@emph{The New S Language.} Chapman & Hall, New York.
This book is often called the ``@emph{Blue Book}''.

John M.@: Chambers and Trevor J.@: Hastie eds. (1992),
@emph{Statistical Models in S.} Chapman & Hall, New York.
This is also called the ``@emph{White Book}''.

Annette J.@: Dobson (1990), @emph{An Introduction to Generalized Linear
Models}, Chapman and Hall, London.

Peter McCullagh and John A.@: Nelder (1989), @emph{Generalized Linear
Models.} Second edition, Chapman and Hall, London.

John A.@ Rice (1995), @emph{Mathematical Statistics and Data Analysis.}
Second edition.  Duxbury Press, Belmont, CA.

S.@: D.@: Silvey (1970), @emph{Statistical Inference.} Penguin, London.

@bye
