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 modelling and graphics.
This manual is for R, version 4.5.3 (2026-03-11).
Copyright © 1990 W. N. Venables
Copyright © 1992 W. N. Venables & D. M. Smith
Copyright © 1997 R. Gentleman & R. Ihaka
Copyright © 1997, 1998 M. Maechler
Copyright © 1999–2026 R Core Team
1 Introduction and preliminaries
1.1 The R environment
R is an integrated suite of software facilities for data
manipulation, calculation and graphical display. Among other things it
has
an effective data handling and storage facility,
a suite of operators for calculations on arrays, in particular matrices,
a large, coherent, integrated collection of intermediate tools for data
analysis,
graphical facilities for data analysis and display either directly at
the computer or on hardcopy, and
a well developed, simple and effective programming language (called ‘S’)
which includes conditionals, loops, user defined recursive functions and
input and output facilities. (Indeed most of the system supplied
functions are themselves written in the S language.)
The term “environment” is intended to characterize it as a fully
planned and coherent system, rather than an incremental accretion of
very specific and inflexible tools, as is frequently the case with other
data analysis software.
R is very much a vehicle for newly developing methods of interactive
data analysis. It has developed rapidly, and has been extended by a
large collection of
packages
. However, most programs written in
R are essentially ephemeral, written for a single piece of data
analysis.
1.3 R and statistics
Our introduction to the R environment did not mention
statistics
, yet many people use R as a statistics system. We
prefer to think of it of an environment within which many classical and
modern statistical techniques have been implemented. A few of these are
built into the base R environment, but many are supplied as
packages
. There are about 25 packages supplied with R (called
“standard” and “recommended” packages) and many more are available
through the
CRAN
family of Internet sites (via
) and elsewhere. More details on
packages are given later (see
Packages
).
Most classical statistics and much of the latest methodology is
available for use with R, but users may need to be prepared to do a
little work to find it.
There is an important difference in philosophy between S (and hence
R) and the other main statistical systems. In S a statistical
analysis is normally done as a series of steps, with intermediate
results being stored in objects. Thus whereas SAS and SPSS will give
copious output from a regression or discriminant analysis, R will
give minimal output and store the results in a fit object for subsequent
interrogation by further R functions.
1.4 R and the window system
The most convenient way to use R is at a graphics workstation running
a windowing system. This guide is aimed at users who have this
facility. In particular we will occasionally refer to the use of R
on an X window system although the vast bulk of what is said applies
generally to any implementation of the R environment.
Most users will find it necessary to interact directly with the
operating system on their computer from time to time. In this guide, we
mainly discuss interaction with the operating system on UNIX machines.
If you are running R under Windows or macOS you will need to make
some small adjustments.
Setting up a workstation to take full advantage of the customizable
features of R is a straightforward if somewhat tedious procedure, and
will not be considered further here. Users in difficulty should seek
local expert help.
1.5 Using R interactively
When you use the R program it issues a prompt when it expects input
commands. The default prompt is ‘
’, which on UNIX might be
the same as the shell prompt, and so it may appear that nothing is
happening. However, as we shall see, it is easy to change to a
different R prompt if you wish. We will assume that the UNIX shell
prompt is ‘
’.
In using R under UNIX the suggested procedure for the first occasion
is as follows:
Create a separate sub-directory, say
work
, to hold data files on
which you will use R for this problem. This will be the working
directory whenever you use R for this particular problem.
Start the R program with the command
At this point R commands may be issued (see later).
To quit the R program the command is
At this point you will be asked whether you want to save the data from
your R session. On some systems this will bring up a dialog box, and
on others you will receive a text prompt to which you can respond
yes
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.
Make
work
the working directory and start the program as before:
Use the R program, terminating with the
q()
command at the end
of the session.
To use R under Windows the procedure to
follow is basically the same. Create a folder as the working directory,
and set that in the
Start In
field in your R shortcut.
Then launch R by double clicking on the icon.
1.6 An introductory session
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
1.7 Getting help with functions and features
R has an inbuilt help facility similar to the
man
facility of
UNIX. To get more information on any specific named function, for
example
solve
, the command is
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
if
for
and
function
Either form of quote mark may be used to escape the other, as in the
string
"It's important"
. Our convention is to use
double quote marks for preference.
On most R installations help is available in
HTML
format by
running
which will launch a Web browser that allows the help pages to be browsed
with hyperlinks. On UNIX, subsequent help requests are sent to the
HTML
-based help system. The ‘Search Engine and Keywords’ link in the
page loaded by
help.start()
is particularly useful as it is
contains a high-level concept list which searches though available
functions. It can be a great way to get your bearings quickly and to
understand the breadth of what R has to offer.
The
help.search
command (alternatively
??
allows searching for help in various
ways. For example,
Try
?help.search
for details and more 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.
1.8 R commands, case sensitivity, etc.
Technically R is an
expression language
with a very simple
syntax. It is
case sensitive
as are most UNIX based packages, so
and
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 letters) plus ‘
’ and
’, with the restriction that a name must start with
’ or a letter, and if it starts with ‘
’ the
second character must not be a digit. Names are effectively
unlimited in length.
Elementary commands consist of either
expressions
or
assignments
. If an expression is given as a command, it is
evaluated, printed (unless specifically made invisible), and the value
is lost. An assignment also evaluates an expression and passes the
value to a variable but the result is not automatically printed.
Commands are separated either by a semi-colon (‘
’), or by a
newline. Elementary commands can be grouped together into one compound
expression by braces (‘
’ and ‘
’).
Comments
can be put almost anywhere,
starting with a hash mark (‘
’), 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.
Command lines entered at the console are limited to about 4095 bytes (not characters).
1.9 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
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
in
R FAQ
for more information.
1.10 Executing commands from or diverting output to a file
If commands are stored in an external
file, say
commands.R
in the working directory
work
, they
may be executed at any time in an R session with the command
For Windows
Source
is also available on the
File
menu. The function
sink
will divert all subsequent output from the console to an external file,
record.lis
. The command
restores it to the console once again.
1.11 Data permanency and removing objects
The entities that R creates and manipulates are known as
objects
. These may be variables, arrays of numbers, character
strings, functions, or more general structures built from such
components.
During an R session, objects are created and stored by name (we
discuss this process in the next section). The R command
(alternatively,
ls()
) can be used to display the names of (most
of) the objects which are currently stored within R. The collection
of objects currently stored is called the
workspace
To remove objects the function
rm
is available:
> rm(x, y, z, ink, junk, temp, foo, bar)
All objects created during an R session can be stored permanently in
a file for use in future R sessions. At the end of each R session
you are given the opportunity to save all the currently available
objects. If you indicate that you want to do this, the objects are
written to a file called
.RData
in the
current directory, and the command lines used in the session are saved
to a file called
.Rhistory
When R is started at later time from the same directory it reloads
the workspace from this file. At the same time the associated commands
history is reloaded.
It is recommended that you should use separate working directories for
analyses conducted with R. It is quite common for objects with names
and
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.
2 Simple manipulations; numbers and vectors
2.1 Vectors and assignment
R operates on named
data structures
. The simplest such
structure is the numeric
vector
, which is a single entity
consisting of an ordered collection of numbers. To set up a vector
named
, 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.
A number occurring by itself in an expression is taken as a vector of
length one.
Notice that the assignment operator (‘
<-
’), which consists
of the two characters ‘
’ (“less than”) and
’ (“minus”) occurring strictly side-by-side and it
‘points’ to the object receiving the value of the expression.
In most contexts the ‘
’ operator can be used as an alternative.
Assignment can also be made using the function
assign()
. An
equivalent way of making the same assignment as above is with:
> 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 lost
. 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
, of course, unchanged).
The further assignment
would create a vector
with 11 entries consisting of two copies
of
with a zero in the middle place.
2.2 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.
Recycling
occurs in each
binary operation: the shorter vector is recycled as often as need be (perhaps
fractionally) until it matches the length of the longer vector. In
particular a constant is simply repeated. So with the above assignments
the command
generates a new vector
of length 11 constructed by adding
together, element by element,
2*x
repeated 2.2 times,
repeated just once, and
repeated 11 times.
The elementary arithmetic operators are the usual
and
for raising to a power.
In addition all of the common arithmetic functions are available.
log
exp
sin
cos
tan
sqrt
and so on, all have their usual meaning.
max
and
min
select the largest and smallest elements of a
vector respectively.
range
is a function whose value is a vector of length two, namely
c(min(x), max(x))
length(x)
is the number of elements in
sum(x)
gives the total of the elements in
and
prod(x)
their product.
Two statistical functions are
mean(x)
which calculates the sample
mean, which is the same as
sum(x)/length(x)
and
var(x)
which gives
sum((x-mean(x))^2)/(length(x)-1)
or sample variance. If the argument to
var()
is an
-by-
matrix the value is a
-by-
sample
covariance matrix got by regarding the rows as independent
-variate sample vectors.
sort(x)
returns a vector of the same size as
with the
elements arranged in increasing order; however there are other more
flexible sorting facilities available (see
order()
or
sort.list()
which produce a permutation to do the sorting).
Note that
max
and
min
select the largest and smallest
values in their arguments, even if they are given several vectors. The
parallel
maximum and minimum functions
pmax
and
pmin
return a vector (of length equal to their longest argument)
that contains in each element the largest (smallest) element in that
position in any of the input vectors.
For most purposes the user will not be concerned if the “numbers” in a
numeric vector are integers, reals or even complex. Internally
calculations are done as double precision real numbers, or double
precision complex numbers if the input data are complex.
To work with complex numbers, supply an explicit complex part. Thus
will give
NaN
and a warning, but
will do the computations as complex numbers.
2.3 Generating regular sequences
R has a number of facilities for generating commonly used sequences
of numbers. For example
1:30
is the vector
c(1, 2,
…, 29, 30)
The colon operator has high priority within an expression, so, for
example
2*1:15
is the vector
c(2, 4, …, 28, 30)
Put
n <- 10
and compare the sequences
1:n-1
and
1:(n-1)
The construction
30:1
may be used to generate a sequence
backwards.
The function
seq()
is a more general facility for generating
sequences. It has five arguments, only some of which may be specified
in any one call. The first two arguments, if given, specify the
beginning and end of the sequence, and if these are the only two
arguments given the result is the same as the colon operator. That is
seq(2,10)
is the same vector as
2:10
Arguments to
seq()
, and to many other R functions, can also
be given in named form, in which case the order in which they appear is
irrelevant. The first two arguments may be named
from=
value
and
to=
value
; thus
seq(1,30)
seq(from=1, to=30)
and
seq(to=30,
from=1)
are all the same as
1:30
. The next two arguments to
seq()
may be named
by=
value
and
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.
For example
> seq(-5, 5, by=.2) -> s3
generates in
s3
the vector
c(-5.0, -4.8, -4.6, …,
4.6, 4.8, 5.0)
. Similarly
> s4 <- seq(length=51, from=-5, by=.2)
generates the same vector in
s4
The fifth argument may be named
along=
vector
, which is
normally used as the only argument to create the sequence
1, 2,
…, length(
vector
, or the empty sequence if the vector is
empty (as it can be).
A related function is
rep()
which can be used for replicating an object in various complicated ways.
The simplest form is
which will put five copies of
end-to-end in
s5
. Another
useful version is
which repeats each element of
five times before moving on to
the next.
2.4 Logical vectors
As well as numerical vectors, R allows manipulation of logical
quantities. The elements of a logical vector can have the values
TRUE
FALSE
, and
NA
(for “not available”, see
below). The first two are often abbreviated as
and
respectively. Note however that
and
are just
variables which are set to
TRUE
and
FALSE
by default, but
are not reserved words and hence can be overwritten by the user. Hence,
you should always use
TRUE
and
FALSE
Logical vectors are generated by
conditions
. For example
sets
temp
as a vector of the same length as
with values
FALSE
corresponding to elements of
where the condition
is
not
met and
TRUE
where it is.
The logical operators are
<=
>=
==
for exact equality and
!=
for inequality.
In addition if
c1
and
c2
are logical expressions, then
c1 & c2
is their intersection (
“and”
),
c1 | c2
is their union (
“or”
), and
!c1
is the negation of
c1
Logical vectors may be used in ordinary arithmetic, in which case they
are
coerced
into numeric vectors,
FALSE
becoming
and
TRUE
becoming
. However there are situations where
logical vectors and their coerced numeric counterparts are not
equivalent, for example see the next subsection.
2.5 Missing values
In some cases the components of a vector may not be completely
known. When an element or value is “not available” or a “missing
value” in the statistical sense, a place within a vector may be
reserved for it by assigning it the special value
NA
In general any operation on an
NA
becomes an
NA
. The
motivation for this rule is simply that if the specification of an
operation is incomplete, the result cannot be known and hence is not
available.
The function
is.na(x)
gives a logical vector of the same size as
with value
TRUE
if and only if the corresponding element
in
is
NA
> z <- c(1:3,NA); ind <- is.na(z)
Notice that the logical expression
x == NA
is quite different
from
is.na(x)
since
NA
is not really a value but a marker
for a quantity that is not available. Thus
x == NA
is a vector
of the same length as
all
of whose values are
NA
as the logical expression itself is incomplete and hence undecidable.
Note that there is a second kind of “missing” values which are
produced by numerical computation, the so-called
Not a Number
NaN
values. Examples are
or
which both give
NaN
since the result cannot be defined sensibly.
In summary,
is.na(xx)
is
TRUE
both
for
NA
and
NaN
values. To differentiate these,
is.nan(xx)
is only
TRUE
for
NaN
s.
Missing values are sometimes printed as

when character
vectors are printed without quotes.
2.6 Character vectors
Character quantities and character vectors are used frequently in R,
for example as plot labels. Where needed they are denoted by a sequence
of characters delimited by the double quote character, e.g.,
"x-values"
"New iteration results"
Character strings are entered using either matching double (
) or
single (
) quotes, but are printed using double quotes (or
sometimes without quotes). They use C-style escape sequences, using
as the escape character, so
is entered and printed as
\\
, and inside double quotes
is entered as
\"
Other useful escape sequences are
\n
, newline,
\t
, tab and
\b
, backspace—see
?Quotes
for a full list.
Character vectors may be concatenated into a vector by the
c()
function; examples of their use will emerge frequently.
The
paste()
function takes an arbitrary number of arguments and
concatenates them one by one into character strings. Any numbers given
among the arguments are coerced into character strings in the evident
way, that is, in the same way they would be if they were printed. The
arguments are by default separated in the result by a single blank
character, but this can be changed by the named argument,
sep=
string
, which changes it to
string
possibly empty.
For example
> labs <- paste(c("X","Y"), 1:10, sep="")
makes
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;
thus
c("X", "Y")
is repeated 5 times to match the sequence
1:10
2.7 Index vectors; selecting and modifying subsets of a data set
Subsets of the elements of a vector may be selected by appending to the
name of the vector an
index vector
in square brackets. More
generally any expression that evaluates to a vector may have subsets of
its elements similarly selected by appending an index vector in square
brackets immediately after the expression.
Such index vectors can be any of four distinct types.
A logical vector
. In this case the index vector is recycled to the
same length as the vector from which elements are to be selected.
Values corresponding to
TRUE
in the index vector are selected and
those corresponding to
FALSE
are omitted. For example
creates (or re-creates) an object
which will contain the
non-missing values of
, in the same order. Note that if
has missing values,
will be shorter than
Also
> (x+1)[(!is.na(x)) & x>0] -> z
creates an object
and places in it the values of the vector
x+1
for which the corresponding value in
was both
non-missing and positive.
A vector of positive integral quantities
. In this case the
values in the index vector must lie in the set {1, 2, …,
length(x)
}. The corresponding elements of the vector are
selected and concatenated,
in that order
, in the result. The
index vector can be of any length and the result is of the same length
as the index vector. For example
x[6]
is the sixth component of
and
selects the first 10 elements of
(assuming
length(x)
is
not 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.
A vector of negative integral quantities
. Such an index vector
specifies the values to be
excluded
rather than included. Thus
gives
all but the first five elements of
A vector of character strings
. This possibility only applies
where an object has a
names
attribute to identify its components.
In this case a sub-vector of the names vector may be used in the same way
as the positive integral labels in item 2 further above.
> fruit <- c(5, 10, 1, 20)
> names(fruit) <- c("orange", "banana", "apple", "peach")
> lunch <- fruit[c("apple","orange")]
The advantage is that alphanumeric
names
are often easier to
remember than
numeric indices
. This option is particularly
useful in connection with data frames, as we shall see later.
An indexed expression can also appear on the receiving end of an
assignment, in which case the assignment operation is performed
only on those elements of the vector
. The expression must be of
the form
vector[
index_vector
as having an arbitrary
expression in place of the vector name does not make much sense here.
For example
replaces any missing values in
by zeros and
has the same effect as
2.8 Other types of objects
Vectors are the most important type of object in R, but there are
several others which we will meet more formally in later sections.
matrices
or more generally
arrays
are multi-dimensional
generalizations of vectors. In fact, they
are
vectors that can
be indexed by two or more indices and will be printed in special ways.
See
Arrays and matrices
factors
provide compact ways to handle categorical data.
See
Ordered and unordered factors
lists
are a general form of vector in which the various elements
need not be of the same type, and are often themselves vectors or lists.
Lists provide a convenient way to return the results of a statistical
computation. See
Lists
data frames
are matrix-like structures, in which the columns can
be of different types. Think of data frames as ‘data matrices’ with one
row per observational unit but with (possibly) both numerical and
categorical variables. Many experiments are best described by data
frames: the treatments are categorical but the response is numeric.
See
Data frames
functions
are themselves objects in R which can be stored in
the project’s workspace. This provides a simple and convenient way to
extend R. See
Writing your own functions
3 Objects, their modes and attributes
3.1 Intrinsic attributes: mode and length
The entities R operates on are technically known as
objects
Examples are vectors of numeric (real) or complex values, vectors of
logical values and vectors of character strings. These are known as
“atomic” structures since their components are all of the same type,
or
mode
, namely
numeric
complex
logical
character
and
raw
Vectors must have their values
all of the same mode
. Thus any
given vector must be unambiguously either
logical
numeric
complex
character
or
raw
. (The
only apparent exception to this rule is the special “value” listed as
NA
for quantities not available, but in fact there are several
types of
NA
). Note that a vector can be empty and still have a
mode. For example the empty character string vector is listed as
character(0)
and the empty numeric vector as
numeric(0)
R also operates on objects called
lists
, which are of mode
list
. These are ordered sequences of objects which individually
can be of any mode.
lists
are known as “recursive” rather than
atomic structures since their components can themselves be lists in
their own right.
The other recursive structures are those of mode
function
and
expression
. Functions are the objects that form part of the R
system along with similar user written functions, which we discuss in
some detail later. Expressions as objects form an
advanced part of R which will not be discussed in this guide, except
indirectly when we discuss
formulae
used with modeling in R.
By the
mode
of an object we mean the basic type of its
fundamental constituents. This is a special case of a “property”
of an object. Another property of every object is its
length
. The
functions
mode(
object
and
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
is a complex vector of length 100, then in an
expression
mode(z)
is the character string
"complex"
and
length(z)
is
100
R caters for changes of mode almost anywhere it could be considered
sensible to do so, (and a few where it might not be). For example with
we could put
> digits <- as.character(z)
after which
digits
is the character vector
c("0", "1", "2",
…, "9")
. A further
coercion
, or change of mode,
reconstructs the numerical vector again:
> d <- as.integer(digits)
Now
and
are the same. There is a
large collection of functions of the form
as.
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.
3.2 Changing the length of an object
An “empty” object may still have a mode. For example
makes
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
now makes
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.) We can
then retain just the first three values by
and vectors can be extended (by missing values) in the same way.
3.3 Getting and setting attributes
The function
attributes(
object
returns a list of all the non-intrinsic attributes currently defined for
that object. The function
attr(
object
name
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
as if it were a 10-by-10 matrix.
3.4 The class of an object
All objects in R have a
class
, reported by the function
class
. For simple vectors this is just the mode, for example
"numeric"
"logical"
"character"
or
"list"
but
"matrix"
"array"
"factor"
and
"data.frame"
are other possible values.
A special attribute known as the
class
of the object is used to
allow for an object-oriented style of
programming in R. For example if an object has class
"data.frame"
, it will be printed in a certain way, the
plot()
function will display it graphically in a certain way, and
other so-called generic functions such as
summary()
will react to
it as an argument in a way sensitive to its class.
To remove temporarily the effects of class, use the function
unclass()
For example if
winter
has the class
"data.frame"
then
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
Classes, generic functions and object orientation
, but only briefly.
4 Ordered and unordered factors
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 a specific example.
4.1 A specific example
Suppose, for example, we have a sample of 30 tax accountants from all
the states and territories of Australia
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.
factor
is similarly created using the
factor()
function:
> statef <- factor(state)
The
print()
function handles factors slightly differently from
other objects:
> statef
[1] tas sa qld nsw nsw nt wa wa qld vic nsw vic qld qld sa
[16] tas sa nt wa vic qld nsw nsw wa sa act nsw vic vic act
Levels: act nsw nt qld sa tas vic wa
To find out the levels of a factor the function
levels()
can be
used.
> levels(statef)
[1] "act" "nsw" "nt" "qld" "sa" "tas" "vic" "wa"
4.2 The function
tapply()
and ragged arrays
To continue the previous example, suppose we have the incomes of the
same tax accountants in another vector (in suitably large units of
money)
> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56,
61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46,
59, 46, 58, 43)
To calculate the sample mean income for each state we can now use the
special function
tapply()
> 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
The function
tapply()
is used to apply a function, here
mean()
, to each group of components of the first argument, here
incomes
, defined by the levels of the second component, here
statef
, as if they were separate vector
structures. The result is a structure of the same length as the levels
attribute of the factor containing the results. The reader should
consult the help document for more details.
Suppose further we needed to calculate the standard errors of the state
income means. To do this we need to write an R function to calculate
the standard error for any given vector. Since there is an builtin
function
var()
to calculate the sample variance, such a function
is a very simple one liner, specified by the assignment:
> stdError <- function(x) sqrt(var(x)/length(x))
(Writing functions will be considered later in
Writing your own functions
. Note that R’s a builtin function
sd()
is something different.)
After this assignment, the standard errors are calculated by
> incster <- tapply(incomes, statef, stdError)
and the values calculated are then
> incster
act nsw nt qld sa tas vic wa
1.5 4.3102 4.5 4.1061 2.7386 0.5 5.244 2.6575
As an exercise you may care to find the usual 95% confidence limits for
the state mean incomes. To do this you could use
tapply()
once
more with the
length()
function to find the sample sizes, and the
qt()
function to find the percentage points of the appropriate
-distributions. (You could also investigate R’s facilities
for
-tests.)
The function
tapply()
can also be used to handle more complicated
indexing of a vector by multiple categories. For example, we might wish
to split the tax accountants by both state and sex. However in this
simple instance (just one factor) what happens can be thought of as
follows. The values in the vector are collected into groups
corresponding to the distinct entries in the factor. The function is
then applied to each of these groups individually. The value is a
vector of function results, labelled by the
levels
attribute of
the factor.
The combination of a vector and a labelling factor is an example of what
is sometimes called a
ragged array
, since the subclass sizes are
possibly irregular. When the subclass sizes are all the same the
indexing may be done implicitly and much more efficiently, as we see in
the next section.
4.3 Ordered factors
The levels of factors are stored in alphabetical order, or in the order
they were specified to
factor
if they were specified explicitly.
Sometimes the levels will have a natural ordering that we want to record
and want our statistical analysis to make use of. The
ordered()
function creates such ordered factors but is otherwise identical to
factor
. For most purposes the only difference between ordered
and unordered factors is that the former are printed showing the
ordering of the levels, but the contrasts generated for them in fitting
linear models are different.
5 Arrays and matrices
5.1 Arrays
An array can be considered as a multiply subscripted collection of data
entries, for example numeric. R allows simple facilities for
creating and handling arrays, and in particular the special case of
matrices.
A dimension vector is a vector of non-negative integers. If its length is
then the array is
-dimensional, e.g. a matrix is a
-dimensional array. The dimensions are indexed from one up to
the values given in the dimension vector.
A vector can be used by R as an array only if it has a dimension
vector as its
dim
attribute. Suppose, for example,
is a
vector of 1500 elements. The assignment
gives it the
dim
attribute that allows it to be treated as a
by
by
100
array.
Other functions such as
matrix()
and
array()
are available
for simpler and more natural looking assignments, as we shall see in
The
array()
function
The 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
, is
c(3,4,2)
then there are 3 * 4 * 2
= 24 entries in
and the data vector holds them in the order
a[1,1,1], a[2,1,1], …, a[2,4,2], a[3,4,2]
Arrays can be one-dimensional: such arrays are usually treated in the
same way as vectors (including when printing), but the exceptions can
cause confusion.
5.2 Array indexing. Subsections of an array
Individual elements of an array may be referenced by giving the name of
the array followed by the subscripts in square brackets, separated by
commas.
More generally, subsections of an array may be specified by giving a
sequence of
index vectors
in place of subscripts; however
if any index position is given an empty index vector, then the
full range of that subscript is taken
Continuing the previous example,
a[2,,]
is a 4 *
2 array with dimension vector
c(4,2)
and data vector containing
the values
c(a[2,1,1], a[2,2,1], a[2,3,1], a[2,4,1],
a[2,1,2], a[2,2,2], a[2,3,2], a[2,4,2])
in that order.
a[,,]
stands for the entire array, which is the
same as omitting the subscripts entirely and using
alone.
For any array, say
, the dimension vector may be referenced
explicitly as
dim(Z)
(on either side of an assignment).
Also, if an array name is given with just
one subscript or index
vector
, then the corresponding values of the data vector only are used;
in this case the dimension vector is ignored. This is not the case,
however, if the single index is not a vector but itself an array, as we
next discuss.
5.3 Index matrices
As well as an index vector in any subscript position, a matrix may be
used with a single
index matrix
in order either to assign a vector
of quantities to an irregular collection of elements in the array, or to
extract an irregular collection as a vector.
A matrix example makes the process clear. In the case of a doubly
indexed array, an index matrix may be given consisting of two columns
and as many rows as desired. The entries in the index matrix are the
row and column indices for the doubly indexed array. Suppose for
example we have a
by
array
and we wish to do
the following:
Extract elements
X[1,3]
X[2,2]
and
X[3,1]
as a
vector structure, and
Replace these entries in the array
by zeroes.
In this case we need a
by
subscript array, as in the
following example.
> x <- array(1:20, dim=c(4,5)) #
Generate a 4 by 5 array.
> x
[,1] [,2] [,3] [,4] [,5]
[1,] 1 5 9 13 17
[2,] 2 6 10 14 18
[3,] 3 7 11 15 19
[4,] 4 8 12 16 20
> i <- array(c(1:3,3:1), dim=c(3,2))
> i #
is a 3 by 2 index array.
[,1] [,2]
[1,] 1 3
[2,] 2 2
[3,] 3 1
> x[i] #
Extract those elements
[1] 9 6 3
> x[i] <- 0 #
Replace those elements by zeros.
> x
[,1] [,2] [,3] [,4] [,5]
[1,] 1 5 0 13 17
[2,] 2 0 10 14 18
[3,] 0 7 11 15 19
[4,] 4 8 12 16 20
Negative indices are not allowed in index matrices.
NA
and zero
values are allowed: rows in the index matrix containing a zero are
ignored, and rows containing an
NA
produce an
NA
in the
result.
As a less trivial example, suppose we wish to generate an (unreduced)
design matrix for a block design defined by factors
blocks
levels) and
varieties
levels). Further
suppose there are
plots in the experiment. We could proceed as
follows:
> Xb <- matrix(0, n, b)
> Xv <- matrix(0, n, v)
> ib <- cbind(1:n, blocks)
> iv <- cbind(1:n, varieties)
> Xb[ib] <- 1
> Xv[iv] <- 1
> X <- cbind(Xb, Xv)
To construct the incidence matrix,
say, we could use
However a simpler direct way of producing this matrix is to use
table()
> N <- table(blocks, varieties)
Index matrices must be numerical: any other form of matrix (e.g. a
logical or character matrix) supplied as a matrix is treated as an
indexing vector.
5.4 The
array()
function
As well as giving a vector structure a
dim
attribute, arrays can
be constructed from vectors by the
array
function, which has the
form
> Z <- array(
data_vector
dim_vector
For example, if the vector
contains 24 or fewer, numbers then
the command
> Z <- array(h, dim=c(3,4,2))
would use
to set up
by
by
array in
. If the size of
is exactly 24 the result is the same as
> Z <- h ; dim(Z) <- c(3,4,2)
However if
is shorter than 24, its values are recycled from the
beginning again to make it up to size 24 (see
Mixed vector and array arithmetic. The recycling rule
but
dim(h) <- c(3,4,2)
would signal an error about mismatching
length.
As an extreme but common example
> Z <- array(0, c(3,4,2))
makes
an array of all zeros.
At this point
dim(Z)
stands for the dimension vector
c(3,4,2)
, and
Z[1:24]
stands for the data vector as it was
in
, and
Z[]
with an empty subscript or
with no
subscript stands for the entire array as an array.
Arrays may be used in arithmetic expressions and the result is an array
formed by element-by-element operations on the data vector. The
dim
attributes of operands generally need to be the same, and
this becomes the dimension vector of the result. So if
and
are all similar arrays, then
makes
a similar array with its data vector being the result of
the given element-by-element operations. However the precise rule
concerning mixed array and vector calculations has to be considered a
little more carefully.
5.4.1 Mixed vector and array arithmetic. The recycling rule
The precise rule affecting element by element mixed calculations with
vectors and arrays is somewhat quirky and hard to find in the
references. From experience we have found the following to be a reliable
guide.
The expression is scanned from left to right.
Any short vector operands are extended by recycling their values until
they match the size of any other operands.
As long as short vectors and arrays
only
are encountered, the
arrays must all have the same
dim
attribute or an error results.
Any vector operand longer than a matrix or array operand generates an error.
If array structures are present and no error or coercion to vector has
been precipitated, the result is an array structure with the common
dim
attribute of its array operands.
5.5 The outer product of two arrays
An important operation on arrays is the
outer product
. If
and
are two numeric arrays, their outer product is an
array whose dimension vector is obtained by concatenating their two
dimension vectors (order is important), and whose data vector is got by
forming all possible products of elements of the data vector of
with those of
. The outer product is formed by the special
operator
%o%
An alternative is
The multiplication function can be replaced by an arbitrary function of
two variables. For example if we wished to evaluate the function
f(x; y) = cos(y)/(1 + x^2)
over a regular grid of values with
- and
-coordinates
defined by the R vectors
and
respectively, we could
proceed as follows:
> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)
In particular the outer product of two ordinary vectors is a doubly
subscripted array (that is a matrix, of rank at most 1). Notice that
the outer product operator is of course non-commutative. Defining your
own R functions will be considered further in
Writing your own functions
An example: Determinants of 2 by 2 single-digit matrices
As an artificial but cute example, consider the determinants of
by
matrices
[a, b; c, d]
where each entry is a
non-negative integer in the range
0, 1, ..., 9
, that is a
digit.
The problem is to find the determinants,
ad - bc
, of all possible
matrices of this form and represent the frequency with which each value
occurs as a
high density
plot. This amounts to finding the
probability distribution of the determinant if each digit is chosen
independently and uniformly at random.
A neat way of doing this uses the
outer()
function twice:
> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(fr, xlab="Determinant", ylab="Frequency")
Notice that
plot()
here uses a histogram like plot method, because
it “sees” that
fr
is of class
"table"
The “obvious” way of doing this problem with
for
loops, to be
discussed in
Grouping, 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.
5.6 Generalized transpose of an array
The function
aperm(a, perm)
may be used to permute an array,
. The argument
perm
must be a permutation of the integers
{1, ..., k}
, where
is the number of subscripts in
. The result of the
function is an array of the same size as
but with old dimension
given by
perm[j]
becoming the new
-th dimension. The
easiest way to think of this operation is as a generalization of
transposition for matrices. Indeed if
is a matrix, (that is, a
doubly subscripted array) then
given by
is just the transpose of
. For this special case a simpler
function
t()
is available, so we could have used
B <- t(A)
5.7 Matrix facilities
As noted above, a matrix is just an array with two subscripts. However
it is such an important special case it needs a separate discussion.
R contains many operators and functions that are available only for
matrices. For example
t(X)
is the matrix transpose function, as
noted above. The functions
nrow(A)
and
ncol(A)
give the
number of rows and columns in the matrix
respectively.
5.7.1 Matrix multiplication
The operator
%*%
is used for matrix multiplication.
An
by
or
by
matrix may of course be
used as an
-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,
and
are square matrices of the same
size, then
is the matrix of element by element products and
is the matrix product. If
is a vector, then
is a quadratic form.
The function
crossprod()
forms “cross products”, meaning that
crossprod(X, y)
is the same as
t(X) %*% y
but the
operation is more efficient. If the second argument to
crossprod()
is omitted it is taken to be the same as the first.
The meaning of
diag()
depends on its argument.
diag(v)
where
is a vector, gives a diagonal matrix with elements of the
vector as the diagonal entries. On the other hand
diag(M)
, where
is a matrix, gives the vector of main diagonal entries of
. This is the same convention as that used for
diag()
in
MATLAB
. Also, somewhat confusingly, if
is a single
numeric value then
diag(k)
is the
by
identity
matrix!
5.7.2 Linear equations and inversion
Solving linear equations is the inverse of matrix multiplication.
When after
only
and
are given, the vector
is the
solution of that linear equation system. In R,
solves the system, returning
(up to some accuracy loss).
Note that in linear algebra, formally
x = A^{-1} %*% b
where
A^{-1}
denotes the
inverse
of
, which can be computed by
but rarely is needed. Numerically, it is both inefficient and
potentially unstable to compute
x <- solve(A) %*% b
instead of
solve(A,b)
The quadratic form
x %*% A^{-1} %*%
which is used in multivariate computations, should be computed by
something like
x %*% solve(A,x)
, rather
than computing the inverse of
5.7.3 Eigenvalues and eigenvectors
The function
eigen(Sm)
calculates the eigenvalues and
eigenvectors of a symmetric matrix
Sm
. The result of this
function is a list of two components named
values
and
vectors
. The assignment
will assign this list to
ev
. Then
ev$val
is the vector of
eigenvalues of
Sm
and
ev$vec
is the matrix of
corresponding eigenvectors. Had we only needed the eigenvalues we could
have used the assignment:
> evals <- eigen(Sm)$values
evals
now holds the vector of eigenvalues and the second
component is discarded. If the expression
is used by itself as a command the two components are printed, with
their names. For large matrices it is better to avoid computing the
eigenvectors if they are not needed by using the expression
> evals <- eigen(Sm, only.values = TRUE)$values
5.7.4 Singular value decomposition and determinants
The function
svd(M)
takes an arbitrary matrix argument,
and calculates the singular value decomposition of
. This
consists of a matrix of orthonormal columns
with the same
column space as
, a second matrix of orthonormal columns
whose column space is the row space of
and a diagonal
matrix of positive entries
such that
M = U %*% D %*%
t(V)
is actually returned as a vector of the diagonal
elements. The result of
svd(M)
is actually a list of three
components named
and
, with evident meanings.
If
is in fact square, then, it is not hard to see that
> absdetM <- prod(svd(M)$d)
calculates the absolute value of the determinant of
. If this
calculation were needed often with a variety of matrices it could be
defined as an R function
> absdet <- function(M) prod(svd(M)$d)
after which we could use
absdet()
as just another R function.
As a further trivial but potentially useful example, you might like to
consider writing a function, say
tr()
, to calculate the trace of
a square matrix. [Hint: You will not need to use an explicit loop.
Look again at the
diag()
function.]
R has a builtin function
det
to calculate a determinant,
including the sign, and another,
determinant
, to give the sign
and modulus (optionally on log scale),
5.7.5 Least squares fitting and the QR decomposition
The function
lsfit()
returns a list giving results of a least
squares fitting procedure. An assignment such as
gives the results of a least squares fit where
is the vector of
observations and
is the design matrix. See the help facility
for more details, and also for the follow-up function
ls.diag()
for, among other things, regression diagnostics. Note that a grand mean
term is automatically included and need not be included explicitly as a
column of
. Further note that you almost always will prefer
using
lm(.)
(see
Linear models
) to
lsfit()
for
regression modelling.
Another closely related function is
qr()
and its allies.
Consider the following assignments
> Xplus <- qr(X)
> b <- qr.coef(Xplus, y)
> fit <- qr.fitted(Xplus, y)
> res <- qr.resid(Xplus, y)
These compute the orthogonal projection of
onto the range of
in
fit
, the projection onto the orthogonal complement in
res
and the coefficient vector for the projection in
that is,
is essentially the result of the
MATLAB
‘backslash’ operator.
It is not assumed that
has full column rank. Redundancies will
be discovered and removed as they are found.
This alternative is the older, low-level way to perform least squares
calculations. Although still useful in some contexts, it would now
generally be replaced by the statistical models features, as will be
discussed in
Statistical models in R
5.8 Forming partitioned matrices,
cbind()
and
rbind()
As we have already seen informally, matrices can be built up from other
vectors and matrices by the functions
cbind()
and
rbind()
Roughly
cbind()
forms matrices by binding together matrices
horizontally, or column-wise, and
rbind()
vertically, or
row-wise.
In the assignment
> X <- cbind(
arg_1
arg_2
arg_3
, ...)
the arguments to
cbind()
must be either vectors of any length, or
matrices with the same column size, that is the same number of rows.
The result is a matrix with the concatenated arguments
arg_1
arg_2
, … forming the columns.
If some of the arguments to
cbind()
are vectors they may be
shorter than the column size of any matrices present, in which case they
are cyclically extended to match the matrix column size (or the length
of the longest vector if no matrices are given).
The function
rbind()
does the corresponding operation for rows.
In this case any vector argument, possibly cyclically extended, are of
course taken as row vectors.
Suppose
X1
and
X2
have the same number of rows. To
combine these by columns into a matrix
, together with an
initial column of
s we can use
The result of
rbind()
or
cbind()
always has matrix status.
Hence
cbind(x)
and
rbind(x)
are possibly the simplest ways
explicitly to allow the vector
to be treated as a column or row
matrix respectively.
5.9 The concatenation function,
c()
, with arrays
It should be noted that whereas
cbind()
and
rbind()
are
concatenation functions that respect
dim
attributes, the basic
c()
function does not, but rather clears numeric objects of all
dim
and
dimnames
attributes. This is occasionally useful
in its own right.
The official way to coerce an array back to a simple vector object is to
use
as.vector()
However a similar result can be achieved by using
c()
with just
one argument, simply for this side-effect:
There are slight differences between the two, but ultimately the choice
between them is largely a matter of style (with the former being
preferable).
5.10 Frequency tables from factors
Recall that a factor defines a partition into groups. Similarly a pair
of factors defines a two way cross classification, and so on.
The function
table()
allows frequency tables to be calculated
from equal length factors. If there are
factor arguments,
the result is a
-way array of frequencies.
Suppose, for example, that
statef
is a factor giving the state
code for each entry in a data vector. The assignment
> statefr <- table(statef)
gives in
statefr
a table of frequencies of each state in the
sample. The frequencies are ordered and labelled by the
levels
attribute of the factor. This simple case is equivalent to, but more
convenient than,
> statefr <- tapply(statef, statef, length)
Further suppose that
incomef
is a factor giving a suitably
defined “income class” for each entry in the data vector, for example
with the
cut()
function:
> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef
Then to calculate a two-way table of frequencies:
> table(incomef,statef)
statef
incomef act nsw nt qld sa tas vic wa
(35,45] 1 1 0 1 0 0 1 0
(45,55] 1 1 1 1 2 0 1 3
(55,65] 0 3 1 3 2 2 2 1
(65,75] 0 1 0 0 0 0 1 0
Extension to higher-way frequency tables is immediate.
6 Lists and data frames
6.1 Lists
An R
list
is an object consisting of an ordered collection of
objects known as its
components
There is no particular need for the components to be of the same mode or
type, and, for example, a list could consist of a numeric vector, a
logical value, a matrix, a complex vector, a character array, a
function, and so on. Here is a simple example of how to make a list:
> Lst <- list(name="Fred", wife="Mary", no.children=3,
child.ages=c(4,7,9))
Components are always
numbered
and may always be referred to as
such. Thus if
Lst
is the name of a list with four components,
these may be individually referred to as
Lst[[1]]
Lst[[2]]
Lst[[3]]
and
Lst[[4]]
. If, further,
Lst[[4]]
is a vector subscripted array then
Lst[[4]][1]
is
its first entry.
If
Lst
is a list, then the function
length(Lst)
gives the
number of (top level) components it has.
Components of lists may also be
named
, and in this case the
component may be referred to either by giving the component name as a
character string in place of the number in double square brackets, or,
more conveniently, by giving an expression of the form
for the same thing.
This is a very useful convention as it makes it easier to get the right
component if you forget the number.
So in the simple example given above:
Lst$name
is the same as
Lst[[1]]
and is the string
"Fred"
Lst$wife
is the same as
Lst[[2]]
and is the string
"Mary"
Lst$child.ages[1]
is the same as
Lst[[4]][1]
and is the
number
Additionally, one can also use the names of the list components in
double square brackets, i.e.,
Lst[["name"]]
is the same as
Lst$name
. This is especially useful, when the name of the
component to be extracted is stored in another variable as in
It is very important to distinguish
Lst[[1]]
from
Lst[1]
[[
]]
’ is the operator used to select a single
element, whereas ‘
’ is a general subscripting
operator. Thus the former is the
first object in the list
Lst
, and if it is a named list the name is
not
included.
The latter is a
sublist of the list
Lst
consisting of the
first entry only. If it is a named list, the names are transferred to
the sublist.
The names of components may be abbreviated down to the minimum number of
letters needed to identify them uniquely. Thus
Lst$coefficients
may be minimally specified as
Lst$coe
and
Lst$covariance
as
Lst$cov
The vector of names is in fact simply an attribute of the list like any
other and may be handled as such. Other structures besides lists may,
of course, similarly be given a
names
attribute also.
6.2 Constructing and modifying lists
New lists may be formed from existing objects by the function
list()
. An assignment of the form
> Lst <- list(
name_1
object_1
...
name_m
object_m
sets up a list
Lst
of
components using
object_1
…,
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
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
> Lst[5] <- list(matrix=Mat)
6.2.1 Concatenating lists
When the concatenation function
c()
is given list arguments, the
result is an object of mode list also, whose components are those of the
argument lists joined together in sequence.
> list.ABC <- c(list.A, list.B, list.C)
Recall that with vector objects as arguments the concatenation function
similarly joined together all arguments into a single vector structure.
In this case all other attributes, such as
dim
attributes, are
discarded.
6.3 Data frames
data frame
is a list with class
"data.frame"
. There are
restrictions on lists that may be made into data frames, namely
The components must be vectors (numeric, character, or logical),
factors, numeric matrices, lists, or other data frames.
Matrices, lists, and data frames provide as many variables to the new
data frame as they have columns, elements, or variables, respectively.
Vector structures appearing as variables of the data frame must all have
the
same length
, and matrix structures must all have the
same number of rows
A data frame may for many purposes be regarded as a matrix with columns
possibly of differing modes and attributes. It may be displayed in
matrix form, and its rows and columns extracted using matrix indexing
conventions.
6.3.1 Making data frames
Objects satisfying the restrictions placed on the columns (components)
of a data frame may be used to form one using the function
data.frame
> accountants <- data.frame(home=statef, loot=incomes, shot=incomef)
A list whose components conform to the restrictions of a data frame may
be
coerced
into a data frame using the function
as.data.frame()
The simplest way to construct a data frame from scratch is to use the
read.table()
function to read an entire data frame from an
external file. This is discussed further in
Reading data from files
6.3.2
attach()
and
detach()
The
notation, such as
accountants$home
, for list
components is not always very convenient. A useful facility would be
somehow to make the components of a list or data frame temporarily
visible as variables under their component name, without the need to
quote the list name explicitly each time.
The
attach()
function takes a ‘database’ such as a list or data
frame as its argument. Thus suppose
lentils
is a
data frame with three variables
lentils$u
lentils$v
lentils$w
. The attach
places the data frame in the search path at position 2, and provided
there are no variables
or
in position 1,
and
are available as variables from the data
frame in their own right. At this point an assignment such as
does not replace the component
of the data frame, but rather
masks it with another variable
in the workspace at
position 1 on the search path. To make a permanent change to the
data frame itself, the simplest way is to resort once again to the
notation:
However the new value of component
is not visible until the
data frame is detached and attached again.
To detach a data frame, use the function
More precisely, this statement detaches from the search path the entity
currently at position 2. Thus in the present context the variables
and
would be no longer visible, except under
the list notation as
lentils$u
and so on. Entities at positions
greater than 2 on the search path can be detached by giving their number
to
detach
, but it is much safer to always use a name, for example
by
detach(lentils)
or
detach("lentils")
Note:
In R lists and data frames can only be attached at position 2 or
above, and what is attached is a
copy
of the original object.
You can alter the attached values
via
assign
, but the
original list or data frame is unchanged.
6.3.3 Working with data frames
A useful convention that allows you to work with many different problems
comfortably together in the same workspace is
gather together all variables for any well defined and separate problem
in a data frame under a suitably informative name;
when working with a problem attach the appropriate data frame at
position 2, and use the workspace at level 1 for
operational quantities and temporary variables;
before leaving a problem, add any variables you wish to keep for future
reference to the data frame using the
form of assignment, and
then
detach()
finally remove all unwanted variables from the workspace and
keep it as clean of left-over temporary variables as possible.
In this way it is quite simple to work with many problems in the same
directory, all of which have variables named
and
, for example.
6.3.4 Attaching arbitrary lists
attach()
is a generic function that allows not only directories
and data frames to be attached to the search path, but other classes of
object as well. In particular any object of mode
"list"
may be
attached in the same way:
Anything that has been attached can be detached by
detach
, by
position number or, preferably, by name.
6.3.5 Managing the search path
The function
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
> search()
[1] ".GlobalEnv" "Autoloads" "package:base"
where
.GlobalEnv
is the workspace.
After
lentils
is attached we have
> search()
[1] ".GlobalEnv" "lentils" "Autoloads" "package:base"
> ls(2)
[1] "u" "v" "w"
and as we see
ls
(or
objects
) can be used to examine the
contents of any position on the search path.
Finally, we detach the data frame and confirm it has been removed from
the search path.
> detach("lentils")
> search()
[1] ".GlobalEnv" "Autoloads" "package:base"
7 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 to fit in with the
requirements of R. Generally this is very simple.
If variables are to be held mainly in data frames, as we strongly
suggest they should be, an entire data frame can be read directly with
the
read.table()
function. There is also a more primitive input
function,
scan()
, that can be called directly.
For more details on importing data into R and also exporting data,
see
R Data Import/Export
7.1 The
read.table()
function
To read an entire data frame directly, the external file will normally
have a special form.
The first line of the file should have a
name
for each variable
in the data frame.
Each additional line of the file has as its first item a
row label
and the values for each variable.
If the file has one fewer item in its first line than in its second, this
arrangement is presumed to be in force. So the first few lines of a file
to be read as a data frame might look as follows.
Input file form with names and row labels:
Price Floor Area Rooms Age Cent.heat
01 52.00 111.0 830 5 6.2 no
02 54.75 128.0 710 5 7.5 no
03 57.50 101.0 1000 5 4.2 no
04 57.50 131.0 690 6 8.8 no
05 59.75 93.0 900 5 1.9 yes
...
By default numeric items (except row labels) are read as numeric
variables and non-numeric variables, such as
Cent.heat
in the
example, as character variables. This can be changed if necessary.
The function
read.table()
can then be used to read the data frame
directly
> HousePrice <- read.table("houses.data")
Often you will want to omit including the row labels directly and use the
default labels. In this case the file may omit the row label column as in
the following.
Input file form without row labels:
Price Floor Area Rooms Age Cent.heat
52.00 111.0 830 5 6.2 no
54.75 128.0 710 5 7.5 no
57.50 101.0 1000 5 4.2 no
57.50 131.0 690 6 8.8 no
59.75 93.0 900 5 1.9 yes
...
The data frame may then be read as
> HousePrice <- read.table("houses.data", header=TRUE)
where the
header=TRUE
option specifies that the first line is a
line of headings, and hence, by implication from the form of the file,
that no explicit row labels are given.
7.2 The
scan()
function
Suppose the data vectors are of equal length and are to be read in
parallel. Further suppose that there are three vectors, the first of
mode character and the remaining two of mode numeric, and the file is
input.dat
. The first step is to use
scan()
to read in the
three vectors as a list, as follows
> inp <- scan("input.dat", list("",0,0))
The second argument is a dummy list structure that establishes the mode
of the three vectors to be read. The result, held in
inp
, is a
list whose components are the three vectors read in. To separate the
data items into three separate vectors, use assignments like
> label <- inp[[1]]; x <- inp[[2]]; y <- inp[[3]]
More conveniently, the dummy list can have named components, in which
case the names can be used to access the vectors read in. For example
> inp <- scan("input.dat", list(id="", x=0, y=0))
If you wish to access the variables separately they may either be
re-assigned to variables in the working frame:
> label <- inp$id; x <- inp$x; y <- inp$y
or the list may be attached at position 2 of the search path
(see
Attaching arbitrary lists
).
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.
> X <- matrix(scan("light.dat", 0), ncol=5, byrow=TRUE)
There are more elaborate input facilities available and these are
detailed in the manuals.
7.3 Accessing builtin datasets
Around 100 datasets are supplied with R (in package
datasets
),
and others are available in packages (including the recommended packages
supplied with R). To see the list of datasets currently available
use
All the datasets supplied with R are available directly by name.
However, many packages still use the obsolete convention in which
data
was also used to load datasets into R, for example
and this can still be used with the standard packages (as in this
example). In most cases this will load an R object of the same name.
However, in a few cases it loads several objects, so see the on-line
help for the object to see what to expect.
7.3.1 Loading data from other R packages
To access data from a particular package, use the
package
argument, for example
data(package="rpart")
data(Puromycin, package="datasets")
If a package has been attached by
library
, its datasets are
automatically included in the search.
User-contributed packages can be a rich source of datasets.
7.4 Editing data
When invoked on a data frame or matrix,
edit
brings up a separate
spreadsheet-like environment for editing. This is useful for making
small changes once a data set has been read. The command
will allow you to edit your data set
xold
, and on completion the
changed object is assigned to
xnew
. If you want to alter the
original dataset
xold
, the simplest way is to use
fix(xold)
, which is equivalent to
xold <- edit(xold)
Use
> xnew <- edit(data.frame())
to enter new data via the spreadsheet interface.
8 Probability distributions
8.1 R as a set of statistical tables
One convenient use of R is to provide a comprehensive set of
statistical tables. Functions are provided to evaluate the cumulative
distribution function P(X <= x),
the probability density function and the quantile function (given
, the smallest
such that P(X <= x) > q),
and to simulate from the distribution.
Distribution
R name
additional arguments
beta
beta
shape1, shape2, ncp
binomial
binom
size, prob
Cauchy
cauchy
location, scale
chi-squared
chisq
df, ncp
exponential
exp
rate
df1, df2, ncp
gamma
gamma
shape, scale
geometric
geom
prob
hypergeometric
hyper
m, n, k
log-normal
lnorm
meanlog, sdlog
logistic
logis
location, scale
negative binomial
nbinom
size, prob
normal
norm
mean, sd
Poisson
pois
lambda
signed rank
signrank
Student’s t
df, ncp
uniform
unif
min, max
Weibull
weibull
shape, scale
Wilcoxon
wilcox
m, n
Prefix the name given here by ‘
’ for the density, ‘
’ for the
CDF, ‘
’ for the quantile function and ‘
’ for simulation
andom deviates). The first argument is
for
xxx
for
xxx
for
xxx
and
for
xxx
(except for
rhyper
rsignrank
and
rwilcox
, for which it is
nn
). In not quite all cases is the non-centrality parameter
ncp
currently available: see the on-line help for details.
The
xxx
and
xxx
functions all have logical
arguments
lower.tail
and
log.p
and the
xxx
ones have
log
. This allows, e.g., getting the cumulative (or
“integrated”)
hazard
function, H(t) = - log(1 - F(t)), by
- p
xxx
(t, ..., lower.tail = FALSE, log.p = TRUE)
or more accurate log-likelihoods (by
xxx
(..., log =
TRUE)
), directly.
In addition there are functions
ptukey
and
qtukey
for the
distribution of the studentized range of samples from a normal
distribution, and
dmultinom
and
rmultinom
for the
multinomial distribution. Further distributions are available in
contributed packages, notably
SuppDists
Here are some examples
> ##
2-tailed p-value for t distribution
> 2*pt(-2.43, df = 13)
> ##
upper 1% point for an F(2, 7) distribution
> qf(0.01, 2, 7, lower.tail = FALSE)
See the on-line help on
RNG
for how random-number generation is
done in R.
8.2 Examining the distribution of a set of data
Given a (univariate) set of data we can examine its distribution in a
large number of ways. The simplest is to examine the numbers. Two
slightly different summaries are given by
summary
and
fivenum
and a display of the numbers by
stem
(a “stem and leaf” plot).
> attach(faithful)
> summary(eruptions)
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.600 2.163 4.000 3.488 4.454 5.100
> fivenum(eruptions)
[1] 1.6000 2.1585 4.0000 4.4585 5.1000
> stem(eruptions)

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

16 | 070355555588
18 | 000022233333335577777777888822335777888
20 | 00002223378800035778
22 | 0002335578023578
24 | 00228
26 | 23
28 | 080
30 | 7
32 | 2337
34 | 250077
36 | 0000823577
38 | 2333335582225577
40 | 0000003357788888002233555577778
42 | 03335555778800233333555577778
44 | 02222335557780000000023333357778888
46 | 0000233357700000023578
48 | 00000022335800333
50 | 0370
A stem-and-leaf plot is like a histogram, and R has a function
hist
to plot histograms.
> hist(eruptions)
##
make the bins smaller, make a plot of density
> hist(eruptions, seq(1.6, 5.2, 0.2), prob=TRUE)
> lines(density(eruptions, bw=0.1))
> rug(eruptions) #
show the actual data points
More elegant density plots can be made by
density
, and we added a
line produced by
density
in this example. The bandwidth
bw
was chosen by trial-and-error as the default gives too much
smoothing (it usually does for “interesting” densities). (Better
automated methods of bandwidth choice are available, and in this example
bw = "SJ"
gives a good result.)
We can plot the empirical cumulative distribution function by using the
function
ecdf
> plot(ecdf(eruptions), do.points=FALSE, verticals=TRUE)
This distribution is obviously far from any standard distribution.
How about the right-hand mode, say eruptions of longer than 3 minutes?
Let us fit a normal distribution and overlay the fitted CDF.
> 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)
Quantile-quantile (Q-Q) plots can help us examine this more carefully.
par(pty="s") # arrange for a square figure region
qqnorm(long); qqline(long)
This shows a reasonable fit but a shorter right tail than one would
expect from a normal distribution. Let us compare this with some
simulated data from a
distribution
x <- rt(250, df = 5)
qqnorm(x); qqline(x)
which will usually (if it is a random sample) show longer tails than
expected for a normal. We can make a Q-Q plot against the generating
distribution by
qqplot(qt(ppoints(250), df = 5), x, xlab = "Q-Q plot for t dsn")
qqline(x)
Finally, we might want a more formal test of agreement with normality
(or not). R provides the Shapiro-Wilk test
> shapiro.test(long)

Shapiro-Wilk normality test

data: long
W = 0.9793, p-value = 0.01052
and the Kolmogorov-Smirnov test
> ks.test(long, "pnorm", mean = mean(long), sd = sqrt(var(long)))

One-sample Kolmogorov-Smirnov test

data: long
D = 0.0661, p-value = 0.4284
alternative hypothesis: two.sided
(Note that the distribution theory is not valid here as we
have estimated the parameters of the normal distribution from the same
sample.)
8.3 One- and two-sample tests
So far we have compared a single sample to a normal distribution. A
much more common operation is to compare aspects of two samples. Note
that in R, all “classical” tests including the ones used below are
in package
stats
which is normally loaded.
Consider the following sets of data on the latent heat of the fusion of
ice (
cal/gm
) from Rice (1995, p.490)
Method A: 79.98 80.04 80.02 80.04 80.03 80.03 80.04 79.97
80.05 80.03 80.02 80.00 80.02
Method B: 80.02 79.94 79.98 79.97 79.97 80.03 79.95 79.97
Boxplots provide a simple graphical comparison of the two samples.
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)
The plot indicates that the first group tends to give higher results than
the second.
To test for the equality of the means of the two examples, we can use
an
unpaired
-test by
> t.test(A, B)

Welch Two Sample t-test

data: A and B
t = 3.2499, df = 12.027, p-value = 0.00694
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
0.01385526 0.07018320
sample estimates:
mean of x mean of y
80.02077 79.97875
which does indicate a significant difference, assuming normality. By
default the R function does not assume equality of variances in the
two samples.
We can use the F test to test for equality in the variances,
provided that the two samples are from normal populations.
> var.test(A, B)

F test to compare two variances

data: A and B
F = 0.5837, num df = 12, denom df = 7, p-value = 0.3938
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.1251097 2.1052687
sample estimates:
ratio of variances
0.5837405
which shows no evidence of a significant difference, and so we can use
the classical
-test that assumes equality of the variances.
> t.test(A, B, var.equal=TRUE)

Two Sample t-test

data: A and B
t = 3.4722, df = 19, p-value = 0.002551
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
0.01669058 0.06734788
sample estimates:
mean of x mean of y
80.02077 79.97875
All these tests assume normality of the two samples. The two-sample
Wilcoxon (or Mann-Whitney) test only assumes a common continuous
distribution under the null hypothesis.
> wilcox.test(A, B)

Wilcoxon rank sum test with continuity correction

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

Warning message:
Cannot compute exact p-value with ties in: wilcox.test(A, B)
Note the warning: there are several ties in each sample, which suggests
strongly that these data are from a discrete distribution (probably due
to rounding).
There are several ways to compare graphically the two samples. We have
already seen a pair of boxplots. The following
> plot(ecdf(A), do.points=FALSE, verticals=TRUE, xlim=range(A, B))
> plot(ecdf(B), do.points=FALSE, verticals=TRUE, add=TRUE)
will show the two empirical CDFs, and
qqplot
will perform a Q-Q
plot of the two samples. The Kolmogorov-Smirnov test is of the maximal
vertical distance between the two
ecdf
s, assuming a common continuous
distribution:
> 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)
10 Writing your own functions
As we have seen informally along the way, the R language allows the
user to create objects of mode
function
. These are true R
functions that are stored in a special internal form and may be used in
further expressions and so on. In the process, the language gains
enormously in power, convenience and elegance, and learning to write
useful functions is one of the main ways to make your use of R
comfortable and productive.
It should be emphasized that most of the functions supplied as part of
the R system, such as
mean()
var()
postscript()
and so on, are themselves written in R and thus
do not differ materially from user written functions.
A function is defined by an assignment of the form
name
<- function(
arg_1
arg_2
, ...)
expression
The
expression
is an R expression, (usually a grouped
expression), that uses the arguments,
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
name
expr_1
expr_2
, …)
and may occur
anywhere a function call is legitimate.
10.1 Simple examples
As a first example, consider a function to calculate the two sample
-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:
> twosam <- function(y1, y2) {
n1 <- length(y1); n2 <- length(y2)
yb1 <- mean(y1); yb2 <- mean(y2)
s1 <- var(y1); s2 <- var(y2)
s <- ((n1-1)*s1 + (n2-1)*s2)/(n1+n2-2)
tst <- (yb1 - yb2)/sqrt(s*(1/n1 + 1/n2))
tst
With this function defined, you could perform two sample
-tests
using a call such as
> tstat <- twosam(data$male, data$female); tstat
As a second example, consider a function to emulate directly the
MATLAB
backslash command, which returns the coefficients of the
orthogonal projection of the vector
onto the column space of
the matrix,
. (This is ordinarily called the least squares
estimate of the regression coefficients.) This would ordinarily be
done with the
qr()
function; however this is sometimes a bit
tricky to use directly and it pays to have a simple function such as the
following to use it safely.
Thus given a
by
vector
and an
by
matrix
then
X \ y
is defined as
(X’X)^{-}X’y,
where
(X’X)^{-}
is a generalized inverse of
X'X
> bslash <- function(X, y) {
X <- qr(X)
qr.coef(X, y)
After this object is created it may be used in statements such as
> regcoeff <- bslash(Xmat, yvar)
and so on.
The classical R function
lsfit()
does this job quite well, and
more. It in turn uses the functions
qr()
and
qr.coef()
in the slightly counterintuitive way above to do this part of the
calculation. Hence there is probably some value in having just this
part isolated in a simple to use function if it is going to be in
frequent use. If so, we may wish to make it a matrix binary operator
for even more convenient use.
10.2 Defining new binary operators
Had we given the
bslash()
function a different name, namely one of
the form
it could have been used as a
binary operator
in expressions
rather than in function form. Suppose, for example, we choose
for the internal character. The function definition would then start as
> "%!%" <- function(X, y) { ... }
(Note the use of quote marks.) The function could then be used as
X %!% y
. (The backslash symbol itself is not a convenient choice
as it presents special problems in this context.)
The matrix multiplication operator,
%*%
, and the outer product
matrix operator
%o%
are other examples of binary operators
defined in this way.
10.3 Named arguments and defaults
As first noted in
Generating regular sequences
, if arguments to
called functions are given in the “
name
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
fun1
defined by
> fun1 <- function(data, data.frame, graph, limit) {
[function body omitted]
then the function may be invoked in several ways, for 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)
are all equivalent.
In many cases arguments can be given commonly appropriate default
values, in which case they may be omitted altogether from the call when
the defaults are appropriate. For example, if
fun1
were defined
as
> fun1 <- function(data, data.frame, graph=TRUE, limit=20) { ... }
it could be called as
which is now equivalent to the three cases above, or as
> ans <- fun1(d, df, limit=10)
which changes one of the defaults.
It is important to note that defaults may be arbitrary expressions, even
involving other arguments to the same function; they are not restricted
to be constants as in our simple example here.
10.4 The ‘
’ argument
Another frequent requirement is to allow one function to pass on
argument settings to another. For example many graphics functions use
the function
par()
and functions like
plot()
allow the
user to pass on graphical parameters to
par()
to control the
graphical output. (See
Permanent changes: The
par()
function
, for more details on the
par()
function.) This can be done by including an extra
argument, literally ‘
’, of the function, which may then be
passed on. An outline example is given below.
fun1 <- function(data, data.frame, graph=TRUE, limit=20, ...) {
[omitted statements]
if (graph)
par(pch="*", ...)
[more omissions]
Less frequently, a function will need to refer to components of
’. The expression
list(...)
evaluates all such
arguments and returns them in a named list, while
..1
..2
, etc. evaluate them one at a time, with ‘
..n
returning the n-th unmatched argument.
10.5 Assignments within functions
Note that
any ordinary assignments done within the function are
local and temporary and are lost after exit from the function
. Thus
the assignment
X <- qr(X)
does not affect the value of the
argument in the calling program.
To understand completely the rules governing the scope of R assignments
the reader needs to be familiar with the notion of an evaluation
frame
. This is a somewhat advanced, though hardly difficult,
topic and is not covered further here.
If global and permanent assignments are intended within a function, then
either the ‘superassignment’ operator,
<<-
or the function
assign()
can be used. See the
help
document for details.
10.6 More advanced examples
10.6.1 Efficiency factors in block designs
As a more complete, if a little pedestrian, example of a function,
consider finding the efficiency factors for a block design. (Some
aspects of this problem have already been discussed in
Index matrices
.)
A block design is defined by two factors, say
blocks
levels) and
varieties
levels). If
and
are the
by
and
by
replications
and
block size
matrices, respectively, and
is the
by
incidence matrix, then the
efficiency factors are defined as the eigenvalues of the matrix
E = I_v - R^{-1/2}N’K^{-1}NR^{-1/2} = I_v - A’A,
where A = K^{-1/2}NR^{-1/2}.
One way to write the function is given below.
> bdeff <- function(blocks, varieties) {
blocks <- as.factor(blocks) #
minor safety move
b <- length(levels(blocks))
varieties <- as.factor(varieties) #
minor safety move
v <- length(levels(varieties))
K <- as.vector(table(blocks)) #
remove dim attr
R <- as.vector(table(varieties)) #
remove dim attr
N <- table(blocks, varieties)
A <- 1/sqrt(K) * N * rep(1/sqrt(R), rep(b, v))
sv <- svd(A)
list(eff=1 - sv$d^2, blockcv=sv$u, varietycv=sv$v)
It is numerically slightly better to work with the singular value
decomposition on this occasion rather than the eigenvalue routines.
The result of the function is a list giving not only the efficiency
factors as the first component, but also the block and variety canonical
contrasts, since sometimes these give additional useful qualitative
information.
10.6.2 Dropping all names in a printed array
For printing purposes with large matrices or arrays, it is often useful
to print them in close block form without the array names or numbers.
Removing the
dimnames
attribute will not achieve this effect, but
rather the array must be given a
dimnames
attribute consisting of
empty strings. For example to print a matrix,
> temp <- X
> dimnames(temp) <- list(rep("", nrow(X)), rep("", ncol(X)))
> temp; rm(temp)
This can be much more conveniently done using a function,
no.dimnames()
, shown below, as a “wrap around” to achieve the
same result. It also illustrates how some effective and useful user
functions can be quite short.
no.dimnames <- function(a) {
##
Remove all dimension names from an array for compact printing.
d <- list()
l <- 0
for(i in dim(a)) {
d[[l <- l + 1]] <- rep("", i)
dimnames(a) <- d
With this function defined, an array may be printed in close format
using
This is particularly useful for large integer arrays, where patterns are
the real interest rather than the values.
10.6.3 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.
area <- function(f, a, b, eps = 1.0e-06, lim = 10) {
fun1 <- function(f, a, b, fa, fb, a0, eps, lim, fun) {
##
function ‘fun1’ is only visible inside ‘area’
d <- (a + b)/2
h <- (b - a)/4
fd <- f(d)
a1 <- h * (fa + fd)
a2 <- h * (fd + fb)
if(abs(a0 - a1 - a2) < eps || lim == 0)
return(a1 + a2)
else {
return(fun(f, a, d, fa, fd, a1, eps, lim - 1, fun) +
fun(f, d, b, fd, fb, a2, eps, lim - 1, fun))
fa <- f(a)
fb <- f(b)
a0 <- ((fa + fb) * (b - a))/2
fun1(f, a, b, fa, fb, a0, eps, lim, fun1)
10.7 Scope
The discussion in this section is somewhat more technical than in other
parts of this document. However, it details one of the major differences
between
S-PLUS
and R.
The symbols which occur in the body of a function can be divided into
three classes; formal parameters, local variables and free variables.
The formal parameters of a function are those occurring in the argument
list of the function. Their values are determined by the process of
binding
the actual function arguments to the formal parameters.
Local variables are those whose values are determined by the evaluation
of expressions in the body of the functions. Variables which are not
formal parameters or local variables are called free variables. Free
variables become local variables if they are assigned to. Consider the
following function definition.
f <- function(x) {
y <- 2*x
print(x)
print(y)
print(z)
In this function,
is a formal parameter,
is a local
variable and
is a free variable.
In R the free variable bindings are resolved by first looking in the
environment in which the function was created. This is called
lexical scope
. First we define a function called
cube
cube <- function(n) {
sq <- function() n*n
n*sq()
The variable
in the function
sq
is not an argument to that
function. Therefore it is a free variable and the scoping rules must be
used to ascertain the value that is to be associated with it. Under static
scope (
S-PLUS
) the value is that associated with a global variable named
. Under lexical scope (R) it is the parameter to the function
cube
since that is the active binding for the variable
at
the time the function
sq
was defined. The difference between
evaluation in R and evaluation in
S-PLUS
is that
S-PLUS
looks for a
global variable called
while R first looks for a variable
called
in the environment created when
cube
was invoked.
##
first evaluation in S
S> cube(2)
Error in sq(): Object "n" not found
Dumped
S> n <- 3
S> cube(2)
[1] 18
##
then the same function evaluated in R
R> cube(2)
[1] 8
Lexical scope can also be used to give functions
mutable state
In the following example we show how R can be used to mimic a bank
account. A functioning bank account needs to have a balance or total, a
function for making withdrawals, a function for making deposits and a
function for stating the current balance. We achieve this by creating
the three functions within
account
and then returning a list
containing them. When
account
is invoked it takes a numerical
argument
total
and returns a list containing the three functions.
Because these functions are defined in an environment which contains
total
, they will have access to its value.
The special assignment operator,
<<-
is used to change the value associated with
total
. This operator
looks back in enclosing environments for an environment that contains
the symbol
total
and when it finds such an environment it
replaces the value, in that environment, with the value of right hand
side. If the global or top-level environment is reached without finding
the symbol
total
then that variable is created and assigned to
there. For most users
<<-
creates a global variable and assigns
the value of the right hand side to it. Only when
<<-
has
been used in a function that was returned as the value of another
function will the special behavior described here occur.
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$withdraw(500)
10.8 Customizing the environment
Users can customize their environment in several different ways. There
is a site initialization file and every directory can have its own
special initialization file. Finally, the special functions
.First
and
.Last
can be used.
The location of the site initialization file is taken from the value of
the
R_PROFILE
environment variable. If that variable is unset,
the file
Rprofile.site
in the R home subdirectory
etc
is
used. This file should contain the commands that you want to execute
every time R is started under your system. A second, personal,
profile file named
.Rprofile
can be placed in any directory. If R is invoked in that
directory then that file will be sourced. This file gives individual
users control over their workspace and allows for different startup
procedures in different working directories. If no
.Rprofile
file is found in the startup directory, then R looks for a
.Rprofile
file in the user’s home directory and uses that (if it
exists). If the environment variable
R_PROFILE_USER
is set, the
file it points to is used instead of the
.Rprofile
files.
Any function named
.First()
in either of the two profile files or
in the
.RData
image has a special status. It is automatically
performed at the beginning of an R session and may be used to
initialize the environment. For example, the definition in the example
below alters the prompt to
and sets up various other useful
things that can then be taken for granted in the rest of the session.
Thus, the sequence in which files are executed is,
Rprofile.site
the user profile,
.RData
and then
.First()
. A definition
in later files will mask definitions in earlier files.
> .First <- function() {
options(prompt="$ ", continue="+\t") #
is the prompt
options(digits=5, length=999) #
custom numbers and printout
x11() #
for graphics
par(pch = "+") #
plotting character
source(file.path(Sys.getenv("HOME"), "R", "mystuff.R"))
my personal functions
library(MASS) #
attach a package
Similarly a function
.Last()
, if defined, is (normally) executed
at the very end of the session. An example is given below.
> .Last <- function() {
graphics.off() #
a small safety measure.
cat(paste(date(),"\nAdios\n")) #
Is it time for lunch?
10.9 Classes, generic functions and object orientation
The class of an object determines how it will be treated by what are
known as
generic
functions. Put the other way round, a generic
function performs a task or action on its arguments
specific to
the class of the argument itself
. If the argument lacks any
class
attribute, or has a class not catered for specifically by the generic
function in question, there is always a
default action
provided.
An example makes things clearer. The class mechanism offers the user
the facility of designing and writing generic functions for special
purposes. Among the other generic functions are
plot()
for
displaying objects graphically,
summary()
for summarizing
analyses of various types, and
anova()
for comparing statistical
models.
The number of generic functions that can treat a class in a specific way
can be quite large. For example, the functions that can accommodate in
some fashion objects of class
"data.frame"
include
[ [[<- any as.matrix
[<- mean plot summary
A currently complete list can be got by using the
methods()
function:
> methods(class="data.frame")
Conversely the number of classes a generic function can handle can also
be quite large. For example the
plot()
function has a default
method and variants for objects of classes
"data.frame"
"density"
"factor"
, and more. A complete list can be got
again by using the
methods()
function:
For many generic functions the function body is quite short, for example
> coef
function (object, ...)
UseMethod("coef")
The presence of
UseMethod
indicates this is a generic function.
To see what methods are available we can use
methods()
> methods(coef)
[1] coef.aov* coef.Arima* coef.default* coef.listof* coef.maov*
[6] coef.nls*
see '?methods' for accessing help and source code
In this example there are six methods, none of which can be seen by
typing its name (as indicated by the asterisk). We can read these by either of
> getAnywhere("coef.aov")
A single object matching 'coef.aov' was found
It was found in the following places
registered S3 method for coef from namespace stats
namespace:stats
with value

function (object, complete = FALSE, ...)
cf <- object$coefficients
if (complete)
cf
else cf[!is.na(cf)]

> getS3method("coef", "aov")
function (object, complete = FALSE, ...)
cf <- object$coefficients
if (complete)
cf
else cf[!is.na(cf)]
A function named
gen
cl
will be invoked by the
generic
gen
for class
cl
, so do not name
functions in this style unless they are intended to be methods.
The reader is referred to the
R Language Definition
for a more
complete discussion of this mechanism.
11 Statistical models in R
This section presumes the reader has some familiarity with statistical
methodology, in particular with regression analysis and the analysis of
variance. Later we make some rather more ambitious presumptions, namely
that something is known about generalized linear models and nonlinear
regression.
The requirements for fitting statistical models are sufficiently well
defined to make it possible to construct general tools that apply in a
broad spectrum of problems.
R provides an interlocking suite of facilities that make fitting
statistical models very simple. As we mention in the introduction, the
basic output is minimal, and one needs to ask for the details by calling
extractor functions.
11.1 Defining statistical models; formulae
The template for a statistical model is a linear regression model with
independent, homoscedastic errors
y_i = sum_{j=0}^p beta_j x_{ij} + e_i,     i = 1, ..., n,
where the e_i are NID(0, sigma^2).
In matrix terms this would be written
where the
is the response vector,
is the
model
matrix
or
design matrix
and has columns
x_0, x_1, ..., x_p
the determining variables. Very often
x_0
will be a column of ones defining an
intercept
term.
Examples
Before giving a formal specification, a few examples may usefully set
the picture.
Suppose
x0
x1
x2
, … are
numeric variables,
is a matrix and
, … are factors. The following formulae on the left
side below specify statistical models as described on the right.
y ~ x
y ~ 1 + x
Both imply the same simple linear regression model of
on
. The first has an implicit intercept term, and the second an
explicit one.
y ~ 0 + x
y ~ -1 + x
y ~ x - 1
Simple linear regression of
on
through the origin
(that is, without an intercept term).
log(y) ~ x1 + x2
Multiple regression of the transformed variable,log(y),
on
x1
and
x2
(with an implicit intercept term).
y ~ poly(x,2)
y ~ 1 + x + I(x^2)
Polynomial regression of
on
of degree 2. The first
form uses orthogonal polynomials, and the second uses explicit powers,
as basis.
y ~ X + poly(x,2)
Multiple regression
with model matrix consisting of the matrix
as well as polynomial terms in
to degree 2.
y ~ A
Single classification analysis of variance model of
, with
classes determined by
y ~ A + x
Single classification analysis of covariance model of
, with
classes determined by
, and with covariate
y ~ A*B
y ~ A + B + A:B
y ~ B %in% A
y ~ A/B
Two factor non-additive model of
on
and
. The
first two specify the same crossed classification and the second two
specify the same nested classification. In abstract terms all four
specify the same model subspace.
y ~ (A + B + C)^2
y ~ A*B*C - A:B:C
Three factor experiment but with a model containing main effects and two
factor interactions only. Both formulae specify the same model.
y ~ A * x
y ~ A/x
y ~ A/(1 + x) - 1
Separate simple linear regression models of
on
within
the levels of
, with different codings. The last form produces
explicit estimates of as many different intercepts and slopes as there
are levels in
y ~ A*B + Error(C)
An experiment with two treatment factors,
and
, and
error strata determined by factor
. For example a split plot
experiment, with whole plots (and hence also subplots), determined by
factor
The operator
is used to define a
model formula
in R.
The form, for an ordinary linear model, is
response
op_1
term_1
op_2
term_2
op_3
term_3
...
where
response
is a vector or matrix, (or expression evaluating to a vector or matrix)
defining the response variable(s).
op_i
is an operator, either
or
, implying the inclusion or
exclusion of a term in the model, (the first is optional).
term_i
is either
a vector or matrix expression, or
a factor, or
formula expression
consisting of factors, vectors or matrices
connected by
formula operators
In all cases each term defines a collection of columns either to be
added to or removed from the model matrix. A
stands for an
intercept column and is by default included in the model matrix unless
explicitly removed.
The
formula operators
are similar in effect to the Wilkinson and
Rogers notation used by such programs as Glim and Genstat. One
inevitable change is that the operator ‘
’ becomes
’ since the period is a valid name character in R.
The notation is summarized below (based on Chambers & Hastie, 1992,
p.29):
is modeled as
M_1
M_2
Include
M_1
and
M_2
M_1
M_2
Include
M_1
leaving out terms of
M_2
M_1
M_2
The tensor product of
M_1
and
M_2
. If both terms are
factors, then the “subclasses” factor.
M_1
%in%
M_2
Similar to
M_1
M_2
, but with a different coding.
M_1
M_2
M_1
M_2
M_1
M_2
M_1
M_2
M_1
M_2
%in%
M_1
All terms in
together with “interactions” up to order
I(
Insulate
. Inside
all operators have their normal
arithmetic meaning, and that term appears in the model matrix.
Note that inside the parentheses that usually enclose function arguments
all operators have their normal arithmetic meaning. The function
I()
is an identity function used to allow terms in model formulae
to be defined using arithmetic operators.
Note particularly that the model formulae specify the
columns
of the model matrix
, the specification of the parameters being
implicit. This is not the case in other contexts, for example in
specifying nonlinear models.
11.1.1 Contrasts
We need at least some idea how the model formulae specify the columns of
the model matrix. This is easy if we have continuous variables, as each
provides one column of the model matrix (and the intercept will provide
a column of ones if included in the model).
What about a
-level factor
? The answer differs for
unordered and ordered factors. For
unordered
factors
k -
columns are generated for the indicators of the second, …,
-th levels of the factor. (Thus the implicit parameterization is
to contrast the response at each level with that at the first.) For
ordered
factors the
k - 1
columns are the orthogonal
polynomials on
1, ..., k
, omitting the constant term.
Although the answer is already complicated, it is not the whole story.
First, if the intercept is omitted in a model that contains a factor
term, the first such term is encoded into
columns giving the
indicators for all the levels. Second, the whole behavior can be
changed by the
options
setting for
contrasts
. The default
setting in R is
options(contrasts = c("contr.treatment", "contr.poly"))
The main reason for mentioning this is that R and S have
different defaults for unordered factors, S using Helmert
contrasts. So if you need to compare your results to those of a textbook
or paper which used
S-PLUS
, you will need to set
options(contrasts = c("contr.helmert", "contr.poly"))
This is a deliberate difference, as treatment contrasts (R’s default)
are thought easier for newcomers to interpret.
We have still not finished, as the contrast scheme to be used can be set
for each term in the model using the functions
contrasts
and
We have not yet considered interaction terms: these generate the
products of the columns introduced for their component terms.
Although the details are complicated, model formulae in R will
normally generate the models that an expert statistician would expect,
provided that marginality is preserved. Fitting, for example, a model
with an interaction but not the corresponding main effects will in
general lead to surprising results, and is for experts only.
11.2 Linear models
The basic function for fitting ordinary multiple models is
lm()
and a streamlined version of the call is as follows:
fitted.model
<- lm(
formula
, data =
data.frame
For example
> fm2 <- lm(y ~ x1 + x2, data = production)
would fit a multiple regression model of
on
x1
and
x2
(with implicit intercept term).
The important (but technically optional) parameter
data =
production
specifies that any variables needed to construct the model
should come first from the
production
data frame
This is the case regardless of whether data frame
production
has been attached on the search path or not
11.4 Analysis of variance and model comparison
The model fitting function
aov(
formula
data=
data.frame
operates at the simplest level in a very similar way to the function
lm()
, and most of the generic functions listed in the table in
Generic functions for extracting model information
apply.
It should be noted that in addition
aov()
allows an analysis of
models with multiple error strata such as split plot experiments, or
balanced incomplete block designs with recovery of inter-block
information. The model formula
response
mean.formula
+ Error(
strata.formula
specifies a multi-stratum experiment with error strata defined by the
strata.formula
. In the simplest case,
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:
> fm <- aov(yield ~ v + n*p*k + Error(farms/blocks), data=farm.data)
would typically be used to describe an experiment with mean model
v + n*p*k
and three error strata, namely “between farms”,
“within farms, between blocks” and “within blocks”.
11.4.1 ANOVA tables
Note also that the analysis of variance table (or tables) are for a
sequence of fitted models. The sums of squares shown are the decrease
in the residual sums of squares resulting from an inclusion of
that term
in the model at
that place
in the sequence.
Hence only for orthogonal experiments will the order of inclusion be
inconsequential.
For multistratum experiments the procedure is first to project the
response onto the error strata, again in sequence, and to fit the mean
model to each projection. For further details, see Chambers & Hastie
(1992).
A more flexible alternative to the default full ANOVA table is to
compare two or more models directly using the
anova()
function.
> anova(
fitted.model.1
fitted.model.2
, ...)
The display is then an ANOVA table showing the differences between the
fitted models when fitted in sequence. The fitted models being compared
would usually be an hierarchical sequence, of course. This does not
give different information to the default, but rather makes it easier to
comprehend and control.
11.5 Updating fitted models
The
update()
function is largely a convenience function that
allows a model to be fitted that differs from one previously fitted
usually by just a few additional or removed terms. Its form is
new.model
<- update(
old.model
new.formula
In the
new.formula
the special name consisting of a period,
’,
only, can be used to stand for “the corresponding part of the old model
formula”. For example,
> fm05 <- lm(y ~ x1 + x2 + x3 + x4 + x5, data = production)
> fm6 <- update(fm05, . ~ . + x6)
> smf6 <- update(fm6, sqrt(.) ~ .)
would fit a five variate multiple regression with variables (presumably)
from the data frame
production
, fit an additional model including
a sixth regressor variable, and fit a variant on the model where the
response had a square root transform applied.
Note especially that if the
data=
argument is specified on the
original call to the model fitting function, this information is passed on
through the fitted model object to
update()
and its allies.
The name ‘
’ can also be used in other contexts, but with slightly
different meaning. For example
> fmfull <- lm(y ~ . , data = production)
would fit a model with response
and regressor variables
all other variables in the data frame
production
Other functions for exploring incremental sequences of models are
add1()
drop1()
and
step()
The names of these give a good clue to their purpose, but for full
details see the on-line help.
11.6 Generalized linear models
Generalized linear modeling is a development of linear models to
accommodate both non-normal response distributions and transformations
to linearity in a clean and straightforward way. A generalized linear
model may be described in terms of the following sequence of
assumptions:
There is a response,
, of interest and stimulus variables
x_1, x_2, …,
whose values influence the distribution of the response.
The stimulus variables influence the distribution of
through
a single linear function, only
. This linear function is called
the
linear predictor
, and is usually written
eta = beta_1 x_1 + beta_2 x_2 + ... + beta_p x_p,
hence
x_i
has no influence on the distribution of
if and
only if beta_i is zero.
The distribution of
is of the form
f_Y(y; mu, phi)
= exp((A/phi) * (y lambda(mu) - gamma(lambda(mu))) + tau(y, phi))
where phi is a
scale parameter
(possibly known),
and is constant for all observations,
represents a prior
weight, assumed known but possibly varying with the observations, and
mu is the mean of
So it is assumed that the distribution of
is determined by its
mean and possibly a scale parameter as well.
The mean, mu, is a smooth invertible function of the linear
predictor:
mu = m(eta), eta = m^{-1}(mu) = ell(mu)
and this inverse function, ell(), is called the
link
function
These assumptions are loose enough to encompass a wide class of models
useful in statistical practice, but tight enough to allow the
development of a unified methodology of estimation and inference, at
least approximately. The reader is referred to any of the current
reference works on the subject for full details, such as McCullagh &
Nelder (1989) or Dobson (1990).
11.6.1 Families
The class of generalized linear models handled by facilities supplied in
R includes
gaussian
binomial
poisson
inverse gaussian
and
gamma
response distributions and also
quasi-likelihood
models where the response distribution is not
explicitly specified. In the latter case the
variance function
must be specified as a function of the mean, but in other cases this
function is implied by the response distribution.
Each response distribution admits a variety of link functions to connect
the mean with the linear predictor. Those automatically available are
shown in the following table:
Family name
Link functions
binomial
logit
probit
log
cloglog
gaussian
identity
log
inverse
Gamma
identity
inverse
log
inverse.gaussian
1/mu^2
identity
inverse
log
poisson
identity
log
sqrt
quasi
logit
probit
cloglog
identity
inverse
log
1/mu^2
sqrt
The combination of a response distribution, a link function and various
other pieces of information that are needed to carry out the modeling
exercise is called the
family
of the generalized linear model.
11.6.2 The
glm()
function
Since the distribution of the response depends on the stimulus variables
through a single linear function
only
, the same mechanism as was
used for linear models can still be used to specify the linear part of a
generalized model. The family has to be specified in a different way.
The R function to fit a generalized linear model is
glm()
which uses the form
fitted.model
<- glm(
formula
, family=
family.generator
, data=
data.frame
The only new feature is the
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
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
quasi
family, the variance function may also be specified in this way.
Some examples make the process clear.
The
gaussian
family
A call such as
> fm <- glm(y ~ x1 + x2, family = gaussian, data = sales)
achieves the same result as
> fm <- lm(y ~ x1+x2, data=sales)
but much less efficiently. Note how the gaussian family is not
automatically provided with a choice of links, so no parameter is
allowed. If a problem requires a gaussian family with a nonstandard
link, this can usually be achieved through the
quasi
family, as
we shall see later.
The
binomial
family
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:
Age:
20
35
45
55
70
No. tested:
50
50
50
50
50
No. blind:
17
26
37
44
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
is the number of blind at age
and
the
number tested, both models have the form
y ~ B(n, F(beta_0 + beta_1 x))
where for the probit case,
F(z) = Phi(z)
is the standard normal distribution function, and in the logit case
(the default),
F(z) = e^z/(1+e^z).
In both cases the LD50 is
LD50 = - beta_0/beta_1
that is, the point at which the argument of the distribution function is
zero.
The first step is to set the data up as a data frame
> kalythos <- data.frame(x = c(20,35,45,55,70), n = rep(50,5),
y = c(6,17,26,37,44))
To fit a binomial model using
glm()
there are three possibilities
for the response:
If the response is a
vector
it is assumed to hold
binary
data, and so must be a
0/1
vector.
If the response is a
two-column matrix
it is assumed that the
first column holds the number of successes for the trial and the second
holds the number of failures.
If the response is a
factor
, its first level is taken as failure
(0) and all other levels as ‘success’ (1).
Here we need the second of these conventions, so we add a matrix to our
data frame:
> kalythos$Ymat <- cbind(kalythos$y, kalythos$n - kalythos$y)
To fit the models we use
> fmp <- glm(Ymat ~ x, family = binomial(link=probit), data = kalythos)
> fml <- glm(Ymat ~ x, family = binomial, data = kalythos)
Since the logit link is the default the parameter may be omitted on the
second call. To see the results of each fit we could use
> summary(fmp)
> summary(fml)
Both models fit (all too) well. To find the LD50 estimate we can use a
simple function:
> ld50 <- function(b) -b[1]/b[2]
> ldp <- ld50(coef(fmp)); ldl <- ld50(coef(fml)); c(ldp, ldl)
The actual estimates from this data are 43.663 years and 43.601 years
respectively.
Poisson models
With the Poisson family the default link is the
log
, and in
practice the major use of this family is to fit surrogate Poisson
log-linear models to frequency data, whose actual distribution is often
multinomial. This is a large and important subject we will not discuss
further here. It even forms a major part of the use of non-gaussian
generalized models overall.
Occasionally genuinely Poisson data arises in practice and in the past
it was often analyzed as gaussian data after either a log or a
square-root transformation. As a graceful alternative to the latter, a
Poisson generalized linear model may be fitted as in the following
example:
> fmod <- glm(y ~ A + B + x, family = poisson(link=sqrt),
data = worm.counts)
Quasi-likelihood models
For all families the variance of the response will depend on the mean
and will have the scale parameter as a multiplier. The form of
dependence of the variance on the mean is a characteristic of the
response distribution; for example for the Poisson distribution
Var(y) = mu.
For quasi-likelihood estimation and inference the precise response
distribution is not specified, but rather only a link function and the
form of the variance function as it depends on the mean. Since
quasi-likelihood estimation uses formally identical techniques to those
for the gaussian distribution, this family provides a way of fitting
gaussian models with non-standard link functions or variance functions,
incidentally.
For example, consider fitting the non-linear regression
y = theta_1 z_1 / (z_2 - theta_2) + e
which may be written alternatively as
y = 1 / (beta_1 x_1 + beta_2 x_2) + e
where
x_1 = z_2/z_1, x_2 = -1/z_1, beta_1 = 1/theta_1, and beta_2 =
theta_2/theta_1.
Supposing a suitable data frame to be set up we could fit this
non-linear regression as
> nlfit <- glm(y ~ x1 + x2 - 1,
family = quasi(link=inverse, variance=constant),
data = biochem)
The reader is referred to the manual and the help document for further
information, as needed.
11.7 Nonlinear least squares and maximum likelihood models
Certain forms of nonlinear model can be fitted by Generalized Linear
Models (
glm()
). But in the majority of cases we have to approach
the nonlinear curve fitting problem as one of nonlinear optimization.
R’s nonlinear optimization routines are
optim()
nlm()
and
nlminb()
We seek the parameter values that minimize some index
of lack-of-fit, and they do this by trying out various parameter values
iteratively. Unlike linear regression for example, there is no
guarantee that the procedure will converge on satisfactory estimates.
All the methods require initial guesses about what parameter values to
try, and convergence may depend critically upon the quality of the
starting values.
11.7.1 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:
> x <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56,
1.10, 1.10)
> y <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200)
The fit criterion to be minimized is:
> fn <- function(p) sum((y - (p[1] * x)/(p[2] + x))^2)
In order to do the fit we need initial estimates of the parameters. One
way to find sensible starting values is to plot the data, guess some
parameter values, and superimpose the model curve using those values.
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 200 * xfit/(0.1 + xfit)
> lines(spline(xfit, yfit))
We could do better, but these starting values of 200 and 0.1 seem
adequate. Now do the fit:
> out <- nlm(fn, p = c(200, 0.1), hessian = TRUE)
After the fitting,
out$minimum
is the SSE, and
out$estimate
are the least squares estimates of the parameters.
To obtain the approximate standard errors (SE) of the estimates we do:
> sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))
The
which is subtracted in the line above represents the number
of parameters. A 95% confidence interval would be the parameter
estimate +/- 1.96 SE. We can superimpose the least squares
fit on a new plot:
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 212.68384222 * xfit/(0.06412146 + xfit)
> lines(spline(xfit, yfit))
The standard package
stats
provides much more extensive facilities
for fitting non-linear models by least squares. The model we have just
fitted is the Michaelis-Menten model, so we can use
> 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
11.7.2 Maximum likelihood
Maximum likelihood is a method of nonlinear model fitting that applies
even if the errors are not normal. The method finds the parameter values
which maximize the log likelihood, or equivalently which minimize the
negative log-likelihood. Here is an example from Dobson (1990), pp.
108–111. This example fits a logistic model to dose-response data,
which clearly could also be fit by
glm()
. The data are:
> x <- c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113,
1.8369, 1.8610, 1.8839)
> y <- c( 6, 13, 18, 28, 52, 53, 61, 60)
> n <- c(59, 60, 62, 56, 63, 59, 62, 60)
The negative log-likelihood to minimize is:
> fn <- function(p)
sum( - (y*(p[1]+p[2]*x) - n*log(1+exp(p[1]+p[2]*x))
+ log(choose(n, y)) ))
We pick sensible starting values and do the fit:
> out <- nlm(fn, p = c(-50,20), hessian = TRUE)
After the fitting,
out$minimum
is the negative log-likelihood,
and
out$estimate
are the maximum likelihood estimates of the
parameters. To obtain the approximate
SE
s of the estimates we do:
> sqrt(diag(solve(out$hessian)))
A 95% confidence interval would be the parameter estimate +/-
1.96 SE.
11.8 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.
Mixed models.
The recommended
nlme
package provides
functions
lme()
and
nlme()
for linear and non-linear mixed-effects models, that is linear and
non-linear regressions in which some of the coefficients correspond to
random effects. These functions make heavy use of formulae to specify
the models.
Local approximating regressions.
The
loess()
function fits a nonparametric regression by using a locally weighted
regression. Such regressions are useful for highlighting a trend in
messy data or for data reduction to give some insight into a large data
set.
Function
loess
is in the standard package
stats
, together
with code for projection pursuit regression.
Robust regression.
There are several functions available for
fitting regression models in a way resistant to the influence of extreme
outliers in the data. Function
lqs
in the recommended package
MASS
provides state-of-art algorithms
for highly-resistant fits. Less resistant but statistically more
efficient methods are available in packages, for example function
rlm
in package
MASS
Additive models.
This technique aims to construct a regression
function from smooth additive functions of the determining variables,
usually one for each determining variable. Functions
avas
and
ace
in package
acepack
and functions
bruto
and
mars
in package
mda
provide some examples of these techniques in
user-contributed packages to R. An extension is
Generalized
Additive Models
, implemented in user-contributed packages
gam
and
mgcv
Tree-based models.
Rather than seek an explicit global linear
model for prediction or interpretation, tree-based models seek to
bifurcate the data, recursively, at critical points of the determining
variables in order to partition the data ultimately into groups that are
as homogeneous as possible within, and as heterogeneous as possible
between. The results often lead to insights that other data analysis
methods tend not to yield.
Models are again specified in the ordinary linear model form. The model
fitting function is
tree()
but many other generic functions such as
plot()
and
text()
are well adapted to displaying the results of a tree-based model fit in
a graphical way.
Tree models are available in R
via
the user-contributed
packages
rpart
and
tree
12 Graphical procedures
Graphical facilities are an important and extremely versatile component
of the R environment. It is possible to use the facilities to
display a wide variety of statistical graphs and also to build entirely
new types of graph.
The graphics facilities can be used in both interactive and batch modes,
but in most cases, interactive use is more productive. Interactive use
is also easy because at startup time R initiates a graphics
device driver
which opens a special
graphics window
for
the display of interactive graphics. Although this is done
automatically, it may useful to know that the command used is
X11()
under UNIX,
windows()
under Windows and
quartz()
under macOS. A new device can always be opened by
dev.new()
Once the device driver is running, R plotting commands can be used to
produce a variety of graphical displays and to create entirely new kinds
of display.
Plotting commands are divided into three basic groups:
High-level
plotting functions create a new plot on the graphics
device, possibly with axes, labels, titles and so on.
Low-level
plotting functions add more information to an
existing plot, such as extra points, lines and labels.
Interactive
graphics functions allow you interactively add
information to, or extract information from, an existing plot, using a
pointing device such as a mouse.
In addition, R maintains a list of
graphical parameters
which
can be manipulated to customize your plots.
This manual only describes what are known as ‘base’ graphics. A
separate graphics sub-system in package
grid
coexists with base –
it is more powerful but harder to use. There is a recommended package
lattice
which builds on
grid
and provides ways to produce
multi-panel plots akin to those in the
Trellis
system in S.
12.1 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.
12.1.1 The
plot()
function
One of the most frequently used plotting functions in R is the
plot()
function. This is a
generic
function: the type of
plot produced is dependent on the type or
class
of the first
argument.
plot(
plot(
xy
If
and
are vectors,
plot(
produces a scatterplot of
against
. The same effect can
be produced by supplying one argument (second form) as either a list
containing two elements
and
or a two-column matrix.
plot(
If
is a time series, this produces a time-series plot. If
is a numeric vector, it produces a plot of the values in the
vector against their index in the vector. If
is a complex
vector, it produces a plot of imaginary versus real parts of the vector
elements.
plot(
plot(
is a factor object,
is a numeric vector. The first form
generates a bar plot of
; the second form produces boxplots of
for each level of
plot(
df
plot(~
expr
plot(
expr
df
is a data frame,
is any object,
expr
is a list
of object names separated by ‘
’ (e.g.,
a + b + c
). The
first two forms produce distributional plots of the variables in a data
frame (first form) or of a number of named objects (second form). The
third form plots
against every object named in
expr
12.1.2 Displaying multivariate data
R provides two very useful functions for representing multivariate
data. If
is a numeric matrix or data frame, the command
produces a pairwise scatterplot matrix of the variables defined by the
columns of
, that is, every column of
is plotted
against every other column of
and the resulting
n(n-1)
plots are arranged in a matrix with plot scales constant over the rows
and columns of the matrix.
When three or four variables are involved a
coplot
may be more
enlightening. If
and
are numeric vectors and
is a numeric vector or factor object (all of the same length), then
the command
produces a number of scatterplots of
against
for given
values of
. If
is a factor, this simply means that
is plotted against
for every level of
. When
is numeric, it is divided into a number of
conditioning
intervals
and for each interval
is plotted against
for values of
within the interval. The number and position of
intervals can be controlled with
given.values=
argument to
coplot()
—the function
co.intervals()
is useful for
selecting intervals. You can also use two
given
variables with a
command like
which produces scatterplots of
against
for every joint
conditioning interval of
and
The
coplot()
and
pairs()
function both take an argument
panel=
which can be used to customize the type of plot which
appears in each panel. The default is
points()
to produce a
scatterplot but by supplying some other low-level graphics function of
two vectors
and
as the value of
panel=
you can
produce any type of plot you wish. An example panel function useful for
coplots is
panel.smooth()
12.1.3 Display graphics
Other high-level graphics functions produce different types of plots.
Some examples are:
qqnorm(x)
qqline(x)
qqplot(x, y)
Distribution-comparison plots. The first form plots the numeric vector
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
against those of
to compare their
respective distributions.
hist(x)
hist(x, nclass=
hist(x, breaks=
, …)
Produces a histogram of the numeric vector
. A sensible number
of classes is usually chosen, but a recommendation can be given with the
nclass=
argument. Alternatively, the breakpoints can be
specified exactly with the
breaks=
argument. If the
probability=TRUE
argument is given, the bars represent relative
frequencies divided by bin width instead of counts.
dotchart(x, …)
Constructs a dot chart of the data in
. In a dot chart the
-axis gives a labelling of the data in
and the
-axis gives its value. For example it allows easy visual
selection of all data entries with values lying in specified ranges.
image(x, y, z, …)
contour(x, y, z, …)
persp(x, y, z, …)
Plots of three variables. The
image
plot draws a grid of rectangles
using different colours to represent the value of
, the
contour
plot draws contour lines to represent the value of
, and the
persp
plot draws a 3D surface.
12.1.4 Arguments to high-level plotting functions
There are a number of arguments which may be passed to high-level
graphics functions, as follows:
add=TRUE
Forces the function to act as a low-level graphics function,
superimposing the plot on the current plot (some functions only).
axes=FALSE
Suppresses generation of axes—useful for adding your own custom axes
with the
axis()
function. The default,
axes=TRUE
, means
include axes.
log="x"
log="y"
log="xy"
Causes the
or both axes to be logarithmic. This will
work for many, but not all, types of plot.
type=
The
type=
argument controls the type of plot produced, as
follows:
type="p"
Plot individual points (the default)
type="l"
Plot lines
type="b"
Plot points connected by lines (
both
type="o"
Plot points overlaid by lines
type="h"
Plot vertical lines from points to the zero axis (
high-density
type="s"
type="S"
Step-function plots. In the first form, the top of the vertical defines
the point; in the second, the bottom.
type="n"
No plotting at all. However axes are still drawn (by default) and the
coordinate system is set up according to the data. Ideal for creating
plots with subsequent low-level graphics functions.
xlab=
string
ylab=
string
Axis labels for the
and
axes. Use these arguments to
change the default labels, usually the names of the objects used in the
call to the high-level plotting function.
main=
string
Figure title, placed at the top of the plot in a large font.
sub=
string
Sub-title, placed just below the
-axis in a smaller font.
12.2 Low-level plotting commands
Sometimes the high-level plotting functions don’t produce exactly the
kind of plot you desire. In this case, low-level plotting commands can
be used to add extra information (such as points, lines or text) to the
current plot.
Some of the more useful low-level plotting functions are:
points(x, y)
lines(x, y)
Adds points or connected lines to the current plot.
plot()
’s
type=
argument can also be passed to these functions (and
defaults to
"p"
for
points()
and
"l"
for
lines()
.)
text(x, y, labels, …)
Add text to a plot at points given by
x, y
. Normally
labels
is an integer or character vector in which case
labels[i]
is plotted at point
(x[i], y[i])
. The default
is
1:length(x)
Note
: This function is often used in the sequence
> plot(x, y, type="n"); text(x, y, names)
The graphics parameter
type="n"
suppresses the points but sets up
the axes, and the
text()
function supplies special characters, as
specified by the character vector
names
for the points.
abline(a, b)
abline(h=
abline(v=
abline(
lm.obj
Adds a line of slope
and intercept
to the current
plot.
h=
may be used to specify
-coordinates for
the heights of horizontal lines to go across a plot, and
v=
similarly for the
-coordinates for vertical
lines. Also
lm.obj
may be list with a
coefficients
component of length 2 (such as the result of model-fitting functions,)
which are taken as an intercept and slope, in that order.
polygon(x, y, …)
Draws a polygon defined by the ordered vertices in (
and (optionally) shade it in with hatch lines, or fill it if the
graphics device allows the filling of figures.
legend(x, y, legend, …)
Adds a legend to the current plot at the specified position. Plotting
characters, line styles, colors etc., are identified with the labels in
the character vector
legend
. At least one other argument
(a vector the same length as
legend
) with the corresponding
values of the plotting unit must also be given, as follows:
legend( , fill=
Colors for filled boxes
legend( , col=
Colors in which points or lines will be drawn
legend( , lty=
Line styles
legend( , lwd=
Line widths
legend( , pch=
Plotting characters (character vector)
title(main, sub)
Adds a title
main
to the top of the current plot in a large font
and (optionally) a sub-title
sub
at the bottom in a smaller font.
axis(side, …)
Adds an axis to the current plot on the side given by the first argument
(1 to 4, counting clockwise from the bottom.) Other arguments control
the positioning of the axis within or beside the plot, and tick
positions and labels. Useful for adding custom axes after calling
plot()
with the
axes=FALSE
argument.
Low-level plotting functions usually require some positioning
information (e.g.,
and
coordinates) to determine where
to place the new plot elements. Coordinates are given in terms of
user coordinates
which are defined by the previous high-level
graphics command and are chosen based on the supplied data.
Where
and
arguments are required, it is also
sufficient to supply a single argument being a list with elements named
and
. Similarly a matrix with two columns is also
valid input. In this way functions such as
locator()
(see below)
may be used to specify positions on a plot interactively.
12.2.1 Mathematical annotation
In some cases, it is useful to add mathematical symbols and formulae to a
plot. This can be achieved in R by specifying an
expression
rather
than a character string in any one of
text
mtext
axis
or
title
. For example, the following code draws the formula for
the Binomial probability function:
> text(x, y, expression(paste(bgroup("(", atop(n, x), ")"), p^x, q^{n-x})))
More information, including a full listing of the features available can
obtained from within R using the commands:
> help(plotmath)
> example(plotmath)
> demo(plotmath)
12.2.2 Hershey vector fonts
It is possible to specify Hershey vector fonts for rendering text when using
the
text
and
contour
functions. There are three reasons for
using the Hershey fonts:
Hershey fonts can produce better
output, especially on a computer screen, for rotated and/or small text.
Hershey fonts
provide certain symbols that may not be available
in the standard fonts. In particular, there are zodiac signs, cartographic
symbols and astronomical symbols.
Hershey fonts provide Cyrillic and Japanese (Kana and Kanji) characters.
More information, including tables of Hershey characters can be obtained from
within R using the commands:
> help(Hershey)
> demo(Hershey)
> help(Japanese)
> demo(Japanese)
12.3 Interacting with graphics
R also provides functions which allow users to extract or add
information to a plot using a mouse. The simplest of these is the
locator()
function:
locator(n, type)
Waits for the user to select locations on the current plot using the
left mouse button. This continues until
(default 512) points
have been selected, or another mouse button is pressed. The
type
argument allows for plotting at the selected points and has
the same effect as for high-level graphics commands; the default is no
plotting.
locator()
returns the locations of the points selected
as a list with two components
and
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
> text(locator(1), "Outlier", adj=0)
may be useful. (
locator()
will be ignored if the current device,
such as
postscript
does not support interactive pointing.)
identify(x, y, labels)
Allow the user to highlight any of the points defined by
and
(using the left mouse button) by plotting the corresponding
component of
labels
nearby (or the index number of the point if
labels
is absent). Returns the indices of the selected points
when another button is pressed.
Sometimes we want to identify particular
points
on a plot, rather
than their positions. For example, we may wish the user to select some
observation of interest from a graphical display and then manipulate
that observation in some way. Given a number of
(x, y)
coordinates in two numeric vectors
and
, we could use
the
identify()
function as follows:
> plot(x, y)
> identify(x, y)
The
identify()
functions performs no plotting itself, but simply
allows the user to move the mouse pointer and click the left mouse
button near a point. If there is a point near the mouse pointer it will
be marked with its index number (that is, its position in the
vectors) plotted nearby. Alternatively, you could use
some informative string (such as a case name) as a highlight by using
the
labels
argument to
identify()
, or disable marking
altogether with the
plot = FALSE
argument. When the process is
terminated (see above),
identify()
returns the indices of the
selected points; you can use these indices to extract the selected
points from the original vectors
and
12.4 Using graphics parameters
When creating graphics, particularly for presentation or publication
purposes, R’s defaults do not always produce exactly that which is
required. You can, however, customize almost every aspect of the
display using
graphics parameters
. R maintains a list of a
large number of graphics parameters which control things such as line
style, colors, figure arrangement and text justification among many
others. Every graphics parameter has a name (such as ‘
col
’,
which controls colors,) and a value (a color number, for example.)
A separate list of graphics parameters is maintained for each active
device, and each device has a default set of parameters when
initialized. Graphics parameters can be set in two ways: either
permanently, affecting all graphics functions which access the current
device; or temporarily, affecting only a single graphics function call.
12.4.1 Permanent changes: The
par()
function
The
par()
function is used to access and modify the list of
graphics parameters for the current graphics device.
par()
Without arguments, returns a list of all graphics parameters and their
values for the current device.
par(c("col", "lty"))
With a character vector argument, returns only the named graphics
parameters (again, as a list.)
par(col=4, lty=2)
With named arguments (or a single list argument), sets the values of
the named graphics parameters, and returns the original values of the
parameters as a list.
Setting graphics parameters with the
par()
function changes the
value of the parameters
permanently
, in the sense that all future
calls to graphics functions (on the current device) will be affected by
the new value. You can think of setting graphics parameters in this way
as setting “default” values for the parameters, which will be used by
all graphics functions unless an alternative value is given.
Note that calls to
par()
always
affect the global values
of graphics parameters, even when
par()
is called from within a
function. This is often undesirable behavior—usually we want to set
some graphics parameters, do some plotting, and then restore the
original values so as not to affect the user’s R session. You can
restore the initial values by saving the result of
par()
when
making changes, and restoring the initial values when plotting is
complete.
> oldpar <- par(col=4, lty=2)
... plotting commands ...
> par(oldpar)
To save and restore
all
settable graphical parameters use
> oldpar <- par(no.readonly=TRUE)
... plotting commands ...
> par(oldpar)
12.4.2 Temporary changes: Arguments to graphics functions
Graphics parameters may also be passed to (almost) any graphics function
as named arguments. This has the same effect as passing the arguments
to the
par()
function, except that the changes only last for the
duration of the function call. For example:
produces a scatterplot using a plus sign as the plotting character,
without changing the default plotting character for future plots.
Unfortunately, this is not implemented entirely consistently and it is
sometimes necessary to set and reset graphics parameters using
par()
12.5 Graphics parameters list
The following sections detail many of the commonly-used graphical
parameters. The R help documentation for the
par()
function
provides a more concise summary; this is provided as a somewhat more
detailed alternative.
Graphics parameters will be presented in the following form:
name
value
A description of the parameter’s effect.
name
is the name of the
parameter, that is, the argument name to use in calls to
par()
or
a graphics function.
value
is a typical value you might use when
setting the parameter.
Note that
axes
is
not
a graphics parameter but an
argument to a few
plot
methods: see
xaxt
and
yaxt
12.5.1 Graphical elements
R plots are made up of points, lines, text and polygons (filled
regions.) Graphical parameters exist which control how these
graphical elements
are drawn, as follows:
pch="+"
Character to be used for plotting points. The default varies with
graphics drivers, but it is usually
a circle.
Plotted points tend to appear slightly above or below the appropriate
position unless you use
"."
as the plotting character, which
produces centered points.
pch=4
When
pch
is given as an integer between 0 and 25 inclusive, a
specialized plotting symbol is produced. To see what the symbols are,
use the command
> legend(locator(1), as.character(0:25), pch = 0:25)
Those from 21 to 25 may appear to duplicate earlier symbols, but can be
coloured in different ways: see the help on
points
and its
examples.
In addition,
pch
can be a character or a number in the range
32:255
representing a character in the current font.
lty=2
Line types. Alternative line styles are not supported on all graphics
devices (and vary on those that do) but line type 1 is always a solid
line, line type 0 is always invisible, and line types 2 and onwards are
dotted or dashed lines, or some combination of both.
lwd=2
Line widths. Desired width of lines, in multiples of the “standard”
line width. Affects axis lines as well as lines drawn with
lines()
, etc. Not all devices support this, and some have
restrictions on the widths that can be used.
col=2
Colors to be used for points, lines, text, filled regions and images.
A number from the current palette (see
?palette
) or a named colour.
col.axis
col.lab
col.main
col.sub
The color to be used for axis annotation,
and
labels,
main and sub-titles, respectively.
font=2
An integer which specifies which font to use for text. If possible,
device drivers arrange so that
corresponds to plain text,
to bold face,
to italic,
to bold italic
and
to a symbol font (which include Greek letters).
font.axis
font.lab
font.main
font.sub
The font to be used for axis annotation,
and
labels,
main and sub-titles, respectively.
adj=-0.1
Justification of text relative to the plotting position.
means
left justify,
means right justify and
0.5
means to
center horizontally about the plotting position. The actual value is
the proportion of text that appears to the left of the plotting
position, so a value of
-0.1
leaves a gap of 10% of the text width
between the text and the plotting position.
cex=1.5
Character expansion. The value is the desired size of text characters
(including plotting characters) relative to the default text size.
cex.axis
cex.lab
cex.main
cex.sub
The character expansion to be used for axis annotation,
and
labels, main and sub-titles, respectively.
12.5.2 Axes and tick marks
Many of R’s high-level plots have axes, and you can construct axes
yourself with the low-level
axis()
graphics function. Axes have
three main components: the
axis line
(line style controlled by the
lty
graphics parameter), the
tick marks
(which mark off unit
divisions along the axis line) and the
tick labels
(which mark the
units.) These components can be customized with the following graphics
parameters.
lab=c(5, 7, 12)
The first two numbers are the desired number of tick intervals on the
and
axes respectively. The third number is the
desired length of axis labels, in characters (including the decimal
point.) Choosing a too-small value for this parameter may result in all
tick labels being rounded to the same number!
las=1
Orientation of axis labels.
means always parallel to axis,
means always horizontal, and
means always
perpendicular to the axis.
mgp=c(3, 1, 0)
Positions of axis components. The first component is the distance from
the axis label to the axis position, in text lines. The second
component is the distance to the tick labels, and the final component is
the distance from the axis position to the axis line (usually zero).
Positive numbers measure outside the plot region, negative numbers
inside.
tck=0.01
Length of tick marks, as a fraction of the size of the plotting region.
When
tck
is small (less than 0.5) the tick marks on the
and
axes are forced to be the same size. A value of 1 gives
grid lines. Negative values give tick marks outside the plotting
region. Use
tck=0.01
and
mgp=c(1,-1.5,0)
for internal
tick marks.
xaxs="r"
yaxs="i"
Axis styles for the
and
axes, respectively. With
styles
"i"
(internal) and
"r"
(the default) tick marks
always fall within the range of the data, however style
"r"
leaves a small amount of space at the edges.
12.5.3 Figure margins
A single plot in R is known as a
figure
and comprises a
plot region
surrounded by margins (possibly containing axis
labels, titles, etc.) and (usually) bounded by the axes themselves.
A typical figure is
Graphics parameters controlling figure layout include:
mai=c(1, 0.5, 0.5, 0)
Widths of the bottom, left, top and right margins, respectively,
measured in inches.
mar=c(4, 2, 2, 1)
Similar to
mai
, except the measurement unit is text lines.
mar
and
mai
are equivalent in the sense that setting one
changes the value of the other. The default values chosen for this
parameter are often too large; the right-hand margin is rarely needed,
and neither is the top margin if no title is being used. The bottom and
left margins must be large enough to accommodate the axis and tick
labels. Furthermore, the default is chosen without regard to the size
of the device surface: for example, using the
postscript()
driver
with the
height=4
argument will result in a plot which is about
50% margin unless
mar
or
mai
are set explicitly. When
multiple figures are in use (see below) the margins are reduced, however
this may not be enough when many figures share the same page.
12.5.4 Multiple figure environment
R allows you to create an
by
array of figures on a
single page. Each figure has its own margins, and the array of figures
is optionally surrounded by an
outer margin
, as shown in the
following figure.
The graphical parameters relating to multiple figures are as follows:
mfcol=c(3, 2)
mfrow=c(2, 4)
Set the size of a multiple figure array. The first value is the number of
rows; the second is the number of columns. The only difference between
these two parameters is that setting
mfcol
causes figures to be
filled by column;
mfrow
fills by rows.
The layout in the Figure could have been created by setting
mfrow=c(3,2)
; the figure shows the page after four plots have
been drawn.
Setting either of these can reduce the base size of symbols and text
(controlled by
par("cex")
and the pointsize of the device). In a
layout with exactly two rows and columns the base size is reduced by a
factor of 0.83: if there are three or more of either rows or columns,
the reduction factor is 0.66.
mfg=c(2, 2, 3, 2)
Position of the current figure in a multiple figure environment. The first
two numbers are the row and column of the current figure; the last two
are the number of rows and columns in the multiple figure array. Set
this parameter to jump between figures in the array. You can even use
different values for the last two numbers than the
true
values
for unequally-sized figures on the same page.
fig=c(4, 9, 1, 4)/10
Position of the current figure on the page. Values are the positions of
the left, right, bottom and top edges respectively, as a percentage of
the page measured from the bottom left corner. The example value would
be for a figure in the bottom right of the page. Set this parameter for
arbitrary positioning of figures within a page. If you want to add a
figure to a current page, use
new=TRUE
as well (unlike S).
oma=c(2, 0, 3, 0)
omi=c(0, 0, 0.8, 0)
Size of outer margins. Like
mar
and
mai
, the first
measures in text lines and the second in inches, starting with the
bottom margin and working clockwise.
Outer margins are particularly useful for page-wise titles, etc. Text
can be added to the outer margins with the
mtext()
function with
argument
outer=TRUE
. There are no outer margins by default,
however, so you must create them explicitly using
oma
or
omi
More complicated arrangements of multiple figures can be produced by the
split.screen()
and
layout()
functions, as well as by the
grid
and
lattice
packages.
12.6 Device drivers
R can generate graphics (of varying levels of quality) on almost any
type of display or printing device. Before this can begin, however,
R needs to be informed what type of device it is dealing with. This
is done by starting a
device driver
. The purpose of a device
driver is to convert graphical instructions from R (“draw a line,”
for example) into a form that the particular device can understand.
Device drivers are started by calling a device driver function. There
is one such function for every device driver: type
help(Devices)
for a list of them all. For example, issuing the command
causes all future graphics output to be sent to the printer in
PostScript format. Some commonly-used device drivers are:
X11()
For use with the X11 window system on Unix-alikes
windows()
For use on Windows
quartz()
For use on macOS
postscript()
For printing on PostScript printers, or creating PostScript graphics
files.
pdf()
Produces a PDF file, which can also be included into PDF files.
png()
Produces a bitmap PNG file. (Not always available: see its help page.)
jpeg()
Produces a bitmap JPEG file, best used for
image
plots.
(Not always available: see its help page.)
When you have finished with a device, be sure to terminate the device
driver by issuing the command
This ensures that the device finishes cleanly; for example in the case
of hardcopy devices this ensures that every page is completed and has
been sent to the printer. (This will happen automatically at the normal
end of a session.)
12.6.1 PostScript diagrams for typeset documents
By passing the
file
argument to the
postscript()
device
driver function, you may store the graphics in PostScript format in a
file of your choice. The plot will be in landscape orientation unless
the
horizontal=FALSE
argument is given, and you can control the
size of the graphic with the
width
and
height
arguments
(the plot will be scaled as appropriate to fit these dimensions.) For
example, the command
> postscript("file.ps", horizontal=FALSE, height=5, pointsize=10)
will produce a file containing PostScript code for a figure five inches
high, perhaps for inclusion in a document. It is important to note that
if the file named in the command already exists, it will be overwritten.
This is the case even if the file was only created earlier in the same
R session.
Many usages of PostScript output will be to incorporate the figure in
another document. This works best when
encapsulated
PostScript
is produced: R always produces conformant output, but only marks the
output as such when the
onefile=FALSE
argument is supplied. This
unusual notation stems from S-compatibility: it really means that
the output will be a single page (which is part of the EPSF
specification). Thus to produce a plot for inclusion use something like
> postscript("plot1.eps", horizontal=FALSE, onefile=FALSE,
height=8, width=6, pointsize=10)
12.6.2 Multiple graphics devices
In advanced use of R it is often useful to have several graphics
devices in use at the same time. Of course only one graphics device can
accept graphics commands at any one time, and this is known as the
current device
. When multiple devices are open, they form a
numbered sequence with names giving the kind of device at any position.
The main commands used for operating with multiple devices, and their
meanings are as follows:
X11()
[UNIX]
windows()
win.printer()
win.metafile()
[Windows]
quartz()
[macOS]
postscript()
pdf()
png()
jpeg()
tiff()
bitmap()
Each new call to a device driver function opens a new graphics device,
thus extending by one the device list. This device becomes the current
device, to which graphics output will be sent.
dev.list()
Returns the number and name of all active devices. The device at
position 1 on the list is always the
null device
which does not
accept graphics commands at all.
dev.next()
dev.prev()
Returns the number and name of the graphics device next to, or previous
to the current device, respectively.
dev.set(which=
Can be used to change the current graphics device to the one at position
of the device list. Returns the number and label of the device.
dev.off(
Terminate the graphics device at point
of the device list. For
some devices, such as
postscript
devices, this will either print
the file immediately or correctly complete the file for later printing,
depending on how the device was initiated.
dev.copy(device, …, which=
dev.print(device, …, which=
Make a copy of the device
. Here
device
is a device
function, such as
postscript
, with extra arguments, if needed,
specified by ‘
’.
dev.print
is similar, but the
copied device is immediately closed, so that end actions, such as
printing hardcopies, are immediately performed.
graphics.off()
Terminate all graphics devices on the list, except the null device.
12.7 Dynamic graphics
R does not have builtin capabilities for dynamic or
interactive graphics, e.g. rotating point clouds or to “brushing”
(interactively highlighting) points. However, extensive dynamic graphics
facilities are available in the system GGobi by Swayne, Cook
and Buja available from
and these can be accessed from R via the package
rggobi
, described at
Also, package
rgl
provides ways to interact with 3D plots, for example
of surfaces.
13 Packages
All R functions and datasets are stored in
packages
. Only
when a package is loaded are its contents available. This is done both
for efficiency (the full list would take more memory and would take
longer to search than a subset), and to aid package developers, who are
protected from name clashes with other code. The process of developing
packages is described in
Creating R packages
in
Writing R Extensions
Here, we will describe them from a user’s point of view.
To see which packages are installed at your site, issue the command
with no arguments. To load a particular package (e.g., the
boot
package containing functions from Davison & Hinkley (1997)), use a
command like
Users connected to the Internet can use the
install.packages()
and
update.packages()
functions (available through the
Packages
menu in the Windows and macOS GUIs, see
Installing
packages
in
R Installation and Administration
) to install
and update packages.
To see which packages are currently loaded, use
to display the search list. Some packages may be loaded but not
available on the search list (see
Namespaces
): these will be
included in the list given by
To see a list of all available help topics in an installed package,
use
to start the
HTML
help system, and then navigate to the package
listing in the
Reference
section.
13.1 Standard packages
The standard (or
base
) packages are considered part of the R
source code. They contain the basic functions that allow R to work,
and the datasets and standard statistical and graphical functions that
are described in this manual. They should be automatically available in
any R installation.
For a complete list, see
Which add-on packages exist for R?
in
R FAQ
13.2 Contributed packages and
CRAN
There are thousands of contributed packages for R, written by many
different authors. Some of these packages implement specialized
statistical methods, others give access to data or hardware, and others
are designed to complement textbooks. Some (the
recommended
packages) are distributed with every binary distribution of R. Most
are available for download from
CRAN
and its mirrors) and other
repositories such as Bioconductor (
).
The
R FAQ
contains a list of CRAN packages current at the time of release, but the
collection of available packages changes very frequently.
13.3 Namespaces
Packages have
namespaces
, which do three things: they allow the
package writer to hide functions and data that are meant only for
internal use, they prevent functions from breaking when a user (or other
package writer) picks a name that clashes with one in the package, and
they provide a way to refer to an object within a particular package.
For example,
t()
is the transpose function in R, but users
might define their own function named
. Namespaces prevent
the user’s definition from taking precedence, and breaking every
function that tries to transpose a matrix.
There are two operators that work with namespaces. The double-colon
operator
::
selects definitions from a particular namespace.
In the example above, the transpose function will always be available
as
base::t
, because it is defined in the
base
package.
Only functions that are exported from the package can be retrieved in
this way.
The triple-colon operator
:::
may be seen in a few places in R
code: it acts like the double-colon operator but also allows access to
hidden objects. Users are more likely to use the
getAnywhere()
function, which searches multiple packages.
Packages are often inter-dependent, and loading one may cause others to
be automatically loaded. The colon operators described above will also
cause automatic loading of the associated package. When packages with
namespaces are loaded automatically they are not added to the search
list.
14 OS facilities
R has quite extensive facilities to access the OS under which it is
running: this allows it to be used as a scripting language and that
ability is much used by R itself, for example to install packages.
Because R’s own scripts need to work across all platforms,
considerable effort has gone into make the scripting facilities as
platform-independent as is feasible.
14.1 Files and directories
There are many functions to manipulate files and directories. Here are
pointers to some of the more commonly used ones.
To create an (empty) file or directory, use
file.create
or
dir.create
. (These are the analogues of the POSIX utilities
touch
and
mkdir
.) For temporary files and
directories in the R session directory see
tempfile
Files can be removed by either
file.remove
or
unlink
: the
latter can remove directory trees.
For directory listings use
list.files
(also available as
dir
) or
list.dirs
. These can select files using a regular
expression: to select by wildcards use
Sys.glob
Many types of information on a filepath (including for example if it is
a file or directory) can be found by
file.info
There are several ways to find out if a file ‘exists’ (a file can
exist on the filesystem and not be visible to the current user).
There are functions
file.exists
file.access
and
file_test
with various versions of this test:
file_test
is
a version of the POSIX
test
command for those familiar with
shell scripting.
Function
file.copy
is the R analogue of the POSIX command
cp
Choosing files can be done interactively by
file.choose
: the
Windows port has the more versatile functions
choose.files
and
choose.dir
and there are similar functions in the
tcltk
package:
tk_choose.files
and
tk_choose.dir
Functions
file.show
and
file.edit
will display and edit
one or more files in a way appropriate to the R port, using the
facilities of a console (such as RGui on Windows or R.app on macOS) if
one is in use.
There is some support for
links
in the filesystem: see functions
file.link
and
Sys.readlink
14.2 Filepaths
With a few exceptions, R relies on the underlying OS functions to
manipulate filepaths. Some aspects of this are allowed to depend on the
OS, and do, even down to the version of the OS. There are POSIX
standards for how OSes should interpret filepaths and many R users
assume POSIX compliance: but Windows does not claim to be compliant and
other OSes may be less than completely compliant.
The following are some issues which have been encountered with filepaths.
POSIX filesystems are case-sensitive, so
foo.png
and
Foo.PNG
are different files. However, the defaults on Windows
and macOS are to be case-insensitive, and FAT filesystems (commonly used
on removable storage) are not normally case-sensitive (and all filepaths
may be mapped to lower case).
Almost all the Windows’ OS services support the use of slash or
backslash as the filepath separator, and R converts the known
exceptions to the form required by Windows.
The behaviour of filepaths with a trailing slash is OS-dependent. Such
paths are not valid on Windows and should not be expected to work.
POSIX-2008 requires such paths to match only directories, but earlier
versions allowed them to also match files. So they are best avoided.
Multiple slashes in filepaths such as
/abc//def
are valid on
POSIX filesystems and treated as if there was only one slash. They are
usually
accepted by Windows’ OS functions. However, leading
double slashes may have a different meaning.
Windows’ UNC filepaths (such as
\\server\dir1\dir2\file
and
\\?\UNC\server\dir1\dir2\file
) are not supported, but they may
work in some R functions. POSIX filesystems are allowed to treat a
leading double slash specially.
Windows allows filepaths containing drives and relative to the current
directory on a drive, e.g.
d:foo/bar
refers to
d:/a/b/c/foo/bar
if the current directory
on drive
d:
is
/a/b/c
. It is intended that these work, but the
use of absolute paths is safer.
Functions
basename
and
dirname
select parts of a file
path: the recommended way to assemble a file path from components is
file.path
. Function
pathexpand
does ‘tilde expansion’,
substituting values for home directories (the current user’s, and
perhaps those of other users).
On filesystems with links, a single file can be referred to by many
filepaths. Function
normalizePath
will find a canonical
filepath.
Windows has the concepts of short (‘8.3’) and long file names:
normalizePath
will return an absolute path using long file names
and
shortPathName
will return a version using short names. The
latter does not contain spaces and uses backslash as the separator, so
is sometimes useful for exporting names from R.
File
permissions
are a related topic. R has support for the
POSIX concepts of read/write/execute permission for owner/group/all but
this may be only partially supported on the filesystem, so for example
on Windows only read-only files (for the account running the R
session) are recognized. Access Control Lists (
ACL
s) are employed on
several filesystems, but do not have an agreed standard and R has no
facilities to control them. Use
Sys.chmod
to change permissions.
14.3 System commands
Functions
system
and
system2
are used to invoke a system
command and optionally collect its output.
system2
is a little
more general but its main advantage is that it is easier to write
cross-platform code using it.
system
behaves differently on Windows from other OSes (because
the API C call of that name does). Elsewhere it invokes a shell to run
the command: the Windows port of R has a function
shell
to do
that.
To find out if the OS includes a command, use
Sys.which
, which
attempts to do this in a cross-platform way (unfortunately it is not a
standard OS service).
Function
shQuote
will quote filepaths as needed for commands in
the current OS.
14.4 Compression and Archives
Recent versions of R have extensive facilities to read and write
compressed files, often transparently. Reading of files in R is to a
very large extent done by
connections
, and the
file
function which is used to open a connection to a file (or a URL) and is
able to identify the compression used from the ‘magic’ header of the
file.
The type of compression which has been supported for longest is
gzip
compression, and that remains a good general compromise.
Files compressed by the earlier Unix
compress
utility can also
be read, but these are becoming rare. Two other forms of compression,
those of the
bzip2
and
xz
utilities are also
available. These generally achieve higher rates of compression
(depending on the file, much higher) at the expense of slower
decompression and much slower compression.
There is some confusion between
xz
and
lzma
compression (see
and
): R can read files
compressed by most versions of either.
File archives are single files which contain a collection of files, the
most common ones being ‘tarballs’ and zip files as used to distribute
R packages. R can list and unpack both (see functions
untar
and
unzip
) and create both (for
zip
with the help of an
external program).
Appendix A A sample session
The following session is intended to introduce to you some features of
the R environment by using them. Many features of the system will be
unfamiliar and puzzling at first, but this puzzlement will soon
disappear.
Start R appropriately for your platform (see
Invoking R
).
The R program begins, with a banner.
(Within R code, the prompt on the left hand side will not be shown to
avoid confusion.)
help.start()
Start the
HTML
interface to on-line help (using a web browser
available at your machine). You should briefly explore the features of
this facility with the mouse.
Iconify the help window and move on to the next part.
x <- rnorm(50)
y <- rnorm(x)
Generate two pseudo-random normal vectors of
- and
-coordinates.
plot(x, y)
Plot the points in the plane. A graphics window will appear automatically.
ls()
See which R objects are now in the R workspace.
rm(x, y)
Remove objects no longer needed. (Clean up).
x <- 1:20
Make
x = (1, 2, ..., 20)
w <- 1 + sqrt(x)/2
A ‘weight’ vector of standard deviations.
dummy <- data.frame(x=x, y= x + rnorm(x)*w)
dummy
Make a
data frame
of two columns,
and
, and look
at it.
fm <- lm(y ~ x, data=dummy)
summary(fm)
Fit a simple linear regression and look at the
analysis. With
to the left of the tilde,
we are modelling
dependent on
fm1 <- lm(y ~ x, data=dummy, weight=1/w^2)
summary(fm1)
Since we know the standard deviations, we can do a weighted regression.
attach(dummy)
Make the columns in the data frame visible as variables.
lrf <- lowess(x, y)
Make a nonparametric local regression function.
plot(x, y)
Standard point plot.
lines(x, lrf$y)
Add in the local regression.
abline(0, 1, lty=3)
The true regression line: (intercept 0, slope 1).
abline(coef(fm))
Unweighted regression line.
abline(coef(fm1), col = "red")
Weighted regression line.
detach()
Remove data frame from the search path.
plot(fitted(fm), resid(fm),
xlab="Fitted values",
ylab="Residuals",
main="Residuals vs Fitted")
A standard regression diagnostic plot to check for heteroscedasticity.
Can you see it?
qqnorm(resid(fm), main="Residuals Rankit Plot")
A normal scores plot to check for skewness, kurtosis and outliers. (Not
very useful here.)
rm(fm, fm1, lrf, x, dummy)
Clean up again.
The next section will look at data from the classical experiment of
Michelson to measure the speed of light. This dataset is available in
the
morley
object, but we will read it to illustrate the
read.table
function.
filepath <- system.file("data", "morley.tab" , package="datasets")
filepath
Get the path to the data file.
file.show(filepath)
Optional. Look at the file.
mm <- read.table(filepath)
mm
Read in the Michelson data as a data frame, and look at it.
There are five experiments (column
Expt
) and each has 20 runs
(column
Run
) and
sl
is the recorded speed of light,
suitably coded.
mm$Expt <- factor(mm$Expt)
mm$Run <- factor(mm$Run)
Change
Expt
and
Run
into factors.
attach(mm)
Make the data frame visible at position 2 (the default).
plot(Expt, Speed, main="Speed of Light Data", xlab="Experiment No.")
Compare the five experiments with simple boxplots.
fm <- aov(Speed ~ Run + Expt, data=mm)
summary(fm)
Analyze as a randomized block, with ‘runs’ and ‘experiments’ as factors.
fm0 <- update(fm, . ~ . - Run)
anova(fm0, fm)
Fit the sub-model omitting ‘runs’, and compare using a formal analysis
of variance.
detach()
rm(fm, fm0)
Clean up before moving on.
We now look at some more graphical features: contour and image plots.
x <- seq(-pi, pi, len=50)
y <- x
is a vector of 50 equally spaced values in
the interval [-pi\, pi].
is the same.
f <- outer(x, y, function(x, y) cos(y)/(1 + x^2))
is a square matrix, with rows and columns indexed by
and
respectively, of values of the function
cos(y)/(1 + x^2).
oldpar <- par(no.readonly = TRUE)
par(pty="s")
Save the plotting parameters and set the plotting region to “square”.
contour(x, y, f)
contour(x, y, f, nlevels=15, add=TRUE)
Make a contour map of
; add in more lines for more detail.
fa <- (f-t(f))/2
fa
is the “asymmetric part” of
. (
t()
is
transpose).
contour(x, y, fa, nlevels=15)
Make a contour plot, …
par(oldpar)
… and restore the old graphics parameters.
image(x, y, f)
image(x, y, fa)
Make some high density image plots, (of which you can get
hardcopies if you wish), …
objects(); rm(x, y, f, fa)
… and clean up before moving on.
R can do complex arithmetic, also.
th <- seq(-pi, pi, len=100)
z <- exp(1i*th)
1i
is used for the complex number
par(pty="s")
plot(z, type="l")
Plotting complex arguments means plot imaginary versus real parts. This
should be a circle.
w <- rnorm(100) + rnorm(100)*1i
Suppose we want to sample points within the unit circle. One method
would be to take complex numbers with standard normal real and imaginary
parts …
w <- ifelse(Mod(w) > 1, 1/w, w)
… and to map any outside the circle onto their reciprocal.
plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+",xlab="x", ylab="y")
lines(z)
All points are inside the unit circle, but the distribution is not
uniform.
w <- sqrt(runif(100))*exp(2*pi*runif(100)*1i)
plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+", xlab="x", ylab="y")
lines(z)
The second method uses the uniform distribution. The points should now
look more evenly spaced over the disc.
rm(th, w, z)
Clean up again.
q()
Quit the R program. You will be asked if you want to save the R
workspace, and for an exploratory session like this, you probably do not
want to save it.
Appendix B Invoking R
Users of R on Windows or macOS should read the OS-specific section
first, but command-line use is also supported.
B.1 Invoking R from the command line
When working at a command line on UNIX or Windows, the command ‘
can be used both for starting the main R program in the form
options
] [
infile
] [
outfile
],
or, via the
R CMD
interface, as a wrapper to various R tools
(e.g., for processing files in R documentation format or manipulating
add-on packages) which are not intended to be called “directly”.
At the Windows command-line,
Rterm.exe
is preferred to
You need to ensure that either the environment variable
TMPDIR
is
unset or it points to a valid place to create temporary files and
directories.
Most options control what happens at the beginning and at the end of an
R session. The startup mechanism is as follows (see also the on-line
help for topic ‘
Startup
’ for more information, and the section below
for some Windows-specific details).
Unless
--no-environ
was given, R searches for user and site
files to process for setting environment variables. The name of the
site file is the one pointed to by the environment variable
R_ENVIRON
; if this is unset,
R_HOME
/etc/Renviron.site
is used (if it exists). The user file is the one pointed to by the
environment variable
R_ENVIRON_USER
if this is set; otherwise,
files
.Renviron
in the current or in the user’s home directory
(in that order) are searched for. These files should contain lines of
the form ‘
name
value
’. (See
help("Startup")
for
a precise description.) Variables you might want to set include
R_PAPERSIZE
(the default paper size),
R_PRINTCMD
(the
default print command) and
R_LIBS
(specifies the list of R
library trees searched for add-on packages).
Then R searches for the site-wide startup profile unless the command
line option
--no-site-file
was given. The name of this file is
taken from the value of the
R_PROFILE
environment variable. If
that variable is unset, the default
R_HOME
/etc/Rprofile.site
is used if this exists.
Then, unless
--no-init-file
was given, R searches for a user
profile and sources it. The name of this file is taken from the
environment variable
R_PROFILE_USER
; if unset, a file called
.Rprofile
in the current directory or in the user’s home
directory (in that order) is searched for.
It also loads a saved workspace from file
.RData
in the current
directory if there is one (unless
--no-restore
or
--no-restore-data
was specified).
Finally, if a function
.First()
exists, it is executed. This
function (as well as
.Last()
which is executed at the end of the
R session) can be defined in the appropriate startup profiles, or
reside in
.RData
In addition, there are options for controlling the memory available to
the R process (see the on-line help for topic ‘
Memory
’ for more
information). Users will not normally need to use these unless they
are trying to limit the amount of memory used by R.
R accepts the following command-line options.
--help
-h
Print short help message to standard output and exit successfully.
--version
Print version information to standard output and exit successfully.
--encoding=
enc
Specify the encoding to be assumed for input from the console or
stdin
. This needs to be an encoding known to
iconv
: see
its help page. (
--encoding
enc
is also accepted.) The
input is re-encoded to the locale R is running in and needs to be
representable in the latter’s encoding (so e.g. you cannot re-encode
Greek text in a French locale unless that locale uses the UTF-8
encoding).
RHOME
Print the path to the R “home directory” to standard output and
exit successfully. Apart from the front-end shell script and the man
page, R installation puts everything (executables, packages, etc.)
into this directory.
--save
--no-save
Control whether data sets should be saved or not at the end of the R
session. If neither is given in an interactive session, the user is
asked for the desired behavior when ending the session with
q()
in non-interactive use one of these must be specified or implied by some
other option (see below).
--no-environ
Do not read any user file to set environment variables.
--no-site-file
Do not read the site-wide profile at startup.
--no-init-file
Do not read the user’s profile at startup.
--restore
--no-restore
--no-restore-data
Control whether saved images (file
.RData
in the directory where
R was started) should be restored at startup or not. The default is
to restore. (
--no-restore
implies all the specific
--no-restore-*
options.)
--no-restore-history
Control whether the history file (normally file
.Rhistory
in the
directory where R was started, but can be set by the environment
variable
R_HISTFILE
) should be restored at startup or not. The
default is to restore.
--no-Rconsole
(Windows only) Prevent loading the
Rconsole
file at startup.
--vanilla
Combine
--no-save
--no-environ
--no-site-file
--no-init-file
and
--no-restore
. Under Windows, this also includes
--no-Rconsole
-f
file
--file=
file
(not
Rgui.exe
) Take input from
file
: ‘
’ means
stdin
. Implies
--no-save
unless
--save
has
been set. On a Unix-alike, shell metacharacters should be avoided in
file
(but spaces are allowed).
-e
expression
(not
Rgui.exe
) Use
expression
as an input line. One or
more
-e
options can be used, but not together with
-f
or
--file
. Implies
--no-save
unless
--save
has been set. (There is a limit of 10,000 bytes on the total length of
expressions used in this way. Expressions containing spaces or shell
metacharacters will need to be quoted.)
--no-readline
(UNIX only) Turn off command-line editing via
readline
. This
is useful when running R from within Emacs using the
ESS
(“Emacs Speaks Statistics”) package. See
The command-line editor
for more information. Command-line editing is enabled for default
interactive use (see
--interactive
). This option also affects
tilde-expansion: see the help for
path.expand
--min-vsize=
--min-nsize=
For expert use only: set the initial trigger sizes for garbage
collection of vector heap (in bytes) and
cons cells
(number)
respectively. Suffix ‘
’ specifies megabytes or millions of cells
respectively. The defaults are 6Mb and 350k respectively and can also
be set by environment variables
R_NSIZE
and
R_VSIZE
--max-ppsize=
Specify the maximum size of the pointer protection stack as
locations. This defaults to 10000, but can be increased to allow
large and complicated calculations to be done. Currently the maximum
value accepted is 100000.
--quiet
--silent
-q
Do not print out the initial copyright and welcome messages.
--no-echo
Make R run as quietly as possible. This option is intended to
support programs which use R to compute results for them. It implies
--quiet
and
--no-save
--interactive
(UNIX only) Assert that R really is being run interactively even if
input has been redirected: use if input is from a FIFO or pipe and fed
from an interactive program. (The default is to deduce that R is
being run interactively if and only if
stdin
is connected to a
terminal or
pty
.) Using
-e
-f
or
--file
asserts non-interactive use even if
--interactive
is given.
Note that this does not turn on command-line editing.
--ess
(Windows only) Set
Rterm
up for use by
R-inferior-mode
in
ESS
, including asserting interactive use (without the
command-line editor) and no buffering of
stdout
--verbose
Print more information about progress, and in particular set R’s
option
verbose
to
TRUE
. R code uses this option to
control the printing of diagnostic messages.
--debugger=
name
-d
name
(UNIX only) Run R through debugger
name
. For most debuggers
(the exceptions are
valgrind
and recent versions of
gdb
), further command line options are disregarded, and should
instead be given when starting the R executable from inside the
debugger.
--gui=
type
-g
type
(UNIX only) Use
type
as graphical user interface (note that this
also includes interactive graphics). Currently, possible values for
type
are ‘
X11
’ (the default) and, provided that ‘
Tcl/Tk
support is available, ‘
Tk
’. (For back-compatibility, ‘
x11
’ and
tk
’ are accepted.)
--arch=
name
(UNIX only) Run the specified sub-architecture.
--args
This flag does nothing except cause the rest of the command line to be
skipped: this can be useful to retrieve values from it with
commandArgs(TRUE)
Note that input and output can be redirected in the usual way (using
’ and ‘
’), but the line length limit of 4095 bytes still
applies. Warning and error messages are sent to the error channel
stderr
).
The command
R CMD
allows the invocation of various tools which
are useful in conjunction with R, but not intended to be called
“directly”. The general form is
where
command
is the name of the tool and
args
the arguments
passed on to it.
Currently, the following tools are available.
BATCH
Run R in batch mode. Runs
R --restore --save
with possibly
further options (see
?BATCH
).
COMPILE
(UNIX only) Compile C, C++, Fortran … files for use with R.
SHLIB
Build shared library for dynamic loading.
INSTALL
Install add-on packages.
REMOVE
Remove add-on packages.
build
Build (that is, package) add-on packages.
check
Check add-on packages.
LINK
(UNIX only) Front-end for creating executable programs.
Rprof
Post-process R profiling files.
Rdconv
Rd2txt
Convert Rd format to various other formats, including
HTML
, LaTeX,
plain text, and extracting the examples.
Rd2txt
can be used as
shorthand for
Rd2conv -t txt
Rd2pdf
Convert Rd format to PDF.
Stangle
Extract S/R code from Sweave or other vignette documentation
Sweave
Process Sweave or other vignette documentation
Rdiff
Diff R output ignoring headers etc
config
Obtain configuration information
javareconf
(Unix only) Update the Java configuration variables
rtags
(Unix only) Create Emacs-style tag files from C, R, and Rd files
open
(Windows only) Open a file via Windows’ file associations
texify
(Windows only) Process (La)TeX files with R’s style files
Use
to obtain usage information for each of the tools accessible via the
R CMD
interface.
In addition, you can use options
--arch=
--no-environ
--no-init-file
--no-site-file
and
--vanilla
between
and
CMD
: these
affect any R processes run by the tools. (Here
--vanilla
is
equivalent to
--no-environ --no-site-file --no-init-file
.)
However, note that
R CMD
does not of itself use any R
startup files (in particular, neither user nor site
Renviron
files), and all of the R processes run by these tools (except
BATCH
) use
--no-restore
. Most use
--vanilla
and so invoke no R startup files: the current exceptions are
INSTALL
REMOVE
Sweave
and
SHLIB
(which uses
--no-site-file --no-init-file
).
for any other executable
cmd
on the path or given by an
absolute filepath: this is useful to have the same environment as R
or the specific commands run under, for example to run
ldd
or
pdflatex
. Under Windows
cmd
can be an executable or a
batch file, or if it has extension
.sh
or
.pl
the
appropriate interpreter (if available) is called to run it.
B.2 Invoking R under Windows
There are two ways to run R under Windows. Within a terminal window
(e.g.
cmd.exe
or a more capable shell), the methods described in
the previous section may be used, invoking by
R.exe
or more
directly by
Rterm.exe
. For interactive use, there is a
console-based GUI (
Rgui.exe
).
The startup procedure under Windows is very similar to that under
UNIX, but references to the ‘home directory’ need to be clarified, as
this is not always defined on Windows. If the environment variable
R_USER
is defined, that gives the home directory. Next, if the
environment variable
is defined, that gives the home
directory. After those two user-controllable settings, R tries to
find system defined home directories. It first tries to use the
Windows "personal" directory (typically
My Documents
in
recent versions of Windows). If that fails, and
environment variables
HOMEDRIVE
and
HOMEPATH
are defined
(and they normally are) these define the home directory. Failing all
those, the home directory is taken to be the starting directory.
You need to ensure that either the environment variables
TMPDIR
TMP
and
TEMP
are either unset or one of them points to a
valid place to create temporary files and directories.
Environment variables can be supplied as ‘
name
value
pairs on the command line.
If there is an argument ending
.RData
(in any case) it is
interpreted as the path to the workspace to be restored: it implies
--restore
and sets the working directory to the parent of the
named file. (This mechanism is used for drag-and-drop and file
association with
RGui.exe
, but also works for
Rterm.exe
If the named file does not exist it sets the working directory
if the parent directory exists.)
The following additional command-line options are available when
invoking
RGui.exe
--mdi
--sdi
--no-mdi
Control whether
Rgui
will operate as an MDI program
(with multiple child windows within one main window) or an SDI application
(with multiple top-level windows for the console, graphics and pager). The
command-line setting overrides the setting in the user’s
Rconsole
file.
--debug
Enable the “Break to debugger” menu item in
Rgui
, and trigger
a break to the debugger during command line processing.
Under Windows with
R CMD
you may also specify your own
.bat
.exe
.sh
or
.pl
file. It will be run
under the appropriate interpreter (Perl for
.pl
) with several
environment variables set appropriately, including
R_HOME
R_OSTYPE
PATH
BSTINPUTS
and
TEXINPUTS
. For
example, if you already have
latex.exe
on your path, then
will run LaTeX on
mydoc.tex
, with the path to R’s
share/texmf
macros appended to
TEXINPUTS
(With the MiKTeX build of LaTeX, using
R CMD texify mydoc
is often more convenient.)
B.3 Invoking R under macOS
There are two ways to run R under macOS. Within a
Terminal.app
window by invoking
, the methods described in the first
subsection apply. There is also console-based GUI (
R.app
) that by
default is installed in the
Applications
folder on your
system. It is a standard double-clickable macOS application.
The startup procedure under macOS is very similar to that under UNIX, but
R.app
does not make use of command-line arguments. The ‘home
directory’ is the one inside the R.framework, but the startup and
current working directory are set as the user’s home directory unless a
different startup directory is given in the Preferences window
accessible from within the GUI.
B.4 Scripting with R
If you just want to run a file
foo.R
of R commands, the
recommended way is to use
R CMD BATCH foo.R
. If you want to
run this in the background or as a batch job use OS-specific facilities
to do so: for example in most shells on Unix-alike OSes
R CMD
BATCH foo.R &
runs a background job.
You can pass parameters to scripts via additional arguments on the
command line: for example (where the exact quoting needed will depend on
the shell in use)
R CMD BATCH "--args arg1 arg2" foo.R &
will pass arguments to a script which can be retrieved as a character
vector by
args <- commandArgs(TRUE)
This is made simpler by the alternative front-end
Rscript
which can be invoked by
and this can also be used to write executable script files like (at
least on Unix-alikes, and in some Windows shells)
#! /path/to/Rscript
args <- commandArgs(TRUE)
...
q(status=)
If this is entered into a text file
runfoo
and this is made
executable (by
chmod 755 runfoo
), it can be invoked for
different arguments by
For further options see
help("Rscript")
. This writes R
output to
stdout
and
stderr
, and this can be redirected in
the usual way for the shell running the command.
If you do not wish to hardcode the path to
Rscript
but have it
in your path (which is normally the case for an installed R except on
Windows, but e.g. macOS users may need to add
/usr/local/bin
to their path), use
#! /usr/bin/env Rscript
...
At least in Bourne and bash shells, the
#!
mechanism does
not
allow extra arguments like
#! /usr/bin/env Rscript --vanilla
One thing to consider is what
stdin()
refers to. It is
commonplace to write R scripts with segments like
chem <- scan(n=24)
2.90 3.10 3.40 3.40 3.70 3.70 2.80 2.50 2.40 2.40 2.70 2.20
5.28 3.37 3.03 3.03 28.95 3.77 3.40 2.20 3.50 3.60 3.70 3.70
and
stdin()
refers to the script file to allow such traditional
usage. If you want to refer to the process’s
stdin
, use
"stdin"
as a
file
connection, e.g.
scan("stdin", ...)
Another way to write executable script files (suggested by
François Pinard) is to use a
here document
like
#!/bin/sh
[environment variables can be set here]
R --no-echo [other options] <

R program goes here...

EOF
but here
stdin()
refers to the program source and
"stdin"
will not be usable.
Short scripts can be passed to
Rscript
on the command-line
via
the
-e
flag. (Empty scripts are not accepted.)
Note that on a Unix-alike the input filename (such as
foo.R
should not contain spaces nor shell metacharacters.
Appendix C The command-line editor
C.1 Preliminaries
When the
GNU
readline
library is available at the
time R is configured for compilation under UNIX, an inbuilt command
line editor allowing recall, editing and re-submission of prior commands
is used. Note that other versions of
readline
exist and may be
used by the inbuilt command line editor: this is most common on macOS.
You can find out which version (if any) is available by running
extSoftVersion()
in an R session.
It can be disabled (useful for usage with
ESS
) using the startup option
--no-readline
Windows versions of R have somewhat simpler command-line editing: see
Console
’ under the ‘
Help
’ menu of the
GUI
, and the
file
README.Rterm
for command-line editing under
Rterm.exe
When using R with GNU
readline
capabilities, the functions described below are available, as well as
others (probably) documented in
man readline
or
info
readline
on your system.
Many of these use either Control or Meta characters. Control
characters, such as
Control-m
, are obtained by holding the
CTRL
down while you press the
key, and are written as
C-m
below. Meta characters, such as
Meta-b
, are typed by
holding down
META
and pressing
, and written as
M-b
in the following. If your terminal does not have a
META
key
enabled, you can still type Meta characters using two-character
sequences starting with
ESC
. Thus, to enter
M-b
, you could
type
ESC
. The
ESC
character sequences are also
allowed on terminals with real Meta keys. Note that case is significant
for Meta characters.
Some but not all versions of
readline
will recognize resizing of the terminal window so this is best avoided.
C.2 Editing actions
The R program keeps a history of the command lines you type,
including the erroneous lines, and commands in your history may be
recalled, changed if necessary, and re-submitted as new commands. In
Emacs-style command-line editing any straight typing you do while in
this editing phase causes the characters to be inserted in the command
you are editing, displacing any characters to the right of the cursor.
In
vi
mode character insertion mode is started by
M-i
or
M-a
, characters are typed and insertion mode is finished by typing
a further
ESC
. (The default is Emacs-style, and only that is
described here: for
vi
mode see the
readline
documentation.)
Pressing the
RET
command at any time causes the command to be
re-submitted.
Other editing actions are summarized in the following table.
C.3 Command-line editor summary
Command recall and vertical motion
C-p
Go to the previous command (backwards in the history).
C-n
Go to the next command (forwards in the history).
C-r
text
Find the last command with the
text
string in it. This can be
cancelled by
C-g
(and on some versions of R by
C-c
).
On most terminals, you can also use the up and down arrow keys instead
of
C-p
and
C-n
, respectively.
Horizontal motion of the cursor
C-a
Go to the beginning of the command.
C-e
Go to the end of the line.
M-b
Go back one word.
M-f
Go forward one word.
C-b
Go back one character.
C-f
Go forward one character.
On most terminals, you can also use the left and right arrow keys
instead of
C-b
and
C-f
, respectively.
Editing and re-submission
text
Insert
text
at the cursor.
C-f
text
Append
text
after the cursor.
DEL
Delete the previous character (left of the cursor).
C-d
Delete the character under the cursor.
M-d
Delete the rest of the word under the cursor, and “save” it.
C-k
Delete from cursor to end of command, and “save” it.
C-y
Insert (yank) the last “saved” text here.
C-t
Transpose the character under the cursor with the next.
M-l
Change the rest of the word to lower case.
M-c
Change the rest of the word to upper case.
RET
Re-submit the command to R.
The final
RET
terminates the command line editing sequence.
The
readline
key bindings can be customized in the usual way
via
~/.inputrc
file. These customizations can be
conditioned on application
, that is by including a section like
$if R
"\C-xd": "q('no')\n"
$endif
References
D. M. Bates and D. G. Watts (1988),
Nonlinear Regression Analysis and Its Applications.
John Wiley & Sons, New York.
Richard A. Becker, John M. Chambers and Allan R. Wilks (1988),
The New S Language.
Chapman & Hall, New York.
This book is often called the “
Blue Book
”.
John M. Chambers and Trevor J. Hastie eds. (1992),
Statistical Models in S.
Chapman & Hall, New York.
This is also called the “
White Book
”.
John M. Chambers (1998)
Programming with Data
Springer, New York.
This is also called the “
Green Book
”.
A. C. Davison and D. V. Hinkley (1997),
Bootstrap Methods and Their Applications
Cambridge University Press.
Annette J. Dobson (1990),
An Introduction to Generalized Linear Models
Chapman and Hall, London.
Peter McCullagh and John A. Nelder (1989),
Generalized Linear Models.
Second edition, Chapman and Hall, London.
John A. Rice (1995),
Mathematical Statistics and Data Analysis.
Second edition. Duxbury Press, Belmont, CA.
S. D. Silvey (1970),
Statistical Inference.
Penguin, London.