An Introduction to R
Notes on R: A Programming Environment for Data Analysis and Graphics
Version 2.11.1 (2010-05-31)
W. N. Venables, D. M. Smith
and the R Development Core Team

Copyright c 1990 W. N. Venables
Copyright c 1992 W. N. Venables & D. M. Smith
Copyright c 1997 R. Gentleman & R. Ihaka
Copyright c 1997, 1998 M. Maechler
Copyright c 1997– R Core Development Team
Copyright c 1999–2010 R Development Core Team
Permission is granted to make and distribute verbatim copies of this manual provided the copy-
right notice and this permission notice are preserved on all copies.
Permission is granted to copy and distribute modified versions of this manual under the condi-
tions for verbatim copying, provided that the entire resulting derived work is distributed under
the terms of a permission notice identical to this one.
Permission is granted to copy and distribute translations of this manual into another language,
under the above conditions for modified versions, except that this permission notice may be
stated in a translation approved by the R Development Core Team.
ISBN 3-900051-12-7

i
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1
Introduction and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1
The R environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2
Related software and documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3
R and statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4
R and the window system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5
Using R interactively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6
An introductory session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
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
2.2
Vector arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3
Generating regular sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4
Logical vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
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
3
Objects, their modes and attributes . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1
Intrinsic attributes: mode and length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2
Changing the length of an object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3
Getting and setting attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4
The class of an object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4
Ordered and unordered factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1
A specific example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2
The function tapply() and ragged arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3
Ordered factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5
Arrays and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.1
Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2
Array indexing. Subsections of an array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3
Index matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.4
The array() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
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

ii
5.7.2
Linear equations and inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.7.3
Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.7.4
Singular value decomposition and determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.7.5
Least squares fitting and the QR decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.8
Forming partitioned matrices, cbind() and rbind() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.9
The concatenation function, c(), with arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.10
Frequency tables from factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6
Lists and data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1
Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2
Constructing and modifying lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2.1
Concatenating lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3
Data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3.1
Making data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3.2
attach() and detach() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3.3
Working with data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3.4
Attaching arbitrary lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.3.5
Managing the search path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7
Reading data from files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.1
The read.table() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.2
The scan() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.3
Accessing builtin datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.3.1
Loading data from other R packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.4
Editing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8
Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.1
R as a set of statistical tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.2
Examining the distribution of a set of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.3
One- and two-sample tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
9
Grouping, loops and conditional execution . . . . . . . . . . . . . . . . . 40
9.1
Grouped expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9.2
Control statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9.2.1
Conditional execution: if statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9.2.2
Repetitive execution: for loops, repeat and while . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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 ‘...’ argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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

iii
11
Statistical models in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11.1
Defining statistical models; formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
11.1.1
Contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
11.2
Linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11.3
Generic functions for extracting model information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11.4
Analysis of variance and model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.4.1
ANOVA tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.5
Updating fitted models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.6
Generalized linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
11.6.1
Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.6.2
The glm() function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.7
Nonlinear least squares and maximum likelihood models . . . . . . . . . . . . . . . . . . . . . . . . . 58
11.7.1
Least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
11.7.2
Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
11.8
Some non-standard models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
12
Graphical procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.1
High-level plotting commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.1.1
The plot() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.1.2
Displaying multivariate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
12.1.3
Display graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
12.1.4
Arguments to high-level plotting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
12.2
Low-level plotting commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.2.1
Mathematical annotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
12.2.2
Hershey vector fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
12.3
Interacting with graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
12.4
Using graphics parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12.4.1
Permanent changes: The par() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12.4.2
Temporary changes: Arguments to graphics functions . . . . . . . . . . . . . . . . . . . . . . . 68
12.5
Graphics parameters list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
12.5.1
Graphical elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
12.5.2
Axes and tick marks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
12.5.3
Figure margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
12.5.4
Multiple figure environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
12.6
Device drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12.6.1
PostScript diagrams for typeset documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12.6.2
Multiple graphics devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
12.7
Dynamic graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
13
Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
13.1
Standard packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
13.2
Contributed packages and CRAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
13.3
Namespaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Appendix A
A sample session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Appendix B
Invoking R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.1
Invoking R from the command line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
B.2
Invoking R under Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.3
Invoking R under Mac OS X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.4
Scripting with R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

iv
Appendix C
The command-line editor . . . . . . . . . . . . . . . . . . . . . . . 87
C.1
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C.2
Editing actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
C.3
Command-line editor summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Appendix D
Function and variable index . . . . . . . . . . . . . . . . . . . . 89
Appendix E
Concept index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Appendix F
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Preface
1
Preface
This introduction to R is derived from an original set of notes describing the S and S-Plus
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
R-core@R-project.org.
Suggestions to the reader
Most R novices will start with the introductory session in Appendix A. This should give some
familiarity with the style of R sessions and more importantly some instant feedback on what
actually happens.
Many users will come to R mainly for its graphical facilities.
In this case, Chapter 12
[Graphics], page 62 on the graphics facilities can be read at almost any time and need not wait
until all the preceding sections have been digested.

Chapter 1: Introduction and preliminaries
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
developed rapidly, and has been extended by a large collection of packages. However, most
programs written in R are essentially ephemeral, written for a single piece of data analysis.
1.2 Related software and documentation
R can be regarded as an implementation of the S language which was developed at Bell Labora-
tories by Rick Becker, John Chambers and Allan Wilks, and also forms the basis of the S-Plus
systems.
The evolution of the S language is characterized by four books by John Chambers and
coauthors. For R, the basic reference is The New S Language: A Programming Environment
for Data Analysis and Graphics by Richard A. Becker, John M. Chambers and Allan R. Wilks.
The new features of the 1991 release of S are covered in Statistical Models in S edited by John
M. Chambers and Trevor J. Hastie. The formal methods and classes of the methods package are
based on those described in Programming with Data by John M. Chambers. See Appendix F
[References], page 94, for precise references.
There are now a number of books which describe how to use R for data analysis and statistics,
and documentation for S/S-Plus can typically be used with R, keeping the differences between
the S implementations in mind. See Section “What documentation exists for R?” in The R
statistical system FAQ.
1.3 R and statistics
Our introduction to the R environment did not mention statistics, yet many people use R as a
statistics system. We prefer to think of it of an environment within which many classical and
modern statistical techniques have been implemented. A few of these are built into the base R
environment, but many are supplied as packages. There are about 25 packages supplied with
R (called “standard” and “recommended” packages) and many more are available through the
CRAN family of Internet sites (via http://CRAN.R-project.org) and elsewhere. More details
on packages are given later (see Chapter 13 [Packages], page 76).
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.

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 Mac OS 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 yes, no or cancel (a single letter abbreviation will do) to save
the data before quitting, quit without saving, or return to the R session. Data which is
saved will be available in future R sessions.
Further R sessions are simple.
1. Make ‘work’ the working directory and start the program as before:
$ cd work
$ R
2. Use the R program, terminating with the q() command at the end of the session.
To use R under Windows the procedure to follow is basically the same. Create a folder as
the working directory, and set that in the ‘Start In’ field in your R shortcut. Then launch R
by double clicking on the icon.

Chapter 1: Introduction and preliminaries
4
1.6 An introductory session
Readers wishing to get a feel for R at a computer before proceeding are strongly advised to work
through the introductory session given in Appendix A [A sample session], page 78.
1.7 Getting help with functions and features
R has an inbuilt help facility similar to the man facility of UNIX. To get more information on
any specific named function, for example solve, the command is
> help(solve)
An alternative is
> ?solve
For a feature specified by special characters, the argument must be enclosed in double or single
quotes, making it a “character string”: This is also necessary for a few words with syntactic
meaning including if, for and function.
> help("[[")
Either form of quote mark may be used to escape the other, as in the string "It’s
important". Our convention is to use double quote marks for preference.
On most R installations help is available in HTML format by running
> help.start()
which will launch a Web browser that allows the help pages to be browsed with hyperlinks. On
UNIX, subsequent help requests are sent to the HTML-based help system. The ‘Search Engine
and Keywords’ link in the page loaded by help.start() is particularly useful as it is contains
a high-level concept list which searches though available functions. It can be a great way to get
your bearings quickly and to understand the breadth of what R has to offer.
The help.search command (alternatively ??) allows searching for help in various ways. For
example,
> ??solve
Try ?help.search for details and more examples.
The examples on a help topic can normally be run by
> example(topic )
Windows versions of R have other optional help systems: use
> ?help
for further details.
1.8 R commands, case sensitivity, etc.
Technically R is an expression language with a very simple syntax. It is case sensitive as are most
UNIX based packages, so A and a are different symbols and would refer to different variables.
The set of symbols which can be used in R names depends on the operating system and country
within which R is being run (technically on the locale in use).
Normally all alphanumeric
symbols are allowed1 (and in some countries this includes accented letters) plus ‘.’ and ‘_’, with
the restriction that a name must start with ‘.’ or a letter, and if it starts with ‘.’ the second
character must not be a digit.
Elementary commands consist of either expressions or assignments. If an expression is given
as a command, it is evaluated, printed (unless specifically made invisible), and the value is lost.
An assignment also evaluates an expression and passes the value to a variable but the result is
not automatically printed.
1 For portable R code (including that to be used in R packages) only A–Za–z0–9 should be used.

Chapter 1: Introduction and preliminaries
5
Commands are separated either by a semi-colon (‘;’), or by a newline. Elementary commands
can be grouped together into one compound expression by braces (‘{’ and ‘}’). Comments can
be put almost2 anywhere, starting with a hashmark (‘#’), everything to the end of the line is a
comment.
If a command is not complete at the end of a line, R will give a different prompt, by default
+
on second and subsequent lines and continue to read input until the command is syntactically
complete. This prompt may be changed by the user. We will generally omit the continuation
prompt and indicate continuation by simple indenting.
Command lines entered at the console are limited3 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 87.
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
If commands4 are stored in an external file, say ‘commands.R’ in the working directory ‘work’,
they may be executed at any time in an R session with the command
> source("commands.R")
For Windows Source is also available on the File menu. The function sink,
> sink("record.lis")
will divert all subsequent output from the console to an external file, ‘record.lis’. The com-
mand
> 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 session). 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.

Chapter 1: Introduction and preliminaries
6
To remove objects the function rm is available:
> rm(x, y, z, ink, junk, temp, foo, bar)
All objects created during an R session can be stored permanently in a file for use in future
R sessions. At the end of each R session you are given the opportunity to save all the currently
available objects. If you indicate that you want to do this, the objects are written to a file called
‘.RData’5 in the current directory, and the command lines used in the session are saved to a file
called ‘.Rhistory’.
When R is started at later time from the same directory it reloads the workspace from this
file. At the same time the associated commands history is reloaded.
It is recommended that you should use separate working directories for analyses conducted
with R. It is quite common for objects with names x and y to be created during an analysis.
Names like this are often meaningful in the context of a single analysis, but it can be quite
hard to decide what they might be when the several analyses have been conducted in the same
directory.
5 The leading “dot” in this file name makes it invisible in normal file listings in UNIX.

Chapter 2: Simple manipulations; numbers and vectors
7
2 Simple manipulations; numbers and vectors
2.1 Vectors and assignment
R operates on named data structures. The simplest such structure is the numeric vector, which
is a single entity consisting of an ordered collection of numbers. To set up a vector named x,
say, consisting of five numbers, namely 10.4, 5.6, 3.1, 6.4 and 21.7, use the R command
> x <- c(10.4, 5.6, 3.1, 6.4, 21.7)
This is an assignment statement using the function c() which in this context can take an
arbitrary number of vector arguments and whose value is a vector got by concatenating its
arguments end to end.1
A number occurring by itself in an expression is taken as a vector of length one.
Notice that the assignment operator (‘<-’), which consists of the two characters ‘<’ (“less
than”) and ‘-’ (“minus”) occurring strictly side-by-side and it ‘points’ to the object receiving
the value of the expression. In most contexts the ‘=’ operator can be used as an alternative.
Assignment can also be made using the function assign(). An equivalent way of making
the same assignment as above is with:
> assign("x", c(10.4, 5.6, 3.1, 6.4, 21.7))
The usual operator, <-, can be thought of as a syntactic short-cut to this.
Assignments can also be made in the other direction, using the obvious change in the assign-
ment operator. So the same assignment could be made using
> c(10.4, 5.6, 3.1, 6.4, 21.7) -> x
If an expression is used as a complete command, the value is printed and lost 2. So now if we
were to use the command
> 1/x
the reciprocals of the five values would be printed at the terminal (and the value of x, of course,
unchanged).
The further assignment
> y <- c(x, 0, x)
would create a vector y with 11 entries consisting of two copies of x with a zero in the middle
place.
2.2 Vector arithmetic
Vectors can be used in arithmetic expressions, in which case the operations are performed element
by element. Vectors occurring in the same expression need not all be of the same length. If
they are not, the value of the expression is a vector with the same length as the longest vector
which occurs in the expression. Shorter vectors in the expression are recycled as often as need be
(perhaps fractionally) until they match the length of the longest vector. In particular a constant
is simply repeated. So with the above assignments the command
> v <- 2*x + y + 1
generates a new vector v of length 11 constructed by adding together, element by element, 2*x
repeated 2.2 times, y repeated just once, and 1 repeated 11 times.
The elementary arithmetic operators are the usual +, -, *, / and ^ for raising to a power. In
addition all of the common arithmetic functions are available. log, exp, sin, cos, tan, sqrt,
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.

Chapter 2: Simple manipulations; numbers and vectors
8
and so on, all have their usual meaning. max and min select the largest and smallest elements of a
vector respectively. range is a function whose value is a vector of length two, namely c(min(x),
max(x)). length(x) is the number of elements in x, sum(x) gives the total of the elements in
x, and prod(x) their product.
Two statistical functions are mean(x) which calculates the sample mean, which is the same
as sum(x)/length(x), and var(x) which gives
sum((x-mean(x))^2)/(length(x)-1)
or sample variance. If the argument to var() is an n-by-p matrix the value is a p-by-p sample
covariance matrix got by regarding the rows as independent p-variate sample vectors.
sort(x) returns a vector of the same size as x with the elements arranged in increasing order;
however there are other more flexible sorting facilities available (see order() or sort.list()
which produce a permutation to do the sorting).
Note that max and min select the largest and smallest values in their arguments, even if they
are given several vectors. The parallel maximum and minimum functions pmax and pmin return
a vector (of length equal to their longest argument) that contains in each element the largest
(smallest) element in that position in any of the input vectors.
For most purposes the user will not be concerned if the “numbers” in a numeric vector
are integers, reals or even complex. Internally calculations are done as double precision real
numbers, or double precision complex numbers if the input data are complex.
To work with complex numbers, supply an explicit complex part. Thus
sqrt(-17)
will give NaN and a warning, but
sqrt(-17+0i)
will do the computations as complex numbers.
2.3 Generating regular sequences
R has a number of facilities for generating commonly used sequences of numbers. For example
1:30 is the vector c(1, 2, ..., 29, 30). The colon operator has high priority within an ex-
pression, so, for example 2*1:15 is the vector c(2, 4, ..., 28, 30). Put n <- 10 and compare
the sequences 1:n-1 and 1:(n-1).
The construction 30:1 may be used to generate a sequence backwards.
The function seq() is a more general facility for generating sequences. It has five arguments,
only some of which may be specified in any one call. The first two arguments, if given, specify
the beginning and end of the sequence, and if these are the only two arguments given the result
is the same as the colon operator. That is seq(2,10) is the same vector as 2:10.
Parameters 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 parameters may be named
from=value and to=value ; thus seq(1,30), seq(from=1, to=30) and seq(to=30, from=1)
are all the same as 1:30. The next two parameters to seq() may be named by=value and
length=value , which specify a step size and a length for the sequence respectively. If neither
of these is given, the default by=1 is assumed.
For example
> seq(-5, 5, by=.2) -> s3
generates in s3 the vector c(-5.0, -4.8, -4.6, ..., 4.6, 4.8, 5.0). Similarly
> s4 <- seq(length=51, from=-5, by=.2)
generates the same vector in s4.

Chapter 2: Simple manipulations; numbers and vectors
9
The fifth parameter may be named along=vector , which if used must be the only parameter,
and creates a sequence 1, 2, ..., length(vector ), or the empty sequence if the vector is
empty (as it can be).
A related function is rep() which can be used for replicating an object in various complicated
ways. The simplest form is
> s5 <- rep(x, times=5)
which will put five copies of x end-to-end in s5. Another useful version is
> s6 <- rep(x, each=5)
which repeats each element of x five times before moving on to the next.
2.4 Logical vectors
As well as numerical vectors, R allows manipulation of logical quantities. The elements of a
logical vector can have the values TRUE, FALSE, and NA (for “not available”, see below). The
first two are often abbreviated as T and F, respectively. Note however that T and F are just
variables which are set to TRUE and FALSE by default, but are not reserved words and hence can
be overwritten by the user. Hence, you should always use TRUE and FALSE.
Logical vectors are generated by conditions. For example
> temp <- x > 13
sets temp as a vector of the same length as x with values FALSE corresponding to elements of x
where the condition is not met and TRUE where it is.
The logical operators are <, <=, >, >=, == for exact equality and != for inequality. In addition
if c1 and c2 are logical expressions, then c1 & c2 is their intersection (“and”), c1 | c2 is their
union (“or”), and !c1 is the negation of c1.
Logical vectors may be used in ordinary arithmetic, in which case they are coerced into
numeric vectors, FALSE becoming 0 and TRUE becoming 1. However there are situations where
logical vectors and their coerced numeric counterparts are not equivalent, for example see the
next subsection.
2.5 Missing values
In some cases the components of a vector may not be completely known. When an element
or value is “not available” or a “missing value” in the statistical sense, a place within a vector
may be reserved for it by assigning it the special value NA. In general any operation on an NA
becomes an NA. The motivation for this rule is simply that if the specification of an operation
is incomplete, the result cannot be known and hence is not available.
The function is.na(x) gives a logical vector of the same size as x with value TRUE if and
only if the corresponding element in x is NA.
> z <- c(1:3,NA);
ind <- is.na(z)
Notice that the logical expression x == NA is quite different from is.na(x) since NA is not
really a value but a marker for a quantity that is not available. Thus x == NA is a vector of the
same length as x all of whose values are NA as the logical expression itself is incomplete and
hence undecidable.
Note that there is a second kind of “missing” values which are produced by numerical com-
putation, the so-called Not a Number, NaN, values. Examples are
> 0/0
or

Chapter 2: Simple manipulations; numbers and vectors
10
> Inf - Inf
which both give NaN since the result cannot be defined sensibly.
In summary, is.na(xx) is TRUE both for NA and NaN values.
To differentiate these,
is.nan(xx) is only TRUE for NaNs.
Missing values are sometimes printed as <NA> when character vectors are printed without
quotes.
2.6 Character vectors
Character quantities and character vectors are used frequently in R, for example as plot labels.
Where needed they are denoted by a sequence of characters delimited by the double quote
character, e.g., "x-values", "New iteration results".
Character strings are entered using either matching double (") or single (’) quotes, but are
printed using double quotes (or sometimes without quotes). They use C-style escape sequences,
using \ as the escape character, so \\ is entered and printed as \\, and inside double quotes "
is entered as \". Other useful escape sequences are \n, newline, \t, tab and \b, backspace—see
?Quotes for a full list.
Character vectors may be concatenated into a vector by the c() function; examples of their
use will emerge frequently.
The paste() function takes an arbitrary number of arguments and concatenates them one by
one into character strings. Any numbers given among the arguments are coerced into character
strings in the evident way, that is, in the same way they would be if they were printed. The
arguments are by default separated in the result by a single blank character, but this can be
changed by the named parameter, sep=string , which changes it to string , possibly empty.
For example
> labs <- paste(c("X","Y"), 1:10, sep="")
makes labs into the character vector
c("X1", "Y2", "X3", "Y4", "X5", "Y6", "X7", "Y8", "X9", "Y10")
Note particularly that recycling of short lists takes place here too; thus c("X", "Y") is
repeated 5 times to match the sequence 1:10.3
2.7 Index vectors; selecting and modifying subsets of a data set
Subsets of the elements of a vector may be selected by appending to the name of the vector an
index vector in square brackets. More generally any expression that evaluates to a vector may
have subsets of its elements similarly selected by appending an index vector in square brackets
immediately after the expression.
Such index vectors can be any of four distinct types.
1. A logical vector. In this case the index vector must be of the same length as the vector
from which elements are to be selected. Values corresponding to TRUE in the index vector
are selected and those corresponding to FALSE are omitted. For example
> y <- x[!is.na(x)]
creates (or re-creates) an object y which will contain the non-missing values of x, in the
same order. Note that if x has missing values, y will be shorter than x. Also
> (x+1)[(!is.na(x)) & x>0] -> z
creates an object z and places in it the values of the vector x+1 for which the corresponding
value in x was both non-missing and positive.
3 paste(..., collapse=ss) joins the arguments into a single character string putting ss in between. There are
more tools for character manipulation, see the help for sub and substring.

Chapter 2: Simple manipulations; numbers and vectors
11
2. A vector of positive integral quantities. In this case the values in the index vector must lie
in the set {1, 2, . . . , length(x)}. The corresponding elements of the vector are selected
and concatenated, in that order, in the result. The index vector can be of any length and the
result is of the same length as the index vector. For example x[6] is the sixth component
of x and
> x[1:10]
selects the first 10 elements of x (assuming length(x) is not less than 10). Also
> c("x","y")[rep(c(1,2,2,1), times=4)]
(an admittedly unlikely thing to do) produces a character vector of length 16 consisting of
"x", "y", "y", "x" repeated four times.
3. A vector of negative integral quantities. Such an index vector specifies the values to be
excluded rather than included. Thus
> y <- x[-(1:5)]
gives y all but the first five elements of x.
4. A vector of character strings. This possibility only applies where an object has a names
attribute to identify its components. In this case a sub-vector of the names vector may be
used in the same way as the positive integral labels in item 2 further above.
> fruit <- c(5, 10, 1, 20)
> names(fruit) <- c("orange", "banana", "apple", "peach")
> lunch <- fruit[c("apple","orange")]
The advantage is that alphanumeric names are often easier to remember than numeric
indices. This option is particularly useful in connection with data frames, as we shall see
later.
An indexed expression can also appear on the receiving end of an assignment, in which case
the assignment operation is performed only on those elements of the vector. The expression
must be of the form vector[index_vector ] as having an arbitrary expression in place of the
vector name does not make much sense here.
The vector assigned must match the length of the index vector, and in the case of a logical
index vector it must again be the same length as the vector it is indexing.
For example
> x[is.na(x)] <- 0
replaces any missing values in x by zeros and
> y[y < 0] <- -y[y < 0]
has the same effect as
> y <- abs(y)
2.8 Other types of objects
Vectors are the most important type of object in R, but there are several others which we will
meet more formally in later sections.
• matrices or more generally arrays are multi-dimensional generalizations of vectors. In fact,
they are vectors that can be indexed by two or more indices and will be printed in special
ways. See Chapter 5 [Arrays and matrices], page 18.
• 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.

Chapter 2: Simple manipulations; numbers and vectors
12
• data frames are matrix-like structures, in which the columns can be of different types. Think
of data frames as ‘data matrices’ with one row per observational unit but with (possibly)
both numerical and categorical variables. Many experiments are best described by data
frames: the treatments are categorical but the response is numeric. See 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.

Chapter 3: Objects, their modes and attributes
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
known as “atomic” structures since their components are all of the same type, or mode, namely
numeric1, complex, logical, character and raw.
Vectors must have their values all of the same mode. Thus any given vector must be un-
ambiguously either logical, numeric, complex, character or raw. (The only apparent exception
to this rule is the special “value” listed as NA for quantities not available, but in fact there are
several types of NA). Note that a vector can be empty and still have a mode. For example
the empty character string vector is listed as character(0) and the empty numeric vector as
numeric(0).
R also operates on objects called lists, which are of mode list. These are ordered sequences
of objects which individually can be of any mode. lists are known as “recursive” rather than
atomic structures since their components can themselves be lists in their own right.
The other recursive structures are those of mode function and expression. Functions are
the objects that form part of the R system along with similar user written functions, which we
discuss in some detail later. Expressions as objects form an advanced part of R which will not
be discussed in this guide, except indirectly when we discuss formulae used with modeling in R.
By the mode of an object we mean the basic type of its fundamental constituents. This is a
special case of a “property” of an object. Another property of every object is its length. The
functions mode(object ) and length(object ) can be used to find out the mode and length of
any defined structure2.
Further properties of an object are usually provided by attributes(object ), see Section 3.3
[Getting and setting attributes], page 14. Because of this, mode and length are also called
“intrinsic attributes” of an object.
For example, if z is a complex vector of length 100, then in an expression mode(z) is the
character string "complex" and length(z) is 100.
R caters for changes of mode almost anywhere it could be considered sensible to do so, (and
a few where it might not be). For example with
> z <- 0:9
we could put
> digits <- as.character(z)
after which digits is the character vector c("0", "1", "2", ..., "9"). A further coercion, or
change of mode, reconstructs the numerical vector again:
> d <- as.integer(digits)
Now d and z are the same.3 There is a large collection of functions of the form as.something ()
for either coercion from one mode to another, or for investing an object with some other attribute
it may not already possess. The reader should consult the different help files to become familiar
with them.
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.

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()
makes e an empty vector structure of mode numeric. Similarly character() is a empty character
vector, and so on. Once an object of any size has been created, new components may be added
to it simply by giving it an index value outside its previous range. Thus
> e[3] <- 17
now makes e a vector of length 3, (the first two components of which are at this point both NA).
This applies to any structure at all, provided the mode of the additional component(s) agrees
with the mode of the object in the first place.
This automatic adjustment of lengths of an object is used often, for example in the scan()
function for input. (see 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
alpha is an object of length 10, then
> alpha <- alpha[2 * 1:5]
makes it an object of length 5 consisting of just the former components with even index. (The
old indices are not retained, of course.) We can then retain just the first three values by
> length(alpha) <- 3
and vectors can be extended (by missing values) in the same way.
3.3 Getting and setting attributes
The function attributes(object ) returns a list of all the non-intrinsic attributes currently
defined for that object. The function attr(object, name ) can be used to select a specific
attribute. These functions are rarely used, except in rather special circumstances when some
new attribute is being created for some particular purpose, for example to associate a creation
date or an operator with an R object. The concept, however, is very important.
Some care should be exercised when assigning or deleting attributes since they are an integral
part of the object system used in R.
When it is used on the left hand side of an assignment it can be used either to associate a
new attribute with object or to change an existing one. For example
> attr(z, "dim") <- c(10,10)
allows R to treat z as if it were a 10-by-10 matrix.
3.4 The class of an object
All objects in R have a class, reported by the function class. For simple vectors this is just the
mode, for example "numeric", "logical", "character" or "list", but "matrix", "array",
"factor" and "data.frame" are other possible values.
A special attribute known as the class of the object is used to allow for an object-oriented
style4 of programming in R. For example if an object has class "data.frame", it will be printed
in a certain way, the plot() function will display it graphically in a certain way, and other
so-called generic functions such as summary() will react to it as an argument in a way sensitive
to its class.
To remove temporarily the effects of class, use the function unclass(). For example if winter
has the class "data.frame" then
4 A different style using ‘formal’ or ‘S4’ classes is provided in package methods.

Chapter 3: Objects, their modes and attributes
15
> winter
will print it in data frame form, which is rather like a matrix, whereas
> unclass(winter)
will print it as an ordinary list. Only in rather special situations do you need to use this facility,
but one is when you are learning to come to terms with the idea of class and generic functions.
Generic functions and classes will be discussed further in Section 10.9 [Object orientation],
page 48, but only briefly.

Chapter 4: Ordered and unordered factors
16
4 Ordered and unordered factors
A factor is a vector object used to specify a discrete classification (grouping) of the components
of other vectors of the same length. R provides both ordered and unordered factors. While the
“real” application of factors is with model formulae (see Section 11.1.1 [Contrasts], page 52), we
here look at a specific example.
4.1 A specific example
Suppose, for example, we have a sample of 30 tax accountants from all the states and territories
of Australia1 and their individual state of origin is specified by a character vector of state
mnemonics as
> state <- c("tas", "sa",
"qld", "nsw", "nsw", "nt",
"wa",
"wa",
"qld", "vic", "nsw", "vic", "qld", "qld", "sa",
"tas",
"sa",
"nt",
"wa",
"vic", "qld", "nsw", "nsw", "wa",
"sa",
"act", "nsw", "vic", "vic", "act")
Notice that in the case of a character vector, “sorted” means sorted in alphabetical order.
A factor is similarly created using the factor() function:
> statef <- factor(state)
The print() function handles factors slightly differently from other objects:
> statef
[1] tas sa
qld nsw nsw nt
wa
wa
qld vic nsw vic qld qld sa
[16] tas sa
nt
wa
vic qld nsw nsw wa
sa
act nsw vic vic act
Levels:
act nsw nt qld sa tas vic wa
To find out the levels of a factor the function levels() can be used.
> levels(statef)
[1] "act" "nsw" "nt"
"qld" "sa"
"tas" "vic" "wa"
4.2 The function tapply() and ragged arrays
To continue the previous example, suppose we have the incomes of the same tax accountants in
another vector (in suitably large units of money)
> incomes <- c(60, 49, 40, 61, 64, 60, 59, 54, 62, 69, 70, 42, 56,
61, 61, 61, 58, 51, 48, 65, 49, 49, 41, 48, 52, 46,
59, 46, 58, 43)
To calculate the sample mean income for each state we can now use the special function
tapply():
> incmeans <- tapply(incomes, statef, mean)
giving a means vector with the components labelled by the levels
act
nsw
nt
qld
sa
tas
vic
wa
44.500 57.333 55.500 53.600 55.000 60.500 56.000 52.250
The function tapply() is used to apply a function, here mean(), to each group of components
of the first argument, here incomes, defined by the levels of the second component, here statef2,
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()).

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:
> stderr <- function(x) sqrt(var(x)/length(x))
(Writing functions will be considered later in Chapter 10 [Writing your own functions], page 42,
and in this case was unnecessary as R also has a builtin function sd().) After this assignment,
the standard errors are calculated by
> incster <- tapply(incomes, statef, stderr)
and the values calculated are then
> incster
act
nsw
nt
qld
sa tas
vic
wa
1.5 4.3102 4.5 4.1061 2.7386 0.5 5.244 2.6575
As an exercise you may care to find the usual 95% confidence limits for the state mean
incomes. To do this you could use tapply() once more with the length() function to find
the sample sizes, and the qt() function to find the percentage points of the appropriate t-
distributions. (You could also investigate R’s facilities for t-tests.)
The function tapply() can also be used to handle more complicated indexing of a vector
by multiple categories. For example, we might wish to split the tax accountants by both state
and sex. However in this simple instance (just one factor) what happens can be thought of as
follows. The values in the vector are collected into groups corresponding to the distinct entries
in the factor. The function is then applied to each of these groups individually. The value is a
vector of function results, labelled by the levels attribute of the factor.
The combination of a vector and a labelling factor is an example of what is sometimes called
a ragged array, since the subclass sizes are possibly irregular. When the subclass sizes are all
the same the indexing may be done implicitly and much more efficiently, as we see in the next
section.
4.3 Ordered factors
The levels of factors are stored in alphabetical order, or in the order they were specified to
factor if they were specified explicitly.
Sometimes the levels will have a natural ordering that we want to record and want our
statistical analysis to make use of. The ordered() function creates such ordered factors but
is otherwise identical to factor. For most purposes the only difference between ordered and
unordered factors is that the former are printed showing the ordering of the levels, but the
contrasts generated for them in fitting linear models are different.

Chapter 5: Arrays and matrices
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.
A dimension vector is a vector of non-negative integers. If its length is k then the array is
k-dimensional, e.g. a matrix is a 2-dimensional array. The dimensions are indexed from one up
to the values given in the dimension vector.
A vector can be used by R as an array only if it has a dimension vector as its dim attribute.
Suppose, for example, z is a vector of 1500 elements. The assignment
> dim(z) <- c(3,5,100)
gives it the dim attribute that allows it to be treated as a 3 by 5 by 100 array.
Other functions such as matrix() and array() are available for simpler and more natural
looking assignments, as we shall see in 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.
For example if the dimension vector for an array, say a, is c(3,4,2) then there are 3 × 4 ×
2 = 24 entries in a and the data vector holds them in the order a[1,1,1], a[2,1,1], ...,
a[2,4,2], a[3,4,2].
Arrays can be one-dimensional: such arrays are usually treated in the same way as vectors
(including when printing), but the exceptions can cause confusion.
5.2 Array indexing. Subsections of an array
Individual elements of an array may be referenced by giving the name of the array followed by
the subscripts in square brackets, separated by commas.
More generally, subsections of an array may be specified by giving a sequence of index vectors
in place of subscripts; however if any index position is given an empty index vector, then the full
range of that subscript is taken.
Continuing the previous example, a[2,,] is a 4 × 2 array with dimension vector c(4,2) and
data vector containing the values
c(a[2,1,1], a[2,2,1], a[2,3,1], a[2,4,1],
a[2,1,2], a[2,2,2], a[2,3,2], a[2,4,2])
in that order. a[,,] stands for the entire array, which is the same as omitting the subscripts
entirely and using a alone.
For any array, say Z, the dimension vector may be referenced explicitly as dim(Z) (on either
side of an assignment).
Also, if an array name is given with just one subscript or index vector, then the corresponding
values of the data vector only are used; in this case the dimension vector is ignored. This is not
the case, however, if the single index is not a vector but itself an array, as we next discuss.

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:
• Extract elements X[1,3], X[2,2] and X[3,1] as a vector structure, and
• Replace these entries in the array X by zeroes.
In this case we need a 3 by 2 subscript array, as in the following example.
> x <- array(1:20, dim=c(4,5))
# Generate a 4 by 5 array.
> x
[,1] [,2] [,3] [,4] [,5]
[1,]
1
5
9
13
17
[2,]
2
6
10
14
18
[3,]
3
7
11
15
19
[4,]
4
8
12
16
20
> i <- array(c(1:3,3:1), dim=c(3,2))
> i
# i is a 3 by 2 index array.
[,1] [,2]
[1,]
1
3
[2,]
2
2
[3,]
3
1
> x[i]
# Extract those elements
[1] 9 6 3
> x[i] <- 0
# Replace those elements by zeros.
> x
[,1] [,2] [,3] [,4] [,5]
[1,]
1
5
0
13
17
[2,]
2
0
10
14
18
[3,]
0
7
11
15
19
[4,]
4
8
12
16
20
>
Negative indices are not allowed in index matrices. NA and zero values are allowed: rows in the
index matrix containing a zero are ignored, and rows containing an NA produce an NA in the
result.
As a less trivial example, suppose we wish to generate an (unreduced) design matrix for a
block design defined by factors blocks (b levels) and varieties (v levels). Further suppose
there are n plots in the experiment. We could proceed as follows:
> Xb <- matrix(0, n, b)
> Xv <- matrix(0, n, v)
> ib <- cbind(1:n, blocks)
> iv <- cbind(1:n, varieties)
> Xb[ib] <- 1
> Xv[iv] <- 1
> X <- cbind(Xb, Xv)
To construct the incidence matrix, N say, we could use
> N <- crossprod(Xb, Xv)

Chapter 5: Arrays and matrices
20
However a simpler direct way of producing this matrix is to use table():
> N <- table(blocks, varieties)
Index matrices must be numerical: any other form of matrix (e.g. a logical or character
matrix) supplied as a matrix is treated as an indexing vector.
5.4 The array() function
As well as giving a vector structure a dim attribute, arrays can be constructed from vectors by
the array function, which has the form
> Z <- array(data_vector, dim_vector )
For example, if the vector h contains 24 or fewer, numbers then the command
> Z <- array(h, dim=c(3,4,2))
would use h to set up 3 by 4 by 2 array in Z. If the size of h is exactly 24 the result is the same
as
> Z <- h ; dim(Z) <- c(3,4,2)
However if h is shorter than 24, its values are recycled from the beginning again to make it
up to size 24 (see Section 5.4.1 [The recycling rule], page 20) but dim(h) <- c(3,4,2) would
signal an error about mismatching length. As an extreme but common example
> Z <- array(0, c(3,4,2))
makes Z an array of all zeros.
At this point dim(Z) stands for the dimension vector c(3,4,2), and Z[1:24] stands for the
data vector as it was in h, and Z[] with an empty subscript or Z with no subscript stands for
the entire array as an array.
Arrays may be used in arithmetic expressions and the result is an array formed by element-
by-element operations on the data vector. The dim attributes of operands generally need to be
the same, and this becomes the dimension vector of the result. So if A, B and C are all similar
arrays, then
> D <- 2*A*B + C + 1
makes D a similar array with its data vector being the result of the given element-by-element
operations. However the precise rule concerning mixed array and vector calculations has to be
considered a little more carefully.
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.

Chapter 5: Arrays and matrices
21
5.5 The outer product of two arrays
An important operation on arrays is the outer product. If a and b are two numeric arrays,
their outer product is an array whose dimension vector is obtained by concatenating their two
dimension vectors (order is important), and whose data vector is got by forming all possible
products of elements of the data vector of a with those of b. The outer product is formed by
the special operator %o%:
> ab <- a %o% b
An alternative is
> ab <- outer(a, b, "*")
The multiplication function can be replaced by an arbitrary function of two variables. For
example if we wished to evaluate the function f (x; y) = cos(y)/(1 + x2) over a regular grid of
values with x- and y-coordinates defined by the R vectors x and y respectively, we could proceed
as follows:
> f <- function(x, y) cos(y)/(1 + x^2)
> z <- outer(x, y, f)
In particular the outer product of two ordinary vectors is a doubly subscripted array (that
is a matrix, of rank at most 1).
Notice that the outer product operator is of course non-
commutative. Defining your own R functions will be considered further in Chapter 10 [Writing
your own functions], page 42.
An example: Determinants of 2 by 2 single-digit matrices
As an artificial but cute example, consider the determinants of 2 by 2 matrices [a, b; c, d] where
each entry is a non-negative integer in the range 0, 1, . . . , 9, that is a digit.
The problem is to find the determinants, ad − bc, of all possible matrices of this form and
represent the frequency with which each value occurs as a high density plot. This amounts to
finding the probability distribution of the determinant if each digit is chosen independently and
uniformly at random.
A neat way of doing this uses the outer() function twice:
> d <- outer(0:9, 0:9)
> fr <- table(outer(d, d, "-"))
> plot(as.numeric(names(fr)), fr, type="h",
xlab="Determinant", ylab="Frequency")
Notice the coercion of the names attribute of the frequency table to numeric in order to
recover the range of the determinant values. The “obvious” way of doing this problem with for
loops, to be discussed in 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) may be used to permute an array, a. The argument perm must be
a permutation of the integers {1, . . . , k}, where k is the number of subscripts in a. The result of
the function is an array of the same size as a but with old dimension given by perm[j] becoming
the new j-th dimension. The easiest way to think of this operation is as a generalization of
transposition for matrices. Indeed if A is a matrix, (that is, a doubly subscripted array) then B
given by
> B <- aperm(A, c(2,1))
is just the transpose of A. For this special case a simpler function t() is available, so we could
have used B <- t(A).

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
available only for matrices. For example t(X) is the matrix transpose function, as noted above.
The functions nrow(A) and ncol(A) give the number of rows and columns in the matrix A
respectively.
5.7.1 Matrix multiplication
The operator %*% is used for matrix multiplication. An n by 1 or 1 by n matrix may of course
be used as an n-vector if in the context such is appropriate. Conversely, vectors which occur in
matrix multiplication expressions are automatically promoted either to row or column vectors,
whichever is multiplicatively coherent, if possible, (although this is not always unambiguously
possible, as we see later).
If, for example, A and B are square matrices of the same size, then
> A * B
is the matrix of element by element products and
> A %*% B
is the matrix product. If x is a vector, then
> x %*% A %*% x
is a quadratic form.1
The function crossprod() forms “crossproducts”, meaning that crossprod(X, y) is the
same as t(X) %*% y but the operation is more efficient. If the second argument to crossprod()
is omitted it is taken to be the same as the first.
The meaning of diag() depends on its argument. diag(v), where v is a vector, gives a
diagonal matrix with elements of the vector as the diagonal entries. On the other hand diag(M),
where M is a matrix, gives the vector of main diagonal entries of M. This is the same convention
as that used for diag() in Matlab. Also, somewhat confusingly, if k is a single numeric value
then diag(k) is the k by k identity matrix!
5.7.2 Linear equations and inversion
Solving linear equations is the inverse of matrix multiplication. When after
> b <- A %*% x
only A and b are given, the vector x is the solution of that linear equation system. In R,
> solve(A,b)
solves the system, returning x (up to some accuracy loss). Note that in linear algebra, formally
x = A−1b where A−1 denotes the inverse of A, which can be computed by
solve(A)
but rarely is needed. Numerically, it is both inefficient and potentially unstable to compute x
<- solve(A) %*% b instead of solve(A,b).
The quadratic form xTA−1x which is used in multivariate computations, should be computed
by something like2 x %*% solve(A,x), rather than computing the inverse of A.
1 Note that x %*% x is ambiguous, as it could mean either xT x or xxT , where x is the column form. In such
cases the smaller matrix seems implicitly to be the interpretation adopted, so the scalar xT x is in this case
the result. The matrix xxT may be calculated either by cbind(x) %*% x or x %*% rbind(x) since the result of
rbind() or cbind() is always a matrix. However, the best way to compute xT x or xxT is crossprod(x) or x
%o% x respectively.
2 Even better would be to form a matrix square root B with A = BBT and find the squared length of the
solution of By = x, perhaps using the Cholesky or eigendecomposition of A.

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
Sm. The result of this function is a list of two components named values and vectors. The
assignment
> ev <- eigen(Sm)
will assign this list to ev. Then ev$val is the vector of eigenvalues of Sm and ev$vec is the
matrix of corresponding eigenvectors. Had we only needed the eigenvalues we could have used
the assignment:
> evals <- eigen(Sm)$values
evals now holds the vector of eigenvalues and the second component is discarded.
If the
expression
> eigen(Sm)
is used by itself as a command the two components are printed, with their names. For large
matrices it is better to avoid computing the eigenvectors if they are not needed by using the
expression
> evals <- eigen(Sm, only.values = TRUE)$values
5.7.4 Singular value decomposition and determinants
The function svd(M) takes an arbitrary matrix argument, M, and calculates the singular value
decomposition of M. This consists of a matrix of orthonormal columns U with the same column
space as M, a second matrix of orthonormal columns V whose column space is the row space
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.
If M 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)
after which we could use absdet() as just another R function. As a further trivial but potentially
useful example, you might like to consider writing a function, say tr(), to calculate the trace
of a square matrix. [Hint: You will not need to use an explicit loop. Look again at the diag()
function.]
R has a builtin function det to calculate a determinant, including the sign, and another,
determinant, to give the sign and modulus (optionally on log scale),
5.7.5 Least squares fitting and the QR decomposition
The function lsfit() returns a list giving results of a least squares fitting procedure. An
assignment such as
> ans <- lsfit(X, y)
gives the results of a least squares fit where y is the vector of observations and X 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 lm(.) (see Section 11.2 [Linear models], page 53) to lsfit() for regression
modelling.
Another closely related function is qr() and its allies. Consider the following assignments

Chapter 5: Arrays and matrices
24
> Xplus <- qr(X)
> b <- qr.coef(Xplus, y)
> fit <- qr.fitted(Xplus, y)
> res <- qr.resid(Xplus, y)
These compute the orthogonal projection of y onto the range of X in fit, the projection onto
the orthogonal complement in res and the coefficient vector for the projection in b, that is, b is
essentially the result of the Matlab ‘backslash’ operator.
It is not assumed that X has full column rank. Redundancies will be discovered and removed
as they are found.
This alternative is the older, low-level way to perform least squares calculations. Although
still useful in some contexts, it would now generally be replaced by the statistical models features,
as will be discussed in Chapter 11 [Statistical models in R], page 50.
5.8 Forming partitioned matrices, cbind() and rbind()
As we have already seen informally, matrices can be built up from other vectors and matrices
by the functions cbind() and rbind(). Roughly cbind() forms matrices by binding together
matrices horizontally, or column-wise, and rbind() vertically, or row-wise.
In the assignment
> X <- cbind(arg_1, arg_2, arg_3, ...)
the arguments to cbind() must be either vectors of any length, or matrices with the same
column size, that is the same number of rows. The result is a matrix with the concatenated
arguments arg 1, arg 2, . . . forming the columns.
If some of the arguments to cbind() are vectors they may be shorter than the column size
of any matrices present, in which case they are cyclically extended to match the matrix column
size (or the length of the longest vector if no matrices are given).
The function rbind() does the corresponding operation for rows. In this case any vector
argument, possibly cyclically extended, are of course taken as row vectors.
Suppose X1 and X2 have the same number of rows. To combine these by columns into a
matrix X, together with an initial column of 1s we can use
> X <- cbind(1, X1, X2)
The result of rbind() or cbind() always has matrix status. Hence cbind(x) and rbind(x)
are possibly the simplest ways explicitly to allow the vector x to be treated as a column or row
matrix respectively.
5.9 The concatenation function, c(), with arrays
It should be noted that whereas cbind() and rbind() are concatenation functions that respect
dim attributes, the basic c() function does not, but rather clears numeric objects of all dim and
dimnames attributes. This is occasionally useful in its own right.
The official way to coerce an array back to a simple vector object is to use as.vector()
> vec <- as.vector(X)
However a similar result can be achieved by using c() with just one argument, simply for
this side-effect:
> vec <- c(X)
There are slight differences between the two, but ultimately the choice between them is
largely a matter of style (with the former being preferable).

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
way cross classification, and so on. The function table() allows frequency tables to be calcu-
lated from equal length factors. If there are k factor arguments, the result is a k-way array of
frequencies.
Suppose, for example, that statef is a factor giving the state code for each entry in a data
vector. The assignment
> statefr <- table(statef)
gives in statefr a table of frequencies of each state in the sample. The frequencies are ordered
and labelled by the levels attribute of the factor. This simple case is equivalent to, but more
convenient than,
> statefr <- tapply(statef, statef, length)
Further suppose that incomef is a factor giving a suitably defined “income class” for each
entry in the data vector, for example with the cut() function:
> factor(cut(incomes, breaks = 35+10*(0:7))) -> incomef
Then to calculate a two-way table of frequencies:
> table(incomef,statef)
statef
incomef
act nsw nt qld sa tas vic wa
(35,45]
1
1
0
1
0
0
1
0
(45,55]
1
1
1
1
2
0
1
3
(55,65]
0
3
1
3
2
2
2
1
(65,75]
0
1
0
0
0
0
1
0
Extension to higher-way frequency tables is immediate.

Chapter 6: Lists and data frames
26
6 Lists and data frames
6.1 Lists
An R list is an object consisting of an ordered collection of objects known as its components.
There is no particular need for the components to be of the same mode or type, and, for
example, a list could consist of a numeric vector, a logical value, a matrix, a complex vector, a
character array, a function, and so on. Here is a simple example of how to make a list:
> Lst <- list(name="Fred", wife="Mary", no.children=3,
child.ages=c(4,7,9))
Components are always numbered and may always be referred to as such. Thus if Lst is
the name of a list with four components, these may be individually referred to as Lst[[1]],
Lst[[2]], Lst[[3]] and Lst[[4]]. If, further, Lst[[4]] is a vector subscripted array then
Lst[[4]][1] is its first entry.
If Lst is a list, then the function length(Lst) gives the number of (top level) components
it has.
Components of lists may also be named, and in this case the component may be referred to
either by giving the component name as a character string in place of the number in double
square brackets, or, more conveniently, by giving an expression of the form
> name $component_name
for the same thing.
This is a very useful convention as it makes it easier to get the right component if you forget
the number.
So in the simple example given above:
Lst$name is the same as Lst[[1]] and is the string "Fred",
Lst$wife is the same as Lst[[2]] and is the string "Mary",
Lst$child.ages[1] is the same as Lst[[4]][1] and is the number 4.
Additionally, one can also use the names of the list components in double square brackets,
i.e., Lst[["name"]] is the same as Lst$name. This is especially useful, when the name of the
component to be extracted is stored in another variable as in
> x <- "name"; Lst[[x]]
It is very important to distinguish Lst[[1]] from Lst[1]. ‘[[...]]’ is the operator used
to select a single element, whereas ‘[...]’ is a general subscripting operator. Thus the former
is the first object in the list Lst, and if it is a named list the name is not included. The latter
is a sublist of the list Lst consisting of the first entry only. If it is a named list, the names are
transferred to the sublist.
The names of components may be abbreviated down to the minimum number of letters needed
to identify them uniquely. Thus Lst$coefficients may be minimally specified as Lst$coe and
Lst$covariance as Lst$cov.
The vector of names is in fact simply an attribute of the list like any other and may be handled
as such. Other structures besides lists may, of course, similarly be given a names attribute also.
6.2 Constructing and modifying lists
New lists may be formed from existing objects by the function list(). An assignment of the
form

Chapter 6: Lists and data frames
27
> Lst <- list(name_1 =object_1, ..., name_m =object_m )
sets up a list Lst of m components using object 1, . . . , object m for the components and giving
them names as specified by the argument names, (which can be freely chosen). If these names
are omitted, the components are numbered only. The components used to form the list are
copied when forming the new list and the originals are not affected.
Lists, like any subscripted object, can be extended by specifying additional components. For
example
> Lst[5] <- list(matrix=Mat)
6.2.1 Concatenating lists
When the concatenation function c() is given list arguments, the result is an object of mode
list also, whose components are those of the argument lists joined together in sequence.
> list.ABC <- c(list.A, list.B, list.C)
Recall that with vector objects as arguments the concatenation function similarly joined
together all arguments into a single vector structure. In this case all other attributes, such as
dim attributes, are discarded.
6.3 Data frames
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.
• Numeric vectors, logicals and factors are included as is, and 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.
6.3.2 attach() and detach()
The $ notation, such as accountants$statef, for list components is not always very convenient.
A useful facility would be somehow to make the components of a list or data frame temporarily
visible as variables under their component name, without the need to quote the list name
explicitly each time.

Chapter 6: Lists and data frames
28
The attach() function takes a ‘database’ such as a list or data frame as its argument. Thus
suppose lentils is a data frame with three variables lentils$u, lentils$v, lentils$w. The
attach
> attach(lentils)
places the data frame in the search path at position 2, and provided there are no variables u, v
or w in position 1, u, v and w are available as variables from the data frame in their own right.
At this point an assignment such as
> u <- v+w
does not replace the component u of the data frame, but rather masks it with another variable
u in the working directory at position 1 on the search path. To make a permanent change to
the data frame itself, the simplest way is to resort once again to the $ notation:
> lentils$u <- v+w
However the new value of component u is not visible until the data frame is detached and
attached again.
To detach a data frame, use the function
> detach()
More precisely, this statement detaches from the search path the entity currently at
position 2. Thus in the present context the variables u, v and w would be no longer visible,
except under the list notation as lentils$u and so on. Entities at positions greater than 2
on the search path can be detached by giving their number to detach, but it is much safer to
always use a name, for example by detach(lentils) or detach("lentils")
Note: In R lists and data frames can only be attached at position 2 or above, and
what is attached is a copy of the original object. You can alter the attached values
via assign, but the original list or data frame is unchanged.
6.3.3 Working with data frames
A useful convention that allows you to work with many different problems comfortably together
in the same 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
data frame using the $ form of assignment, and then 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
have variables named x, y and z, for example.
6.3.4 Attaching arbitrary lists
attach() is a generic function that allows not only directories and data frames to be attached
to the search path, but other classes of object as well. In particular any object of mode "list"
may be attached in the same way:
> attach(any.old.list)
Anything that has been attached can be detached by detach, by position number or, prefer-
ably, by name.

Chapter 6: Lists and data frames
29
6.3.5 Managing the search path
The function search shows the current search path and so is a very useful way to keep track of
which data frames and lists (and packages) have been attached and detached. Initially it gives
> search()
[1] ".GlobalEnv"
"Autoloads"
"package:base"
where .GlobalEnv is the workspace.1
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"
1 See the on-line help for autoload for the meaning of the second term.

Chapter 7: Reading data from files
30
7 Reading data from files
Large data objects will usually be read as values from external files rather than entered during
an R session at the keyboard. R input facilities are simple and their requirements are fairly
strict and even rather inflexible. There is a clear presumption by the designers of R that you
will be able to modify your input files using other tools, such as file editors or Perl1 to fit in
with the requirements of R. Generally this is very simple.
If variables are to be held mainly in data frames, as we strongly suggest they should be, an
entire data frame can be read directly with the read.table() function. There is also a more
primitive input function, scan(), that can be called directly.
For more details on importing data into R and also exporting data, see the R Data Im-
port/Export manual.
7.1 The read.table() function
To read an entire data frame directly, the external file will normally have a special form.
• The first line of the file should have a name for each variable in the data frame.
• Each additional line of the file has as its first item a row label and the values for each
variable.
If the file has one fewer item in its first line than in its second, this arrangement is presumed
to be in force. So the first few lines of a file to be read as a data frame might look as follows.

¨
Input file form with names and row labels:
Price
Floor
Area
Rooms
Age
Cent.heat
01
52.00
111.0
830
5
6.2
no
02
54.75
128.0
710
5
7.5
no
03
57.50
101.0
1000
5
4.2
no
04
57.50
131.0
690
6
8.8
no
05
59.75
93.0
900
5
1.9
yes
...

©
By default numeric items (except row labels) are read as numeric variables and non-numeric
variables, such as Cent.heat in the example, as factors. This can be changed if necessary.
The function read.table() can then be used to read the data frame directly
> HousePrice <- read.table("houses.data")
Often you will want to omit including the row labels directly and use the default labels. In
this case the file may omit the row label column as in the following.

¨
Input file form without row labels:
Price
Floor
Area
Rooms
Age
Cent.heat
52.00
111.0
830
5
6.2
no
54.75
128.0
710
5
7.5
no
57.50
101.0
1000
5
4.2
no
57.50
131.0
690
6
8.8
no
59.75
93.0
900
5
1.9
yes
...

©
1 Under UNIX, the utilities Sed or Awk can be used.

Chapter 7: Reading data from files
31
The data frame may then be read as
> HousePrice <- read.table("houses.data", header=TRUE)
where the header=TRUE option specifies that the first line is a line of headings, and hence, by
implication from the form of the file, that no explicit row labels are given.
7.2 The scan() function
Suppose the data vectors are of equal length and are to be read in parallel. Further suppose
that there are three vectors, the first of mode character and the remaining two of mode numeric,
and the file is ‘input.dat’. The first step is to use scan() to read in the three vectors as a list,
as follows
> inp <- scan("input.dat", list("",0,0))
The second argument is a dummy list structure that establishes the mode of the three vectors
to be read. The result, held in inp, is a list whose components are the three vectors read in. To
separate the data items into three separate vectors, use assignments like
> label <- inp[[1]]; x <- inp[[2]]; y <- inp[[3]]
More conveniently, the dummy list can have named components, in which case the names
can be used to access the vectors read in. For example
> inp <- scan("input.dat", list(id="", x=0, y=0))
If you wish to access the variables separately they may either be re-assigned to variables in
the working frame:
> label <- inp$id; x <- inp$x; y <- inp$y
or the list may be attached at position 2 of the search path (see Section 6.3.4 [Attaching arbitrary
lists], page 28).
If the second argument is a single value and not a list, a single vector is read in, all components
of which must be of the same mode as the dummy value.
> X <- matrix(scan("light.dat", 0), ncol=5, byrow=TRUE)
There are more elaborate input facilities available and these are detailed in the manuals.
7.3 Accessing builtin datasets
Around 100 datasets are supplied with R (in package datasets), and others are available in
packages (including the recommended packages supplied with R). To see the list of datasets
currently available use
data()
As from R version 2.0.0 all the datasets supplied with R are available directly by name. However,
many packages still use the earlier convention in which data was also used to load datasets into
R, for example
data(infert)
and this can still be used with the standard packages (as in this example). In most cases this
will load an R object of the same name. However, in a few cases it loads several objects, so see
the on-line help for the object to see what to expect.
7.3.1 Loading data from other R packages
To access data from a particular package, use the package argument, for example
data(package="rpart")
data(Puromycin, package="datasets")
If a package has been attached by library, its datasets are automatically included in the
search.
User-contributed packages can be a rich source of datasets.

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.

Chapter 8: Probability distributions
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
provided to evaluate the cumulative distribution function P (X ≤ x), the probability density
function and the quantile function (given q, the smallest x such that P (X ≤ x) > q), and to
simulate from the distribution.
Distribution
R name
additional arguments
beta
beta
shape1, shape2, ncp
binomial
binom
size, prob
Cauchy
cauchy
location, scale
chi-squared
chisq
df, ncp
exponential
exp
rate
F
f
df1, df2, ncp
gamma
gamma
shape, scale
geometric
geom
prob
hypergeometric
hyper
m, n, k
log-normal
lnorm
meanlog, sdlog
logistic
logis
location, scale
negative binomial
nbinom
size, prob
normal
norm
mean, sd
Poisson
pois
lambda
Student’s t
t
df, ncp
uniform
unif
min, max
Weibull
weibull
shape, scale
Wilcoxon
wilcox
m, n
Prefix the name given here by ‘d’ for the density, ‘p’ for the CDF, ‘q’ for the quantile function
and ‘r’ for simulation (r andom deviates). The first argument is x for dxxx , q for pxxx , p for
qxxx and n for rxxx (except for rhyper and rwilcox, for which it is nn). In not quite all cases
is the non-centrality parameter ncp are currently available: see the on-line help for details.
The pxxx and qxxx functions all have logical arguments lower.tail and log.p and the
dxxx ones have log. This allows, e.g., getting the cumulative (or “integrated”) hazard function,
H(t) = − log(1 − F (t)), by
- pxxx (t, ..., lower.tail = FALSE, log.p = TRUE)
or more accurate log-likelihoods (by dxxx (..., log = TRUE)), directly.
In addition there are functions ptukey and qtukey for the distribution of the studentized
range of samples from a normal distribution.
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)
8.2 Examining the distribution of a set of data
Given a (univariate) set of data we can examine its distribution in a large number of ways. The
simplest is to examine the numbers. Two slightly different summaries are given by summary and
fivenum and a display of the numbers by stem (a “stem and leaf” plot).

Chapter 8: Probability distributions
34
> 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

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 bw = "SJ" gives a good result.)
Histogram of eruptions
0.7
0.6
0.5
0.4
0.3
Relative Frequency
0.2
0.1
0.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
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
3.0
3.5
4.0
4.5
5.0
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)

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 t distribution
Normal Q−Q Plot
5.0
4.5
4.0
Sample Quantiles
3.5
3.0
−2
−1
0
1
2
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)

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

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)
will show the two empirical CDFs, and qqplot will perform a Q-Q plot of the two samples. The
Kolmogorov-Smirnov test is of the maximal vertical distance between the two ecdf’s, assuming
a common continuous distribution:
> ks.test(A, B)
Two-sample Kolmogorov-Smirnov test
data:
A and B
D = 0.5962, p-value = 0.05919
alternative hypothesis: two-sided

Chapter 8: Probability distributions
39
Warning message:
cannot compute correct p-values with ties in: ks.test(A, B)

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

Chapter 9: Grouping, loops and conditional execution
41
Warning: for() loops are used in R code much less often than in compiled languages.
Code that takes a ‘whole object’ view is likely to be both clearer and faster in R.
Other looping facilities include the
> repeat expr
statement and the
> while (condition ) expr
statement.
The break statement can be used to terminate any loop, possibly abnormally. This is the
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.

Chapter 10: Writing your own functions
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
as mean(), var(), postscript() and so on, are themselves written in R and thus do not differ
materially from user written functions.
A function is defined by an assignment of the form
> name <- function(arg_1, arg_2, ...) expression
The expression is an R expression, (usually a grouped expression), that uses the arguments,
arg i, to calculate a value. The value of the expression is the value returned for the function.
A call to the function then usually takes the form name (expr_1, expr_2, ...) and may
occur anywhere a function call is legitimate.
10.1 Simple examples
As a first example, consider a function to calculate the two sample t-statistic, showing “all the
steps”. This is an artificial example, of course, since there are other, simpler ways of achieving
the same end.
The function is defined as follows:
> twosam <- function(y1, y2) {
n1
<- length(y1); n2
<- length(y2)
yb1 <- mean(y1);
yb2 <- mean(y2)
s1
<- var(y1);
s2
<- var(y2)
s <- ((n1-1)*s1 + (n2-1)*s2)/(n1+n2-2)
tst <- (yb1 - yb2)/sqrt(s*(1/n1 + 1/n2))
tst
}
With this function defined, you could perform two sample t-tests using a call such as
> tstat <- twosam(data$male, data$female); tstat
As a second example, consider a function to emulate directly the Matlab backslash com-
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
coefficients.) This would ordinarily be done with the qr() function; however this is sometimes
a bit tricky to use directly and it pays to have a simple function such as the following to use it
safely.
Thus given a n by 1 vector y and an n by p matrix X then X y is defined as (XT X)−XT y,
where (XT X)− is a generalized inverse of X X.
> bslash <- function(X, y) {
X <- qr(X)
qr.coef(X, y)
}
After this object is created it may be used in statements such as
> regcoeff <- bslash(Xmat, yvar)
and so on.

Chapter 10: Writing your own functions
43
The classical R function lsfit() does this job quite well, and more1. It in turn uses the
functions qr() and qr.coef() in the slightly counterintuitive way above to do this part of the
calculation. Hence there is probably some value in having just this part isolated in a simple to
use function if it is going to be in frequent use. If so, we may wish to make it a matrix binary
operator for even more convenient use.
10.2 Defining new binary operators
Had we given the bslash() function a different name, namely one of the form
%anything %
it could have been used as a binary operator in expressions rather than in function form. Suppose,
for example, we choose ! for the internal character. The function definition would then start as
> "%!%" <- function(X, y) { ... }
(Note the use of quote marks.) The function could then be used as X %!% y. (The backslash
symbol itself is not a convenient choice as it presents special problems in this context.)
The matrix multiplication operator, %*%, and the outer product matrix operator %o% are
other examples of binary operators defined in this way.
10.3 Named arguments and defaults
As first noted in Section 2.3 [Generating regular sequences], page 8, if arguments to called
functions are given in the “name =object ” form, they may be given in any order. Furthermore
the argument sequence may begin in the unnamed, positional form, and specify named arguments
after the positional arguments.
Thus if there is a function fun1 defined by
> fun1 <- function(data, data.frame, graph, limit) {
[function body omitted]
}
then the function may be invoked in several ways, for example
> ans <- fun1(d, df, TRUE, 20)
> ans <- fun1(d, df, graph=TRUE, limit=20)
> ans <- fun1(data=d, limit=20, graph=TRUE, data.frame=df)
are all equivalent.
In many cases arguments can be given commonly appropriate default values, in which case
they may be omitted altogether from the call when the defaults are appropriate. For example,
if fun1 were defined as
> fun1 <- function(data, data.frame, graph=TRUE, limit=20) { ... }
it could be called as
> ans <- fun1(d, df)
which is now equivalent to the three cases above, or as
> ans <- fun1(d, df, limit=10)
which changes one of the defaults.
It is important to note that defaults may be arbitrary expressions, even involving other
arguments to the same function; they are not restricted to be constants as in our simple example
here.
1 See also the methods described in Chapter 11 [Statistical models in R], page 50

Chapter 10: Writing your own functions
44
10.4 The ‘...’ argument
Another frequent requirement is to allow one function to pass on argument settings to another.
For example many graphics functions use the function par() and functions like plot() allow the
user to pass on graphical parameters to par() to control the graphical output. (See Section 12.4.1
[The par() function], page 67, for more details on the par() function.) This can be done by
including an extra argument, literally ‘...’, of the function, which may then be passed on. An
outline example is given below.
fun1 <- function(data, data.frame, graph=TRUE, limit=20, ...) {
[omitted statements]
if (graph)
par(pch="*", ...)
[more omissions]
}
10.5 Assignments within functions
Note that any ordinary assignments done within the function are local and temporary and are
lost after exit from the function. Thus the assignment X <- qr(X) does not affect the value of
the argument in the calling program.
To understand completely the rules governing the scope of R assignments the reader needs
to be familiar with the notion of an evaluation frame. This is a somewhat advanced, though
hardly difficult, topic and is not covered further here.
If global and permanent assignments are intended within a function, then either the “su-
perassignment” operator, <<- or the function assign() can be used. See the help document for
details. S-Plus users should be aware that <<- has different semantics in R. These are discussed
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 K are the v by v and b by b replications and block size matrices, respectively, and N is the
b by v incidence matrix, then the efficiency factors are defined as the eigenvalues of the matrix
E = Iv − R−1/2N T K−1N R−1/2 = Iv − AT A,
where A = K−1/2N R−1/2. One way to write the function is given below.
> bdeff <- function(blocks, varieties) {
blocks <- as.factor(blocks)
# minor safety move
b <- length(levels(blocks))
varieties <- as.factor(varieties)
# minor safety move
v <- length(levels(varieties))
K <- as.vector(table(blocks))
# remove dim attr
R <- as.vector(table(varieties))
# remove dim attr
N <- table(blocks, varieties)
A <- 1/sqrt(K) * N * rep(1/sqrt(R), rep(b, v))
sv <- svd(A)
list(eff=1 - sv$d^2, blockcv=sv$u, varietycv=sv$v)

Chapter 10: Writing your own functions
45
}
It is numerically slightly better to work with the singular value decomposition on this occasion
rather than the eigenvalue routines.
The result of the function is a list giving not only the efficiency factors as the first component,
but also the block and variety canonical contrasts, since sometimes these give additional useful
qualitative information.
10.6.2 Dropping all names in a printed array
For printing purposes with large matrices or arrays, it is often useful to print them in close block
form without the array names or numbers. Removing the dimnames attribute will not achieve
this effect, but rather the array must be given a dimnames attribute consisting of empty strings.
For example to print a matrix, X
> 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) {
fun1 <- function(f, a, b, fa, fb, a0, eps, lim, fun) {
## function ‘fun1’ is only visible inside ‘area’
d <- (a + b)/2

Chapter 10: Writing your own functions
46
h <- (b - a)/4
fd <- f(d)
a1 <- h * (fa + fd)
a2 <- h * (fd + fb)
if(abs(a0 - a1 - a2) < eps || lim == 0)
return(a1 + a2)
else {
return(fun(f, a, d, fa, fd, a1, eps, lim - 1, fun) +
fun(f, d, b, fd, fb, a2, eps, lim - 1, fun))
}
}
fa <- f(a)
fb <- f(b)
a0 <- ((fa + fb) * (b - a))/2
fun1(f, a, b, fa, fb, a0, eps, lim, fun1)
}
10.7 Scope
The discussion in this section is somewhat more technical than in other parts of this document.
However, it details one of the major differences between S-Plus and R.
The symbols which occur in the body of a function can be divided into three classes; formal
parameters, local variables and free variables. The formal parameters of a function are those
occurring in the argument list of the function. Their values are determined by the process of
binding the actual function arguments to the formal parameters. Local variables are those whose
values are determined by the evaluation of expressions in the body of the functions. Variables
which are not formal parameters or local variables are called free variables. Free variables become
local variables if they are assigned to. Consider the following function definition.
f <- function(x) {
y <- 2*x
print(x)
print(y)
print(z)
}
In this function, x is a formal parameter, y is a local variable and z is a free variable.
In R the free variable bindings are resolved by first looking in the environment in which the
function was created. This is called lexical scope. First we define a function called cube.
cube <- function(n) {
sq <- function() n*n
n*sq()
}
The variable n in the function sq is not an argument to that function. Therefore it is a free
variable and the scoping rules must be used to ascertain the value that is to be associated with
it. Under static scope (S-Plus) the value is that associated with a global variable named n.
Under lexical scope (R) it is the parameter to the function cube since that is the active binding
for the variable n at the time the function sq was defined. The difference between evaluation
in R and evaluation in S-Plus is that S-Plus looks for a global variable called n while R first
looks for a variable called n in the environment created when cube was invoked.
## first evaluation in S
S> cube(2)
Error in sq(): Object "n" not found

Chapter 10: Writing your own functions
47
Dumped
S> n <- 3
S> cube(2)
[1] 18
## then the same function evaluated in R
R> cube(2)
[1] 8
Lexical scope can also be used to give functions mutable state. In the following example
we show how R can be used to mimic a bank account. A functioning bank account needs to
have a balance or total, a function for making withdrawals, a function for making deposits and
a function for stating the current balance. We achieve this by creating the three functions
within account and then returning a list containing them. When account is invoked it takes
a numerical argument total and returns a list containing the three functions. Because these
functions are defined in an environment which contains total, they will have access to its value.
The special assignment operator, <<-, is used to change the value associated with total.
This operator looks back in enclosing environments for an environment that contains the symbol
total and when it finds such an environment it replaces the value, in that environment, with
the value of right hand side. If the global or top-level environment is reached without finding
the symbol total then that variable is created and assigned to there. For most users <<- creates
a global variable and assigns the value of the right hand side to it2. Only when <<- has been
used in a function that was returned as the value of another function will the special behavior
described here occur.
open.account <- function(total) {
list(
deposit = function(amount) {
if(amount <= 0)
stop("Deposits must be positive!\n")
total <<- total + amount
cat(amount, "deposited.
Your balance is", total, "\n\n")
},
withdraw = function(amount) {
if(amount > total)
stop("You don’t have that much money!\n")
total <<- total - amount
cat(amount, "withdrawn.
Your balance is", total, "\n\n")
},
balance = function() {
cat("Your balance is", total, "\n\n")
}
)
}
ross <- open.account(100)
robert <- open.account(200)
ross$withdraw(30)
ross$balance()
robert$balance()
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.

Chapter 10: Writing your own functions
48
ross$deposit(50)
ross$balance()
ross$withdraw(500)
10.8 Customizing the environment
Users can customize their environment in several different ways. There is a site initialization
file and every directory can have its own special initialization file. Finally, the special functions
.First and .Last can be used.
The location of the site initialization file is taken from the value of the R_PROFILE environment
variable. If that variable is unset, the file ‘Rprofile.site’ in the R home subdirectory ‘etc’
is used.
This file should contain the commands that you want to execute every time R is
started under your system. A second, personal, profile file named ‘.Rprofile’3 can be placed
in any directory. If R is invoked in that directory then that file will be sourced. This file
gives individual users control over their workspace and allows for different startup procedures in
different working directories. If no ‘.Rprofile’ file is found in the startup directory, then R looks
for a ‘.Rprofile’ file in the user’s home directory and uses that (if it exists). If the environment
variable R_PROFILE_USER is set, the file it points to is used instead of the ‘.Rprofile’ files.
Any function named .First() in either of the two profile files or in the ‘.RData’ image has
a special status. It is automatically performed at the beginning of an R session and may be
used to initialize the environment. For example, the definition in the example below alters the
prompt to $ and sets up various other useful things that can then be taken for granted in the
rest of the session.
Thus, the sequence in which files are executed is, ‘Rprofile.site’, the user profile, ‘.RData’
and then .First(). A definition in later files will mask definitions in earlier files.
> .First <- function() {
options(prompt="$ ", continue="+\t")
# $ is the prompt
options(digits=5, length=999)
# custom numbers and printout
x11()
# for graphics
par(pch = "+")
# plotting character
source(file.path(Sys.getenv("HOME"), "R", "mystuff.R"))
# my personal functions
library(MASS)
# attach a package
}
Similarly a function .Last(), if defined, is (normally) executed at the very end of the session.
An example is given below.
> .Last <- function() {
graphics.off()
# a small safety measure.
cat(paste(date(),"\nAdios\n"))
# Is it time for lunch?
}
10.9 Classes, generic functions and object orientation
The class of an object determines how it will be treated by what are known as generic functions.
Put the other way round, a generic function performs a task or action on its arguments specific
to the class of the argument itself. If the argument lacks any class attribute, or has a class
not catered for specifically by the generic function in question, there is always a default action
provided.
An example makes things clearer. The class mechanism offers the user the facility of designing
and writing generic functions for special purposes. Among the other generic functions are plot()
3 So it is hidden under UNIX.

Chapter 10: Writing your own functions
49
for displaying objects graphically, summary() for summarizing analyses of various types, and
anova() for comparing statistical models.
The number of generic functions that can treat a class in a specific way can be quite large.
For example, the functions that can accommodate in some fashion objects of class "data.frame"
include
[
[[<-
any
as.matrix
[<-
mean
plot
summary
A currently complete list can be got by using the methods() function:
> methods(class="data.frame")
Conversely the number of classes a generic function can handle can also be quite large.
For example the plot() function has a default method and variants for objects of classes
"data.frame", "density", "factor", and more. A complete list can be got again by using
the methods() function:
> methods(plot)
For many generic functions the function body is quite short, for example
> coef
function (object, ...)
UseMethod("coef")
The presence of UseMethod indicates this is a generic function. To see what methods are available
we can use methods()
> methods(coef)
[1] coef.aov*
coef.Arima*
coef.default*
coef.listof*
[5] coef.nls*
coef.summary.nls*
Non-visible functions are asterisked
In this example there are six methods, none of which can be seen by typing its name. We can
read these by either of
> getAnywhere("coef.aov")
A single object matching ’coef.aov’ was found
It was found in the following places
registered S3 method for coef from namespace stats
namespace:stats
with value
function (object, ...)
{
z <- object$coef
z[!is.na(z)]
}
> getS3method("coef", "aov")
function (object, ...)
{
z <- object$coef
z[!is.na(z)]
}
The reader is referred to the R Language Definition for a more complete discussion of this
mechanism.

Chapter 11: Statistical models in R
50
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
yi =
βjxij + ei,
ei ∼ NID(0, σ2),
i = 1, . . . , n
j=0
In matrix terms this would be written
y = Xβ + e
where the y is the response vector, X is the model matrix or design matrix and has columns
x0, x1, . . . , xp, the determining variables. Very often x0 will be a column of ones defining an
intercept term.
Examples
Before giving a formal specification, a few examples may usefully set the picture.
Suppose y, x, x0, x1, x2, . . . are numeric variables, X is a matrix and A, B, C, . . . are factors.
The following formulae on the left side below specify statistical models as described on the right.
y ~ x
y ~ 1 + x
Both imply the same simple linear regression model of y on x. The first has an
implicit intercept term, and the second an explicit one.
y ~ 0 + x
y ~ -1 + x
y ~ x - 1
Simple linear regression of y on x through the origin (that is, without an intercept
term).
log(y) ~ x1 + x2
Multiple regression of the transformed variable, log(y), on x1 and x2 (with an
implicit intercept term).
y ~ poly(x,2)
y ~ 1 + x + I(x^2)
Polynomial regression of y on x of degree 2. The first form uses orthogonal 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.

Chapter 11: Statistical models in R
51
y ~ A
Single classification analysis of variance model of y, with classes determined by A.
y ~ A + x
Single classification analysis of covariance model of y, with classes determined by
A, and with covariate x.
y ~ A*B
y ~ A + B + A:B
y ~ B %in% A
y ~ A/B
Two factor non-additive model of y on A and B. The first two specify the same
crossed classification and the second two specify the same nested classification. In
abstract terms all four specify the same model subspace.
y ~ (A + B + C)^2
y ~ A*B*C - A:B:C
Three factor experiment but with a model containing main effects and two factor
interactions only. Both formulae specify the same model.
y ~ A * x
y ~ A/x
y ~ A/(1 + x) - 1
Separate simple linear regression models of y on x within the levels of A, with
different codings. The last form produces explicit estimates of as many different
intercepts and slopes as there are levels in A.
y ~ A*B + Error(C)
An experiment with two treatment factors, A and B, and error strata determined
by factor C. For example a split plot experiment, with whole plots (and hence also
subplots), determined by factor C.
The operator ~ is used to define a model formula in R. The form, for an ordinary linear
model, is
response ~ op_1 term_1 op_2 term_2 op_3 term_3 ...
where
response
is a vector or matrix, (or expression evaluating to a vector or matrix) defining the
response variable(s).
op i
is an operator, either + or -, implying the inclusion or exclusion of a term in the
model, (the first is optional).
term i
is either
• a vector or matrix expression, or 1,
• a factor, or
• a formula expression consisting of factors, vectors or matrices connected by
formula operators.
In all cases each term defines a collection of columns either to be added to or
removed from the model matrix. A 1 stands for an intercept column and is by
default included in the model matrix unless explicitly removed.
The formula operators are similar in effect to the Wilkinson and Rogers notation used by
such programs as Glim and Genstat. One inevitable change is that the operator ‘.’ becomes ‘:’
since the period is a valid name character in R.
The notation is summarized below (based on Chambers & Hastie, 1992, p.29):
Y ~ M
Y is modeled as M.
M_1 + M_2 Include M 1 and M 2.

Chapter 11: Statistical models in R
52
M_1 - M_2 Include M 1 leaving out terms of M 2.
M_1 : M_2 The tensor product of M 1 and M 2. If both terms are factors, then the “subclasses”
factor.
M_1 %in% M_2
Similar to M_1 :M_2 , but with a different coding.
M_1 * M_2 M_1 + M_2 + M_1 :M_2 .
M_1 / M_2 M_1 + M_2 %in% M_1 .
M ^n
All terms in M together with “interactions” up to order n
I(M )
Insulate M. Inside M all operators have their normal arithmetic meaning, and that
term appears in the model matrix.
Note that inside the parentheses that usually enclose function arguments all operators have
their normal arithmetic meaning. The function I() is an identity function used to allow terms
in model formulae to be defined using arithmetic operators.
Note particularly that the model formulae specify the columns of the model matrix, the
specification of the parameters being implicit. This is not the case in other contexts, for example
in specifying nonlinear models.
11.1.1 Contrasts
We need at least some idea how the model formulae specify the columns of the model matrix.
This is easy if we have continuous variables, as each provides one column of the model matrix
(and the intercept will provide a column of ones if included in the model).
What about a k-level factor A? The answer differs for unordered and ordered factors. For
unordered factors k − 1 columns are generated for the indicators of the second, . . . , kth levels
of the factor. (Thus the implicit parameterization is to contrast the response at each level with
that at the first.) For ordered factors the k − 1 columns are the orthogonal polynomials on
1, . . . , k, omitting the constant term.
Although the answer is already complicated, it is not the whole story. First, if the intercept
is omitted in a model that contains a factor term, the first such term is encoded into k columns
giving the indicators for all the levels. Second, the whole behavior can be changed by the
options setting for contrasts. The default setting in R is
options(contrasts = c("contr.treatment", "contr.poly"))
The main reason for mentioning this is that R and S have different defaults for unordered factors,
S using Helmert contrasts. So if you need to compare your results to those of a textbook or
paper which used S-Plus, you will need to set
options(contrasts = c("contr.helmert", "contr.poly"))
This is a deliberate difference, as treatment contrasts (R’s default) are thought easier for 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.

Chapter 11: Statistical models in R
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11.2 Linear models
The basic function for fitting ordinary multiple models is lm(), and a streamlined version of the
call is as follows:
> fitted.model <- lm(formula, data = data.frame )
For example
> fm2 <- lm(y ~ x1 + x2, data = production)
would fit a multiple regression model of y on x1 and x2 (with implicit intercept term).
The important (but technically optional) parameter data = production specifies that any
variables needed to construct the model should come first from the production data frame.
This is the case regardless of whether data frame production has been attached on the search
path or not.
11.3 Generic functions for extracting model information
The value of lm() is a fitted model object; technically a list of results of class "lm". Information
about the fitted model can then be displayed, extracted, plotted and so on by using generic
functions that orient themselves to objects of class "lm". These include
add1
deviance
formula
predict
step
alias
drop1
kappa
print
summary
anova
effects
labels
proj
vcov
coef
family
plot
residuals
A brief description of the most commonly used ones is given below.
anova(object_1, object_2 )
Compare a submodel with an outer model and produce an analysis of variance table.
coef(object )
Extract the regression coefficient (matrix).
Long form: coefficients(object ).
deviance(object )
Residual sum of squares, weighted if appropriate.
formula(object )
Extract the model formula.
plot(object )
Produce four plots, showing residuals, fitted values and some diagnostics.
predict(object, newdata=data.frame )
The data frame supplied must have variables specified with the same labels as the
original. The value is a vector or matrix of predicted values corresponding to the
determining variable values in data.frame.
print(object )
Print a concise version of the object. Most often used implicitly.
residuals(object )
Extract the (matrix of) residuals, weighted as appropriate.
Short form: resid(object ).
step(object )
Select a suitable model by adding or dropping terms and preserving hierarchies. The
model with the smallest value of AIC (Akaike’s An Information Criterion) discovered
in the stepwise search is returned.

Chapter 11: Statistical models in R
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summary(object )
Print a comprehensive summary of the results of the regression analysis.
vcov(object )
Returns the variance-covariance matrix of the main parameters of a fitted model
object.
11.4 Analysis of variance and model comparison
The model fitting function aov(formula, data=data.frame ) operates at the simplest level in
a very similar way to the function lm(), and most of the generic functions listed in the table in
Section 11.3 [Generic functions for extracting model information], page 53 apply.
It should be noted that in addition aov() allows an analysis of models with multiple error
strata such as split plot experiments, or balanced incomplete block designs with recovery of
inter-block information. The model formula
response ~ mean.formula + Error(strata.formula )
specifies a multi-stratum experiment with error strata defined by the strata.formula. In the
simplest case, strata.formula is simply a factor, when it defines a two strata experiment, namely
between and within the levels of the factor.
For example, with all determining variables factors, a model formula such as that in:
> fm <- aov(yield ~ v + n*p*k + Error(farms/blocks), data=farm.data)
would typically be used to describe an experiment with mean model v + n*p*k and three error
strata, namely “between farms”, “within farms, between blocks” and “within blocks”.
11.4.1 ANOVA tables
Note also that the analysis of variance table (or tables) are for a sequence of fitted models.
The sums of squares shown are the decrease in the residual sums of squares resulting from an
inclusion of that term in the model at that place in the sequence. Hence only for orthogonal
experiments will the order of inclusion be inconsequential.
For multistratum experiments the procedure is first to project the response onto the error
strata, again in sequence, and to fit the mean model to each projection. For further details, see
Chambers & Hastie (1992).
A more flexible alternative to the default full ANOVA table is to compare two or more models
directly using the anova() function.
> anova(fitted.model.1, fitted.model.2, ...)
The display is then an ANOVA table showing the differences between the fitted models when
fitted in sequence. The fitted models being compared would usually be an hierarchical sequence,
of course. This does not give different information to the default, but rather makes it easier to
comprehend and control.
11.5 Updating fitted models
The update() function is largely a convenience function that allows a model to be fitted that
differs from one previously fitted usually by just a few additional or removed terms. Its form is
> new.model <- update(old.model, new.formula )
In the new.formula the special name consisting of a period, ‘.’, only, can be used to stand
for “the corresponding part of the old model formula”. For example,
> fm05 <- lm(y ~ x1 + x2 + x3 + x4 + x5, data = production)
> fm6
<- update(fm05, . ~ . + x6)
> smf6 <- update(fm6, sqrt(.) ~ .)

Chapter 11: Statistical models in R
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would fit a five variate multiple regression with variables (presumably) from the data frame
production, fit an additional model including a sixth regressor variable, and fit a variant on
the model where the response had a square root transform applied.
Note especially that if the data= argument is specified on the original call to the model
fitting function, this information is passed on through the fitted model object to update() and
its allies.
The name ‘.’ can also be used in other contexts, but with slightly different meaning. For
example
> fmfull <- lm(y ~ . , data = production)
would fit a model with response y and regressor variables all other variables in the data frame
production.
Other functions for exploring incremental sequences of models are add1(), drop1() and
step(). The names of these give a good clue to their purpose, but for full details see the on-line
help.
11.6 Generalized linear models
Generalized linear modeling is a development of linear models to accommodate both non-normal
response distributions and transformations to linearity in a clean and straightforward way. A
generalized linear model may be described in terms of the following sequence of assumptions:
• There is a response, y, of interest and stimulus variables x1, x2, . . . , whose values influence
the distribution of the response.
• The stimulus variables influence the distribution of y through a single linear function, only.
This linear function is called the linear predictor, and is usually written
η = β1x1 + β2x2 + · · · + βpxp,
hence xi has no influence on the distribution of y if and only if βi = 0.
• The distribution of y is of the form
A
fY (y; µ, ϕ) = exp
{yλ(µ) − γ (λ(µ))} + τ (y, ϕ)
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
µ is the mean of y. So it is assumed that the distribution of y is determined by its mean
and possibly a scale parameter as well.
• The mean, µ, is a smooth invertible function of the linear predictor:
µ = m(η),
η = m−1(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).

Chapter 11: Statistical models in R
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11.6.1 Families
The class of generalized linear models handled by facilities supplied in R includes gaussian,
binomial, poisson, inverse gaussian and gamma response distributions and also quasi-likelihood
models where the response distribution is not explicitly specified. In the latter case the variance
function must be specified as a function of the mean, but in other cases this function is implied
by the response distribution.
Each response distribution admits a variety of link functions to connect the mean with the
linear predictor. Those automatically available are shown in the following table:
Family name
Link functions
binomial
logit, probit, log, cloglog
gaussian
identity, log, inverse
Gamma
identity, inverse, log
inverse.gaussian
1/mu^2, identity, inverse, log
poisson
identity, log, sqrt
quasi
logit, probit, cloglog, identity, inverse,
log, 1/mu^2, sqrt
The combination of a response distribution, a link function and various other pieces of infor-
mation that are needed to carry out the modeling exercise is called the family of the generalized
linear model.
11.6.2 The glm() function
Since the distribution of the response depends on the stimulus variables through a single linear
function only, the same mechanism as was used for linear models can still be used to specify the
linear part of a generalized model. The family has to be specified in a different way.
The R function to fit a generalized linear model is glm() which uses the form
> fitted.model <- glm(formula, family=family.generator, data=data.frame )
The only new feature is the family.generator, which is the instrument by which the family is
described. It is the name of a function that generates a list of functions and expressions that
together define and control the model and estimation process. Although this may seem a little
complicated at first sight, its use is quite simple.
The names of the standard, supplied family generators are given under “Family Name” in
the table in Section 11.6.1 [Families], page 56. 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).

Chapter 11: Statistical models in R
57
On the Aegean island of Kalythos the male inhabitants suffer from a congenital eye disease,
the effects of which become more marked with increasing age. Samples of islander males of
various ages were tested for blindness and the results recorded. The data is shown below:
Age:
20
35
45
55
70
No. tested:
50
50
50
50
50
No. blind:
6
17
26
37
44
The problem we consider is to fit both logistic and probit models to this data, and to estimate
for each model the LD50, that is the age at which the chance of blindness for a male inhabitant
is 50%.
If y is the number of blind at age x and n the number tested, both models have the form
y ∼ B(n, F (β0 + β1x))
where for the probit case, F (z) = Φ(z) is the standard normal distribution function, and in the
logit case (the default), F (z) = ez/(1 + ez). In both cases the LD50 is
LD50 = −β0/β1
that is, the point at which the argument of the distribution function is zero.
The first step is to set the data up as a data frame
> kalythos <- data.frame(x = c(20,35,45,55,70), n = rep(50,5),
y = c(6,17,26,37,44))
To fit a binomial model using glm() there are three possibilities for the response:
• If the response is a vector it is assumed to hold binary data, and so must be a 0/1 vector.
• If the response is a two-column matrix it is assumed that the first column holds the number
of successes for the trial and the second holds the number of failures.
• If the response is a factor, its first level is taken as failure (0) and all other levels as ‘success’
(1).
Here we need the second of these conventions, so we add a matrix to our data frame:
> kalythos$Ymat <- cbind(kalythos$y, kalythos$n - kalythos$y)
To fit the models we use
> fmp <- glm(Ymat ~ x, family = binomial(link=probit), data = kalythos)
> fml <- glm(Ymat ~ x, family = binomial, data = kalythos)
Since the logit link is the default the parameter may be omitted on the second call. To see
the results of each fit we could use
> summary(fmp)
> summary(fml)
Both models fit (all too) well. To find the LD50 estimate we can use a simple function:
> ld50 <- function(b) -b[1]/b[2]
> ldp <- ld50(coef(fmp)); ldl <- ld50(coef(fml)); c(ldp, ldl)
The actual estimates from this data are 43.663 years and 43.601 years respectively.
Poisson models
With the Poisson family the default link is the log, and in practice the major use of this family
is to fit surrogate Poisson log-linear models to frequency data, whose actual distribution is often
multinomial. This is a large and important subject we will not discuss further here. It even
forms a major part of the use of non-gaussian generalized models overall.
Occasionally genuinely Poisson data arises in practice and in the past it was often analyzed
as gaussian data after either a log or a square-root transformation. As a graceful alternative to
the latter, a Poisson generalized linear model may be fitted as in the following example:

Chapter 11: Statistical models in R
58
> fmod <- glm(y ~ A + B + x, family = poisson(link=sqrt),
data = worm.counts)
Quasi-likelihood models
For all families the variance of the response will depend on the mean and will have the scale
parameter as a multiplier. The form of dependence of the variance on the mean is a characteristic
of the response distribution; for example for the poisson distribution Var[y] = µ.
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
θ
y =
1z1
+ e
z2 − θ2
which may be written alternatively as
1
y =
+ e
β1x1 + β2x2
where x1 = z2/z1, x2 = −1/z1, β1 = 1/θ1 and β2 = θ2/θ1. Supposing a suitable data frame to
be set up we could fit this non-linear regression as
> nlfit <- glm(y ~ x1 + x2 - 1,
family = quasi(link=inverse, variance=constant),
data = biochem)
The reader is referred to the manual and the help document for further information, as
needed.
11.7 Nonlinear least squares and maximum likelihood models
Certain forms of nonlinear model can be fitted by Generalized Linear Models (glm()). But
in the majority of cases we have to approach the nonlinear curve fitting problem as one of
nonlinear optimization. R’s nonlinear optimization routines are optim(), nlm() and (from R
2.2.0) nlminb(), which provide the functionality (and more) of S-Plus’s ms() and nlminb().
We seek the parameter values that minimize some index of lack-of-fit, and they do this by
trying out various parameter values iteratively. Unlike linear regression for example, there is no
guarantee that the procedure will converge on satisfactory estimates. All the methods require
initial guesses about what parameter values to try, and convergence may depend critically upon
the quality of the starting values.
11.7.1 Least squares
One way to fit a nonlinear model is by minimizing the sum of the squared errors (SSE) or
residuals. This method makes sense if the observed errors could have plausibly arisen from a
normal distribution.
Here is an example from Bates & Watts (1988), page 51. The data are:
> x <- c(0.02, 0.02, 0.06, 0.06, 0.11, 0.11, 0.22, 0.22, 0.56, 0.56,
1.10, 1.10)
> y <- c(76, 47, 97, 107, 123, 139, 159, 152, 191, 201, 207, 200)
The fit criterion to be minimized is:

Chapter 11: Statistical models in R
59
> fn <- function(p) sum((y - (p[1] * x)/(p[2] + x))^2)
In order to do the fit we need initial estimates of the parameters. One way to find sensible
starting values is to plot the data, guess some parameter values, and superimpose the model
curve using those values.
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 200 * xfit/(0.1 + xfit)
> lines(spline(xfit, yfit))
We could do better, but these starting values of 200 and 0.1 seem adequate. Now do the fit:
> out <- nlm(fn, p = c(200, 0.1), hessian = TRUE)
After the fitting, out$minimum is the SSE, and out$estimate are the least squares estimates
of the parameters. To obtain the approximate standard errors (SE) of the estimates we do:
> sqrt(diag(2*out$minimum/(length(y) - 2) * solve(out$hessian)))
The 2 in the line above represents the number of parameters. A 95% confidence interval
would be the parameter estimate ± 1.96 SE. We can superimpose the least squares fit on a new
plot:
> plot(x, y)
> xfit <- seq(.02, 1.1, .05)
> yfit <- 212.68384222 * xfit/(0.06412146 + xfit)
> lines(spline(xfit, yfit))
The standard package stats provides much more extensive facilities for fitting non-linear
models by least squares. The model we have just fitted is the Michaelis-Menten model, so we
can use
> df <- data.frame(x=x, y=y)
> fit <- nls(y ~ SSmicmen(x, Vm, K), df)
> fit
Nonlinear regression model
model:
y ~ SSmicmen(x, Vm, K)
data:
df
Vm
K
212.68370711
0.06412123
residual sum-of-squares:
1195.449
> summary(fit)
Formula: y ~ SSmicmen(x, Vm, K)
Parameters:
Estimate Std. Error t value Pr(>|t|)
Vm 2.127e+02
6.947e+00
30.615 3.24e-11
K
6.412e-02
8.281e-03
7.743 1.57e-05
Residual standard error: 10.93 on 10 degrees of freedom
Correlation of Parameter Estimates:
Vm
K 0.7651
11.7.2 Maximum likelihood
Maximum likelihood is a method of nonlinear model fitting that applies even if the errors are
not normal. The method finds the parameter values which maximize the log likelihood, or

Chapter 11: Statistical models in R
60
equivalently which minimize the negative log-likelihood. Here is an example from Dobson (1990),
pp. 108–111. This example fits a logistic model to dose-response data, which clearly could also
be fit by glm(). The data are:
> x <- c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113,
1.8369, 1.8610, 1.8839)
> y <- c( 6, 13, 18, 28, 52, 53, 61, 60)
> n <- c(59, 60, 62, 56, 63, 59, 62, 60)
The negative log-likelihood to minimize is:
> fn <- function(p)
sum( - (y*(p[1]+p[2]*x) - n*log(1+exp(p[1]+p[2]*x))
+ log(choose(n, y)) ))
We pick sensible starting values and do the fit:
> out <- nlm(fn, p = c(-50,20), hessian = TRUE)
After the fitting, out$minimum is the negative log-likelihood, and out$estimate are the maxi-
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 nlme package provides functions lme() and nlme() for
linear and non-linear mixed-effects models, that is linear and non-linear regressions in which
some of the coefficients correspond to random effects. These functions make heavy use of
formulae to specify the models.
• Local approximating regressions. The loess() function fits a nonparametric regression by
using a locally weighted regression. Such regressions are useful for highlighting a trend in
messy data or for data reduction to give some insight into a large data set.
Function loess is in the standard package stats, together with code for projection pursuit
regression.
• Robust regression. There are several functions available for fitting regression models in a
way resistant to the influence of extreme outliers in the data. Function lqs in the rec-
ommended package MASS provides state-of-art algorithms for highly-resistant fits. Less
resistant but statistically more efficient methods are available in packages, for example
function rlm in package MASS.
• Additive models. This technique aims to construct a regression function from smooth
additive functions of the determining variables, usually one for each determining variable.
Functions avas and ace in package acepack and functions bruto and mars in package mda
provide some examples of these techniques in user-contributed packages to R. An extension
is Generalized Additive Models, implemented in user-contributed packages gam and mgcv.
• Tree-based models. Rather than seek an explicit global linear model for prediction or
interpretation, tree-based models seek to bifurcate the data, recursively, at critical points
of the determining variables in order to partition the data ultimately into groups that are
as homogeneous as possible within, and as heterogeneous as possible between. The results
often lead to insights that other data analysis methods tend not to yield.
Models are again specified in the ordinary linear model form. The model fitting function is
tree(), but many other generic functions such as plot() and text() are well adapted to
displaying the results of a tree-based model fit in a graphical way.

Chapter 11: Statistical models in R
61
Tree models are available in R via the user-contributed packages rpart and tree.

Chapter 12: Graphical procedures
62
12 Graphical procedures
Graphical facilities are an important and extremely versatile component of the R environment.
It is possible to use the facilities to display a wide variety of statistical graphs and also to build
entirely new types of graph.
The graphics facilities can be used in both interactive and batch modes, but in most cases,
interactive use is more productive. Interactive use is also easy because at startup time R initiates
a graphics device driver which opens a special graphics window for the display of interactive
graphics. Although this is done automatically, it is useful to know that the command used is
X11() under UNIX, windows() under Windows and quartz() under Mac OS X.
Once the device driver is running, R plotting commands can be used to produce a variety of
graphical displays and to create entirely new kinds of display.
Plotting commands are divided into three basic groups:
• High-level plotting functions create a new plot on the graphics device, possibly with axes,
labels, titles and so on.
• Low-level plotting functions add more information to an existing plot, such as extra points,
lines and labels.
• Interactive graphics functions allow you interactively add information to, or extract infor-
mation from, an existing plot, using a pointing device such as a mouse.
In addition, R maintains a list of graphical parameters which can be manipulated to customize
your plots.
This manual only describes what are known as ‘base’ graphics. A separate graphics sub-
system in package grid coexists with base – it is more powerful but harder to use. There is a
recommended package lattice which builds on grid and provides ways to produce multi-panel
plots akin to those in the Trellis system in S.
12.1 High-level plotting commands
High-level plotting functions are designed to generate a complete plot of the data passed as 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.
plot(x, y )
plot(xy )
If x and y are vectors, plot(x, y ) produces a scatterplot of y against x. The same
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
produces a plot of the values in the vector against their index in the vector. If x
is a complex vector, it produces a plot of imaginary versus real parts of the vector
elements.
plot(f )
plot(f, y )
f is a factor object, y is a numeric vector. The first form generates a bar plot of f ;
the second form produces boxplots of y for each level of f.

Chapter 12: Graphical procedures
63
plot(df )
plot(~ expr )
plot(y ~ expr )
df is a data frame, y is any object, expr is a list of object names separated by ‘+’
(e.g., a + b + c). The first two forms produce distributional plots of the variables in
a data frame (first form) or of a number of named objects (second form). The third
form plots y against every object named in expr.
12.1.2 Displaying multivariate data
R provides two very useful functions for representing multivariate data. If X 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,
every column of X is plotted against every other column of X and the resulting n(n − 1) plots
are arranged in a matrix with plot scales constant over the rows and columns of the matrix.
When three or four variables are involved a coplot may be more enlightening. If a and b are
numeric vectors and c is a numeric vector or factor object (all of the same length), then the
command
> coplot(a ~ b | c)
produces a number of scatterplots of a against b for given values of c. If c is a factor, this
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=
argument to coplot()—the function co.intervals() is useful for selecting intervals. You can
also use two given variables with a command like
> coplot(a ~ b | c + d)
which produces scatterplots of a against b for every joint conditioning interval of c and d.
The coplot() and pairs() function both take an argument panel= which can be used to
customize the type of plot which appears in each panel. The default is points() to produce a
scatterplot but by supplying some other low-level graphics function of two vectors 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.

Chapter 12: Graphical procedures
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If the probability=TRUE argument is given, the bars represent relative frequencies
divided by bin width instead of counts.
dotchart(x, ...)
Constructs a dotchart of the data in x. In a dotchart the y-axis gives a labelling
of the data in x and the 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
colours to represent the value of z, the contour plot draws contour lines to represent
the value of z, and the persp plot draws a 3D surface.
12.1.4 Arguments to high-level plotting functions
There are a number of arguments which may be passed to high-level graphics functions, as
follows:
add=TRUE
Forces the function to act as a low-level graphics function, superimposing the plot
on the current plot (some functions only).
axes=FALSE
Suppresses generation of axes—useful for adding your own custom axes with the
axis() function. The default, axes=TRUE, means include axes.
log="x"
log="y"
log="xy"
Causes the x, y or both axes to be logarithmic. This will work for many, but not
all, types of plot.
type=
The type= argument controls the type of plot produced, as follows:
type="p"
Plot individual points (the default)
type="l"
Plot lines
type="b"
Plot points connected by lines (both)
type="o"
Plot points overlaid by lines
type="h"
Plot vertical lines from points to the zero axis (high-density)
type="s"
type="S"
Step-function plots. In the first form, the top of the vertical defines the
point; in the second, the bottom.
type="n"
No plotting at all. However axes are still drawn (by default) and the
coordinate system is set up according to the data. Ideal for creating
plots with subsequent low-level graphics functions.
xlab=string
ylab=string
Axis labels for the 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.

Chapter 12: Graphical procedures
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12.2 Low-level plotting commands
Sometimes the high-level plotting functions don’t produce exactly the kind of plot you desire.
In this case, low-level plotting commands can be used to add extra information (such as points,
lines or text) to the current plot.
Some of the more useful low-level plotting functions are:
points(x, y)
lines(x, y)
Adds points or connected lines to the current plot. plot()’s type= argument can
also be passed to these functions (and defaults to "p" for points() and "l" for
lines().)
text(x, y, labels, ...)
Add text to a plot at points given by x, y. Normally labels is an integer or
character vector in which case labels[i] is plotted at point (x[i], y[i]). The
default is 1:length(x).
Note: This function is often used in the sequence
> plot(x, y, type="n"); text(x, y, names)
The graphics parameter type="n" suppresses the points but sets up the axes, and
the text() function supplies special characters, as specified by the character vector
names for the points.
abline(a, b)
abline(h=y )
abline(v=x )
abline(lm.obj )
Adds a line of slope b and intercept a to the current plot. h=y may be used to
specify y-coordinates for the heights of horizontal lines to go across a plot, and
v=x similarly for the x-coordinates for vertical lines. Also lm.obj may be list with a
coefficients component of length 2 (such as the result of model-fitting functions,)
which are taken as an intercept and slope, in that order.
polygon(x, y, ...)
Draws a polygon defined by the ordered vertices in (x, y) and (optionally) shade it
in with hatch lines, or fill it if the graphics device allows the filling of figures.
legend(x, y, legend, ...)
Adds a legend to the current plot at the specified position. Plotting characters, line
styles, colors etc., are identified with the labels in the character vector legend. At
least one other argument v (a vector the same length as legend) with the corre-
sponding values of the plotting unit must also be given, as follows:
legend( , fill=v )
Colors for filled boxes
legend( , col=v )
Colors in which points or lines will be drawn
legend( , lty=v )
Line styles
legend( , lwd=v )
Line widths
legend( , pch=v )
Plotting characters (character vector)

Chapter 12: Graphical procedures
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title(main, sub)
Adds a title main to the top of the current plot in a large font and (optionally) a
sub-title sub at the bottom in a smaller font.
axis(side, ...)
Adds an axis to the current plot on the side given by the first argument (1 to 4,
counting clockwise from the bottom.) Other arguments control the positioning of
the axis within or beside the plot, and tick positions and labels. Useful for adding
custom axes after calling plot() with the axes=FALSE argument.
Low-level plotting functions usually require some positioning information (e.g., x and y co-
ordinates) to determine where to place the new plot elements. Coordinates are given in terms of
user coordinates which are defined by the previous high-level graphics command and are chosen
based on the supplied data.
Where x and y arguments are required, it is also sufficient to supply a single argument being
a list with elements named x and y. Similarly a matrix with two columns is also valid input.
In this way functions such as locator() (see below) may be used to specify positions on a plot
interactively.
12.2.1 Mathematical annotation
In some cases, it is useful to add mathematical symbols and formulae to a plot. This can be
achieved in R by specifying an expression rather than a character string in any one of text,
mtext, axis, or title. For example, the following code draws the formula for the Binomial
probability function:
> text(x, y, expression(paste(bgroup("(", atop(n, x), ")"), p^x, q^{n-x})))
More information, including a full listing of the features available can obtained from within
R using the commands:
> help(plotmath)
> example(plotmath)
> demo(plotmath)
12.2.2 Hershey vector fonts
It is possible to specify Hershey vector fonts for rendering text when using the text and contour
functions. There are three reasons for using the Hershey fonts:
• Hershey fonts can produce better output, especially on a computer screen, for rotated
and/or small text.
• Hershey fonts provide certain symbols that may not be available in the standard fonts. In
particular, there are zodiac signs, cartographic symbols and astronomical symbols.
• Hershey fonts provide cyrillic and japanese (Kana and Kanji) characters.
More information, including tables of Hershey characters can be obtained from within R
using the commands:
> help(Hershey)
> demo(Hershey)
> help(Japanese)
> demo(Japanese)
12.3 Interacting with graphics
R also provides functions which allow users to extract or add information to a plot using a
mouse. The simplest of these is the locator() function:

Chapter 12: Graphical procedures
67
locator(n, type)
Waits for the user to select locations on the current plot using the left mouse button.
This continues until n (default 512) points have been selected, or another mouse
button is pressed. The type argument allows for plotting at the selected points and
has the same effect as for high-level graphics commands; the default is no plotting.
locator() returns the locations of the points selected as a list with two components
x and y.
locator() is usually called with no arguments. It is particularly useful for interactively
selecting positions for graphic elements such as legends or labels when it is difficult to calculate
in advance where the graphic should be placed. For example, to place some informative text
near an outlying point, the command
> text(locator(1), "Outlier", adj=0)
may be useful. (locator() will be ignored if the current device, such as postscript does not
support interactive pointing.)
identify(x, y, labels)
Allow the user to highlight any of the points defined by x and y (using the left mouse
button) by plotting the corresponding component of labels nearby (or the index
number of the point if labels is absent). Returns the indices of the selected points
when another button is pressed.
Sometimes we want to identify particular points on a plot, rather than their positions. For
example, we may wish the user to select some observation of interest from a graphical display
and then manipulate that observation in some way. Given a number of (x, y) coordinates in two
numeric vectors x and y, we could use the identify() function as follows:
> plot(x, y)
> identify(x, y)
The identify() functions performs no plotting itself, but simply allows the user to move
the mouse pointer and click the left mouse button near a point. If there is a point near the
mouse pointer it will be marked with its index number (that is, its position in the x/y vectors)
plotted nearby. Alternatively, you could use some informative string (such as a case name) as a
highlight by using the labels argument to identify(), or disable marking altogether with the
plot = FALSE argument. When the process is terminated (see above), identify() returns the
indices of the selected points; you can use these indices to extract the selected points from the
original vectors x and y.
12.4 Using graphics parameters
When creating graphics, particularly for presentation or publication purposes, R’s defaults do
not always produce exactly that which is required. You can, however, customize almost every
aspect of the display using graphics parameters. R maintains a list of a large number of graphics
parameters which control things such as line style, colors, figure arrangement and text 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.

Chapter 12: Graphical procedures
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par()
Without arguments, returns a list of all graphics parameters and their values for
the current device.
par(c("col", "lty"))
With a character vector argument, returns only the named graphics parameters
(again, as a list.)
par(col=4, lty=2)
With named arguments (or a single list argument), sets the values of the named
graphics parameters, and returns the original values of the parameters as a list.
Setting graphics parameters with the par() function changes the value of the parameters
permanently, in the sense that all future calls to graphics functions (on the current device) will
be affected by the new value. You can think of setting graphics parameters in this way as
setting “default” values for the parameters, which will be used by all graphics functions unless
an alternative value is given.
Note that calls to par() always affect the global values of graphics parameters, even when
par() is called from within a function. This is often undesirable behavior—usually we want to
set some graphics parameters, do some plotting, and then restore the original values so as not
to affect the user’s R session. You can restore the initial values by saving the result of par()
when making changes, and restoring the initial values when plotting is complete.
> oldpar <- par(col=4, lty=2)
. . . plotting commands . . .
> par(oldpar)
To save and restore all settable1 graphical parameters use
> oldpar <- par(no.readonly=TRUE)
. . . plotting commands . . .
> par(oldpar)
12.4.2 Temporary changes: Arguments to graphics functions
Graphics parameters may also be passed to (almost) any graphics function as named arguments.
This has the same effect as passing the arguments to the par() function, except that the changes
only last for the duration of the function call. For example:
> 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 par() function provides a more concise summary; this is provided as a
somewhat more detailed alternative.
Graphics parameters will be presented in the following form:
name =value
A description of the parameter’s effect. name is the name of the parameter, that
is, the argument name to use in calls to par() or a graphics function. value is a
typical value you might use when setting the parameter.
Note that axes is not a graphics parameter but an argument to a few plot methods: see
xaxt and yaxt.
1 Some graphics parameters such as the size of the current device are for information only.

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

Chapter 12: Graphical procedures
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cex.axis
cex.lab
cex.main
cex.sub
The character expansion to be used for axis annotation, x and y labels, main and
sub-titles, respectively.
12.5.2 Axes and tick marks
Many of R’s high-level plots have axes, and you can construct axes yourself with the low-level
axis() graphics function. Axes have three main components: the axis line (line style controlled
by the lty graphics parameter), the tick marks (which mark off unit divisions along the axis
line) and the tick labels (which mark the units.) These components can be customized with the
following graphics parameters.
lab=c(5, 7, 12)
The first two numbers are the desired number of tick intervals on the x and y axes
respectively. The third number is the desired length of axis labels, in characters
(including the decimal point.) Choosing a too-small value for this parameter may
result in all tick labels being rounded to the same number!
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
is small (less than 0.5) the tick marks on the x and y axes are forced to be the
same size. A value of 1 gives grid lines. Negative values give tick marks outside the
plotting region. Use tck=0.01 and mgp=c(1,-1.5,0) for internal tick marks.
xaxs="r"
yaxs="i"
Axis styles for the x and y axes, respectively. With styles "i" (internal) and "r"
(the default) tick marks always fall within the range of the data, however style "r"
leaves a small amount of space at the edges. (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.

Chapter 12: Graphical procedures
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A typical figure is
−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−
mar[3]
−−−−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−−−−
3.0
Plot region
1.5
y
0.0
mai[2]
−1.5
−3.0
−3.0
−1.5
0.0
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)
Similar to mai, except the measurement unit is text lines.
mar and mai are equivalent in the sense that setting one changes the value of the other. The
default values chosen for this parameter are often too large; the right-hand margin is rarely
needed, and neither is the top margin if no title is being used. The bottom and left margins
must be large enough to accommodate the axis and tick labels. Furthermore, the default is
chosen without regard to the size of the device surface: for example, using the postscript()
driver with the height=4 argument will result in a plot which is about 50% margin unless mar
or mai are set explicitly. When multiple figures are in use (see below) the margins are reduced,
however this may not be enough when many figures share the same page.

Chapter 12: Graphical procedures
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12.5.4 Multiple figure environment
R allows you to create an n by m array of figures on a single page. Each figure has its own
margins, and the array of figures is optionally surrounded by an outer margin, as shown in the
following figure.
−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−
−−−−−−−−−−−−−−−
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
par("cex") and the pointsize of the device). In a layout with exactly two rows and
columns the base size is reduced by a factor of 0.83: if there are three or more of
either rows or columns, the reduction factor is 0.66.
mfg=c(2, 2, 3, 2)
Position of the current figure in a multiple figure environment. The first two numbers
are the row and column of the current figure; the last two are the number of rows
and columns in the multiple figure array. Set this parameter to jump between figures
in the array. You can even use different values for the last two numbers than the
true values for unequally-sized figures on the same page.
fig=c(4, 9, 1, 4)/10
Position of the current figure on the page. Values are the positions of the left, right,
bottom and top edges respectively, as a percentage of the page measured from the
bottom left corner. The example value would be for a figure in the bottom right of
the page. Set this parameter for arbitrary positioning of figures within a page. If
you want to add a figure to a current page, use new=TRUE as well (unlike S).
oma=c(2, 0, 3, 0)
omi=c(0, 0, 0.8, 0)
Size of outer margins. Like mar and mai, the first measures in text lines and the
second in inches, starting with the bottom margin and working clockwise.

Chapter 12: Graphical procedures
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Outer margins are particularly useful for page-wise titles, etc. Text can be added to the outer
margins with the mtext() function with argument outer=TRUE. There are no outer margins by
default, however, so you must create them explicitly using oma or omi.
More complicated arrangements of multiple figures can be produced by the split.screen()
and layout() functions, as well as by the grid and lattice packages.
12.6 Device drivers
R can generate graphics (of varying levels of quality) on almost any type of display or printing
device. Before this can begin, however, R needs to be informed what type of device it is dealing
with. This is done by starting a device driver. The purpose of a device driver is to convert
graphical instructions from R (“draw a line,” for example) into a form that the particular device
can understand.
Device drivers are started by calling a device driver function. There is one such function
for every device driver: type help(Devices) for a list of them all. For example, issuing the
command
> postscript()
causes all future graphics output to be sent to the printer in PostScript format. Some commonly-
used device drivers are:
X11()
For use with the X11 window system on Unix-alikes
windows()
For use on Windows
quartz()
For use on Mac OS X
postscript()
For printing on PostScript printers, or creating PostScript graphics files.
pdf()
Produces a PDF file, which can also be included into PDF files.
png()
Produces a bitmap PNG file. (Not always available: see its help page.)
jpeg()
Produces a bitmap JPEG file, best used for image plots. (Not always available: see
its help page.)
When you have finished with a device, be sure to terminate the device driver by issuing the
command
> dev.off()
This ensures that the device finishes cleanly; for example in the case of hardcopy devices
this ensures that every page is completed and has been sent to the printer. (This will happen
automatically at the normal end of a session.)
12.6.1 PostScript diagrams for typeset documents
By passing the file argument to the postscript() device driver function, you may store the
graphics in PostScript format in a file of your choice. The plot will be in landscape orientation
unless the horizontal=FALSE argument is given, and you can control the size of the graphic with
the width and height arguments (the plot will be scaled as appropriate to fit these dimensions.)
For example, the command
> postscript("file.ps", horizontal=FALSE, height=5, pointsize=10)
will produce a file containing PostScript code for a figure five inches high, perhaps for inclusion
in a document. It is important to note that if the file named in the command already exists,
it will be overwritten. This is the case even if the file was only created earlier in the same R
session.

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

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

Chapter 13: Packages
76
13 Packages
All R functions and datasets are stored in packages. Only when a package is loaded are its
contents available. This is done both for efficiency (the full list would take more memory and
would take longer to search than a subset), and to aid package developers, who are protected
from name clashes with other code. The process of developing packages is described in Section
“Creating R packages” in Writing R Extensions. Here, we will describe them from a user’s point
of view.
To see which packages are installed at your site, issue the command
> library()
with no arguments. To load a particular package (e.g., the boot package containing functions
from Davison & Hinkley (1997)), use a command like
> library(boot)
Users connected to the Internet can use the install.packages() and update.packages()
functions (available through the Packages menu in the Windows and RAqua GUIs, see Section
“Installing packages” in R Installation and Administration) to install and update packages.
To see which packages are currently loaded, use
> search()
to display the search list. Some packages may be loaded but not available on the search list (see
Section 13.3 [Namespaces], page 76): these will be included in the list given by
> loadedNamespaces()
To see a list of all available help topics in an installed package, use
> help.start()
to start the HTML help system, and then navigate to the package listing in the Reference
section.
13.1 Standard packages
The standard (or base) packages are considered part of the R source code. They contain the
basic functions that allow R to work, and the datasets and standard statistical and graphical
functions that are described in this manual. They should be automatically available in any R
installation. See Section “R packages” in R FAQ, for a complete list.
13.2 Contributed packages and CRAN
There are hundreds of contributed packages for R, written by many different authors. Some of
these packages implement specialized statistical methods, others give access to data or hard-
ware, and others are designed to complement textbooks. Some (the recommended packages) are
distributed with every binary distribution of R. Most are available for download from CRAN
(http://CRAN.R-project.org/ and its mirrors), and other repositories such as Bioconductor
(http://www.bioconductor.org/). The R FAQ contains a list that was current at the time of
release, but the collection of available packages changes frequently.
13.3 Namespaces
Packages can have namespaces, and currently all of the base and recommended packages do
except the datasets package. Namespaces do three things: they allow the package writer
to hide functions and data that are meant only for internal use, they prevent functions from
breaking when a user (or other package writer) picks a name that clashes with one in the package,
and they provide a way to refer to an object within a particular package.

Chapter 13: Packages
77
For example, t() is the transpose function in R, but users might define their own function
named t. Namespaces prevent the user’s definition from taking precedence, and breaking every
function that tries to transpose a matrix.
There are two operators that work with namespaces. The double-colon operator :: selects
definitions from a particular namespace. In the example above, the transpose function will
always be available as base::t, because it is defined in the base package. Only functions that
are exported from the package can be retrieved in this way.
The triple-colon operator ::: may be seen in a few places in R code: it acts like the
double-colon operator but also allows access to hidden objects. Users are more likely to use
the getAnywhere() function, which searches multiple packages.
Packages are often inter-dependent, and loading one may cause others to be automatically
loaded. The colon operators described above will also cause automatic loading of the associated
package. When packages with namespaces are loaded automatically they are not added to the
search list.

Appendix A: A sample session
78
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.
Login, start your windowing system.
$ R
Start R as appropriate for your platform.
The R program begins, with a banner.
(Within R, the prompt on the left hand side will not be shown to avoid confusion.)
help.start()
Start the HTML interface to on-line help (using a web browser available at your
machine). You should briefly explore the features of this facility with the mouse.
Iconify the help window and move on to the next part.
x <- rnorm(50)
y <- rnorm(x)
Generate two pseudo-random normal vectors of x- and y-coordinates.
plot(x, y)
Plot the points in the plane. A graphics window will appear automatically.
ls()
See which R objects are now in the R workspace.
rm(x, y)
Remove objects no longer needed. (Clean up).
x <- 1:20
Make x = (1, 2, . . . , 20).
w <- 1 + sqrt(x)/2
A ‘weight’ vector of standard deviations.
dummy <- data.frame(x=x, y= x + rnorm(x)*w)
dummy
Make a data frame of two columns, x and y, and look at it.
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.

Appendix A: A sample session
79
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 Michaelson and Morley
to measure the speed of light. This dataset is available in the morley object, but we will read
it to illustrate the read.table function.
filepath <- system.file("data", "morley.tab" , package="datasets")
filepath
Get the path to the data file.
file.show(filepath)
Optional. Look at the file.
mm <- read.table(filepath)
mm
Read in the Michaelson and Morley data as a data frame, and look at it. There are
five experiments (column Expt) and each has 20 runs (column Run) and sl is the
recorded speed of light, suitably coded.
mm$Expt <- factor(mm$Expt)
mm$Run <- factor(mm$Run)
Change Expt and Run into factors.
attach(mm)
Make the data frame visible at position 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 −π ≤ x ≤ π. y is the same.

Appendix A: A sample session
80
f <- outer(x, y, function(x, y) cos(y)/(1 + x^2))
f is a square matrix, with rows and columns indexed by x and y respectively, of
values of the function cos(y)/(1 + x2).
oldpar <- par(no.readonly = TRUE)
par(pty="s")
Save the plotting parameters and set the plotting region to “square”.
contour(x, y, f)
contour(x, y, f, nlevels=15, add=TRUE)
Make a contour map of f ; add in more lines for more detail.
fa <- (f-t(f))/2
fa is the “asymmetric part” of f . (t() is transpose).
contour(x, y, fa, nlevels=15)
Make a contour plot, . . .
par(oldpar)
. . . and restore the old graphics parameters.
image(x, y, f)
image(x, y, fa)
Make some high density image plots, (of which you can get hardcopies if you wish),
. . .
objects(); rm(x, y, f, fa)
. . . and clean up before moving on.
R can do complex arithmetic, also.
th <- seq(-pi, pi, len=100)
z <- exp(1i*th)
1i is used for the complex number 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.

Appendix B: Invoking R
81
Appendix B Invoking R
B.1 Invoking R from the command line
When working in UNIX or at a command line in Windows, the command ‘R’ can be used both
for starting the main R program in the form
R [options] [<infile] [>outfile],
or, via the R CMD interface, as a wrapper to various R tools (e.g., for processing files in R
documentation format or manipulating add-on packages) which are not intended to be called
“directly”.
You need to ensure that either the environment variable TMPDIR is unset or it points to a
valid place to create temporary files and directories.
Most options control what happens at the beginning and at the end of an R session. The
startup mechanism is as follows (see also the on-line help for topic ‘Startup’ for more informa-
tion, and the section below for some Windows-specific details).
• Unless ‘--no-environ’ was given, R searches for user and site files to process for setting
environment variables. The name of the site file is the one pointed to by the environment
variable R_ENVIRON; if this is unset, ‘R_HOME /etc/Renviron.site’ is used (if it exists).
The user file is the one pointed to by the environment variable R_ENVIRON_USER if this
is set; otherwise, files ‘.Renviron’ in the current or in the user’s home directory (in that
order) are searched for. These files should contain lines of the form ‘name =value ’. (See
help("Startup") for a precise description.) Variables you might want to set include R_
PAPERSIZE (the default paper size), R_PRINTCMD (the default print command) and R_LIBS
(specifies the list of R library trees searched for add-on packages).
• Then R searches for the site-wide startup profile unless the command line option
‘--no-site-file’ was given.
The name of this file is taken from the value
of the R_PROFILE environment variable.
If that variable is unset,
the default
‘R_HOME /etc/Rprofile.site’ is used if this exists.
• Then, unless ‘--no-init-file’ was given, R searches for a user profile and sources it. The
name of this file is taken from the environment variable R_PROFILE_USER; if unset, a file
called ‘.Rprofile’ in the current directory or in the user’s home directory (in that order)
is searched for.
• It also loads a saved image from ‘.RData’ if there is one (unless ‘--no-restore’ or
‘--no-restore-data’ was specified).
• Finally, if a function .First exists, it is executed. This function (as well as .Last which is
executed at the end of the R session) can be defined in the appropriate startup profiles, or
reside in ‘.RData’.
In addition, there are options for controlling the memory available to the R process (see the
on-line help for topic ‘Memory’ for more information). Users will not normally need to use these
unless they are trying to limit the amount of memory used by R.
R accepts the following command-line options.
‘--help’
‘-h’
Print short help message to standard output and exit successfully.
‘--version’
Print version information to standard output and exit successfully.
‘--encoding=enc ’
Specify the encoding to be assumed for input from the console or stdin. This needs
to be an encoding known to iconv: see its help page.

Appendix B: Invoking R
82
‘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 ‘.RData’ in the directory where R was started)
should be restored at startup or not. The default is to restore. (‘--no-restore’
implies all the specific ‘--no-restore-*’ options.)
‘--no-restore-history’
Control whether the history file (normally file ‘.Rhistory’ in the directory where
R was started, but can be set by the environment variable R_HISTFILE) should be
restored at startup or not. The default is to restore.
‘--no-Rconsole’
(Windows only) Prevent loading the ‘Rconsole’ file at startup.
‘--vanilla’
Combine ‘--no-save’, ‘--no-environ’, ‘--no-site-file’, ‘--no-init-file’ and
‘--no-restore’. Under Windows, this also includes ‘--no-Rconsole’.
‘-f file ’
‘--file=file ’
Take input from file: ‘-’ means stdin. Implies ‘--no-save’ unless ‘--save’ has
been set.
‘-e expression ’
Use expression as an input line. One or more ‘-e’ options can be used, but not
together with ‘-f’ or ‘--file’. Implies ‘--no-save’ unless ‘--save’ has been set.
(There is a limit of 10,000 bytes on the total length of expressions used in this way.)
‘--no-readline’
(UNIX only) Turn off command-line editing via readline. This is useful when run-
ning R from within Emacs using the ESS (“Emacs Speaks Statistics”) package. See
Appendix C [The command-line editor], page 87, for more information. Command-
line editing is enabled by default interactive use (see ‘--interactive’). This option
also affects tilde-expansion: see the help for path.expand.
‘--min-vsize=N ’
‘--max-vsize=N ’
Specify the minimum or maximum amount of memory used for variable size objects
by setting the “vector heap” size to N bytes. Here, N must either be an integer

Appendix B: Invoking R
83
or an integer ending with ‘G’, ‘M’, ‘K’, or ‘k’, meaning ‘Giga’ (2^30), ‘Mega’ (2^20),
(computer) ‘Kilo’ (2^10), or regular ‘kilo’ (1000).
‘--min-nsize=N ’
‘--max-nsize=N ’
Specify the amount of memory used for fixed size objects by setting the number of
“cons cells” to N. See the previous option for details on N. A cons cell takes 28 bytes
on a 32-bit machine, and usually 56 bytes on a 64-bit machine.
‘--max-ppsize=N ’
Specify the maximum size of the pointer protection stack as N locations. This
defaults to 10000, but can be increased to allow large and complicated calculations
to be done. Currently the maximum value accepted is 100000.
‘--max-mem-size=N ’
(Windows only) Specify a limit for the amount of memory to be used both for R
objects and working areas. This is set by default to the smaller of 1.5Gb1 and
the amount of physical RAM in the machine, and must be between 32Mb and the
maximum allowed on that version of Windows.
‘--quiet’
‘--silent’
‘-q’
Do not print out the initial copyright and welcome messages.
‘--slave’
Make R run as quietly as possible. This option is intended to support programs
which use R to compute results for them. It implies ‘--quiet’ and ‘--no-save’.
‘--interactive’
(UNIX only) Assert that R really is being run interactively even if input has been
redirected: use if input is from a FIFO or pipe and fed from an interactive program.
(The default is to deduce that R is being run interactively if and only if ‘stdin’ is
connected to a terminal or pty.) Using ‘-e’, ‘-f’ or ‘--file’ asserts non-interactive
use even if ‘--interactive’ is given.
‘--ess’
(Windows only) Set Rterm up for use by R-inferior-mode in ESS, including assert-
ing interactive use without the command-line editor.
‘--verbose’
Print more information about progress, and in particular set R’s option verbose to
TRUE. R code uses this option to control the printing of diagnostic messages.
‘--debugger=name ’
‘-d name ’
(UNIX only) Run R through debugger name. For most debuggers (the exceptions are
valgrind and recent versions of gdb), further command line options are disregarded,
and should instead be given when starting the R executable from inside the debugger.
‘--gui=type ’
‘-g type ’
(UNIX only) Use type as graphical user interface (note that this also includes in-
teractive graphics). Currently, possible values for type are ‘X11’ (the default) and,
provided that ‘Tcl/Tk’ support is available, ‘Tk’. (For back-compatibility, ‘x11’ and
‘tk’ are accepted.)
‘--args’
This flag does nothing except cause the rest of the command line to be skipped:
this can be useful to retrieve values from it with commandArgs(TRUE).
1 2.5Gb on versions of Windows that support 3Gb per process and have the support enabled: see the ‘rw-FAQ’
Q2.9; 3.5Gb on some 64-bit versions of Windows.

Appendix B: Invoking R
84
Note that input and output can be redirected in the usual way (using ‘<’ and ‘>’), but the
line length limit of 4095 bytes still applies. Warning and error messages are sent to the error
channel (stderr).
The command R CMD allows the invocation of various tools which are useful in conjunction
with R, but not intended to be called “directly”. The general form is
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
Run R in batch mode.
COMPILE
(UNIX only) Compile files for use with R.
SHLIB
Build shared library for dynamic loading.
INSTALL
Install add-on packages.
REMOVE
Remove add-on packages.
build
Build (that is, package) add-on packages.
check
Check add-on packages.
LINK
(UNIX only) Front-end for creating executable programs.
Rprof
Post-process R profiling files.
Rdconv
Rd2txt
Convert Rd format to various other formats, including HTML, LATEX, plain text,
and extracting the examples. Rd2txt can be used as shorthand for Rd2txt -t txt.
Rd2dvi
Rd2pdf
Convert Rd format to DVI/PDF. Rd2pdf can be used as shorthand for Rd2dvi
--pdf.
Sd2Rd
Convert S documentation to Rd format.
Stangle
Extract S/R code from Sweave documentation
Sweave
Process Sweave documentation
Rdiff
Diff R output ignoring headers etc
config
Obtain configuration information.
javareconf
(Unix only) Update the Java configuration variables
Use
R CMD command --help
to obtain usage information for each of the tools accessible via the R CMD interface.
In addition, you can use
R CMD cmd args
for any other executable cmd on the path: this is useful to have the same environment as R or
the specific commands run under, for example to run ldd or pdflatex. Under a Unix-alike, if
cmd is ‘perl’ or ‘awk’ it is replaced by the full path to the Perl or AWK command found when
R was configured.

Appendix B: Invoking R
85
B.2 Invoking R under Windows
There are two ways to run R under Windows. Within a terminal window (e.g. cmd.exe or
command.com or a more capable shell), the methods described in the previous section may be
used, invoking by R.exe or more directly by Rterm.exe. (These are principally intended for
batch use.) For interactive use, there is a console-based GUI (Rgui.exe).
The startup procedure under Windows is very similar to that under UNIX, but references
to the ‘home directory’ need to be clarified, as this is not always defined on Windows. If the
environment variable R_USER is defined, that gives the home directory. Next, if the environment
variable HOME 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 C:\Documents and Settings\username\My Documents in Windows XP). If
that fails, and environment variables HOMEDRIVE and HOMEPATH are defined (and they normally
are) these define the home directory. Failing all those, the home directory is taken to be the
starting directory.
You need to ensure that either the environment variables TMPDIR, TMP and TEMP are either
unset or one of them points to a valid place to create temporary files and directories.
Environment variables can be supplied as ‘name =value ’ pairs on the command line.
If there is an argument ending ‘.RData’ (in any case) it is interpreted as the path to the
workspace to be restored: it implies ‘--restore’ and sets the working directory to the parent of
the named file. (This mechanism is used for drag-and-drop and file association with RGui.exe,
but also works for Rterm.exe. If the named file does not exist it sets the working directory if
the parent directory exists.)
The following additional command-line options are available when invoking RGui.exe.
‘--mdi’
‘--sdi’
‘--no-mdi’
Control whether Rgui will operate as an MDI program (with multiple child windows
within one main window) or an SDI application (with multiple top-level windows for
the console, graphics and pager). The command-line setting overrides the setting in
the user’s ‘Rconsole’ file.
‘--debug’
Enable the “Break to debugger” menu item in Rgui, and trigger a break to the
debugger during command line processing.
In Windows with R CMD you may also specify your own ‘.bat’, ‘.exe’, ‘.sh’ or ‘.pl’ file. It will
be run under the appropriate interpreter (Perl for ‘.pl’) with several environment variables set
appropriately, including R_HOME, R_VERSION, R_CMD, R_OSTYPE, PATH, PERL5LIB, and TEXINPUTS.
For example, if you already have ‘latex.exe’ on your path, then
R CMD latex.exe mydoc
will run LATEX on ‘mydoc.tex’, with the path to R’s ‘share/texmf’ macros appended to
TEXINPUTS. (Unfortunately, this does not help with the MiKTeX build of LATEX.)
B.3 Invoking R under Mac OS X
There are two ways to run R under Mac OS X. Within a Terminal.app window by invoking R,
the methods described in the first subsection apply. There is also console-based GUI (R.app)
that by default is installed in the Applications folder on your system. It is a standard double-
clickable Mac OS X application.
The startup procedure under Mac OS X is very similar to that under UNIX. 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.

Appendix B: Invoking R
86
B.4 Scripting with R
If you just want to run a file ‘foo.R’ of R commands, the recommended way is to use R CMD
BATCH foo.R.
If you want to run this in the background or as a batch job use OS-specific
facilities to do so: for example in most shells on Unix-alike OSes R CMD BATCH foo.R & runs a
background job.
You can pass parameters to scripts via additional arguments on the command line: for
example
R CMD BATCH --args arg1 arg2 foo.R &
will pass arguments to a script which can be retrieved as a character vector by
args <- commandArgs(TRUE)
This is made simpler by the alternative front-end Rscript, which can be invoked by
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=<exit status code>)
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.
If you do not wish to hardcode the path to Rscript but have it in your path (which is
normally the case for an installed R except on Windows), use
#! /usr/bin/env Rscript
...
At least in Bourne and bash shells, the #! mechanism does not allow extra arguments like #!
/usr/bin/env Rscript --vanilla.
One thing to consider is what stdin() refers to. It is commonplace to write R scripts with
segments like
chem <- scan(n=24)
2.90 3.10 3.40 3.40 3.70 3.70 2.80 2.50 2.40 2.40 2.70 2.20
5.28 3.37 3.03 3.03 28.95 3.77 3.40 2.20 3.50 3.60 3.70 3.70
and stdin() refers to the script file to allow such traditional usage. If you want to refer to the
process’s ‘stdin’, use "stdin" as a file connection, e.g. scan("stdin", ...).
Another way to write executable script files (suggested by Fran¸
cois Pinard) is to use a here
document like
#!/bin/sh
[environment variables can be set here]
R --slave [other options] <<EOF
R program goes here...
EOF
but here stdin() refers to the program source and "stdin" will not be usable.
Very short scripts can be passed to Rscript on the command-line via the ‘-e’ flag.

Appendix C: The command-line editor
87
Appendix C The command-line editor
C.1 Preliminaries
When the GNU readline library is available at the time R is configured for compilation un-
der UNIX, an inbuilt command line editor allowing recall, editing and re-submission of prior
commands is used. Note that other versions of readline exist and may be used by the inbuilt
command line editor: this may happen on Mac OS X.
It can be disabled (useful for usage with ESS1) using the startup option ‘--no-readline’.
Windows versions of R have somewhat simpler command-line editing: see ‘Console’ under the
‘Help’ menu of the GUI, and the file ‘README.Rterm’ for command-line editing under Rterm.exe.
When using R with readline capabilities, the functions described below are available.
Many of these use either Control or Meta characters. Control characters, such as Control-m,
are obtained by holding the CTRL down while you press the M key, and are written as C-m
below. Meta characters, such as Meta-b, are typed by holding down META2 and pressing B,
and written as M-b in the following. If your terminal does not have a META key, you can still
type Meta characters using two-character sequences starting with ESC. Thus, to enter M-b, you
could type ESCB. The ESC character sequences are also allowed on terminals with real Meta
keys. Note that case is significant for Meta characters.
C.2 Editing actions
The R program keeps a history of the command lines you type, including the erroneous lines,
and commands in your history may be recalled, changed if necessary, and re-submitted as new
commands.
In Emacs-style command-line editing any straight typing you do while in this
editing phase causes the characters to be inserted in the command you are editing, displacing
any characters to the right of the cursor. In vi mode character insertion mode is started by M-i
or M-a, characters are typed and insertion mode is finished by typing a further ESC.
Pressing the RET command at any time causes the command to be re-submitted.
Other editing actions are summarized in the following table.
C.3 Command-line editor summary
Command recall and vertical motion
C-p
Go to the previous command (backwards in the history).
C-n
Go to the next command (forwards in the history).
C-r text
Find the last command with the text string in it.
On most terminals, you can also use the up and down arrow keys instead of C-p and C-n,
respectively.
Horizontal motion of the cursor
C-a
Go to the beginning of the command.
C-e
Go to the end of the line.
M-b
Go back one word.
1 The ‘Emacs Speaks Statistics’ package; see the URL http://ESS.R-project.org
2 On a PC keyboard this is usually the Alt key, occasionally the ‘Windows’ key.

Appendix C: The command-line editor
88
M-f
Go forward one word.
C-b
Go back one character.
C-f
Go forward one character.
On most terminals, you can also use the left and right arrow keys instead of C-b and C-f,
respectively.
Editing and re-submission
text
Insert text at the cursor.
C-f text
Append text after the cursor.
DEL
Delete the previous character (left of the cursor).
C-d
Delete the character under the cursor.
M-d
Delete the rest of the word under the cursor, and “save” it.
C-k
Delete from cursor to end of command, and “save” it.
C-y
Insert (yank) the last “saved” text here.
C-t
Transpose the character under the cursor with the next.
M-l
Change the rest of the word to lower case.
M-c
Change the rest of the word to upper case.
RET
Re-submit the command to R.
The final RET terminates the command line editing sequence.

Appendix D: Function and variable index
89
Appendix D Function and variable index
!
?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
!= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
^
^ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
%
%*% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
|
%o% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
|| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
&
& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
~
&& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
*
A
* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
abline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
ace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
+
add1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
anova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53, 54
+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
aov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
aperm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
-
array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
as.data.frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
as.vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
attach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
.
attr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
avas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
.First . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
.Last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
B
/
boxplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
bruto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
:
C
: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
:: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 10, 24, 27
::: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
cbind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
<
coef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
< . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
<<- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
<= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
coplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
=
crossprod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19, 22
cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
== . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
D
>
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
data.frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
>= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
det . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
?
detach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
dev.list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Appendix D: Function and variable index
90
dev.next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
dev.off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
dev.prev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
dev.set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
lm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
deviance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
lme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
diag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
locator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
dim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
loess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
dotchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
drop1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
lqs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
lsfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
E
M
ecdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
edit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
eigen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
else . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
F
N
F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
NA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
NaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
FALSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ncol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
fivenum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
next . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
nlm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58, 59, 60
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
nlme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
nlminb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
nrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
G
getAnywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
O
getS3method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
optim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
glm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ordered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
H
outer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
P
help.search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
help.start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
hist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34, 63
par . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
I
pdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
persp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
identify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53, 62
if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
pmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ifelse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
pmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
png . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
is.na . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
is.nan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
J
predict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
jpeg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
prod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
K
Q
ks.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
qqline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35, 63
qqnorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35, 63
L
qqplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
qr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 13

Appendix D: Function and variable index
91
R
T
range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
rbind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
read.table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
t.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 25
repeat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
resid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
tapply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
rlm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
TRUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
S
U
scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
unclass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
sd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
seq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
shapiro.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
V
sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8, 17
sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
var.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
vcov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
W
sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
stem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
while . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53, 55
wilcox.test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 53
svd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
X
X11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Appendix E: Concept index
92
Appendix E Concept index
A
K
Accessing builtin datasets . . . . . . . . . . . . . . . . . . . . . . . . 31
Kolmogorov-Smirnov test . . . . . . . . . . . . . . . . . . . . . . . . 36
Additive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Analysis of variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Arithmetic functions and operators . . . . . . . . . . . . . . . . 7
L
Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Least squares fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B
Local approximating regressions . . . . . . . . . . . . . . . . . . 60
Loops and conditional execution . . . . . . . . . . . . . . . . . . 40
Binary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Box plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
M
Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
C
Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Character vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14, 48
Missing values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Concatenating lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Mixed models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Control statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
N
CRAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Customizing the environment . . . . . . . . . . . . . . . . . . . . 48
Named arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Namespace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Nonlinear least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
D
Data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
O
Default values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Object orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Density estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
One- and two-sample tests . . . . . . . . . . . . . . . . . . . . . . . 36
Diverting input and output . . . . . . . . . . . . . . . . . . . . . . . . 5
Ordered factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 52
Dynamic graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Outer products of arrays . . . . . . . . . . . . . . . . . . . . . . . . . 21
E
P
Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . 23
Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2, 76
Empirical CDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Probability distributions . . . . . . . . . . . . . . . . . . . . . . . . . 33
F
Q
Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 52
QR decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Quantile-quantile plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
R
G
Reading data from files . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Generalized linear models . . . . . . . . . . . . . . . . . . . . . . . . 55
Recycling rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 20
Generalized transpose of an array . . . . . . . . . . . . . . . . 21
Regular sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Generic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Removing objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Graphics device drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Robust regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Graphics parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Grouped expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
S
I
Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Search path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Indexing of and by arrays . . . . . . . . . . . . . . . . . . . . . . . . 18
Shapiro-Wilk test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Indexing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Singular value decomposition . . . . . . . . . . . . . . . . . . . . . 23
Statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Appendix E: Concept index
93
Student’s t test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
V
Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
T
Tabulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Tree-based models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
W
Wilcoxon test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
U
Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Updating fitted models . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Writing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Appendix F: References
94
Appendix F References
D. M. Bates and D. G. Watts (1988), Nonlinear Regression Analysis and Its Applications. John
Wiley & Sons, New York.
Richard A. Becker, John M. Chambers and Allan R. Wilks (1988), The New S Language. Chap-
man & Hall, New York. This book is often called the “Blue Book ”.
John M. Chambers and Trevor J. Hastie eds. (1992), Statistical Models in S. Chapman & Hall,
New York. This is also called the “White Book ”.
John M. Chambers (1998) Programming with Data. Springer, New York. This is also called the
“Green Book ”.
A. C. Davison and D. V. Hinkley (1997), Bootstrap Methods and Their Applications, Cambridge
University Press.
Annette J. Dobson (1990), An Introduction to Generalized Linear Models, Chapman and Hall,
London.
Peter McCullagh and John A. Nelder (1989), Generalized Linear Models. Second edition, 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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