# R intro

## Transcript

1 An Introduction to R Notes on R: A Programming Environment for Data Analysis and Graphics Version 3.6.0 (2019-04-26) W. N. Venables, D. M. Smith and the R Core Team

3 i Table of Contents ... 1 Preface 2 ... 1 Introduction and preliminaries ... 2 1.1 The R environment 1.2 Related software and documentation ... 2 ... 2 1.3 R and statistics 3 1.4 R and the window system ... 3 1.5 Using R interactively ... 4 1.6 An introductory session ... 1.7 Getting help with functions and features ... 4 1.8 R commands, case sensitivity, etc. ... 4 1.9 Recall and correction of previous commands 5 ... 1.10 Executing commands from or diverting output to a file 5 ... ... 1.11 Data permanency and removing objects 5 2 Simple manipulations; numbers and vectors ... 7 2.1 Vectors and assignment ... 7 ... 7 2.2 Vector arithmetic 2.3 Generating regular sequences ... 8 ... 9 2.4 Logical vectors 2.5 Missing values 9 ... 2.6 Character vectors ... 10 2.7 Index vectors; selecting and modifying subsets of a data set ... 10 2.8 Other types of objects ... 11 ... 13 3 Objects, their modes and attributes 13 3.1 Intrinsic attributes: mode and length ... 3.2 Changing the length of an object ... 14 3.3 Getting and setting attributes ... 14 ... 14 3.4 The class of an object ... 4 Ordered and unordered factors 16 ... 4.1 A specific example 16 tapply() and ragged arrays 4.2 The function 16 ... 4.3 Ordered factors ... 17 5 Arrays and matrices ... 18 5.1 Arrays ... 18 ... 5.2 Array indexing. Subsections of an array 18 ... 19 5.3 Index matrices array() function ... 20 5.4 The 5.4.1 Mixed vector and array arithmetic. The recycling rule ... 20 5.5 The outer product of two arrays 21 ... 5.6 Generalized transpose of an array ... 21 5.7 Matrix facilities ... 22 5.7.1 Matrix multiplication ... 22

4 ii 5.7.2 Linear equations and inversion 22 ... ... 23 5.7.3 Eigenvalues and eigenvectors 23 ... 5.7.4 Singular value decomposition and determinants 5.7.5 Least squares fitting and the QR decomposition 23 ... and ... 24 cbind() 5.8 Forming partitioned matrices, rbind() , with arrays ... 24 5.9 The concatenation function, c() 25 ... 5.10 Frequency tables from factors 26 6 Lists and data frames ... ... 26 6.1 Lists 6.2 Constructing and modifying lists ... 27 ... 27 6.2.1 Concatenating lists ... 6.3 Data frames 27 6.3.1 Making data frames 27 ... attach() and ... 28 6.3.2 detach() ... 6.3.3 Working with data frames 28 6.3.4 Attaching arbitrary lists ... 28 ... 29 6.3.5 Managing the search path 7 Reading data from files ... 30 read.table() ... 30 7.1 The function scan() 7.2 The ... 31 function 7.3 Accessing builtin datasets ... 31 7.3.1 Loading data from other R packages ... 31 7.4 Editing data ... 32 ... 33 8 Probability distributions 33 8.1 R as a set of statistical tables ... ... 34 8.2 Examining the distribution of a set of data 8.3 One- and two-sample tests ... 36 ... 40 9 Grouping, loops and conditional execution ... 40 9.1 Grouped expressions ... 9.2 Control statements 40 if ... 40 9.2.1 Conditional execution: statements for 9.2.2 Repetitive execution: repeat and while ... 40 loops, 10 Writing your own functions ... 42 10.1 Simple examples ... 42 ... 10.2 Defining new binary operators 43 10.3 Named arguments and defaults ... 43 ... 10.4 The ‘ ... 44 ’ argument 10.5 Assignments within functions ... 44 10.6 More advanced examples ... 44 10.6.1 Efficiency factors in block designs 44 ... 10.6.2 Dropping all names in a printed array ... 45 10.6.3 Recursive numerical integration ... 45 10.7 Scope ... 46 10.8 Customizing the environment ... 48 10.9 Classes, generic functions and object orientation ... 48

5 iii 11 Statistical models in R 51 ... ... 51 11.1 Defining statistical models; formulae 53 ... 11.1.1 Contrasts ... 11.2 Linear models 54 11.3 Generic functions for extracting model information 54 ... 11.4 Analysis of variance and model comparison ... 55 55 ... 11.4.1 ANOVA tables 55 11.5 Updating fitted models ... ... 11.6 Generalized linear models 56 57 ... 11.6.1 Families 11.6.2 The function ... 57 glm() ... 59 11.7 Nonlinear least squares and maximum likelihood models 11.7.1 Least squares ... 59 ... 61 11.7.2 Maximum likelihood 61 11.8 Some non-standard models ... 12 Graphical procedures ... 63 12.1 High-level plotting commands ... 63 plot() function ... 63 12.1.1 The ... 12.1.2 Displaying multivariate data 64 ... 12.1.3 Display graphics 64 ... 65 12.1.4 Arguments to high-level plotting functions 12.2 Low-level plotting commands 66 ... 12.2.1 Mathematical annotation ... 67 12.2.2 Hershey vector fonts ... 67 12.3 Interacting with graphics ... 67 ... 68 12.4 Using graphics parameters function 68 12.4.1 Permanent changes: The ... par() 12.4.2 Temporary changes: Arguments to graphics functions ... 69 ... 69 12.5 Graphics parameters list ... 70 12.5.1 Graphical elements ... 12.5.2 Axes and tick marks 71 ... 12.5.3 Figure margins 71 ... 73 12.5.4 Multiple figure environment 12.6 Device drivers 74 ... 12.6.1 PostScript diagrams for typeset documents ... 74 12.6.2 Multiple graphics devices ... 75 12.7 Dynamic graphics ... 76 ... 13 Packages 77 13.1 Standard packages ... 77 13.2 Contributed packages and ... 77 CRAN 13.3 Namespaces ... 78 14 OS facilities ... 79 14.1 Files and directories 79 ... 14.2 Filepaths ... 79 14.3 System commands ... 80 14.4 Compression and Archives ... 80 Appendix A A sample session ... 82

6 iv Appendix B Invoking R 85 ... ... B.1 Invoking R from the command line 85 ... 89 B.2 Invoking R under Windows B.3 Invoking R under macOS 90 ... B.4 Scripting with R ... 90 Appendix C The command-line editor ... 92 C.1 Preliminaries 92 ... C.2 Editing actions ... 92 C.3 Command-line editor summary ... 92 Appendix D Function and variable index ... 94 Appendix E Concept index ... 97 Appendix F References ... 99

7 1 Preface S-Plus This introduction to R is derived from an original set of notes describing the S and environments written in 1990–2 by Bill Venables and David M. Smith when at the University of Adelaide. We have made a number of small changes to reflect differences between the R and S programs, and expanded some of the material. We would like to extend warm thanks to Bill Venables (and David Smith) for granting permission to distribute this modified version of the notes in this way, and for being a supporter of R from way back. Comments and corrections are always welcome. Please address email correspondence to [email protected] . Suggestions to the reader Most R novices will start with the introductory session in Appendix A. This should give some familiarity with the style of R sessions and more importantly some instant feedback on what actually happens. Many users will come to R mainly for its graphical facilities. See Chapter 12 [Graphics], page 63 , which can be read at almost any time and need not wait until all the preceding sections have been digested.

8 2 1 Introduction and preliminaries 1.1 The R environment R is an integrated suite of software facilities for data manipulation, calculation and graphical display. Among other things it has • an effective data handling and storage facility, • a suite of operators for calculations on arrays, in particular matrices, • a large, coherent, integrated collection of intermediate tools for data analysis, • graphical facilities for data analysis and display either directly at the computer or on hard- copy, 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 packages developed rapidly, and has been extended by a large collection of . 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 Labora- S-Plus tories by Rick Becker, John Chambers and Allan Wilks, and also forms the basis of the 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 Programming with Data based on those described in by John M. Chambers. See Appendix F [References], page 99 , 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/ can typically be used with R, keeping the differences between S-Plus Section “What documentation exists for R?” in The R the S implementations in mind. See . statistical system FAQ 1.3 R and statistics Our introduction to the R environment did not mention , yet many people use R as a statistics 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 Chapter 13 [Packages], page 77 ). 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.

9 Chapter 1: Introduction and preliminaries 3 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 straight- forward if somewhat tedious procedure, and will not be considered further here. Users in diffi- culty 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 , no or cancel (a single letter abbreviation will do) to save yes 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.

11 Chapter 1: Introduction and preliminaries 5 ; Commands are separated either by a semi-colon (‘ ’), or by a newline. Elementary commands ’ and ‘ ’). Comments can can be grouped together into one compound expression by braces (‘ } { 2 # anywhere, starting with a hashmark (‘ be put almost ’), 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. 3 Command lines entered at the console are limited to about 4095 bytes (not characters). 1.9 Recall and correction of previous commands Under many versions of UNIX and on Windows, R provides a mechanism for recalling and re- executing previous commands. The vertical arrow keys on the keyboard can be used to scroll forward and backward through 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 Appendix C [The command-line editor], page 92 . 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 Section “R and Emacs” in The R statistical system FAQ . 1.10 Executing commands from or diverting output to a file 4 If commands are stored in an external file, say commands.R in the working directory work , they may be executed at any time in an R session with the command > source("commands.R") For Windows is also available on the File menu. The function sink , Source > 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 2 not inside strings, nor within the argument list of a function definition 3 some of the consoles will not allow you to enter more, and amongst those which do some will silently discard the excess and some will use it as the start of the next line. 4 of unlimited length.

12 6 To remove objects the function is available: rm > 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 5 .RData in the current directory, and the command lines used in the session are saved to a file called .Rhistory . When R is started at later time from the same directory it reloads the workspace from this file. At the same time the associated commands history is reloaded. It is recommended that you should use separate working directories for analyses conducted x and y to be created during an analysis. with R. It is quite common for objects with names 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. 5 The leading “dot” in this file name makes it invisible in normal file listings in UNIX, and in default GUI file listings on macOS and Windows.

13 7 2 Simple manipulations; numbers and vectors 2.1 Vectors and assignment data structures , which . The simplest such structure is the numeric R operates on named vector x is a single entity consisting of an ordered collection of numbers. To set up a vector named , say, consisting of five numbers, namely 10.4, 5.6, 3.1, 6.4 and 21.7, use the R command > x <- c(10.4, 5.6, 3.1, 6.4, 21.7) This is an function c() which in this context can take an assignment statement using the arguments and whose value is a vector got by concatenating its arbitrary number of vector 1 arguments end to end. A number occurring by itself in an expression is taken as a vector of length one. <- ’), which consists of the two characters ‘ < ’ (“less Notice that the assignment operator (‘ - ’ (“minus”) occurring strictly side-by-side and it ‘points’ to the object receiving than”) and ‘ 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)) <- , can be thought of as a syntactic short-cut to this. The usual operator, Assignments can also be made in the other direction, using the obvious change in the assign- ment operator. So the same assignment could be made using > c(10.4, 5.6, 3.1, 6.4, 21.7) -> x 2 and lost If an expression is used as a complete command, the value is printed . So now if we were to use the command > 1/x x the reciprocals of the five values would be printed at the terminal (and the value of , of course, unchanged). The further assignment > y <- c(x, 0, x) y with 11 entries consisting of two copies of x with a zero in the middle would create a vector 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 of length 11 constructed by adding together, element by element, 2*x v repeated 2.2 times, y repeated just once, and 1 repeated 11 times. 1 With other than vector types of argument, such as list mode arguments, the action of c() is rather different. See Section 6.2.1 [Concatenating lists], page 27 . 2 Actually, it is still available as .Last.value before any other statements are executed.

14 Chapter 2: Simple manipulations; numbers and vectors 8 + , , * , / and ^ for raising to a power. In The elementary arithmetic operators are the usual - , sqrt , sin , cos , tan , log , addition all of the common arithmetic functions are available. exp and select the largest and smallest elements of a max and so on, all have their usual meaning. min is a function whose value is a vector of length two, namely range c(min(x), vector respectively. length(x) is the number of elements in max(x)) , sum(x) gives the total of the elements in . x x prod(x) their product. , and Two statistical functions are which calculates the sample mean, which is the same mean(x) sum(x)/length(x) var(x) which gives as , and sum((x-mean(x))^2)/(length(x)-1) var() is an n -by- p matrix the value is a p -by- p sample or sample variance. If the argument to covariance matrix got by regarding the rows as independent p -variate sample vectors. returns a vector of the same size as with the elements arranged in increasing order; sort(x) x order() or sort.list() however there are other more flexible sorting facilities available (see 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 and a warning, but NaN 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 is the vector c(1, 2, ..., 29, 30) . The colon operator has high priority within an ex- 1:30 2*1:15 is the vector pression, so, for example . Put n <- 10 and compare c(2, 4, ..., 28, 30) the sequences and 1:(n-1) . 1:n-1 30:1 The construction 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 seq(1,30) and to= from= ; thus value , seq(from=1, to=30) and seq(to=30, from=1) value are all the same as 1:30 . The next two arguments to seq() may be named by= value and length= , which specify a step size and a length for the sequence respectively. If neither value 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)

15 Chapter 2: Simple manipulations; numbers and vectors 9 s4 . generates the same vector in vector along= , which is normally used as the only argu- The fifth argument may be named ) ment to create the sequence , or the empty sequence if the vector 1, 2, ..., length( vector is empty (as it can be). rep() A related function is which can be used for replicating an object in various complicated ways. The simplest form is > s5 <- rep(x, times=5) end-to-end in which will put five copies of . Another useful version is x s5 > s6 <- rep(x, each=5) five times before moving on to the next. which repeats each element of x 2.4 Logical vectors As well as numerical vectors, R allows manipulation of logical quantities. The elements of a logical vector can have the values TRUE FALSE , and NA (for “not available”, see below). The , first two are often abbreviated as and F , respectively. Note however that T and F are just T variables which are set to and FALSE by default, but are not reserved words and hence can TRUE TRUE FALSE . be overwritten by the user. Hence, you should always use and Logical vectors are generated by conditions . For example > temp <- x > 13 temp as a vector of the same length as sets with values FALSE corresponding to elements of x 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 c2 are logical expressions, then c1 & c2 is their intersection ( “and” ), c1 | c2 is their and “or” !c1 is the negation of c1 . union ( ), and coerced into Logical vectors may be used in ordinary arithmetic, in which case they are FALSE TRUE 0 and numeric vectors, becoming 1 . However there are situations where becoming 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 gives a logical vector of the same size as x with value TRUE if and is.na(x) x is NA . only if the corresponding element in > 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 is a vector of the x == NA 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 com- putation, the so-called Not a Number , NaN , values. Examples are > 0/0 or > Inf - Inf

16 Chapter 2: Simple manipulations; numbers and vectors 10 NaN since the result cannot be defined sensibly. which both give NaN is for NA and both values. To differentiate these, In summary, TRUE is.na(xx) TRUE for NaN is.nan(xx) is only s. Missing values are sometimes printed as when character vectors are printed without quotes. 2.6 Character vectors Character quantities and character vectors are used frequently in R, for example as plot labels. Where needed they are denoted by a sequence of characters delimited by the double quote "x-values" , "New iteration results" . character, e.g., " Character strings are entered using either matching double ( ’ ) quotes, but are ) or single ( printed using double quotes (or sometimes without quotes). They use C-style escape sequences, \ \\ is entered and printed as \\ , and inside double quotes " using as the escape character, so \" . Other useful escape sequences are \n , newline, \t , tab and is entered as , backspace—see \b ?Quotes for a full list. c() function; examples of their Character vectors may be concatenated into a vector by the use will emerge frequently. paste() function takes an arbitrary number of arguments and concatenates them one by The 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="") labs into the character vector makes c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10") c("X", "Y") is Note particularly that recycling of short lists takes place here too; thus 3 repeated 5 times to match the sequence . 1:10 2.7 Index vectors; selecting and modifying subsets of a data set Subsets of the elements of a vector may be selected by appending to the name of the vector an in square brackets. More generally any expression that evaluates to a vector may index vector 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. . In this case the index vector is recycled to the same length as the vector A logical vector TRUE from which elements are to be selected. Values corresponding to 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 y will be shorter than x . Also has missing values, > (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. 3 ss ss ) joins the arguments into a single character string putting ss in between, e.g., paste(..., collapse= <- "|" . There are more tools for character manipulation, see the help for sub and substring .

17 Chapter 2: Simple manipulations; numbers and vectors 11 A vector of positive integral quantities . In this case the values in the index vector must lie 2. 1, 2, . . . , } . The corresponding elements of the vector are selected and in the set length(x) { concatenated, in that order , in the result. The index vector can be of any length and the is the sixth component result is of the same length as the index vector. For example x[6] and x of > 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. A vector of negative integral quantities 3. . Such an index vector specifies the values to be excluded rather than included. Thus > y <- x[-(1:5)] y all but the first five elements of x . gives A vector of character strings . This possibility only applies where an object has a names 4. 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 expression the assignment operation is performed only on those elements of the vector vector[ ] as having an arbitrary expression in place of the must be of the form index_vector 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 Chapter 5 [Arrays and matrices], page 18 . ways. See factors provide compact ways to handle categorical data. See • Chapter 4 [Factors], page 16 . • 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 Section 6.1 [Lists], page 26 . • 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)

18 Chapter 2: Simple manipulations; numbers and vectors 12 both numerical and categorical variables. Many experiments are best described by data frames: the treatments are categorical but the response is numeric. See Section 6.3 [Data frames], page 27 . • 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 Chapter 10 [Writing your own functions], page 42 .

19 13 3 Objects, their modes and attributes 3.1 Intrinsic attributes: mode and length The entities R operates on are technically known as objects . Examples are vectors of numeric (real) or complex values, vectors of logical values and vectors of character strings. These are mode known as “atomic” structures since their components are all of the same type, or , namely 1 complex , logical , character and raw numeric . , . Thus any given vector must be un- all of the same mode Vectors must have their values character , , complex ambiguously either numeric or raw . (The only apparent exception logical , NA to this rule is the special “value” listed as for quantities not available, but in fact there are several types of ). Note that a vector can be empty and still have a mode. For example NA character(0) and the empty numeric vector as the empty character string vector is listed as numeric(0) . R also operates on objects called list . These are ordered sequences lists , which are of mode are known as “recursive” rather than lists of objects which individually can be of any mode. atomic structures since their components can themselves be lists in their own right. function The other recursive structures are those of mode expression . Functions are and 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 of an object we mean the basic type of its fundamental constituents. This is a mode 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 2 any defined structure . attributes( object , see Further properties of an object are usually provided by ) Section 3.3 . Because of this, [Getting and setting attributes], page 14 and length are also called mode “intrinsic attributes” of an object. z is a complex vector of length 100, then in an expression For example, if is the mode(z) character string and length(z) is 100 "complex" . 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) 3 something z are the same. d There is a large collection of functions of the form Now and () as. 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. 1 numeric mode is actually an amalgam of two distinct modes, namely integer and double precision, as explained in the manual. 2 Note however that length( object ) does not always contain intrinsic useful information, e.g., when object is a function. 3 In general, coercion from numeric to character and back again will not be exactly reversible, because of roundoff errors in the character representation.

20 Chapter 3: Objects, their modes and attributes 14 3.2 Changing the length of an object An “empty” object may still have a mode. For example > e <- numeric() an empty vector structure of mode numeric. Similarly is a empty character makes character() e 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 a vector of length 3, (the first two components of which are at this point both NA now makes e ). 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 Section 7.2 [The scan() function], page 31 .) Conversely to truncate the size of an object requires only an assignment to do so. Hence if is an object of length 10, then alpha > 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 attr( object defined for that object. The function name ) can be used to select a specific , attribute. These functions are rarely used, except in rather special circumstances when some new attribute is being created for some particular purpose, for example to associate a creation date or an operator with an R object. The concept, however, is very important. Some care should be exercised when assigning or deleting attributes since they are an integral part of the object system used in R. When it is used on the left hand side of an assignment it can be used either to associate a new attribute with object or to change an existing one. For example > attr(z, "dim") <- c(10,10) allows R to treat as if it were a 10-by-10 matrix. z 3.4 The class of an object All objects in R have a , reported by the function class . For simple vectors this is just the class "numeric" , "logical" , "character" mode, for example "list" , but "matrix" , "array" , or "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 4 style "data.frame" , it will be printed of programming in R. For example if an object has class plot() in a certain way, the 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 4 A different style using ‘formal’ or ‘S4’ classes is provided in package methods .

21 Chapter 3: Objects, their modes and attributes 15 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 Section 10.9 [Object orientation], page 48 , but only briefly.

22 16 4 Ordered and unordered factors factor is a vector object used to specify a discrete classification (grouping) of the components A and factors. While the of other vectors of the same length. R provides both unordered ordered “real” application of factors is with model formulae (see Section 11.1.1 [Contrasts], page 53 ), we here look at a specific example. 4.1 A specific example Suppose, for example, we have a sample of 30 tax accountants from all the states and territories 1 and their individual state of origin is specified by a character vector of state of Australia mnemonics as > state <- c("tas", "sa", "qld", "nsw", "nsw", "nt", "wa", "wa", "qld", "vic", "nsw", "vic", "qld", "qld", "sa", "tas", "sa", "nt", "wa", "vic", "qld", "nsw", "nsw", "wa", "sa", "act", "nsw", "vic", "vic", "act") Notice that in the case of a character vector, “sorted” means sorted in alphabetical order. factor factor() function: A is similarly created using the > statef <- factor(state) print() function handles factors slightly differently from other objects: The > 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 2 incomes , defined by the levels of the second component, here statef of the first argument, here , 1 Readers should note that there are eight states and territories in Australia, namely the Australian Capital Territory, New South Wales, the Northern Territory, Queensland, South Australia, Tasmania, Victoria and Western Australia. 2 Note that tapply() also works in this case when its second argument is not a factor, e.g., ‘ tapply(incomes, state) ’, and this is true for quite a few other functions, since arguments are coerced to factors when necessary (using as.factor() ).

23 Chapter 4: Ordered and unordered factors 17 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 Chapter 10 [Writing your own functions], page 42 . sd() Note that R’s a builtin function 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 tapply() once more with the length() function to find incomes. To do this you could use 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.) tapply() The function 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.

24 18 5 Arrays and matrices 5.1 Arrays An array can be considered as a multiply subscripted collection of data entries, for example numeric. R allows simple facilities for creating and handling arrays, and in particular the special case of matrices. k then the array is A dimension vector is a vector of non-negative integers. If its length 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. z is a vector of 1500 elements. The assignment Suppose, for example, > dim(z) <- c(3,5,100) dim attribute that allows it to be treated as a 3 by 5 by 100 array. gives it the matrix() Other functions such as array() are available for simpler and more natural and looking assignments, as we shall see in Section 5.4 [The array() function], page 20 . 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. a For example if the dimension vector for an array, say c(3,4,2) then there are 3 × 4 × , is 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.

25 Chapter 5: Arrays and matrices 19 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: X[1,3] , X[2,2] and X[3,1] as a vector structure, and • Extract elements • X by zeroes. Replace these entries in the array In this case we need a 3 by 2 subscript array, as in the following example. Generate a 4 by 5 array. > x <- array(1:20, dim=c(4,5)) # > 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 > NA Negative indices are not allowed in index matrices. 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 plots in the experiment. We could proceed as follows: n > 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)

26 Chapter 5: Arrays and matrices 20 table() : However a simpler direct way of producing this matrix is to use > 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. array() function 5.4 The As well as giving a vector structure a dim attribute, arrays can be constructed from vectors by function, which has the form the array , dim_vector ) data_vector > Z <- array( For example, if the vector h contains 24 or fewer, numbers then the command > Z <- array(h, dim=c(3,4,2)) h would use Z . If the size of h is exactly 24 the result is the same to set up 3 by 4 by 2 array in as > Z <- h ; dim(Z) <- c(3,4,2) h is shorter than 24, its values are recycled from the beginning again to make it However if Section 5.4.1 [The recycling rule], page 20 ) but dim(h) <- c(3,4,2) would up to size 24 (see 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 Z[] with an empty subscript or Z with no subscript stands for , and the entire array as an array. Arrays may be used in arithmetic expressions and the result is an array formed by element- dim attributes of operands generally need to be by-element operations on the data vector. The A , B the same, and this becomes the dimension vector of the result. So if C are all similar and 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. 5.4.1 Mixed vector and array arithmetic. The recycling rule The precise rule affecting element by element mixed calculations with vectors and arrays is somewhat quirky and hard to find in the references. From experience we have found the following to be a reliable guide. • The expression is scanned from left to right. • Any short vector operands are extended by recycling their values until they match the size of any other operands. • As long as short vectors and arrays only are encountered, the arrays must all have the same dim attribute or an error results. • Any vector operand longer than a matrix or array operand generates an error. • If array structures are present and no error or coercion to vector has been precipitated, the result is an array structure with the common dim attribute of its array operands.

27 Chapter 5: Arrays and matrices 21 5.5 The outer product of two arrays An important operation on arrays is the a and b are two numeric arrays, . If outer product 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 with those of b . The outer product is formed by products of elements of the data vector of a : %o% the special operator > 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 2 ( x ; y ) = cos( y ) / (1 + example if we wished to evaluate the function f ) over a regular grid of x values with - and y -coordinates defined by the R vectors x and y respectively, we could proceed x 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 Chapter 10 [Writing . your own functions], page 42 An example: Determinants of 2 by 2 single-digit matrices a,b ; c,d As an artificial but cute example, consider the determinants of 2 by 2 matrices [ ] 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. function twice: A neat way of doing this uses the outer() > d <- outer(0:9, 0:9) > fr <- table(outer(d, d, "-")) > plot(fr, xlab="Determinant", ylab="Frequency") plot() here uses a histogram like plot method, because it “sees” that fr is of Notice that "table" class for loops, to be discussed in . The “obvious” way of doing this problem with Chapter 9 [Loops and conditional execution], page 40 , 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) a . The argument perm must be may be used to permute an array, 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 but with old dimension given by perm[j] becoming a 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) .

28 Chapter 5: Arrays and matrices 22 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 is the matrix transpose function, as noted above. t(X) available only for matrices. For example ncol(A) A nrow(A) give the number of rows and columns in the matrix and The functions respectively. 5.7.1 Matrix multiplication n by 1 or 1 by n matrix may of course %*% The operator is used for matrix multiplication. An -vector if in the context such is appropriate. Conversely, vectors which occur in be used as an n matrix multiplication expressions are automatically promoted either to row or column vectors, whichever is multiplicatively coherent, if possible, (although this is not always unambiguously possible, as we see later). and are square matrices of the same size, then If, for example, A B > A * B is the matrix of element by element products and > A %*% B is a vector, then is the matrix product. If x > x %*% A %*% x 1 is a quadratic form. crossprod() forms “crossproducts”, meaning that crossprod(X, y) is the The function t(X) %*% y same as crossprod() but the operation is more efficient. If the second argument to is omitted it is taken to be the same as the first. The meaning of diag(v) , where v is a vector, gives a diag() depends on its argument. diag(M) diagonal matrix with elements of the vector as the diagonal entries. On the other hand , where M is a matrix, gives the vector of main diagonal entries of M . This is the same convention diag() in Matlab . Also, somewhat confusingly, if as that used for is a single numeric value k then is the k by k identity matrix! diag(k) 5.7.2 Linear equations and inversion Solving linear equations is the inverse of matrix multiplication. When after > b <- A %*% x A and only are given, the vector x is the solution of that linear equation system. In R, b > solve(A,b) solves the system, returning x (up to some accuracy loss). Note that in linear algebra, formally − 1 − 1 x where A = b denotes the inverse of A , which can be computed by A solve(A) but rarely is needed. Numerically, it is both inefficient and potentially unstable to compute x instead of solve(A,b) . <- solve(A) %*% b − 1 T x A The quadratic form which is used in multivariate computations, should be computed x 2 by something like x %*% solve(A,x) , rather than computing the inverse of A . 1 T T is ambiguous, as it could mean either x x x or xx Note that , where x %*% x is the column form. In such T cases the smaller matrix seems implicitly to be the interpretation adopted, so the scalar x is in this case x T xx the result. The matrix may be calculated either by cbind(x) %*% x or x %*% rbind(x) since the result of T T crossprod(x) cbind() or x or xx rbind() is x or x is always a matrix. However, the best way to compute respectively. %o% x 2 T B with A = BB Even better would be to form a matrix square root and find the squared length of the solution of By = x , perhaps using the Cholesky or eigen decomposition of A .

29 Chapter 5: Arrays and matrices 23 5.7.3 Eigenvalues and eigenvectors The function eigen(Sm) calculates the eigenvalues and eigenvectors of a symmetric matrix values vectors . The . The result of this function is a list of two components named and Sm assignment > ev <- eigen(Sm) . Then ev\$val is the vector of eigenvalues of Sm will assign this list to ev\$vec is the ev and matrix of corresponding eigenvectors. Had we only needed the eigenvalues we could have used the assignment: > evals <- eigen(Sm)\$values evals now holds the vector of eigenvalues and the second component is discarded. If the expression > eigen(Sm) is used by itself as a command the two components are printed, with their names. For large matrices it is better to avoid computing the eigenvectors if they are not needed by using the expression > evals <- eigen(Sm, only.values = TRUE)\$values 5.7.4 Singular value decomposition and determinants takes an arbitrary matrix argument, M , and calculates the singular value The function svd(M) . This consists of a matrix of orthonormal columns U with the same column decomposition of M M space as V whose column space is the row space , a second matrix of orthonormal columns of M and a diagonal matrix of positive entries D such that M = U %*% D %*% t(V) . D is actually returned as a vector of the diagonal elements. The result of svd(M) is actually a list of three components named d u and v , with evident meanings. , M If is in fact square, then, it is not hard to see that > absdetM <- prod(svd(M)\$d) calculates the absolute value of the determinant of M . If this calculation were needed often with a variety of matrices it could be defined as an R function > absdet <- function(M) prod(svd(M)\$d) absdet() as just another R function. As a further trivial but potentially after which we could use tr() useful example, you might like to consider writing a function, say , to calculate the trace diag() of a square matrix. [Hint: You will not need to use an explicit loop. Look again at the function.] det to calculate a determinant, including the sign, and another, R has a builtin function determinant , to give the sign and modulus (optionally on log scale), 5.7.5 Least squares fitting and the QR decomposition The function lsfit() returns a list giving results of a least squares fitting procedure. An assignment such as > ans <- lsfit(X, y) y is the vector of observations and X gives the results of a least squares fit where is the design matrix. See the help facility for more details, and also for the follow-up function ls.diag() for, among other things, regression diagnostics. Note that a grand mean term is automatically in- cluded and need not be included explicitly as a column of X . Further note that you almost always will prefer using (see lm(.) Section 11.2 [Linear models], page 54 ) to lsfit() for regression modelling. Another closely related function is qr() and its allies. Consider the following assignments > Xplus <- qr(X)

30 Chapter 5: Arrays and matrices 24 > b <- qr.coef(Xplus, y) > fit <- qr.fitted(Xplus, y) > res <- qr.resid(Xplus, y) onto the range of in fit , the projection onto These compute the orthogonal projection of y X and the coefficient vector for the projection in b , that is, b is the orthogonal complement in res ‘backslash’ operator. Matlab essentially the result of the has full column rank. Redundancies will be discovered and removed It is not assumed that X as they are found. This alternative is the older, low-level way to perform least squares calculations. Although still useful in some contexts, it would now generally be replaced by the statistical models features, as will be discussed in Chapter 11 [Statistical models in R], page 51 . 5.8 Forming partitioned matrices, and rbind() cbind() As we have already seen informally, matrices can be built up from other vectors and matrices cbind() and . Roughly cbind() forms matrices by binding together rbind() by the functions vertically, or row-wise. rbind() matrices horizontally, or column-wise, and In the assignment arg_1 > X <- cbind( 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). does the corresponding operation for rows. In this case any vector The function rbind() argument, possibly cyclically extended, are of course taken as row vectors. Suppose and X2 have the same number of rows. To combine these by columns into a X1 X matrix 1 s we can use , together with an initial column of > 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. c() , with arrays 5.9 The concatenation function, 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).

31 Chapter 5: Arrays and matrices 25 5.10 Frequency tables from factors Recall that a factor defines a partition into groups. Similarly a pair of factors defines a two table() allows frequency tables to be calcu- way cross classification, and so on. The function k factor arguments, the result is a k -way array of lated from equal length factors. If there are 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.

32 26 6 Lists and data frames 6.1 Lists An R components . list is an object consisting of an ordered collection of objects known as its 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)) and may always be referred to as such. Thus if Lst is Components are always numbered Lst[[1]] , the name of a list with four components, these may be individually referred to as Lst[[3]] and Lst[[2]] . If, further, Lst[[4]] is a vector subscripted array then , Lst[[4]] Lst[[4]][1] is its first entry. If is a list, then the function length(Lst) gives the number of (top level) components Lst it has. Components of lists may also be , and in this case the component may be referred to named 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 and is the number 4 . Lst[[4]][1] Additionally, one can also use the names of the list components in double square brackets, i.e., is the same as Lst\$name . This is especially useful, when the name of the Lst[["name"]] component to be extracted is stored in another variable as in > x <- "name"; Lst[[x]] It is very important to distinguish from Lst[1] . ‘ [[ ... ]] ’ is the operator used Lst[[1]] to select a single element, whereas ‘ [ ... ] ’ is a general subscripting operator. Thus the former is the first object in the list , and if it is a named list the name is not included. The latter Lst is a Lst consisting of the first entry only. If it is a named list, the names are sublist of the list 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.

33 Chapter 6: Lists and data frames 27 6.2 Constructing and modifying lists New lists may be formed from existing objects by the function list() . An assignment of the form name_m object_1 , > Lst <- list( , = = object_m ) name_1 ... m object 1 for the components and giving , . . . , object m of Lst sets up a list components using 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 when forming the new list and the originals are not affected. copied Lists, like any subscripted object, can be extended by specifying additional components. For example > Lst[5] <- list(matrix=Mat) 6.2.1 Concatenating lists When the concatenation function is given list arguments, the result is an object of mode c() list also, whose components are those of the argument lists joined together in sequence. > list.ABC <- c(list.A, list.B, list.C) Recall that with vector objects as arguments the concatenation function similarly joined together all arguments into a single vector structure. In this case all other attributes, such as dim attributes, are discarded. 6.3 Data frames 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. 1 • Numeric vectors, logicals and factors are included as is, and by default character vectors are coerced to be factors, whose levels are the unique values appearing in the vector. Vector structures appearing as variables of the data frame must all have the • , same length and matrix structures must all have the same row size . A data frame may for many purposes be regarded as a matrix with columns possibly of differing modes and attributes. It may be displayed in matrix form, and its rows and columns extracted using matrix indexing conventions. 6.3.1 Making data frames Objects satisfying the restrictions placed on the columns (components) of a data frame may be used to form one using the function : data.frame > accountants <- data.frame(home=statef, loot=incomes, shot=incomef) A list whose components conform to the restrictions of a data frame may be coerced into a data frame using the function as.data.frame() The simplest way to construct a data frame from scratch is to use the read.table() function to read an entire data frame from an external file. This is discussed further in Chapter 7 [Reading data from files], page 30 . 1 Conversion of character columns to factors is overridden using the stringsAsFactors argument to the data.frame() function.

34 Chapter 6: Lists and data frames 28 attach() and 6.3.2 detach() notation, such as The accountants\$home \$ , for list components is not always very convenient. A useful facility would be somehow to make the components of a list or data frame temporarily visible as variables under their component name, without the need to quote the list name explicitly each time. attach() function takes a ‘database’ such as a list or data frame as its argument. Thus The suppose lentils\$u , lentils\$v , lentils\$w . The lentils is a data frame with three variables attach > attach(lentils) u , places the data frame in the search path at position 2, and provided there are no variables v or in position 1, u , v and w are available as variables from the data frame in their own right. w At this point an assignment such as > u <- v+w u of the data frame, but rather masks it with another variable does not replace the component in the working directory at position 1 on the search path. To make a permanent change to u \$ notation: the data frame itself, the simplest way is to resort once again to the > lentils\$u <- v+w However the new value of component is not visible until the data frame is detached and u attached again. To detach a data frame, use the function > detach() More precisely, this statement detaches from the search path the entity currently at position 2. Thus in the present context the variables u v and w would be no longer visible, , lentils\$u except under the list notation as and so on. Entities at positions greater than 2 on the search path can be detached by giving their number to detach , but it is much safer to detach(lentils) or detach("lentils") always use a name, for example by Note: In R lists and data frames can only be attached at position 2 or above, and copy of the original object. You can alter the attached values what is attached is a via , but the original list or data frame is unchanged. assign 6.3.3 Working with data frames A useful convention that allows you to work with many different problems comfortably together in the same working directory is • gather together all variables for any well defined and separate problem in a data frame under a suitably informative name; • when working with a problem attach the appropriate data frame at position 2, and use the working directory at level 1 for operational quantities and temporary variables; • before leaving a problem, add any variables you wish to keep for future reference to the \$ form of assignment, and then data frame using the ; detach() • finally remove all unwanted variables from the working directory and keep it as clean of left-over temporary variables as possible. In this way it is quite simple to work with many problems in the same directory, all of which , for example. have variables named , y and z x 6.3.4 Attaching arbitrary lists attach() is a generic function that allows not only directories and data frames to be attached to the search path, but other classes of object as well. In particular any object of mode "list" may be attached in the same way:

35 Chapter 6: Lists and data frames 29 > attach(any.old.list) Anything that has been attached can be detached by , by position number or, prefer- detach ably, by name. 6.3.5 Managing the search path search shows the current search path and so is a very useful way to keep track of The function which data frames and lists (and packages) have been attached and detached. Initially it gives > search() [1] ".GlobalEnv" "Autoloads" "package:base" 2 where .GlobalEnv is the workspace. After lentils is attached we have > search() [1] ".GlobalEnv" "lentils" "Autoloads" "package:base" > ls(2) [1] "u" "v" "w" and as we see ls (or objects ) can be used to examine the contents of any position on the search path. Finally, we detach the data frame and confirm it has been removed from the search path. > detach("lentils") > search() [1] ".GlobalEnv" "Autoloads" "package:base" 2 See the on-line help for autoload for the meaning of the second term.

38 Chapter 7: Reading data from files 32 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.

39 33 8 Probability distributions 8.1 R as a set of statistical tables One convenient use of R is to provide a comprehensive set of statistical tables. Functions are ( X ≤ x ), the probability density provided to evaluate the cumulative distribution function P , the smallest ) such that P ( X ≤ x q > q ), and to function and the quantile function (given x simulate from the distribution. R name Distribution additional arguments beta shape1, shape2, ncp beta binomial binom size, prob cauchy location, scale Cauchy chi-squared chisq df, ncp exp rate exponential f df1, df2, ncp F gamma shape, scale gamma geometric geom prob hyper m, n, k hypergeometric log-normal lnorm meanlog, sdlog logistic logis location, scale negative binomial nbinom size, prob norm mean, sd normal Poisson pois lambda signed rank signrank n Student’s t t df, ncp unif min, max uniform weibull shape, scale Weibull Wilcoxon wilcox m, n Prefix the name given here by ‘ ’ for the density, ‘ p ’ for the CDF, ‘ q ’ for the quantile function d and ‘ r ’ for simulation ( r andom deviates). The first argument is x for d xxx , q for p xxx , p for q xxx n for r xxx (except for rhyper , rsignrank and rwilcox , for which it is nn ). In not and ncp quite all cases is the non-centrality parameter currently available: see the on-line help for details. p xxx and q xxx functions all have logical arguments lower.tail The log.p and the and d xxx ones have log . This allows, e.g., getting the cumulative (or “integrated”) hazard function, H ( t ) = − log(1 − F ( t )), by - p xxx (t, ..., lower.tail = FALSE, log.p = TRUE) d xxx (..., log = TRUE) or more accurate log-likelihoods (by ), directly. In addition there are functions ptukey and qtukey for the distribution of the studentized range of samples from a normal distribution, and dmultinom and rmultinom for the multinomial distribution. Further distributions are available in contributed packages, notably SuppDists . ( https://CRAN.R-project.org/package=SuppDists ) Here are some examples > ## 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.

40 Chapter 8: Probability distributions 34 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 and simplest is to examine the numbers. Two slightly different summaries are given by summary stem and a display of the numbers by fivenum (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

41 Chapter 8: Probability distributions 35 too much smoothing (it usually does for “interesting” densities). (Better automated methods of bandwidth choice are available, and in this example gives a good result.) bw = "SJ" Histogram of eruptions 0.7 0.6 0.5 0.4 0.3 Relative Frequency 0.2 0.1 0.0 2.0 2.5 3.0 3.5 1.5 4.5 5.0 4.0 eruptions 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) ecdf(long) 1.0 0.8 0.6 Fn(x) 0.4 0.2 0.0 4.5 5.0 3.0 3.5 4.0 x 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)

42 Chapter 8: Probability distributions 36 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 distribution t Normal Q−Q Plot 5.0 4.5 4.0 Sample Quantiles 3.5 3.0 0 −1 −2 2 1 Theoretical Quantiles 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)

43 Chapter 8: Probability distributions 37 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. 80.04 80.02 80.00 79.98 79.96 79.94 1 2 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 (in contrast to the similar S-Plus t.test function). We can use the F test to test for equality in the variances, provided that the two samples are from normal populations. > var.test(A, B) F test to compare two variances

44 Chapter 8: Probability distributions 38 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) qqplot will perform a Q-Q plot of the two samples. The will show the two empirical CDFs, and 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

45 39 Warning message: cannot compute correct p-values with ties in: ks.test(A, B)

46 40 9 Grouping, loops and conditional execution 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. ; ... ; expr_m } expr_1 { Commands may be grouped together in braces, , 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 statements 9.2.1 Conditional execution: if The language has available a conditional construction of the form expr_1 ) expr_2 expr_3 > if ( else where expr 1 must evaluate to a single logical value and the result of the entire expression is then evident. && || are often used as part of the condition in an if The “short-circuit” operators and & and | apply element-wise to vectors, && and || apply to vectors of length statement. Whereas one, and only evaluate their second argument if necessary. There is a vectorized version of the / else construct, the ifelse if function. This has the form ifelse(condition, a, b) and returns a vector of the same length as condition , with elements a[i] if condition[i] is true, otherwise b[i] (where a and b are recycled as necessary). 9.2.2 Repetitive execution: for repeat and while loops, for loop construction which has the form There is also a in expr_1 ) > for ( name expr_2 where name is the loop variable. expr is a vector expression, (often a sequence like 1:20 1 ), and expr is often a grouped expression with its sub-expressions written in terms of the dummy 2 . expr 2 name name ranges through the values in the vector result of is repeatedly evaluated as expr 1 . As an example, suppose ind is a vector of class indicators and we wish to produce separate 1 plots of x within classes. One possibility here is to use coplot() , y which will produce versus an array of plots corresponding to each level of the factor. Another way to do this, now putting all plots on the one display, is as follows: > xc <- split(x, ind) > yc <- split(y, ind) > for (i in 1:length(yc)) { plot(xc[[i]], yc[[i]]) abline(lsfit(xc[[i]], yc[[i]])) } (Note the function which produces a list of vectors obtained by splitting a larger split() vector according to the classes specified by a factor. This is a useful function, mostly used in connection with boxplots. See the help facility for further details.) 1 . xyplot from package lattice ( https://CRAN.R-project.org/package=lattice ) to be discussed later, or use

47 Chapter 9: Grouping, loops and conditional execution 41 : loops are used in R code much less often than in compiled languages. Warning for() Code that takes a ‘whole object’ view is likely to be both clearer and faster in R. Other looping facilities include the > repeat expr statement and the > while ( condition ) expr statement. The statement can be used to terminate any loop, possibly abnormally. This is the break only way to terminate repeat loops. The next statement can be used to discontinue one particular cycle and skip to the “next”. Control statements are most often used in connection with functions which are discussed in Chapter 10 [Writing your own functions], page 42 , and where more examples will emerge.

48 42 10 Writing your own functions As we have seen informally along the way, the R language allows the user to create objects of mode function . These are true R functions that are stored in a special internal form and may be used in further expressions and so on. In the process, the language gains enormously in power, convenience and elegance, and learning to write useful functions is one of the main ways to make your use of R comfortable and productive. It should be emphasized that most of the functions supplied as part of the R system, such , var() , postscript() and so on, are themselves written in R and thus do not differ as mean() materially from user written functions. A function is defined by an assignment of the form > name <- function( arg_1 , arg_2 , ...) expression expression is an R expression, (usually a grouped expression), that uses the arguments, The i arg , to calculate a value. The value of the expression is the value returned for the function. name expr_1 , expr_2 , ...) and may A call to the function then usually takes the form ( occur anywhere a function call is legitimate. 10.1 Simple examples t -statistic, showing “all the As a first example, consider a function to calculate the two sample 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 backslash com- Matlab As a second example, consider a function to emulate directly the mand, 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 qr() function; however this is sometimes coefficients.) This would ordinarily be done with the a bit tricky to use directly and it pays to have a simple function such as the following to use it safely. T − T y and an n by p matrix X then X y is defined as ( X Thus given a X ) n X by 1 vector y , T − ′ X ) where ( is a generalized inverse of X 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.

49 Chapter 10: Writing your own functions 43 1 lsfit() . It in turn uses the The classical R function does this job quite well, and more and in the slightly counterintuitive way above to do this part of the qr() functions qr.coef() 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 in expressions rather than in function form. Suppose, binary operator ! for example, we choose for the internal character. The function definition would then start as > "%!%" <- function(X, y) { ... } X %!% y . (The backslash (Note the use of quote marks.) The function could then be used as symbol itself is not a convenient choice as it presents special problems in this context.) %*% , and the outer product matrix operator %o% are The matrix multiplication operator, other examples of binary operators defined in this way. 10.3 Named arguments and defaults As first noted in Section 2.3 [Generating regular sequences], page 8 , if arguments to called name object ” form, they may be given in any order. Furthermore functions are given in the “ = the argument sequence may begin in the unnamed, positional form, and specify named arguments after the positional arguments. fun1 Thus if there is a function 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. 1 See also the methods described in Chapter 11 [Statistical models in R], page 51

50 Chapter 10: Writing your own functions 44 ... ’ argument 10.4 The ‘ Another frequent requirement is to allow one function to pass on argument settings to another. par() allow the and functions like For example many graphics functions use the function plot() Section 12.4.1 to control the graphical output. (See user to pass on graphical parameters to par() function.) This can be done by [The par() function], page 68 , for more details on the par() ’, of the function, which may then be passed on. An ... including an extra argument, literally ‘ outline example is given below. fun1 <- function(data, data.frame, graph=TRUE, limit=20, ...) { [omitted statements] if (graph) par(pch="*", ...) [more omissions] } ... ’. The expression Less frequently, a function will need to refer to components of ‘ list(...) , ..2 , evaluates all such arguments and returns them in a named list, while ..1 ..n etc. evaluate them one at a time, with ‘ ’ returning the n’th unmatched argument. 10.5 Assignments within functions any ordinary assignments done within the function are local and temporary and are Note that X <- qr(X) does not affect the value of lost after exit from the function . Thus the assignment the argument in the calling program. To understand completely the rules governing the scope of R assignments the reader needs frame . This is a somewhat advanced, though to be familiar with the notion of an evaluation hardly difficult, topic and is not covered further here. If global and permanent assignments are intended within a function, then either the “su- <<- or the function assign() can be used. See the perassignment” operator, document for help details. users should be aware that <<- has different semantics in R. These are discussed S-Plus further in Section 10.7 [Scope], page 46 . 10.6 More advanced examples 10.6.1 Efficiency factors in block designs As a more complete, if a little pedestrian, example of a function, consider finding the effi- ciency factors for a block design. (Some aspects of this problem have already been discussed in Section 5.3 [Index matrices], page 19 .) A block design is defined by two factors, say blocks ( b levels) and varieties ( v levels). If R and and v by v and b by b replications are the block size matrices, respectively, and N is the K b by v incidence matrix, then the efficiency factors are defined as the eigenvalues of the matrix T 2 / 1 − 1 − 1 / 2 − T = K NR − A, I = R I E − A N v v 2 / 1 − − 1 / 2 NR = A where . One way to write the function is given below. K > 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

51 Chapter 10: Writing your own functions 45 N <- table(blocks, varieties) A <- 1/sqrt(K) * N * rep(1/sqrt(R), rep(b, v)) sv <- svd(A) list(eff=1 - sv\$d^2, blockcv=sv\$u, varietycv=sv\$v) } It is numerically slightly better to work with the singular value decomposition on this occasion rather than the eigenvalue routines. The result of the function is a list giving not only the efficiency factors as the first component, but also the block and variety canonical contrasts, since sometimes these give additional useful qualitative information. 10.6.2 Dropping all names in a printed array For printing purposes with large matrices or arrays, it is often useful to print them in close block dimnames form without the array names or numbers. Removing the attribute will not achieve this effect, but rather the array must be given a attribute consisting of empty strings. dimnames X For example to print a matrix, > temp <- X > dimnames(temp) <- list(rep("", nrow(X)), rep("", ncol(X))) > temp; rm(temp) This can be much more conveniently done using a function, no.dimnames() , shown below, as a “wrap around” to achieve the same result. It also illustrates how some effective and useful user functions can be quite short. no.dimnames <- function(a) { ## Remove all dimension names from an array for compact printing. d <- list() l <- 0 for(i in dim(a)) { d[[l <- l + 1]] <- rep("", i) } dimnames(a) <- d a } With this function defined, an array may be printed in close format using > no.dimnames(X) This is particularly useful for large integer arrays, where patterns are the real interest rather than the values. 10.6.3 Recursive numerical integration Functions may be recursive, and may themselves define functions within themselves. Note, however, that such functions, or indeed variables, are not inherited by called functions in higher evaluation frames as they would be if they were on the search path. The example below shows a naive way of performing one-dimensional numerical integration. The integrand is evaluated at the end points of the range and in the middle. If the one-panel trapezium rule answer is close enough to the two panel, then the latter is returned as the value. Otherwise the same process is recursively applied to each panel. The result is an adaptive integration process that concentrates function evaluations in regions where the integrand is farthest from linear. There is, however, a heavy overhead, and the function is only competitive with other algorithms when the integrand is both smooth and very difficult to evaluate. The example is also given partly as a little puzzle in R programming. area <- function(f, a, b, eps = 1.0e-06, lim = 10) {

52 Chapter 10: Writing your own functions 46 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. S-Plus However, it details one of the major differences between 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 . First we define a function called cube . lexical scope 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 since that is the active binding cube 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.

53 Chapter 10: Writing your own functions 47 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 . In the following example mutable state Lexical scope can also be used to give functions 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 account within account is invoked it takes and then returning a list containing them. When a numerical argument and returns a list containing the three functions. Because these total total functions are defined in an environment which contains , they will have access to its value. <<- , is used to change the value associated with total The special assignment operator, . 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 2 a global variable and assigns the value of the right hand side to it <<- has been . Only when 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) 2 In some sense this mimics the behavior in S-Plus since in S-Plus this operator always creates or assigns to a global variable.

54 Chapter 10: Writing your own functions 48 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. environment The location of the site initialization file is taken from the value of the R_PROFILE Rprofile.site in the R home subdirectory etc is used. variable. If that variable is unset, the file This file should contain the commands that you want to execute every time R is started under 3 your system. A second, personal, profile file named .Rprofile can be placed in any directory. If R is invoked in that directory then that file will be sourced. This file gives individual users control over their workspace and allows for different startup procedures in different working directories. .Rprofile file is found in the startup directory, then R looks for a .Rprofile file in the If no 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 files. .Rprofile 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, , the user profile, .RData Rprofile.site .First() and then . A definition in later files will mask definitions in earlier files. > .First <- function() { options(prompt="\$ ", continue="+\t") # \$ is the prompt custom numbers and printout options(digits=5, length=999) # 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 3 So it is hidden under UNIX.

55 Chapter 10: Writing your own functions 49 default action not catered for specifically by the generic function in question, there is always a provided. An example makes things clearer. The class mechanism offers the user the facility of designing plot() and writing generic functions for special purposes. Among the other generic functions are summary() for summarizing analyses of various types, and for displaying objects graphically, anova() for comparing statistical models. The number of generic functions that can treat a class in a specific way can be quite large. "data.frame" For example, the functions that can accommodate in some fashion objects of class include [ [[<- any as.matrix [<- mean plot summary A currently complete list can be got by using the function: methods() > methods(class="data.frame") Conversely the number of classes a generic function can handle can also be quite large. plot() function has a default method and variants for objects of classes For example the "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 methods() we can use > 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)]

56 50 } 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.

57 51 11 Statistical models in R This section presumes the reader has some familiarity with statistical methodology, in particular with regression analysis and the analysis of variance. Later we make some rather more ambitious presumptions, namely that something is known about generalized linear models and nonlinear regression. The requirements for fitting statistical models are sufficiently well defined to make it possible to construct general tools that apply in a broad spectrum of problems. R provides an interlocking suite of facilities that make fitting statistical models very simple. As we mention in the introduction, the basic output is minimal, and one needs to ask for the details by calling extractor functions. 11.1 Defining statistical models; formulae The template for a statistical model is a linear regression model with independent, homoscedastic errors p ∑ 2 + ) β = 1 x , i ,...,n e , e ,σ ∼ NID(0 = y i i ij j i =0 j In matrix terms this would be written = Xβ + e y y model matrix X is the where the or design matrix and has columns is the response vector, x ,x will be a column of ones defining an ,...,x x , the determining variables. Very often p 0 1 0 intercept term. Examples Before giving a formal specification, a few examples may usefully set the picture. Suppose , x , x0 , x1 , x2 , . . . are numeric variables, X is a matrix and A , y , C , . . . are factors. B 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 . The first has an x 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 y ), on x 1 and Multiple regression of the transformed variable, log( 2 (with an x 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 polyno- mials, 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.

58 Chapter 11: Statistical models in R 52 Single classification analysis of variance model of y A . y ~ A , with classes determined by y Single classification analysis of covariance model of y ~ A + x , with classes determined by A , and with covariate x . y ~ A*B y ~ A + B + A:B y ~ B %in% A . The first two specify the same Two factor non-additive model of A and B on y ~ A/B y 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 on x Separate simple linear regression models of A , with y within the levels of different codings. The last form produces explicit estimates of as many different A . intercepts and slopes as there are levels in y ~ A*B + Error(C) An experiment with two treatment factors, and B , and error strata determined A by factor . For example a split plot experiment, with whole plots (and hence also C 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 + - , implying the inclusion or exclusion of a term in the or model, (the first is optional). term i is either a vector or matrix expression, or , • 1 • a factor, or • formula expression consisting of factors, vectors or matrices connected by a 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 are similar in effect to the Wilkinson and Rogers notation used by formula operators . 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 . 1 and M 2 . M_1 + M_2 Include M

59 Chapter 11: Statistical models in R 53 - M_2 M 1 leaving out terms of M 2 . M_1 Include and 1 M 2 . If both terms are factors, then the “subclasses” M M_2 : M_1 The tensor product of factor. %in% M_2 M_1 Similar to M_1 : M_2 , but with a different coding. * M_2 M_1 M_1 M_2 + M_1 : M_2 . + M_1 M_2 M_1 + M_2 %in% M_1 . / ^ M All terms in M together with “interactions” up to order n n M I( 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 , the columns of the model matrix specification of the parameters being implicit. This is not the case in other contexts, for example in specifying nonlinear models. 11.1.1 Contrasts We need at least some idea how the model formulae specify the columns of the model matrix. This is easy if we have continuous variables, as each provides one column of the model matrix (and the intercept will provide a column of ones if included in the model). k -level factor A ? The answer differs for unordered and ordered factors. For What about a factors unordered − 1 columns are generated for the indicators of the second, . . . , k th levels k of the factor. (Thus the implicit parameterization is to contrast the response at each level with ordered factors the k − 1 columns are the orthogonal polynomials on that at the first.) For 1 ,...,k , omitting the constant term. Although the answer is already complicated, it is not the whole story. First, if the intercept is omitted in a model that contains a factor term, the first such term is encoded into columns k giving the indicators for all the levels. Second, the whole behavior can be changed by the options contrasts . The default setting in R is setting for options(contrasts = c("contr.treatment", "contr.poly")) The main reason for mentioning this is that R and S have different defaults for unordered factors, S using Helmert contrasts. So if you need to compare your results to those of a textbook or paper which used S-Plus , you will need to set options(contrasts = c("contr.helmert", "contr.poly")) This is a deliberate difference, as treatment contrasts (R’s default) are thought easier for new- comers to interpret. We have still not finished, as the contrast scheme to be used can be set for each term in the model using the functions contrasts and C . We have not yet considered interaction terms: these generate the products of the columns introduced for their component terms. Although the details are complicated, model formulae in R will normally generate the models that an expert statistician would expect, provided that marginality is preserved. Fitting, for example, a model with an interaction but not the corresponding main effects will in general lead to surprising results, and is for experts only.

60 Chapter 11: Statistical models in R 54 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 formula , data = data.frame ) > <- lm( For example > fm2 <- lm(y ~ x1 + x2, data = production) would fit a multiple regression model of x 1 and x 2 (with implicit intercept term). y on data = production specifies that any The important (but technically optional) parameter variables needed to construct the model should come first from the data frame . production This is the case regardless of whether data frame has been attached on the search production . path or not 11.3 Generic functions for extracting model information lm() is a fitted model object; technically a list of results of class "lm" . Information The value of about the fitted model can then be displayed, extracted, plotted and so on by using generic "lm" . These include functions that orient themselves to objects of class 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: object ) . coefficients( object ) deviance( Residual sum of squares, weighted if appropriate. object ) formula( 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 data.frame . determining variable values in 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: object ) . resid( 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.

61 Chapter 11: Statistical models in R 55 object ) summary( Print a comprehensive summary of the results of the regression analysis. object ) vcov( Returns the variance-covariance matrix of the main parameters of a fitted model object. 11.4 Analysis of variance and model comparison formula , data= The model fitting function ) operates at the simplest level in aov( data.frame lm() , and most of the generic functions listed in the table in a very similar way to the function Section 11.3 [Generic functions for extracting model information], page 54, apply. It should be noted that in addition allows an analysis of models with multiple error aov() 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 ) ~ strata.formula . In the specifies a multi-stratum experiment with error strata defined by the strata.formula is simply a factor, when it defines a two strata experiment, namely simplest case, between and within the levels of the factor. For example, with all determining variables factors, a model formula such as that in: > fm <- aov(yield ~ v + n*p*k + Error(farms/blocks), data=farm.data) would typically be used to describe an experiment with mean model v + n*p*k and three error strata, namely “between farms”, “within farms, between blocks” and “within blocks”. 11.4.1 ANOVA tables Note also that the analysis of variance table (or tables) are for a sequence of fitted models. The sums of squares shown are the decrease in the residual sums of squares resulting from an inclusion of in the model at that place in the sequence. Hence only for orthogonal that term experiments will the order of inclusion be inconsequential. For multistratum experiments the procedure is first to project the response onto the error strata, again in sequence, and to fit the mean model to each projection. For further details, see Chambers & Hastie (1992). A more flexible alternative to the default full ANOVA table is to compare two or more models anova() function. directly using the > anova( fitted.model.1 , fitted.model.2 , ...) The display is then an ANOVA table showing the differences between the fitted models when fitted in sequence. The fitted models being compared would usually be an hierarchical sequence, of course. This does not give different information to the default, but rather makes it easier to comprehend and control. 11.5 Updating fitted models The update() function is largely a convenience function that allows a model to be fitted that differs from one previously fitted usually by just a few additional or removed terms. Its form is > new.model <- update( old.model , new.formula ) In the new.formula the special name consisting of a period, ‘ . ’, only, can be used to stand for “the corresponding part of the old model formula”. For example, > fm05 <- lm(y ~ x1 + x2 + x3 + x4 + x5, data = production) > fm6 <- update(fm05, . ~ . + x6) > smf6 <- update(fm6, sqrt(.) ~ .)

62 Chapter 11: Statistical models in R 56 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. data= argument is specified on the original call to the model Note especially that if the fitting function, this information is passed on through the fitted model object to and update() its allies. The name ‘ ’ can also be used in other contexts, but with slightly different meaning. For . example > fmfull <- lm(y ~ . , data = production) and regressor variables all other variables in the data frame would fit a model with response y production . , drop1() and Other functions for exploring incremental sequences of models are add1() . The names of these give a good clue to their purpose, but for full details see the on-line step() 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 • , whose values influence , x ... , x 2 1 the distribution of the response. • y through a single linear function, only . The stimulus variables influence the distribution of This linear function is called the linear predictor , and is usually written = β , x η + β x x β + ··· + 2 p 2 p 1 1 hence x = 0. has no influence on the distribution of y if and only if β i i The distribution of y • is of the form [ ] A ( y μ,φ ) = exp f ; { yλ ( ) ) − γ ( λ ( μ )) } + τ ( y,φ μ Y φ where φ 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 y . So it is assumed that the distribution of y is determined by its mean is the mean of μ and possibly a scale parameter as well. The mean, μ , is a smooth invertible function of the linear predictor: • − 1 m ( η ) , η = m μ = ( μ ) = ` ( μ ) and this inverse function, ` (), 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).

63 Chapter 11: Statistical models in R 57 11.6.1 Families The class of generalized linear models handled by facilities supplied in R includes gaussian , poisson inverse gaussian and gamma response distributions and also quasi-likelihood , , binomial models where the response distribution is not explicitly specified. In the latter case the variance must be specified as a function of the mean, but in other cases this function is implied function by the response distribution. Each response distribution admits a variety of link functions to connect the mean with the linear predictor. Those automatically available are shown in the following table: Family name Link functions , probit , binomial logit , cloglog log gaussian identity log , inverse , , , log Gamma identity inverse , identity , inverse.gaussian 1/mu^2 , log inverse poisson identity , log , sqrt quasi logit , probit , cloglog , identity , inverse , log , , sqrt 1/mu^2 The combination of a response distribution, a link function and various other pieces of infor- mation that are needed to carry out the modeling exercise is called the of the generalized family linear model. 11.6.2 The glm() function Since the distribution of the response depends on the stimulus variables through a single linear only , the same mechanism as was used for linear models can still be used to specify the function linear part of a generalized model. The family has to be specified in a different way. The R function to fit a generalized linear model is which uses the form glm() fitted.model <- glm( > , family= family.generator , data= data.frame ) formula The only new feature is the family.generator , which is the instrument by which the family is described. It is the name of a function that generates a list of functions and expressions that together define and control the model and estimation process. Although this may seem a little complicated at first sight, its use is quite simple. The names of the standard, supplied family generators are given under “Family Name” in the table in Section 11.6.1 [Families], page 57 . Where there is a choice of links, the name of the link may also be supplied with the family name, in parentheses as a parameter. In the case of the quasi family, the variance function may also be specified in this way. Some examples make the process clear. The gaussian family A call such as > fm <- glm(y ~ x1 + x2, family = gaussian, data = sales) achieves the same result as > fm <- lm(y ~ x1+x2, data=sales) but much less efficiently. Note how the gaussian family is not automatically provided with a choice of links, so no parameter is allowed. If a problem requires a gaussian family with a nonstandard link, this can usually be achieved through the quasi family, as we shall see later. The binomial family Consider a small, artificial example, from Silvey (1970).

64 Chapter 11: Statistical models in R 58 On the Aegean island of Kalythos the male inhabitants suffer from a congenital eye disease, the effects of which become more marked with increasing age. Samples of islander males of various ages were tested for blindness and the results recorded. The data is shown below: 20 35 45 55 70 Age: 50 50 50 No. tested: 50 50 6 17 26 37 44 No. blind: The problem we consider is to fit both logistic and probit models to this data, and to estimate for each model the LD50, that is the age at which the chance of blindness for a male inhabitant is 50%. y is the number of blind at age x and n the number tested, both models have the form If )) ∼ n,F ( β y + β B( x 1 0 F ( z where for the probit case, z ) is the standard normal distribution function, and in the ) = Φ( z z ( z ) = e logit case (the default), / (1 + e F ). In both cases the LD50 is LD50 = − β /β 1 0 that is, the point at which the argument of the distribution function is zero. The first step is to set the data up as a data frame > kalythos <- data.frame(x = c(20,35,45,55,70), n = rep(50,5), y = c(6,17,26,37,44)) To fit a binomial model using glm() there are three possibilities for the response: If the response is a • it is assumed to hold binary data, and so must be a 0 / 1 vector. vector • If the response is a two-column matrix it is assumed that the first column holds the number of successes for the trial and the second holds the number of failures. • If the response is a , its first level is taken as failure (0) and all other levels as ‘success’ factor (1). Here we need the second of these conventions, so we add a matrix to our data frame: > kalythos\$Ymat <- cbind(kalythos\$y, kalythos\$n - kalythos\$y) To fit the models we use > fmp <- glm(Ymat ~ x, family = binomial(link=probit), data = kalythos) > fml <- glm(Ymat ~ x, family = binomial, data = kalythos) Since the logit link is the default the parameter may be omitted on the second call. To see the results of each fit we could use > summary(fmp) > summary(fml) Both models fit (all too) well. To find the LD50 estimate we can use a simple function: > ld50 <- function(b) -b[1]/b[2] > ldp <- ld50(coef(fmp)); ldl <- ld50(coef(fml)); c(ldp, ldl) The actual estimates from this data are 43.663 years and 43.601 years respectively. Poisson models With the Poisson family the default link is the log , and in practice the major use of this family is to fit surrogate Poisson log-linear models to frequency data, whose actual distribution is often multinomial. This is a large and important subject we will not discuss further here. It even forms a major part of the use of non-gaussian generalized models overall.

65 Chapter 11: Statistical models in R 59 Occasionally genuinely Poisson data arises in practice and in the past it was often analyzed as gaussian data after either a log or a square-root transformation. As a graceful alternative to the latter, a Poisson generalized linear model may be fitted as in the following example: > fmod <- glm(y ~ A + B + x, family = poisson(link=sqrt), data = worm.counts) Quasi-likelihood models For all families the variance of the response will depend on the mean and will have the scale parameter as a multiplier. The form of dependence of the variance on the mean is a characteristic ] = μ of the response distribution; for example for the poisson distribution Var[ y . For quasi-likelihood estimation and inference the precise response distribution is not specified, but rather only a link function and the form of the variance function as it depends on the mean. Since quasi-likelihood estimation uses formally identical techniques to those for the gaussian distribution, this family provides a way of fitting gaussian models with non-standard link functions or variance functions, incidentally. For example, consider fitting the non-linear regression z θ 1 1 y = + e − θ z 2 2 which may be written alternatively as 1 = e + y β + x x β 2 2 1 1 where x . Supposing a suitable data frame to = z /θ /z θ , x = = − 1 /z β , β and = 1 /θ 1 1 2 2 1 2 2 1 1 1 be set up we could fit this non-linear regression as > nlfit <- glm(y ~ x1 + x2 - 1, family = quasi(link=inverse, variance=constant), data = biochem) The reader is referred to the manual and the help document for further information, as needed. 11.7 Nonlinear least squares and maximum likelihood models Certain forms of nonlinear model can be fitted by Generalized Linear Models ( glm() ). But in the majority of cases we have to approach the nonlinear curve fitting problem as one of nonlinear optimization. R’s nonlinear optimization routines are optim() , nlm() and nlminb() , which provide the functionality (and more) of ’s ms() and nlminb() . We seek the parameter S-Plus values that minimize some index of lack-of-fit, and they do this by trying out various parameter values iteratively. Unlike linear regression for example, there is no guarantee that the procedure will converge on satisfactory estimates. All the methods require initial guesses about what parameter values to try, and convergence may depend critically upon the quality of the starting values. 11.7.1 Least squares One way to fit a nonlinear model is by minimizing the sum of the squared errors (SSE) or residuals. This method makes sense if the observed errors could have plausibly arisen from a normal distribution. Here is an example from Bates & Watts (1988), page 51. The data are: > x <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56,

66 Chapter 11: Statistical models in R 60 1.10, 1.10) > y <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200) The fit criterion to be minimized is: > fn <- function(p) sum((y - (p[1] * x)/(p[2] + x))^2) In order to do the fit we need initial estimates of the parameters. One way to find sensible starting values is to plot the data, guess some parameter values, and superimpose the model curve using those values. > plot(x, y) > xfit <- seq(.02, 1.1, .05) > yfit <- 200 * xfit/(0.1 + xfit) > lines(spline(xfit, yfit)) We could do better, but these starting values of 200 and 0.1 seem adequate. Now do the fit: > out <- nlm(fn, p = c(200, 0.1), hessian = TRUE) After the fitting, is the SSE, and out\$estimate are the least squares estimates out\$minimum of the parameters. To obtain the approximate standard errors (SE) of the estimates we do: > sqrt(diag(2*out\$minimum/(length(y) - 2) * solve(out\$hessian))) The 2 which is subtracted in the line above represents the number of parameters. A 95% confidence interval would be the parameter estimate ± 1.96 SE. We can superimpose the least squares fit on a new plot: > plot(x, y) > xfit <- seq(.02, 1.1, .05) > yfit <- 212.68384222 * xfit/(0.06412146 + xfit) > lines(spline(xfit, yfit)) stats provides much more extensive facilities for fitting non-linear The standard package models by least squares. The model we have just fitted is the Michaelis-Menten model, so we can use > df <- data.frame(x=x, y=y) > fit <- nls(y ~ SSmicmen(x, Vm, K), df) > fit Nonlinear regression model model: y ~ SSmicmen(x, Vm, K) data: df Vm K 212.68370711 0.06412123 residual sum-of-squares: 1195.449 > summary(fit) Formula: y ~ SSmicmen(x, Vm, K) Parameters: Estimate Std. Error t value Pr(>|t|) Vm 2.127e+02 6.947e+00 30.615 3.24e-11 K 6.412e-02 8.281e-03 7.743 1.57e-05 Residual standard error: 10.93 on 10 degrees of freedom Correlation of Parameter Estimates: Vm K 0.7651

67 Chapter 11: Statistical models in R 61 11.7.2 Maximum likelihood Maximum likelihood is a method of nonlinear model fitting that applies even if the errors are not normal. The method finds the parameter values which maximize the log likelihood, or equivalently which minimize the negative log-likelihood. Here is an example from Dobson (1990), pp. 108–111. This example fits a logistic model to dose-response data, which clearly could also be fit by glm() . The data are: > x <- c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113, 1.8369, 1.8610, 1.8839) > y <- c( 6, 13, 18, 28, 52, 53, 61, 60) > n <- c(59, 60, 62, 56, 63, 59, 62, 60) The negative log-likelihood to minimize is: > fn <- function(p) sum( - (y*(p[1]+p[2]*x) - n*log(1+exp(p[1]+p[2]*x)) + log(choose(n, y)) )) We pick sensible starting values and do the fit: > out <- nlm(fn, p = c(-50,20), hessian = TRUE) out\$minimum out\$estimate are the maxi- After the fitting, is the negative log-likelihood, and mum likelihood estimates of the parameters. To obtain the approximate SEs of the estimates we do: > sqrt(diag(solve(out\$hessian))) A 95% confidence interval would be the parameter estimate ± 1.96 SE. 11.8 Some non-standard models We conclude this chapter with just a brief mention of some of the other facilities available in R for special regression and data analysis problems. Mixed models. The recommended • https://CRAN.R-project.org/package=nlme ) nlme ( package provides functions and nlme() for linear and non-linear mixed-effects models, lme() 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. • loess() function fits a nonparametric regression by The 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. loess is in the standard package stats , together with code for projection pursuit Function 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 recom- MASS https://CRAN.R-project.org/package=MASS ) provides state- ( mended package of-art algorithms for highly-resistant fits. Less resistant but statistically more efficient rlm in package MASS methods are available in packages, for example function https:// ( . CRAN.R-project.org/package=MASS ) • Additive models. This technique aims to construct a regression function from smooth addi- tive functions of the determining variables, usually one for each determining variable. Func- ) https://CRAN.R-project.org/package=acepack ( acepack in package ace avas tions and bruto and functions mars in package mda ( https: / / CRAN . R-project . org / and package=mda ) provide some examples of these techniques in user-contributed packages to R. An extension is , implemented in user-contributed pack- Generalized Additive Models ages gam ( https://CRAN.R-project.org/package=gam ) and mgcv ( https://CRAN. R-project.org/package=mgcv ) .

68 Chapter 11: Statistical models in R 62 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 , but many other generic functions such as plot() and text() are well adapted to tree() displaying the results of a tree-based model fit in a graphical way. Tree models are available in R the user-contributed packages via rpart ( https: / / CRAN . R-project . org / package=rpart ) and tree ( https: / / CRAN . R-project . org / package=tree ) .

69 63 12 Graphical procedures Graphical facilities are an important and extremely versatile component of the R environment. It is possible to use the facilities to display a wide variety of statistical graphs and also to build entirely new types of graph. The graphics facilities can be used in both interactive and batch modes, but in most cases, interactive use is more productive. Interactive use is also easy because at startup time R initiates graphics window for the display of interactive device driver a graphics which opens a special graphics. Although this is done automatically, it may useful to know that the command used is under UNIX, under Windows and quartz() under macOS. A new device can X11() windows() always be opened by dev.new() . Once the device driver is running, R plotting commands can be used to produce a variety of graphical displays and to create entirely new kinds of display. Plotting commands are divided into three basic groups: High-level plotting functions create a new plot on the graphics device, possibly with axes, • labels, titles and so on. Low-level plotting functions add more information to an existing plot, such as extra points, • lines and labels. • graphics functions allow you interactively add information to, or extract infor- Interactive mation from, an existing plot, using a pointing device such as a mouse. graphical parameters In addition, R maintains a list of which can be manipulated to customize your plots. This manual only describes what are known as ‘base’ graphics. A separate graphics sub- system in package grid coexists with base – it is more powerful but harder to use. There is a recommended package ( https://CRAN.R-project.org/package=lattice ) which builds lattice grid on Trellis system in S. and provides ways to produce multi-panel plots akin to those in the 12.1 High-level plotting commands High-level plotting functions are designed to generate a complete plot of the data passed as ar- guments to the function. Where appropriate, axes, labels and titles are automatically generated (unless you request otherwise.) High-level plotting commands always start a new plot, erasing the current plot if necessary. 12.1.1 The plot() function One of the most frequently used plotting functions in R is the plot() function. This is a generic function: the type of plot produced is dependent on the type or class of the first argument. x y ) plot( , xy ) If x and y are vectors, plot( x , y ) produces a scatterplot of y against x . The same plot( effect can be produced by supplying one argument (second form) as either a list containing two elements x and y or a two-column matrix. plot( x If x is a time series, this produces a time-series plot. If x is a numeric vector, it ) x produces a plot of the values in the vector against their index in the vector. If is a complex vector, it produces a plot of imaginary versus real parts of the vector elements. plot( f ) plot( f , y ) ; f y is a numeric vector. The first form generates a bar plot of f is a factor object, the second form produces boxplots of y for each level of f .

70 Chapter 12: Graphical procedures 64 df ) plot( ) expr plot(~ expr y ) ~ plot( is any object, df is a list of object names separated by ‘ + ’ is a data frame, y expr ). The first two forms produce distributional plots of the variables in (e.g., a + b + c a data frame (first form) or of a number of named objects (second form). The third y expr . form plots against every object named in 12.1.2 Displaying multivariate data X R provides two very useful functions for representing multivariate data. If is a numeric matrix or data frame, the command > pairs(X) produces a pairwise scatterplot matrix of the variables defined by the columns of X , that is, X is plotted against every other column of every column of and the resulting n ( n − 1) plots X are arranged in a matrix with plot scales constant over the rows and columns of the matrix. coplot may be more enlightening. If and b are When three or four variables are involved a a is a numeric vector or factor object (all of the same length), then the numeric vectors and c command > coplot(a ~ b | c) a against b produces a number of scatterplots of c . If c is a factor, this for given values of simply means that a is plotted against b for every level of c . When c is numeric, it is divided into a number of conditioning intervals and for each interval a is plotted against b for values of c within the interval. The number and position of intervals can be controlled with given.values= coplot() co.intervals() is useful for selecting intervals. You can argument to —the function also use two given variables with a command like > coplot(a ~ b | c + d) a against b for every joint conditioning interval of c and d which produces scatterplots of . The and pairs() function both take an argument panel= which can be used to coplot() customize the type of plot which appears in each panel. The default is to produce a points() scatterplot but by supplying some other low-level graphics function of two vectors x and y as the value of panel= you can produce any type of plot you wish. An example panel function useful for coplots is panel.smooth() . 12.1.3 Display graphics Other high-level graphics functions produce different types of plots. Some examples are: qqnorm(x) qqline(x) qqplot(x, y) Distribution-comparison plots. The first form plots the numeric vector x against the expected Normal order scores (a normal scores plot) and the second adds a straight line to such a plot by drawing a line through the distribution and data quartiles. The third form plots the quantiles of x against those of y to compare their respective distributions. hist(x) hist(x, nclass= n ) hist(x, breaks= b , ...) Produces a histogram of the numeric vector x . A sensible number of classes is usually chosen, but a recommendation can be given with the nclass= argument. Alternatively, the breakpoints can be specified exactly with the breaks= argument.

71 Chapter 12: Graphical procedures 65 probability=TRUE argument is given, the bars represent relative frequencies If the divided by bin width instead of counts. dotchart(x, ...) Constructs a dotchart of the data in . In a dotchart the y -axis gives a labelling x and the x of the data in x -axis gives its value. For example it allows easy visual selection of all data entries with values lying in specified ranges. image(x, y, z, ...) contour(x, y, z, ...) persp(x, y, z, ...) Plots of three variables. The image plot draws a grid of rectangles using different z colours to represent the value of contour plot draws contour lines to represent , the the value of , and the persp plot draws a 3D surface. z 12.1.4 Arguments to high-level plotting functions There are a number of arguments which may be passed to high-level graphics functions, as follows: add=TRUE Forces the function to act as a low-level graphics function, superimposing the plot on the current plot (some functions only). axes=FALSE Suppresses generation of axes—useful for adding your own custom axes with the function. The default, axes=TRUE , means include axes. axis() log="x" log="y" log="xy" Causes the x , y or both axes to be logarithmic. This will work for many, but not all, types of plot. type= type= argument controls the type of plot produced, as follows: The Plot individual points (the default) type="p" Plot lines type="l" Plot points connected by lines ( both ) type="b" type="o" Plot points overlaid by lines type="h" Plot vertical lines from points to the zero axis ( high-density ) type="s" type="S" Step-function plots. In the first form, the top of the vertical defines the point; in the second, the bottom. type="n" No plotting at all. However axes are still drawn (by default) and the coordinate system is set up according to the data. Ideal for creating plots with subsequent low-level graphics functions. xlab= string ylab= string Axis labels for the x and y axes. Use these arguments to change the default labels, usually the names of the objects used in the call to the high-level plotting function. main= string Figure title, placed at the top of the plot in a large font. sub= string Sub-title, placed just below the x -axis in a smaller font.

72 Chapter 12: Graphical procedures 66 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 for points() and also be passed to these functions (and defaults to for "p" "l" .) lines() text(x, y, labels, ...) Add text to a plot at points given by x, y . Normally labels is an integer or labels[i] character vector in which case (x[i], y[i]) . The is plotted at point default is . 1:length(x) : This function is often used in the sequence Note > 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 for the points. names 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 x x -coordinates for vertical lines. Also lm.obj may be list with a v= similarly for the 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 corre- sponding values of the plotting unit must also be given, as follows: v ) legend( , fill= 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)

74 Chapter 12: Graphical procedures 68 locator(n, type) Waits for the user to select locations on the current plot using the left mouse button. (default 512) points have been selected, or another mouse This continues until n argument allows for plotting at the selected points and type button is pressed. The 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 y . x and is usually called with no arguments. It is particularly useful for interactively locator() 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) locator() will be ignored if the current device, such as may be useful. ( does not postscript support interactive pointing.) identify(x, y, labels) Allow the user to highlight any of the points defined by and y (using the left mouse x button) by plotting the corresponding component of labels nearby (or the index labels number of the point if 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) identify() The 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 argument to identify() , or disable marking altogether with the labels 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 justifica- tion among many others. Every graphics parameter has a name (such as ‘ col ’, which controls colors,) and a value (a color number, for example.) A separate list of graphics parameters is maintained for each active device, and each device has a default set of parameters when initialized. Graphics parameters can be set in two ways: either permanently, affecting all graphics functions which access the current device; or temporarily, affecting only a single graphics function call. 12.4.1 Permanent changes: The par() function The par() function is used to access and modify the list of graphics parameters for the current graphics device.

75 Chapter 12: Graphical procedures 69 Without arguments, returns a list of all graphics parameters and their values for par() the current device. par(c("col", "lty")) With a character vector argument, returns only the named graphics parameters (again, as a list.) par(col=4, lty=2) With named arguments (or a single list argument), sets the values of the named graphics parameters, and returns the original values of the parameters as a list. function changes the value of the parameters Setting graphics parameters with the par() , in the sense that all future calls to graphics functions (on the current device) will permanently be affected by the new value. You can think of setting graphics parameters in this way as setting “default” values for the parameters, which will be used by all graphics functions unless an alternative value is given. par() always affect the global values of graphics parameters, even when Note that calls to is called from within a function. This is often undesirable behavior—usually we want to par() set some graphics parameters, do some plotting, and then restore the original values so as not to affect the user’s R session. You can restore the initial values by saving the result of par() when making changes, and restoring the initial values when plotting is complete. > oldpar <- par(col=4, lty=2) . . . plotting commands . . . > par(oldpar) 1 To save and restore all graphical parameters use settable > oldpar <- par(no.readonly=TRUE) . . . plotting commands . . . > par(oldpar) 12.4.2 Temporary changes: Arguments to graphics functions Graphics parameters may also be passed to (almost) any graphics function as named arguments. This has the same effect as passing the arguments to the par() function, except that the changes only last for the duration of the function call. For example: > plot(x, y, pch="+") produces a scatterplot using a plus sign as the plotting character, without changing the default plotting character for future plots. Unfortunately, this is not implemented entirely consistently and it is sometimes necessary to set and reset graphics parameters using par() . 12.5 Graphics parameters list The following sections detail many of the commonly-used graphical parameters. The R help documentation for the function provides a more concise summary; this is provided as a par() 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 or a graphics function. value is a par() 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 . 1 Some graphics parameters such as the size of the current device are for information only.

76 Chapter 12: Graphical procedures 70 12.5.1 Graphical elements R plots are made up of points, lines, text and polygons (filled regions.) Graphical parameters are drawn, as follows: exist which control how these graphical elements Character to be used for plotting points. The default varies with graphics drivers, pch="+" but it is usually ‘ ’. Plotted points tend to appear slightly above or below the ◦ as the plotting character, which produces appropriate position unless you use "." centered points. pch=4 pch is given as an integer between 0 and 25 inclusive, a specialized plotting When symbol is produced. To see what the symbols are, use the command > legend(locator(1), as.character(0:25), pch = 0:25) Those from 21 to 25 may appear to duplicate earlier symbols, but can be coloured points and its examples. in different ways: see the help on pch In addition, 32:255 representing can be a character or a number in the range a character in the current font. lty=2 Line types. Alternative line styles are not supported on all graphics devices (and vary on those that do) but line type 1 is always a solid line, line type 0 is always invis- ible, and line types 2 and onwards are dotted or dashed lines, or some combination of both. Line widths. Desired width of lines, in multiples of the “standard” line width. lwd=2 lines() Affects axis lines as well as lines drawn with , etc. Not all devices support this, and some have restrictions on the widths that can be used. col=2 Colors to be used for points, lines, text, filled regions and images. A number from the current palette (see ?palette ) or a named colour. col.axis col.lab col.main col.sub x and y labels, main and sub-titles, re- The color to be used for axis annotation, spectively. font=2 An integer which specifies which font to use for text. If possible, device drivers 1 corresponds to plain text, 2 to bold face, 3 arrange so that 4 to bold to italic, italic and 5 to a symbol font (which include Greek letters). font.axis font.lab font.main font.sub x and y labels, main and sub-titles, respec- The font to be used for axis annotation, tively. adj=-0.1 Justification of text relative to the plotting position. 0 means left justify, 1 means right justify and means to center horizontally about the plotting position. The 0.5 actual value is the proportion of text that appears to the left of the plotting position, so a value of -0.1 leaves a gap of 10% of the text width between the text and the plotting position. cex=1.5 Character expansion. The value is the desired size of text characters (including plotting characters) relative to the default text size.

77 Chapter 12: Graphical procedures 71 cex.axis cex.lab cex.main x y labels, main and cex.sub and The character expansion to be used for axis annotation, sub-titles, respectively. 12.5.2 Axes and tick marks Many of R’s high-level plots have axes, and you can construct axes yourself with the low-level graphics function. Axes have three main components: the axis line (line style controlled axis() lty graphics parameter), the tick marks (which mark off unit divisions along the axis by the tick labels line) and the (which mark the units.) These components can be customized with the following graphics parameters. lab=c(5, 7, 12) x and y axes The first two numbers are the desired number of tick intervals on the respectively. The third number is the desired length of axis labels, in characters (including the decimal point.) Choosing a too-small value for this parameter may result in all tick labels being rounded to the same number! las=1 Orientation of axis labels. 0 means always parallel to axis, 1 means always horizon- tal, and 2 means always perpendicular to the axis. mgp=c(3, 1, 0) Positions of axis components. The first component is the distance from the axis label to the axis position, in text lines. The second component is the distance to the tick labels, and the final component is the distance from the axis position to the axis line (usually zero). Positive numbers measure outside the plot region, negative numbers inside. tck=0.01 Length of tick marks, as a fraction of the size of the plotting region. When tck x and is small (less than 0.5) the tick marks on the axes are forced to be the y same size. A value of 1 gives grid lines. Negative values give tick marks outside the plotting region. Use tck=0.01 and mgp=c(1,-1.5,0) for internal tick marks. xaxs="r" yaxs="i" x and y axes, respectively. With styles "i" (internal) and "r" Axis styles for the (the default) tick marks always fall within the range of the data, however style "r" leaves a small amount of space at the edges. (S has other styles not implemented in R.) 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.

78 Chapter 12: Graphical procedures 72 A typical figure is −−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−− mar[3] −−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−− 3.0 Plot region 1.5 y 0.0 mai[2] −1.5 −3.0 0.0 3.0 1.5 −1.5 −3.0 x mai[1] Margin 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) mai , except the measurement unit is text lines. Similar to 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.

79 Chapter 12: Graphical procedures 73 12.5.4 Multiple figure environment R allows you to create an m array of figures on a single page. Each figure has its own n by outer margin margins, and the array of figures is optionally surrounded by an , as shown in the following figure. −−−−−−−−−−−−−−− −−−−−−−−−−−−−−− oma[3] −−−−−−−−−−−−−−− −−−−−−−−−−−−−−− −−−−−−−−−−−−−−− omi[4] mfg=c(3,2,3,2) omi[1] mfrow=c(3,2) 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 and the pointsize of the device). In a layout with exactly two rows and par("cex") 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.

80 Chapter 12: Graphical procedures 74 Outer margins are particularly useful for page-wise titles, etc. Text can be added to the outer margins with the outer=TRUE . There are no outer margins by mtext() function with argument or . oma default, however, so you must create them explicitly using omi More complicated arrangements of multiple figures can be produced by the split.screen() functions, as well as by the grid and and layout() ( https://CRAN.R-project.org/ lattice ) package=lattice packages. 12.6 Device drivers R can generate graphics (of varying levels of quality) on almost any type of display or printing device. Before this can begin, however, R needs to be informed what type of device it is dealing with. This is done by starting a device driver . The purpose of a device driver is to convert graphical instructions from R (“draw a line,” for example) into a form that the particular device can understand. Device drivers are started by calling a device driver function. There is one such function help(Devices) for a list of them all. For example, issuing the for every device driver: type 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 For use on macOS quartz() 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.) Produces a bitmap JPEG file, best used for jpeg() plots. (Not always available: see image 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.) 12.6.1 PostScript diagrams for typeset documents file argument to the postscript() device driver function, you may store the By passing the graphics in PostScript format in a file of your choice. The plot will be in landscape orientation unless the horizontal=FALSE argument is given, and you can control the size of the graphic with the width and height arguments (the plot will be scaled as appropriate to fit these dimensions.) For example, the command > postscript("file.ps", horizontal=FALSE, height=5, pointsize=10) will produce a file containing PostScript code for a figure five inches high, perhaps for inclusion in a document. It is important to note that if the file named in the command already exists,

81 Chapter 12: Graphical procedures 75 it will be overwritten. This is the case even if the file was only created earlier in the same R session. Many usages of PostScript output will be to incorporate the figure in another document. This PostScript is produced: R always produces conformant output, works best when encapsulated onefile=FALSE argument is supplied. This unusual but only marks the output as such when the notation stems from S-compatibility: it really means that the output will be a single page (which is part of the EPSF specification). Thus to produce a plot for inclusion use something like > postscript("plot1.eps", horizontal=FALSE, onefile=FALSE, height=8, width=6, pointsize=10) 12.6.2 Multiple graphics devices In advanced use of R it is often useful to have several graphics devices in use at the same time. Of course only one graphics device can accept graphics commands at any one time, and this is current device known as the . When multiple devices are open, they form a numbered sequence with names giving the kind of device at any position. The main commands used for operating with multiple devices, and their meanings are as follows: X11() [UNIX] windows() win.printer() win.metafile() [Windows] quartz() [macOS] postscript() pdf() png() jpeg() tiff() bitmap() Each new call to a device driver function opens a new graphics device, thus extending ... by one the device list. This device becomes the current device, to which graphics output will be sent. dev.list() Returns the number and name of all active devices. The device at position 1 on the null device which does not accept graphics commands at all. list is always the dev.next() dev.prev() Returns the number and name of the graphics device next to, or previous to the current device, respectively. dev.set(which= ) k Can be used to change the current graphics device to the one at position k of the device list. Returns the number and label of the device. dev.off( k ) Terminate the graphics device at point k of the device list. For some devices, such as postscript devices, this will either print the file immediately or correctly complete the file for later printing, depending on how the device was initiated.

82 Chapter 12: Graphical procedures 76 k dev.copy(device, ..., which= ) dev.print(device, ..., which= k ) k . Here device is a device function, such as Make a copy of the device , postscript with extra arguments, if needed, specified by ‘ ... ’. dev.print is similar, but the copied device is immediately closed, so that end actions, such as printing hardcopies, are immediately performed. graphics.off() Terminate all graphics devices on the list, except the null device. 12.7 Dynamic graphics R does not have builtin capabilities for dynamic or interactive graphics, e.g. rotating point clouds or to “brushing” (interactively highlighting) points. However, extensive dynamic graphics facilities are available in the system GGobi by Swayne, Cook and Buja available from http://www.ggobi.org/ and these can be accessed from R via the package rggobi ( https://CRAN.R-project.org/ package=rggobi ) , described at http://www.ggobi.org/rggobi . Also, package rgl ( https://CRAN.R-project.org/package=rgl ) provides ways to interact with 3D plots, for example of surfaces.

84 Chapter 13: Packages 78 13.3 Namespaces Packages have , which do three things: they allow the package writer to hide functions namespaces 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, is the transpose function in R, but users might define their own function t() 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 base::t , because it is defined in the base package. Only functions that always be available as 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.

85 79 14 OS facilities R has quite extensive facilities to access the OS under which it is running: this allows it to be used as a scripting language and that ability is much used by R itself, for example to install packages. Because R’s own scripts need to work across all platforms, considerable effort has gone into make the scripting facilities as platform-independent as is feasible. 14.1 Files and directories There are many functions to manipulate files and directories. Here are pointers to some of the more commonly used ones. or dir.create . (These are the To create an (empty) file or directory, use file.create and mkdir .) For temporary files and directories in the analogues of the POSIX utilities touch R session directory see tempfile . file.remove or unlink Files can be removed by either : the latter can remove directory trees. list.files dir ) or list.dirs . These can select For directory listings use (also available as 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 file.info . be found by 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 is the R analogue of the POSIX command cp . file.copy file.choose Choosing files can be done interactively by : the Windows port has the more choose.files and choose.dir and there are similar functions in the tcltk versatile functions tk_choose.files and tk_choose.dir . package: Functions and file.edit will display and edit one or more files in a way appro- file.show priate 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 file.link and Sys.readlink . in the filesystem: see functions 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. • foo.png and Foo.PNG are different files. However, POSIX filesystems are case-sensitive, so the defaults on Windows and macOS are to be case-insensitive, and FAT filesystems (com- monly 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.

87 Chapter 14: OS facilities 81 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 https://en.wikipedia.org/wiki/LZMA ): R can read files compressed by and 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).

88 82 Appendix A A sample session The following session is intended to introduce to you some features of the R environment by using them. Many features of the system will be unfamiliar and puzzling at first, but this puzzlement will soon disappear. Start R appropriately for your platform (see ). Appendix B [Invoking R], page 85 The R program begins, with a banner. (Within R code, the prompt on the left hand side will not be shown to avoid con- fusion.) help.start() interface to on-line help (using a web browser available at your HTML Start the machine). You should briefly explore the features of this facility with the mouse. Iconify the help window and move on to the next part. x <- rnorm(50) y <- rnorm(x) x y -coordinates. Generate two pseudo-random normal vectors of - and plot(x, y) Plot the points in the plane. A graphics window will appear automatically. ls() See which R objects are now in the R workspace. rm(x, y) Remove objects no longer needed. (Clean up). x <- 1:20 Make x = (1 , 2 ,..., 20). w <- 1 + sqrt(x)/2 A ‘weight’ vector of standard deviations. dummy <- data.frame(x=x, y= x + rnorm(x)*w) , and look at it. data frame of two columns, x and dummy Make a y fm <- lm(y ~ x, data=dummy) summary(fm) Fit a simple linear regression and look at the analysis. With y to the left of the tilde, we are modelling y dependent on x . fm1 <- lm(y ~ x, data=dummy, weight=1/w^2) summary(fm1) Since we know the standard deviations, we can do a weighted regression. attach(dummy) Make the columns in the data frame visible as variables. lrf <- lowess(x, y) Make a nonparametric local regression function. plot(x, y) Standard point plot. lines(x, lrf\$y) Add in the local regression. abline(0, 1, lty=3) The true regression line: (intercept 0, slope 1). abline(coef(fm)) Unweighted regression line.

89 Appendix A: A sample session 83 abline(coef(fm1), col = "red") Weighted regression line. detach() Remove data frame from the search path. plot(fitted(fm), resid(fm), xlab="Fitted values", ylab="Residuals", main="Residuals vs Fitted") A standard regression diagnostic plot to check for heteroscedasticity. Can you see it? qqnorm(resid(fm), main="Residuals Rankit Plot") A normal scores plot to check for skewness, kurtosis and outliers. (Not very useful here.) rm(fm, fm1, lrf, x, dummy) Clean up again. The next section will look at data from the classical experiment of Michelson to measure the morley speed of light. This dataset is available in the object, but we will read it to illustrate read.table the function. filepath <- system.file("data", "morley.tab" , package="datasets") Get the path to the data file. filepath file.show(filepath) Optional. Look at the file. mm <- read.table(filepath) mm Read in the Michelson data as a data frame, and look at it. There are five exper- iments (column Expt ) and each has 20 runs (column Run ) and sl is the recorded speed of light, suitably coded. mm\$Expt <- factor(mm\$Expt) mm\$Run <- factor(mm\$Run) Change and Run into factors. Expt attach(mm) Make the data frame visible at position 3 (the default). plot(Expt, Speed, main="Speed of Light Data", xlab="Experiment No.") Compare the five experiments with simple boxplots. fm <- aov(Speed ~ Run + Expt, data=mm) summary(fm) Analyze as a randomized block, with ‘runs’ and ‘experiments’ as factors. fm0 <- update(fm, . ~ . - Run) anova(fm0, fm) Fit the sub-model omitting ‘runs’, and compare using a formal analysis of variance. detach() rm(fm, fm0) Clean up before moving on. We now look at some more graphical features: contour and image plots. x <- seq(-pi, pi, len=50) y <- x x is a vector of 50 equally spaced values in is the same. π ≤ x ≤ π . y −

90 84 f <- outer(x, y, function(x, y) cos(y)/(1 + x^2)) is a square matrix, with rows and columns indexed by x y respectively, of f and 2 ) / y x ). values of the function cos( (1 + oldpar <- par(no.readonly = TRUE) par(pty="s") Save the plotting parameters and set the plotting region to “square”. contour(x, y, f) contour(x, y, f, nlevels=15, add=TRUE) f Make a contour map of ; add in more lines for more detail. fa <- (f-t(f))/2 is the “asymmetric part” of f . ( fa is transpose). t() contour(x, y, fa, nlevels=15) Make a contour plot, . . . par(oldpar) . . . and restore the old graphics parameters. image(x, y, f) image(x, y, fa) Make some high density image plots, (of which you can get hardcopies if you wish), . . . objects(); rm(x, y, f, fa) . . . and clean up before moving on. R can do complex arithmetic, also. th <- seq(-pi, pi, len=100) z <- exp(1i*th) is used for the complex number 1i . i par(pty="s") plot(z, type="l") Plotting complex arguments means plot imaginary versus real parts. This should be a circle. w <- rnorm(100) + rnorm(100)*1i Suppose we want to sample points within the unit circle. One method would be to take complex numbers with standard normal real and imaginary parts . . . w <- ifelse(Mod(w) > 1, 1/w, w) . . . and to map any outside the circle onto their reciprocal. plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+",xlab="x", ylab="y") lines(z) All points are inside the unit circle, but the distribution is not uniform. w <- sqrt(runif(100))*exp(2*pi*runif(100)*1i) plot(w, xlim=c(-1,1), ylim=c(-1,1), pch="+", xlab="x", ylab="y") lines(z) The second method uses the uniform distribution. The points should now look more evenly spaced over the disc. rm(th, w, z) Clean up again. q() Quit the R program. You will be asked if you want to save the R workspace, and for an exploratory session like this, you probably do not want to save it.

92 Appendix B: Invoking R 86 enc --encoding= . This needs Specify the encoding to be assumed for input from the console or stdin : see its help page. ( enc is also iconv to be an encoding known to --encoding accepted.) The input is re-encoded to the locale R is running in and needs to be representable in the latter’s encoding (so e.g. you cannot re-encode Greek text in a French locale unless that locale uses the UTF-8 encoding). RHOME Print the path to the R “home directory” to standard output and exit success- fully. Apart from the front-end shell script and the man page, R installation puts everything (executables, packages, etc.) into this directory. --save --no-save Control whether data sets should be saved or not at the end of the R session. If neither is given in an interactive session, the user is asked for the desired behavior when ending the session with q() ; in non-interactive use one of these must be specified or implied by some other option (see below). --no-environ Do not read any user file to set environment variables. --no-site-file Do not read the site-wide profile at startup. --no-init-file Do not read the user’s profile at startup. --restore --no-restore --no-restore-data Control whether saved images (file in the directory where R was started) .RData should be restored at startup or not. The default is to restore. ( --no-restore implies all the specific options.) --no-restore-* --no-restore-history Control whether the history file (normally file .Rhistory in the directory where R was started, but can be set by the environment variable R_HISTFILE ) should be restored at startup or not. The default is to restore. --no-Rconsole (Windows only) Prevent loading the Rconsole file at startup. --vanilla Combine , --no-environ , --no-site-file , --no-init-file and --no-save --no- restore . Under Windows, this also includes --no-Rconsole . -f file --file= file (not Rgui.exe ) Take input from file : ‘ - ’ means stdin . Implies --no-save unless --save has been set. On a Unix-alike, shell metacharacters should be avoided in file (but spaces are allowed). -e expression (not Rgui.exe ) Use expression as an input line. One or more -e options can be unless used, but not together with or --file . Implies --no-save -f --save has been set. (There is a limit of 10,000 bytes on the total length of expressions used in this way. Expressions containing spaces or shell metacharacters will need to be quoted.)

94 Appendix B: Invoking R 88 Tcl/Tk ’ support is available, ‘ ’. (For back-compatibility, ‘ x11 ’ and provided that ‘ Tk ’ are accepted.) tk ‘ name --arch= (UNIX only) Run the specified sub-architecture. --args This flag does nothing except cause the rest of the command line to be skipped: commandArgs(TRUE) this can be useful to retrieve values from it with . Note that input and output can be redirected in the usual way (using ‘ < ’ and ‘ > ’), but the line length limit of 4095 bytes still applies. Warning and error messages are sent to the error channel ( ). stderr The command allows the invocation of various tools which are useful in conjunction R CMD with R, but not intended to be called “directly”. The general form is R CMD command args where command is the name of the tool and args the arguments passed on to it. Currently, the following tools are available. BATCH R --restore --save with possibly further options (see Run R in batch mode. Runs ?BATCH ). COMPILE (UNIX only) Compile C, C ++ , Fortran . . . files for use with R. SHLIB Build shared library for dynamic loading. INSTALL Install add-on packages. Remove add-on packages. REMOVE Build (that is, package) add-on packages. build check Check add-on packages. LINK (UNIX only) Front-end for creating executable programs. Post-process R profiling files. Rprof Rdconv A Rd2txt Convert Rd format to various other formats, including HTML , L T X, plain text, E and extracting the examples. Rd2txt can be used as shorthand for Rd2conv -t txt . Rd2pdf Convert Rd format to PDF. Extract S/R code from Sweave or other vignette documentation Stangle Sweave Process Sweave or other vignette documentation Rdiff Diff R output ignoring headers etc config Obtain configuration information javareconf (Unix only) Update the Java configuration variables rtags (Unix only) Create Emacs-style tag files from C, R, and Rd files open (Windows only) Open a file via Windows’ file associations texify (Windows only) Process (La)TeX files with R’s style files Use R CMD command --help to obtain usage information for each of the tools accessible via the R CMD interface.

95 Appendix B: Invoking R 89 --arch= , , --no-init-file , --no-site- In addition, you can use options --no-environ --vanilla file and CMD : these affect any R processes run by the tools. (Here between R and .) However, note --no-environ --no-site-file --no-init-file --vanilla is equivalent to does not of itself use any R startup files (in particular, neither user nor site Renviron that R CMD ) use files), and all of the R processes run by these tools (except . Most BATCH --no-restore and so invoke no R startup files: the current exceptions are --vanilla , REMOVE , use INSTALL and SHLIB (which uses --no-site-file --no-init-file Sweave ). R CMD cmd args cmd on the path or given by an absolute filepath: this is useful to have for any other executable or the same environment as R or the specific commands run under, for example to run ldd . Under Windows cmd can be an executable or a batch file, or if it has extension .sh pdflatex the appropriate interpreter (if available) is called to run it. or .pl B.2 Invoking R under Windows There are two ways to run R under Windows. Within a terminal window (e.g. cmd.exe or a more capable shell), the methods described in the previous section may be used, invoking by R.exe or more directly by . For interactive use, there is a console-based GUI ( Rgui.exe ). Rterm.exe The startup procedure under Windows is very similar to that under UNIX, but references to the ‘home directory’ need to be clarified, as this is not always defined on Windows. If the R_USER is defined, that gives the home directory. Next, if the environment environment variable HOME variable is defined, that gives the home directory. After those two user-controllable settings, R tries to find system defined home directories. It first tries to use the Windows " personal " directory (typically My Documents in recent versions of Windows). If that fails, and environment variables HOMEDRIVE HOMEPATH are defined (and they normally are) these define the home and directory. Failing all those, the home directory is taken to be the starting directory. You need to ensure that either the environment variables TMP and TEMP are either TMPDIR , unset or one of them points to a valid place to create temporary files and directories. name = value ’ pairs on the command line. Environment variables can be supplied as ‘ .RData (in any case) it is interpreted as the path to the If there is an argument ending --restore workspace to be restored: it implies and sets the working directory to the parent of RGui.exe the named file. (This mechanism is used for drag-and-drop and file association with , Rterm.exe . If the named file does not exist it sets the working directory if but also works for the parent directory exists.) RGui.exe . The following additional command-line options are available when invoking --mdi --sdi --no-mdi Control whether Rgui will operate as an MDI program (with multiple child windows within one main window) or an SDI application (with multiple top-level windows for the console, graphics and pager). The command-line setting overrides the setting in the user’s Rconsole file. --debug Enable the “Break to debugger” menu item in Rgui , and trigger a break to the debugger during command line processing. Under Windows with R CMD you may also specify your own .bat , .exe , .sh or .pl file. It will be run under the appropriate interpreter (Perl for ) with several environment variables .pl set appropriately, including R_HOME , R_OSTYPE , PATH , BSTINPUTS and TEXINPUTS . For example, if you already have latex.exe on your path, then R CMD latex.exe mydoc

96 Appendix B: Invoking R 90 A T X on mydoc.tex , with the path to R’s share/texmf macros appended to TEXINPUTS . will run L E A X, but R CMD texify mydoc (Unfortunately, this does not help with the MiKTeX build of L T E will work in that case.) B.3 Invoking R under macOS Terminal.app window by invoking R , the There are two ways to run R under macOS. Within a methods described in the first subsection apply. There is also console-based GUI ( ) that by R.app default is installed in the folder on your system. It is a standard double-clickable Applications macOS application. The startup procedure under macOS is very similar to that under UNIX, but does not R.app make use of command-line arguments. The ‘home directory’ is the one inside the R.framework, but the startup and current working directory are set as the user’s home directory unless a different startup directory is given in the Preferences window accessible from within the GUI. B.4 Scripting with R foo.R If you just want to run a file R CMD BATCH of R commands, the recommended way is to use foo.R . If you want to run this in the background or as a batch job use OS-specific facilities to do so: for example in most shells on Unix-alike OSes R CMD BATCH foo.R & runs a background job. You can pass parameters to scripts via additional arguments on the command line: for example (where the exact quoting needed will depend on the shell in use) R CMD BATCH "--args arg1 arg2" foo.R & will pass arguments to a script which can be retrieved as a character vector by args <- commandArgs(TRUE) Rscript , which can be invoked by This is made simpler by the alternative front-end Rscript foo.R arg1 arg2 and this can also be used to write executable script files like (at least on Unix-alikes, and in some Windows shells) #! /path/to/Rscript args <- commandArgs(TRUE) ... q(status=) If this is entered into a text file runfoo and this is made executable (by chmod 755 runfoo ), it can be invoked for different arguments by runfoo arg1 arg2 For further options see help("Rscript") . This writes R output to stdout and stderr , and this can be redirected in the usual way for the shell running the command. Rscript but have it in your path (which is If you do not wish to hardcode the path to normally the case for an installed R except on Windows, but e.g. macOS users may need to add /usr/local/bin to their path), use #! /usr/bin/env Rscript ... At least in Bourne and bash shells, the #! mechanism does not allow extra arguments like #! /usr/bin/env Rscript --vanilla . One thing to consider is what stdin() refers to. It is commonplace to write R scripts with segments like chem <- scan(n=24)

97 91 2.90 3.10 3.40 3.40 3.70 3.70 2.80 2.50 2.40 2.40 2.70 2.20 5.28 3.37 3.03 3.03 28.95 3.77 3.40 2.20 3.50 3.60 3.70 3.70 and refers to the script file to allow such traditional usage. If you want to refer to the stdin() stdin , use "stdin" as a file process’s scan("stdin", ...) . connection, e.g. Another way to write executable script files (suggested by Fran ̧cois Pinard) is to use a here document like #!/bin/sh [environment variables can be set here] R --slave [other options] <

99 Appendix C: The command-line editor 93 C-p and , On most terminals, you can also use the up and down arrow keys instead of C-n respectively. Horizontal motion of the cursor C-a Go to the beginning of the command. Go to the end of the line. C-e M-b Go back one word. M-f Go forward one word. C-b Go back one character. C-f Go forward one character. C-b C-f , On most terminals, you can also use the left and right arrow keys instead of and respectively. Editing and re-submission text Insert text at the cursor. C-f text Append text after the cursor. DEL Delete the previous character (left of the cursor). Delete the character under the cursor. C-d Delete the rest of the word under the cursor, and “save” it. M-d C-k Delete from cursor to end of command, and “save” it. Insert (yank) the last “saved” text here. C-y Transpose the character under the cursor with the next. C-t M-l Change the rest of the word to lower case. M-c Change the rest of the word to upper case. RET Re-submit the command to R. The final terminates the command line editing sequence. RET The readline key bindings can be customized in the usual way via a ~/.inputrc file. These customizations can be conditioned on application R , that is by including a section like \$if R "\C-xd": "q(’no’)\n" \$endif

100 94 Appendix D Function and variable index ? ! 9 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ? 9 != . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 % ^ 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . %*% ^ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 %o% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 | & 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 9 & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . || . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . && 40 * ~ 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 + A 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . abline ace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . add1 56 – 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . anova , 55 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 aov 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aperm array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . as.data.frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . as.vector 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .First . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 attach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 48 .Last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . attr 14 attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 avas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 / 67 axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 boxplot : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 :: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 bruto 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ::: C < 27 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 , 10 , 24 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . < 9 cbind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 <<- 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coef 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <= 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coefficients contour 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . contrasts coplot 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . == 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 , crossprod cut 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . > 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . >= 9

101 Appendix D: Function and variable index 95 K D 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . data 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ks.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 data.frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . density 34 L 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . det 28 detach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . legend 66 determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 length 13 8 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dev.list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . levels 16 dev.next 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dev.off 75 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . list 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dev.prev 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lm dev.set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 lme 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . locator 68 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . diag loess 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 log 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dotchart 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lqs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . drop1 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lsfit 23 E M 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ecdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 mars 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . edit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 eigen mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 else . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 55 Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mode exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 N F 9 NaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 NA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 FALSE 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ncol fivenum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for nlm 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 61 , 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 formula 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nlme 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nlminb 59 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nrow G O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 getAnywhere 59 optim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . getS3method 49 order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 glm 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ordered 17 outer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 H help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 P 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . help.search 64 pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 help.start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 par . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . hist 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 paste 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pdf 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . persp I plot 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 , identify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8 pmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8 pmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ifelse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . png . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . image 65 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 is.na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 polygon 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . is.nan postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . predict 54 print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 J 8 prod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . jpeg 74

102 Appendix D: Function and variable index 96 34 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q , 54 svd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 64 qqline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 35 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 64 qqnorm qqplot 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qr 23 t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 quartz t.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 25 table , 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 tan R 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tapply 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rbind 24 67 title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . read.table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rep 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T 9 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . repeat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 TRUE resid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 U 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rlm rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 unclass 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . update S V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scan 31 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . var 17 , 8 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . search 38 var.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 seq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vcov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 shapiro.test 7 vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sin 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sink W solve 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . while 41 source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 wilcox.test 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . split 40 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . windows sqrt 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 stem X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 , 54 step X11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sum 74

103 97 Appendix E Concept index A K Accessing builtin datasets . . . . . . . . . . . . . . . . . . . . . . . . 31 36 Kolmogorov-Smirnov test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Additive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Analysis of variance Arithmetic functions and operators 7 . . . . . . . . . . . . . . . . L 18 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Least squares fitting 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assignment 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear equations 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear models 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 B 61 . . . . . . . . . . . . . . . . . . Local approximating regressions 40 Loops and conditional execution . . . . . . . . . . . . . . . . . . Binary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 37 Box plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M C 18 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Character vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix multiplication 22 48 , 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum likelihood 61 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concatenating lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Missing values 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contrasts 53 Mixed models 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control statements 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CRAN 77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 . . . . . . . . . . . . . . . . . . . . . Customizing the environment N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Named arguments 43 D Namespace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Nonlinear least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Default values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determinants O Diverting input and output . . . . . . . . . . . . . . . . . . . . . . . . 5 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Object orientation Dynamic graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objects . . . . . . . . . . . . . . . . . . . . . . . . One- and two-sample tests 36 Ordered factors , 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 E Outer products of arrays . . . . . . . . . . . . . . . . . . . . . . . . . 21 23 . . . . . . . . . . . . . . . . . . . . . . Eigenvalues and eigenvectors Empirical CDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 P F 77 , 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packages 33 . . . . . . . . . . . . . . . . . . . . . . . . . . Probability distributions Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 53 , 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Families Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QR decomposition 23 G Quantile-quantile plots 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized linear models . . . . . . . . . . . . . . . . . . . . . . . . . 56 Generalized transpose of an array . . . . . . . . . . . . . . . . . 21 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generic functions R 74 . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphics device drivers 68 Graphics parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading data from files Grouped expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7 , 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recycling rule Regular sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6 Removing objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Robust regression 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Indexing of and by arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indexing vectors 10

104 Appendix E: Concept index 98 U S 55 Updating fitted models . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Search path 29 Shapiro-Wilk test 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . Singular value decomposition V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical models 51 7 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Student’s t test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W T Wilcoxon test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tree-based models Writing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

105 99 Appendix F References Nonlinear Regression Analysis and Its Applications. D. M. Bates and D. G. Watts (1988), John Wiley & Sons, New York. Richard A. Becker, John M. Chambers and Allan R. Wilks (1988), The New S Language. Chap- Blue Book ”. man & Hall, New York. This book is often called the “ Statistical Models in S. Chapman & Hall, John M. Chambers and Trevor J. Hastie eds. (1992), New York. This is also called the “ White Book ”. John M. Chambers (1998) Programming with Data . Springer, New York. This is also called the “ Green Book ”. A. C. Davison and D. V. Hinkley (1997), , Cambridge Bootstrap Methods and Their Applications University Press. Annette J. Dobson (1990), An Introduction to Generalized Linear Models , Chapman and Hall, London. Peter McCullagh and John A. Nelder (1989), Generalized Linear Models. Second edition, Chap- man and Hall, London. John A. Rice (1995), Mathematical Statistics and Data Analysis. Second edition. Duxbury Press, Belmont, CA. S. D. Silvey (1970), Statistical Inference. Penguin, London.

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