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.

1.2 Related software and documentation

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

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

There are now a number of books which describe how to use R for data analysis and statistics, and documentation for S/S-PLUS can typically be used with R, keeping the differences between the S implementations in mind. See What documentation exists for R? in The R statistical system FAQ.

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 https://CRAN.R-project.org) 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:

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

   $ mkdir work
   $ cd work
   

2. Start the R program with the command

   $ R
   

3. At this point R commands may be issued (see later).
4. To quit the R program the command is

   > q()
   

   At this point you will be asked whether you want to save the data from your R session. On some systems this will bring up a dialog box, and on others you will receive a text prompt to which you can respond yes, 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.

1. Make work the working directory and start the program as before:

   $ cd work
   $ R
   

2. Use the R program, terminating with the q() command at the end of the session.

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

1.6 An introductory session

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

> help(solve)

An alternative is

> ?solve

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

> help("[[")

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

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

> help.start()

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

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

> ??solve

Try ?help.search for details and more examples.

The examples on a help topic can normally be run by

> example(topic)

Windows versions of R have other optional help systems: use

> ?help

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 A and a are different symbols and would refer to different variables. The set of symbols which can be used in R names depends on the operating system and country within which R is being run (technically on the locale in use). Normally all alphanumeric symbols are allowed² (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 almost³ anywhere, starting with a hashmark (‘#’), everything to the end of the line is a comment.

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

+

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

Command lines entered at the console are limited⁴ 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 a command history. Once a command is located in this way, the cursor can be moved within the command using the horizontal arrow keys, and characters can be removed with the DEL key or added with the other keys. More details are provided later: see The command-line editor.

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

Alternatively, the Emacs text editor provides more general support mechanisms (via ESS, Emacs Speaks Statistics) for working interactively with R. See R and Emacs in The R statistical system FAQ.

1.10 Executing commands from or diverting output to a file

If commands⁵ 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

> source("commands.R")

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

> sink("record.lis")

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

> sink()

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

> objects()

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

To remove objects the function rm is available:

> 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 .RData⁶ in the current directory, and the command lines used in the session are saved to a file called .Rhistory.

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

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

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 x, say, consisting of five numbers, namely 10.4, 5.6, 3.1, 6.4 and 21.7, use the R command

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

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

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

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

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

> 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 lost⁸. So now if we were to use the command

> 1/x

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

The further assignment

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

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

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. Shorter vectors in the expression are recycled as often as need be (perhaps fractionally) until they match the length of the longest vector. In particular a constant is simply repeated. So with the above assignments the command

> v <- 2*x + y + 1

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

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

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

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

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

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

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

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

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

sqrt(-17)

will give NaN and a warning, but

sqrt(-17+0i)

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

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

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

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

which repeats each element of x 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 T and F, respectively. Note however that T and F are just variables which are set to TRUE and FALSE by default, but are not reserved words and hence can be overwritten by the user. Hence, you should always use TRUE and FALSE.

Logical vectors are generated by conditions. For example

> temp <- x > 13

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

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

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

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 x with value TRUE if and only if the corresponding element in x 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 x all of whose values are NA as the logical expression itself is incomplete and hence undecidable.

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

> 0/0

or

> Inf - Inf

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

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

Missing values are sometimes printed as <NA> 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. ⁹

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.

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

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

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

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

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

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

   > x[1:10]
   

   selects the first 10 elements of x (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.

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

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

   gives y all but the first five elements of x.

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

   > 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

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

replaces any missing values in x by zeros and

> y[y < 0] <- -y[y < 0]

has the same effect as

> y <- abs(y)

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.

3.1 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 numeric¹⁰, 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 ¹¹.

Further properties of an object are usually provided by attributes(object), see Getting and setting attributes. Because of this, mode and length are also called “intrinsic attributes” of an object.

For example, if z is a complex vector of length 100, then in an 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

> z <- 0:9

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 d and z are the same.¹² 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

> e <- numeric()

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

> e[3] <- 17

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

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

> length(alpha) <- 3

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 z 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 style¹³ 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

> winter

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

> unclass(winter)

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

Generic functions and classes will be discussed further in Classes, generic functions and object orientation, but only briefly.

4.1 A specific example

Suppose, for example, we have a sample of 30 tax accountants from all the states and territories of Australia¹⁴ and their individual state of origin is specified by a character vector of state mnemonics as

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

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

A factor is similarly created using the 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 statef¹⁵, 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 t-distributions. (You could also investigate R’s facilities for t-tests.)

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

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

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.

5.1 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 k then the array is k-dimensional, e.g. a matrix is a 2-dimensional array. The dimensions are indexed from one up to the values given in the dimension vector.

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

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

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

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

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

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

Arrays can be one-dimensional: such arrays are usually treated in the same way as vectors (including when printing), but the exceptions can cause confusion.

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 a alone.

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

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

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 4 by 5 array X and we wish to do the following:

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

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

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

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

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

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

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

> N <- crossprod(Xb, Xv)

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 h contains 24 or fewer, numbers then the command

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

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

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

However if h is shorter than 24, its values are recycled from the beginning again to make it up to size 24 (see 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 Z an array of all zeros.

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

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

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

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

• Mixed vector and array arithmetic. The recycling rule

5.5 The outer product of two arrays

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

> ab <- a %o% b

An alternative is

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

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

> 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 2 by 2 matrices [a, b; c, d] where each entry is a non-negative integer in the range 0, 1, …, 9, that is a digit.

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

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

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

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

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

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 A respectively.

• Matrix multiplication
• Linear equations and inversion
• Eigenvalues and eigenvectors
• Singular value decomposition and determinants
• Least squares fitting and the QR decomposition

> A * B

> A %*% B

> x %*% A %*% x

> b <- A %*% x

> solve(A,b)

solve(A)

> ev <- eigen(Sm)

> evals <- eigen(Sm)$values

> eigen(Sm)

> evals <- eigen(Sm, only.values = TRUE)$values

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

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

> ans <- lsfit(X, y)

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

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 X, together with an initial column of 1s we can use

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

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

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()

> vec <- as.vector(X)

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

> vec <- c(X)

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

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 k factor arguments, the result is a k-way array of frequencies.

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

> 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.

6.1 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

> name$component_name

for the same thing.

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

So in the simple example given above:

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

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

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

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

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

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 m 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)

• Concatenating lists

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

6.3 Data frames

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

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

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

• Making data frames
• attach() and detach()
• Working with data frames
• Attaching arbitrary lists
• Managing the search path

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

> attach(lentils)

> u <- v+w

> lentils$u <- v+w

> detach()

> attach(any.old.list)

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

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

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

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

data()

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

data(infert)

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

• Loading data from other R packages

data(package="rpart")
data(Puromycin, package="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

> xnew <- edit(xold)

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

Use

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

to enter new data via the spreadsheet interface.

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 q, the smallest x 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
F	f	df1, df2, ncp
gamma	gamma	shape, scale
geometric	geom	prob
hypergeometric	hyper	m, n, k
log-normal	lnorm	meanlog, sdlog
logistic	logis	location, scale
negative binomial	nbinom	size, prob
normal	norm	mean, sd
Poisson	pois	lambda
signed rank	signrank	n
Student’s t	t	df, ncp
uniform	unif	min, max
Weibull	weibull	shape, scale
Wilcoxon	wilcox	m, n

Prefix the name given here by ‘d’ for the density, ‘p’ for the CDF, ‘q’ for the quantile function and ‘r’ for simulation (random deviates). The first argument is x for dxxx, q for pxxx, p for qxxx and n for rxxx (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 pxxx and qxxx functions all have logical arguments lower.tail and log.p and the dxxx ones have log. This allows, e.g., getting the cumulative (or “integrated”) hazard function, H(t) = - log(1 - F(t)), by

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

or more accurate log-likelihoods (by dxxx(..., 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)

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

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)

which 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 t-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 t-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)

9.1 Grouped expressions

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

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

9.2 Control statements

• Conditional execution: if statements
• Repetitive execution: for loops, repeat and while

> if (expr_1) expr_2 else expr_3

> for (name in expr_1) expr_2

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

> repeat expr

> while (condition) expr

10.1 Simple examples

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

The function is defined as follows:

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

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

> 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 y onto the column space of the matrix, X. (This is ordinarily called the least squares estimate of the regression coefficients.) This would ordinarily be done with the qr() function; however this is sometimes a bit tricky to use directly and it pays to have a simple function such as the following to use it safely.

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

> 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 more²¹. 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

%anything%

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

> "%!%" <- 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

> ans <- fun1(d, df)

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

• Efficiency factors in block designs
• Dropping all names in a printed array
• Recursive numerical integration

> 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)
}

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

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

> no.dimnames(X)

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, x is a formal parameter, y is a local variable and z is a free variable.

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

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

The variable n in the function sq is not an argument to that function. Therefore it is a free variable and the scoping rules must be used to ascertain the value that is to be associated with it. Under static scope (S-PLUS) the value is that associated with a global variable named n. Under lexical scope (R) it is the parameter to the function cube since that is the active binding for the variable n at the time the function sq was defined. The difference between evaluation in R and evaluation in S-PLUS is that S-PLUS looks for a global variable called n while R first looks for a variable called n 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 it²². 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 .Rprofile²³ 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:

> methods(plot)

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*
[5] coef.nls*         coef.summary.nls*

   Non-visible functions are asterisked

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

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

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

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

A function named gen.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.

11.1 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

y = X  beta + e

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

Examples

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

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

y ~ x

y ~ 1 + x

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

y ~ 0 + x

y ~ -1 + x

y ~ x - 1

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

log(y) ~ x1 + x2

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

y ~ poly(x,2)

y ~ 1 + x + I(x^2)

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

y ~ X + poly(x,2)

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

y ~ A

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

y ~ A + x

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

y ~ A*B

y ~ A + B + A:B

y ~ B %in% A

y ~ A/B

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

y ~ (A + B + C)^2

y ~ A*B*C - A:B:C

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

y ~ A * x

y ~ A/x

y ~ A/(1 + x) - 1

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

y ~ A*B + Error(C)

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

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

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 1,
• a factor, or
• a formula expression consisting of factors, vectors or matrices connected by formula operators.

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

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

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

Y ~ M

Y is modeled as M.

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.

M^n

All terms in M together with “interactions” up to order n

I(M)

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

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

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

• Contrasts

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

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

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 y on x1 and x2 (with implicit intercept term).

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

11.3 Generic functions for extracting model information

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

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

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

anova(object_1, object_2)

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

coef(object)

Extract the regression coefficient (matrix).

Long form: coefficients(object).

deviance(object)

Residual sum of squares, weighted if appropriate.

formula(object)

Extract the model formula.

plot(object)

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

predict(object, newdata=data.frame)

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

print(object)

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

residuals(object)

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

Short form: resid(object).

step(object)

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

summary(object)

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

vcov(object)

Returns the variance-covariance matrix of the main parameters of a fitted model object.

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”.

• ANOVA tables

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

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 y and regressor variables all other variables in the data frame production.

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

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, y, of interest and stimulus variables x_1, x_2, …, whose values influence the distribution of the response.
• The stimulus variables influence the distribution of y through a single linear function, only. This linear function is called the linear predictor, and is usually written

  eta = beta_1 x_1 + beta_2 x_2 + … + beta_p x_p,
  

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

• The distribution of y is of the form

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

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

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

  mu = m(eta),    eta = m^{-1}(mu) = ell(mu)
  

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

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

• Families
• The glm() function

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

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

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

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

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

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

> summary(fmp)
> summary(fml)

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

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

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

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.

• Least squares
• Maximum likelihood

> 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)

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

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

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

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

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

> 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

> 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)

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

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

> sqrt(diag(solve(out$hessian)))

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.

12.1 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.

• The plot() function
• Displaying multivariate data
• Display graphics
• Arguments to high-level plotting functions

> pairs(X)

> coplot(a ~ b | c)

> coplot(a ~ b | c + d)

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=y)

abline(v=x)

abline(lm.obj)

Adds a line of slope b and intercept a to the current plot. h=y may be used to specify y-coordinates for the heights of horizontal lines to go across a plot, and v=x similarly for the x-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 (x, y) and (optionally) shade it in with hatch lines, or fill it if the graphics device allows the filling of figures.

legend(x, y, legend, …) ¶

Adds a legend to the current plot at the specified position. Plotting characters, line styles, colors etc., are identified with the labels in the character vector legend. At least one other argument v (a vector the same length as legend) with the corresponding values of the plotting unit must also be given, as follows:

legend( , fill=v)

Colors for filled boxes

legend( , col=v)

Colors in which points or lines will be drawn

legend( , lty=v)

Line styles

legend( , lwd=v)

Line widths

legend( , pch=v)

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., x and y coordinates) to determine where to place the new plot elements. Coordinates are given in terms of user coordinates which are defined by the previous high-level graphics command and are chosen based on the supplied data.

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

• Mathematical annotation
• Hershey vector fonts

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

> help(plotmath)
> example(plotmath)
> demo(plotmath)

> 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 n (default 512) points have been selected, or another mouse button is pressed. The type argument allows for plotting at the selected points and has the same effect as for high-level graphics commands; the default is no plotting. locator() returns the locations of the points selected as a list with two components x and y.

locator() is usually called with no arguments. It is particularly useful for interactively selecting positions for graphic elements such as legends or labels when it is difficult to calculate in advance where the graphic should be placed. For example, to place some informative text near an outlying point, the command

> 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 x and y (using the left mouse button) by plotting the corresponding component of labels nearby (or the index number of the point if labels is absent). Returns the indices of the selected points when another button is pressed.

Sometimes we want to identify particular points on a plot, rather than their positions. For example, we may wish the user to select some observation of interest from a graphical display and then manipulate that observation in some way. Given a number of (x, y) coordinates in two numeric vectors x and y, we could use the identify() function as follows:

> plot(x, y)
> identify(x, y)

The identify() functions performs no plotting itself, but simply allows the user to move the mouse pointer and click the left mouse button near a point. If there is a point near the mouse pointer it will be marked with its index number (that is, its position in the x/y vectors) plotted nearby. Alternatively, you could use some informative string (such as a case name) as a highlight by using the labels argument to identify(), or disable marking altogether with the plot = FALSE argument. When the process is terminated (see above), identify() returns the indices of the selected points; you can use these indices to extract the selected points from the original vectors x and y.

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.

• Permanent changes: The par() function
• Temporary changes: Arguments to graphics functions

> oldpar <- par(col=4, lty=2)
  … plotting commands …
> par(oldpar)

> oldpar <- par(no.readonly=TRUE)
  … plotting commands …
> par(oldpar)

> plot(x, y, pch="+")

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.

• Graphical elements
• Axes and tick marks
• Figure margins
• Multiple figure environment

Next: Multiple figure environment, Previous: Axes and tick marks, Up: Graphics parameters list   [Contents][Index]

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.

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Previous: Figure margins, Up: Graphics parameters list   [Contents][Index]

12.5.4 Multiple figure environment

R allows you to create an n by m array of figures on a single page. Each figure has its own margins, and the array of figures is optionally surrounded by an outer margin, as shown in the following figure.

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.

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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

> postscript()

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

> dev.off()

This ensures that the device finishes cleanly; for example in the case of hardcopy devices this ensures that every page is completed and has been sent to the printer. (This will happen automatically at the normal end of a session.)

• PostScript diagrams for typeset documents
• Multiple graphics devices

> postscript("file.ps", horizontal=FALSE, height=5, pointsize=10)

> postscript("plot1.eps", horizontal=FALSE, onefile=FALSE,
             height=8, width=6, pointsize=10)

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

http://ggobi.org/

and these can be accessed from R via the package rggobi, described at http://ggobi.org/rggobi.html.

Also, package rgl provides ways to interact with 3D plots, for example of surfaces.

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. See R packages in R FAQ, for a complete list.

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 (https://CRAN.R-project.org/ and its mirrors) and other repositories such as Bioconductor (https://www.bioconductor.org/). The R FAQ contains a list of CRAN packages current at the time of release, but the collection of available packages changes very frequently.

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 t. Namespaces prevent the user’s definition from taking precedence, and breaking every function that tries to transpose a matrix.

There are two operators that work with namespaces. The double-colon operator :: selects definitions from a particular namespace. In the example above, the transpose function will always be available as base::t, because it is defined in the base package. Only functions that are exported from the package can be retrieved in this way.

The triple-colon operator ::: may be seen in a few places in R code: it acts like the double-colon operator but also allows access to hidden objects. Users are more likely to use the getAnywhere() function, which searches multiple packages.

Packages are often inter-dependent, and loading one may cause others to be automatically loaded. The colon operators described above will also cause automatic loading of the associated package. When packages with namespaces are loaded automatically they are not added to the search list.

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 (ACLs) are employed on several filesystems, but do not have an agreed standard and R has no facilities to control them. Use Sys.chmod to change permissions.

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 vey large extent done by connections, and the file function which is used to open a connection to a file (or a URL) and is able to identify the compression used from the ‘magic’ header of the file.

The type of compression which has been supported for longest is gzip compression, and that remains a good general compromise. Files compressed by the earlier Unix compress utility can also be read, but these are becoming rare. Two other forms of compression, those of the bzip2 and xz utilities are also available. These generally achieve higher rates of compression (depending on the file, much higher) at the expense of slower decompression and much slower compression.

There is some confusion between xz and lzma compression (see https://en.wikipedia.org/wiki/Xz and https://en.wikipedia.org/wiki/LZMA): R can read files compressed by most versions of either.

File archives are single files which contain a collection of files, the most common ones being ‘tarballs’ and zip files as used to distribute R packages. R can list and unpack both (see functions untar and unzip) and create both (for zip with the help of an external program).