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 1.7.1 (2003-06-16). ISBN 3-901167-55-2
This introduction to R is derived from an original set of notes describing the S and S-PLUS 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 S programs, and expanded some of the material.
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 Remail@example.com.
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, Graphics on the graphics facilities can be read at almost any time and need not wait until all the preceding sections have been digested.
R is an integrated suite of software facilities for data manipulation, calculation and graphical display. Among other things it has
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.
R can be regarded as an implementation of the S language which was developed at Bell Laboratories by Rick Becker, John Chambers and Allan Wilks, and also forms the basis of the S-PLUS systems.
The evolution of the S language is characterized by four books by John Chambers and coauthors. For R, the basic reference is The New S Language: A Programming Environment for Data Analysis and Graphics by Richard A. Becker, John M. Chambers and Allan R. Wilks. The new features of the 1991 release of S (S version 3) are covered in Statistical Models in S edited by John M. Chambers and Trevor J. Hastie. See References, for precise references.
In addition, documentation for S/S-PLUS can typically be used with R, keeping the differences between the S implementations in mind. See What documentation exists for R?.
Our introduction to the R environment did not mention statistics, yet many people use R as a statistics system. We prefer to think of it of an environment within which many classical and modern statistical techniques have been implemented. Some of these are built into the base R environment, but many are supplied as 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 CRAN family of Internet sites (via 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 S (and hence R) and the other main statistical systems. In S a statistical analysis is normally done as a series of steps, with intermediate results being stored in objects. Thus whereas SAS and SPSS will give copious output from a regression or discriminant analysis, R will give minimal output and store the results in a fit object for subsequent interrogation by further R functions.
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.
When you use the R program it issues a prompt when it expects input
commands. The default prompt is
, which on UNIX might be
the same as the shell prompt, and so it may appear that nothing is
happening. However, as we shall see, it is easy to change to a
different R prompt if you wish. We will assume that the UNIX shell
In using R under UNIX the suggested procedure for the first occasion is as follows:
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.
$ mkdir work $ cd work
At this point you will be asked whether you want to save the data from your R session. You can respond yes, no or 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.
Further R sessions are simple.
workthe working directory and start the program as before:
$ cd work $ R
q()command at the end of the session.
To use R under Windows the procedure to
follow is basically the same. Create a folder as the working directory,
and set that in the
Start In field in your R shortcut.
Then launch R by double clicking on the icon.
Readers wishing to get a feel for R at a computer before proceeding are strongly advised to work through the introductory session given in A sample session.
R has an inbuilt help facility similar to the
man facility of
UNIX. To get more information on any specific named function, for
solve, the command is
An alternative is
For a feature specified by special characters, the argument must be
enclosed in double or single quotes, making it a "character string":
This is also necessary for a few words with syntactic meaning including
Either form of quote mark may be used to escape the other, as in the
"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
which will launch a Web browser (
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
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
help.search command allows searching for help in various
?help.search for details and examples.
The examples on a help topic can normally be run by
Windows versions of R have other optional help systems: use
for further details.
Technically R is an expression language with a very simple
syntax. It is case sensitive as are most UNIX based packages, so
a are different symbols and would refer to different
variables. The set of symbols which can be used in R names depends
on the operating system and country within which R is being run
(technically on the locale in use). Normally all alphanumeric
symbols are allowed (and in some countries this includes accented
1, with the restriction that a name
cannot start with a digit.
Elementary commands consist of either expressions or 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 (
), or by a
newline. Elementary commands can be grouped together into one compound
expression by braces (
Comments can be put almost2 anywhere,
starting with a hashmark (
), everything to the end of the
line is a comment.
If a command is not complete at the end of a line, R will give a different prompt, by default
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.
Under many versions of UNIX and on Windows, R provides a mechanism for recalling and re-executing previous commands. The vertical arrow keys on the keyboard can be used to scroll forward and backward through a command history. Once a command is located in this way, the cursor can be moved within the command using the horizontal arrow keys, and characters can be removed with the <DEL> key or added with the other keys. More details are provided later: see The command line editor.
The recall and editing capabilities under UNIX are highly customizable. You can find out how to do this by reading the manual entry for the readline library.
Alternatively, the Emacs text editor provides more general support mechanisms (via ESS, Emacs Speaks Statistics) for working interactively with R. See R and Emacs.
If commands are stored on an external file, say
commands.R in the
work, they may be executed at any time in an
R session with the command
For Windows Source is also available on the
File menu. The function
will divert all subsequent output from the console to an external file,
record.lis. The command
restores it to the console once again.
The entities that R creates and manipulates are known as objects. These may be variables, arrays of numbers, character strings, functions, or more general structures built from such components.
During an R session, objects are created and stored by name (we discuss this process in the next session). The R command
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 workspace.
To remove objects the function
rm is available:
> rm(x, y, z, ink, junk, temp, foo, bar)
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
.RData3 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
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.
R operates on named data structures. The simplest such
structure is the numeric vector, which is a single entity
consisting of an ordered collection of numbers. To set up a vector
x, say, consisting of five numbers, namely 10.4, 5.6, 3.1,
6.4 and 21.7, use the R command
> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)
This is an assignment statement using the function
c() which in this context can take an arbitrary number of vector
arguments and whose value is a vector got by concatenating its
arguments end to end.4
A number occurring by itself in an expression is taken as a vector of length one.
Notice that the assignment operator (
) is not
operator, which is reserved for another
purpose. It consists of the two characters
("minus") occurring strictly side-by-side
and it `points' to the object receiving the value of the expression.
Assignment can also be made using the function
equivalent way of making the same assignment as above is with:
> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))
The usual operator,
<-, can be thought of as a syntactic
short-cut to this.
Assignments can also be made in the other direction, using the obvious change in the assignment operator. So the same assignment could be made using
> c(10.4, 5.6, 3.1, 6.4, 21.7) -> x
If an expression is used as a complete command, the value is printed and lost6. So now if we were to use the command
the reciprocals of the five values would be printed at the terminal (and
the value of
x, of course, unchanged).
The further assignment
> y <- c(x, 0, x)
would create a vector
y with 11 entries consisting of two copies
x with a zero in the middle place.
Vectors can be used in arithmetic expressions, in which case the operations are performed element by element. Vectors occurring in the same expression need not all be of the same length. If they are not, the value of the expression is a vector with the same length as the longest vector which occurs in the expression. Shorter vectors in the expression are recycled as often as need be (perhaps fractionally) until they match the length of the longest vector. In particular a constant is simply repeated. So with the above assignments the command
> v <- 2*x + y + 1
generates a new vector
v of length 11 constructed by adding
together, element by element,
2*x repeated 2.2 times,
repeated just once, and
1 repeated 11 times.
The elementary arithmetic operators are the usual
^ for raising to a power.
In addition all of the common arithmetic functions are available.
and so on, all have their usual meaning.
min select the largest and smallest elements of a
range is a function whose value is a vector of length two, namely
length(x) is the number of elements in
sum(x) gives the total of the elements in
prod(x) their product.
Two statistical functions are
mean(x) which calculates the sample
mean, which is the same as
var(x) which gives
or sample variance. If the argument to
var() is an
n-by-p matrix the value is a p-by-p sample
covariance matrix got by regarding the rows as independent
p-variate sample vectors.
sort(x) returns a vector of the same size as
x with the
elements arranged in increasing order; however there are other more
flexible sorting facilities available (see
sort.list() which produce a permutation to do the sorting).
min select the largest and smallest
values in their arguments, even if they are given several vectors. The
parallel maximum and minimum functions
pmin return a vector (of length equal to their longest argument)
that contains in each element the largest (smallest) element in that
position in any of the input vectors.
For most purposes the user will not be concerned if the "numbers" in a numeric vector are integers, reals or even complex. Internally calculations are done as double precision real numbers, or double precision complex numbers if the input data are complex.
To work with complex numbers, supply an explicit complex part. Thus
NaN and a warning, but
will do the computations as complex numbers.
R has a number of facilities for generating commonly used sequences
of numbers. For example
1:30 is the vector
..., 29, 30).
The colon operator has highest priority within an expression, so, for
2*1:15 is the vector
c(2, 4, ..., 28, 30).
n <- 10 and compare the sequences
30:1 may be used to generate a sequence
seq() is a more general facility for generating
sequences. It has five arguments, only some of which may be specified
in any one call. The first two arguments, if given, specify the
beginning and end of the sequence, and if these are the only two
arguments given the result is the same as the colon operator. That is
seq(2,10) is the same vector as
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
seq(from=1, to=30) and
from=1) are all the same as
1:30. The next two parameters to
seq() may be named
length=value, which specify a step size and a length for
the sequence respectively. If neither of these is given, the default
by=1 is assumed.
> seq(-5, 5, by=.2) -> s3
s3 the vector
c(-5.0, -4.8, -4.6, ...,
4.6, 4.8, 5.0). Similarly
> s4 <- seq(length=51, from=-5, by=.2)
generates the same vector in
The fifth parameter may be named
along=vector, which if
used must be the only parameter, and creates a sequence
..., length(vector), or the empty sequence if the vector is
empty (as it can be).
A related function is
which can be used for replicating an object in various complicated ways.
The simplest form is
> s5 <- rep(x, times=5)
which will put five copies of
x end-to-end in
As well as numerical vectors, R allows manipulation of logical
quantities. The elements of a logical vectors can have the values
NA (for "not available", see
below). The first two are often abbreviated as
respectively. Note however that
F are just
variables which are set to
FALSE by default, but
are not reserved words and hence can be overwritten by the user. Hence,
you should always use
Logical vectors are generated by conditions. For example
> temp <- x > 13
temp as a vector of the same length as
x with values
FALSE corresponding to elements of
x where the condition
is not met and
TRUE where it is.
The logical operators are
== for exact equality and
!= for inequality.
In addition if
c2 are logical expressions, then
c1 & c2 is their intersection ("and"),
c1 | c2
is their union ("or"), and
!c1 is the negation of
Logical vectors may be used in ordinary arithmetic, in which case they
are coerced into numeric vectors,
1. However there are situations where
logical vectors and their coerced numeric counterparts are not
equivalent, for example see the next subsection.
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
In general any operation on an
NA becomes an
motivation for this rule is simply that if the specification of an
operation is incomplete, the result cannot be known and hence is not
is.na(x) gives a logical vector of the same size as
x with value
TRUE if and only if the corresponding element
> z <- c(1:3,NA); ind <- is.na(z)
Notice that the logical expression
x == NA is quite different
NA is not really a value but a marker
for a quantity that is not available. Thus
x == NA is a vector
of the same length as
x all of whose values are
as the logical expression itself is incomplete and hence undecidable.
Note that there is a second kind of "missing" values which are
produced by numerical computation, the so-called Not a Number,
values. Examples are
> Inf - Inf
which both give
NaN since the result cannot be defined sensibly.
TRUE both for
NaN values. To differentiate these,
is.nan(xx) is only
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.,
"New iteration results".
Character strings are entered using either double (
") or single
') quotes, but are printed using double quotes (or sometimes
without quotes). They use C-style escape sequences, using
the escape character, so
\\ is entered and printed as
and inside double quotes
" is entered as
useful escape sequences are
\t, tab and
Character vectors may be concatenated into a vector by the
function; examples of their use will emerge frequently.
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,
sep=string, which changes it to
> labs <- paste(c("X","Y"), 1:10, sep="")
labs into the character vector
c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10")
Note particularly that recycling of short lists takes place here too;
c("X", "Y") is repeated 5 times to match the sequence
Subsets of the elements of a vector may be selected by appending to the name of the vector an index vector in square brackets. More generally any expression that evaluates to a vector may have subsets of its elements similarly selected by appending an index vector in square brackets immediately after the expression.
Such index vectors can be any of four distinct types.
TRUEin the index vector are selected and those corresponding to
FALSEomitted. For example
> y <- x[!is.na(x)]
creates (or re-creates) an object
y which will contain the
non-missing values of
x, in the same order. Note that if
x has missing values,
y will be shorter than
> (x+1)[(!is.na(x)) & x>0] -> z
creates an object
z and places in it the values of the vector
x+1 for which the corresponding value in
x was both
non-missing and positive.
length(x)}. The corresponding elements of the vector are selected and concatenated, in that order, in the result. The index vector can be of any length and the result is of the same length as the index vector. For example
xis the sixth component of
selects the first 10 elements of
no less than 10). Also
> c("x","y")[rep(c(1,2,2,1), times=4)]
(an admittedly unlikely thing to do) produces a character vector of
length 16 consisting of
"x", "y", "y", "x" repeated four times.
> y <- x[-(1:5)]
y all but the first five elements of
namesattribute 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.
> fruit <- c(5, 10, 1, 20) > names(fruit) <- c("orange", "banana", "apple", "peach") > lunch <- fruit[c("apple","orange")]
The advantage is that alphanumeric names are often easier to remember than numeric indices. This option is particularly useful in connection with data frames, as we shall see later.
An indexed expression can also appear on the receiving end of an
assignment, in which case the assignment operation is performed
only on those elements of the vector. The expression must be of
vector[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.
> x[is.na(x)] <- 0
replaces any missing values in
x by zeros and
> y[y < 0] <- -y[y < 0]
has the same effect as
> y <- abs(y)
Vectors are the most important type of object in R, but there are several others which we will meet more formally in later sections.
The entities R operates on are technically known as objects. Examples are vectors of numeric (real) or complex values, vectors of logical values and vectors of character strings. These are known as "atomic" structures since their components are all of the same type, or mode, namely numeric8, complex, logical and character respectively.
Vectors must have their values all of the same mode. Thus any
given vector must be unambiguously either logical,
numeric, complex or character. The only mild
exception to this rule is the special "value" listed as
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
character(0) and the empty numeric vector as
R also operates on objects called lists, which are of mode list. These are ordered sequences of objects which individually can be of any mode. lists are known as "recursive" rather than atomic structures since their components can themselves be lists in their own right.
The other recursive structures are those of mode function and expression. Functions are the objects that form part of the R system along with similar user written functions, which we discuss in some detail later. Expressions as objects form an advanced part of R which will not be discussed in this guide, except indirectly when we discuss formulae used with modeling in R.
By the mode of an object we mean the basic type of its
fundamental constituents. This is a special case of a "property"
of an object. Another property of every object is its length. The
length(object) can be
used to find out the mode and length of any defined structure
Further properties of an object are usually provided by
attributes(object), see Getting and setting attributes.
Because of this, mode and length are also called "intrinsic
attributes" of an object.
For example, if
z is a complex vector of length 100, then in an
mode(z) is the character string
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
> z <- 0:9
we could put
> digits <- as.character(z)
digits is the character vector
c("0", "1", "2",
..., "9"). A further coercion, or change of mode,
reconstructs the numerical vector again:
> d <- as.integer(digits)
z are the same.10 There is a
large collection of functions of the form
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.
An "empty" object may still have a mode. For example
> e <- numeric()
e an empty vector structure of mode numeric. Similarly
character() is a empty character vector, and so on. Once an
object of any size has been created, new components may be added to it
simply by giving it an index value outside its previous range. Thus
> e <- 17
e a vector of length 3, (the first two components of
which are at this point both
NA). This applies to any structure
at all, provided the mode of the additional component(s) agrees with the
mode of the object in the first place.
This automatic adjustment of lengths of an object is used often, for
example in the
scan() function for input. (See The scan() function.)
Conversely to truncate the size of an object requires only an assignment
to do so. Hence if
alpha is an object of length 10, then
> alpha <- alpha[2 * 1:5]
makes it an object of length 5 consisting of just the former components with even index. The old indices are not retained, of course.
gives a list of all the non-intrinsic attributes currently defined for
that object. The function
can be used to select a specific attribute. These functions are rarely
used, except in rather special circumstances when some new attribute is
being created for some particular purpose, for example to associate a
creation date or an operator with an R object. The concept, however,
is very important.
Some care should be exercised when assigning or deleting attributes since they are an integral part of the object system used in R.
When it is used on the left hand side of an assignment it can be used
either to associate a new attribute with
object or to
change an existing one. For example
> attr(z,"dim") <- c(10,10)
allows R to treat
z as if it were a 10-by-10 matrix.
A special attribute known as the class of the object is used to allow for an object oriented style of programming in R.
For example if an object has class
"data.frame", it will be
printed in a certain way, the
plot() function will display it
graphically in a certain way, and other so-called generic functions such
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
For example if
winter has the class
will print it in data frame form, which is rather like a matrix, whereas
will print it as an ordinary list. Only in rather special situations do you need to use this facility, but one is when you are learning to come to terms with the idea of class and generic functions.
Generic functions and classes will be discussed further in Object orientation, but only briefly.
A factor is a vector object used to specify a discrete classification (grouping) of the components of other vectors of the same length. R provides both ordered and unordered factors. While the "real" application of factors is with model formulae (see Contrasts), we here look at
Suppose, for example, we have a sample of 30 tax accountants from all the states and territories of Australia11 and their individual state of origin is specified by a character vector of state mnemonics as
> state <- c("tas", "sa", "qld", "nsw", "nsw", "nt", "wa", "wa", "qld", "vic", "nsw", "vic", "qld", "qld", "sa", "tas", "sa", "nt", "wa", "vic", "qld", "nsw", "nsw", "wa", "sa", "act", "nsw", "vic", "vic", "act")
Notice that in the case of a character vector, "sorted" means sorted in alphabetical order.
A factor is similarly created using the
> statef <- factor(state)
print() function handles factors slightly differently from
> statef  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 Levels: act nsw nt qld sa tas vic wa
To find out the levels of a factor the function
levels() can be
> levels(statef)  "act" "nsw" "nt" "qld" "sa" "tas" "vic" "wa"
tapply()and ragged arrays
To continue the previous example, suppose we have the incomes of the same tax accountants in another vector (in suitably large units of money)
> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56, 61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46, 59, 46, 58, 43)
To calculate the sample mean income for each state we can now use the
> incmeans <- tapply(incomes, statef, mean)
giving a means vector with the components labelled by the levels
act nsw nt qld sa tas vic wa 44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250
tapply() is used to apply a function, here
mean(), to each group of components of the first argument, here
incomes, defined by the levels of the second component, here
statef12, 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