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Date Created: 09/18/15
An Introduction to R Notes on R A Programming Environment for Data Analysis and Graphics Version 251 2007 06 27 W N Venables D M Smith and the R Development Core Team Copyright 1990 W N Venables Copyright 1992 W N Venables amp D M Smith Copyright 1997 R Gentleman amp R lhaka Copyright 19977 1998 M Maechler Copyright 199972006 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 modi ed versions of this manual under the condi tions for verbatim copying7 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 language7 under the above conditions for modi ed versions7 except that this permission notice may be stated in a translation approved by the R Development Core Team ISBN 3 900051127 Table of Contents Preface 1 1 Introduction and preliminaries 2 11 The R environment 2 12 Related software and documentation 2 13 R and statistics 2 14 R and the window system 3 15 Using R interactively 3 16 An introductory session 4 17 Getting help with functions and features 4 18 R commands7 case sensitivity7 etc 4 19 Recall and correction of previous commands 5 110 Executing commands from or diverting output to a le 5 111 Data permanency and removing objects 5 2 Simple manipulations numbers and vectors 7 21 Vectors and assignment 7 22 Vector arithmetic 7 23 Generating regular sequences 8 24 Logical vectors 9 25 Missing values 9 26 Character vectors 10 27 Index vectors selecting and modifying subsets of a data set 10 28 Other types of objects 11 3 Objects their modes and attributes 13 31 Intrinsic attributes mode and length 13 32 Changing the length of an object 14 33 Getting and setting attributes 14 34 The class of an object 14 4 Ordered and unordered factors 16 41 A speci c example 16 42 The function tapply and ragged arrays 16 43 Ordered factors 17 5 Arrays and matrices 18 51 Arrays 18 52 Array indexing Subsections of an array 18 53 Index matrices 19 54 The array function 20 541 Mixed vector and array arithmetic The recycling rule 20 55 The outer product of two arrays 21 56 Generalized transpose of an array 21 57 Matrix facilities 22 571 Matrix multiplication 22 ii 572 Linear equations and inversion 22 573 Eigenvalues and eigenvectors 23 574 Singular value decomposition and determinants 23 575 Least squares tting and the QR decomposition 23 58 Forming partitioned matrices7 cbindO and rbindO 24 59 The concatenation function7 c 7 with arrays 24 510 Frequency tables from factors 25 6 Lists and data frames 26 61 Lists 26 62 Constructing and modifying lists 26 621 Concatenating lists 27 63 Data frames 27 631 Making data frames 27 632 attach and detach 27 633 Working with data frames 28 634 Attaching arbitrary lists 28 635 Managing the search path 29 7 Reading data from les 30 71 The readtable function 30 72 The scan function 31 73 Accessing builtin datasets 31 731 Loading data from other R packages 31 74 Editing data 32 8 Probability distributions 33 81 R as a set of statistical tables 33 82 Examining the distribution of a set of data 33 83 One and two sample tests 36 9 Grouping loops and conditional execution 40 91 Grouped expressions 40 92 Control statements iii 11 Statistical models in R 50 111 De ning statistical models formulae 50 1111 Contrasts 52 112 Linear models 53 113 Generic functions for extracting model information 53 114 Analysis of variance and model comparison 54 1141 ANOVA tables 54 115 Updating tted models 54 116 Generalized linear models 55 1161 Families 56 1162 The glmO function 56 117 Nonlinear least squares and maximum likelihood models 58 1171 Least squares 58 1172 Maximum likelihood 59 118 Some non standard models 60 12 Graphical procedures 62 121 High level plotting commands 62 1211 The plot function 62 1212 Displaying multivariate data 63 1213 Display graphics 63 1214 Arguments to high level plotting functions 64 122 Low level plotting commands 65 1221 Mathematical annotation 66 1222 Hershey vector fonts 66 123 Interacting with graphics 66 124 Using graphics parameters 67 1241 Permanent changes The par function 67 1242 Temporary changes Arguments to graphics functions 68 125 Graphics parameters list 68 1251 Graphical elements 69 1252 Axes and tick marks 70 1253 Figure margins 70 1254 Multiple gure environment 71 126 Device drivers 73 1261 PostScript diagrams for typeset documents 73 1262 Multiple graphics devices 74 127 Dynamic graphics 75 13 Packages 76 131 Standard packages 76 132 Contributed packages and ORAN 76 133 Namespaces 76 Appendix A A sample session 78 Appendix B Invoking R 81 B1 lnvoking R from the command line 81 B2 lnvoking R under Windows 84 B3 lnvoking R under Mac OS X 85 B4 Scripting with R 85 iV Appendix C The commandline editor 87 01 Preliminaries 87 02 Editing actions 87 03 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 by Bill Venables and David M Smith Insightful Corporation We have made a number of small changes to re ect 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 modi ed 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 Rcore Rprojectorg 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 11 The R environment R is an integrated suite of software facilities for data manipulation calculation and graphical display Among other things it has 0 an effective data handling and storage facility 0 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 de ned recursive functions and input and output facilities Indeed most of the system supplied functions are themselves written in the S language The term environmen 7 is intended to characterize it as a fully planned and coherent system rather than an incremental accretion of very speci c and in exible 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 12 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 T M 39 A P g 39 g r39 39 t 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 SS PLUS can typically be used with R keeping the differences between the S implementations in mind See section What documentation exists for R77 in The R statistical system FAQ 13 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 CRMIRprojevtorg 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 nd 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 t object for subsequent interrogation by further R functions 14 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 nd it necessary to interact directly with the operating system on their computer from time to time In this guide we mainly discuss interaction with the operating system on UNIX machines If you are running R under Windows or MacOS you will need to make some small adjustments Setting up a workstation to take full advantage of the customizable features of R is a straight forward if somewhat tedious procedure and will not be considered further here Users in dif culty should seek local expert help 15 Using R interactively When you use the R program it issues a prompt when it expects input commands The default prompt is gt 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 rst occasion is as follows 1 Create a separate sub directory say work to hold data les 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 Start the R program with the command R At this point R commands may be issued see later to F To quit the R program the command is gt 10 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 In7 eld in your R shortcut Then launch R by double clicking on the icon Chapter 1 Introduction and preliminaries 4 16 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 17 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 speci c named function for example solve the command is gt helpsolve An alternative is gt solve For a feature speci ed 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 gt helpquot IIquot Either form of quote mark may be used to escape the other as in the string quotIt 5 important Our convention is to use double quote marks for preference On most R installations help is available in HTML format by running gt 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 Keywords7 link in the page loaded by helpstart 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 allows searching for help in various ways try help search for details and examples The examples on a help topic can normally be run by gt example topic Windows versions of R have other optional help systems use gt help for further details 18 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 allowed 1 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 p 39 or 39 If an e piession is given as a command it is evaluated printed unless speci cally made invisible and the value is lost An assignment also evaluates an expression and passes the value to a variable but the result is not automatically printed Commands are separated either by a semi colon 7 or by a newline Elementary commands can be grouped together into one compound expression by braces 7 and Comments can 1 For portable R code including that to be used in R packages only A7Za72079 should be used Chapter 1 Introduction and preliminaries 5 be put almost 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 1024 bytes not characters 19 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 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 nd 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 Emacs77 in The R statistical system FAQ 110 Executing commands from or diverting output to a le If commands4 are stored in an external le say commandsR in the working directory work they may be executed at any time in an R session with the command gt sourcequotcommandsRquot For Windows Source is also available on the File menu The function sink gt sinkquotrecordlisquot will divert all subsequent output from the console to an external le recordlis The com mand gt sinkO restores it to the console once again 111 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 gt ob j acts 0 alternatively ls can be used to display the names of most of the objects which are currently stored within R The collection of objects currently stored is called the workspace To remove objects the function rm is available 2 not inside strings nor within the argument list of a function de nition 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 gt rmx y z ink junk temp foo bar All objects created during an R sessions can be stored permanently in a le 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 this7 the objects are written to a le called RData75 in the current directory7 and the command lines used in the session are saved to a le called Rhistory When R is started at later time from the same directory it reloads the workspace from this le 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 analysis7 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 le name makes it invisible in normal le listings in UNIX Chapter 2 Simple manipulations numbers and vectors 7 2 Simple manipulations numbers and vectors 21 Vectors and assignment R operates on named data structures The simplest such structure is the numeric uector which is a single entity consisting of an ordered collection of numbers To set up a vector named x say consisting of ve numbers namely 104 56 31 64 and 217 use the R command gt x lt c104 56 31 64 217 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 end1 A number occurring by itself in an expression is taken as a vector of length one Notice that the assignment operator lt which consists of the two characters lt less than and minus occurring strictly sideby side and it points to the object receiving the value of the expression In most contexts the operator can be used as a alternative Assignment can also be made using the function assign An equivalent way of making the same assignment as above is with gt assignquotxquot c104 56 31 64 217 The usual operator lt 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 gt c104 56 31 64 217 gt x If an expression is used as a complete command the value is printed and lost2 So now if we were to use the command gt 1 the reciprocals of the ve values would be printed at the terminal and the value of x of course unchanged The further assignment gt y lt Cx 0 X would create a vector y with 11 entries consisting of two copies of x with a zero in the middle place 22 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 gt v lt 2x y 1 generates a new vector v of length 11 constructed by adding together element by element 2 repeated 22 times y repeated just once and 1 repeated 11 times The elementary arithmetic operators are the usual and quot 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 CO is rather different See Section 621 Concatenating lists page 27 2 Actually it is still available as Lastvalue 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 6 minx maxx lengthx is the number of elements in x sumx gives the total of the elements in x and prodx their product Two statistical functions are meanx which calculates the sample mean which is the same as sumxlengthx and varx which gives sum xmeanx quot2 lengthx 1 or sample variance If the argument to varO 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 exible sorting facilities available see order or sortlist 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 lnternally 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 170i will do the computations as complex numbers 23 Generating regular sequences R has a number of facilities for generating commonly used sequences of numbers For example 130 is the vector Cl 2 29 30 The colon operator has high priority within an ex pression so for example 21 15 is the vector C2 4 28 30 Put n lt 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 ve arguments only some of which may be speci ed in any one call The rst 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 seq2 10 is the same vector as 2 10 Parameters to ser 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 rst two parameters may be named fromva1ue and toVa1ue thus seq130 seqfrom1 to30 and seqto30 from1 are all the same as 130 The next two parameters to seq may be named byva1ue and lengthva1ue which specify a step size and a length for the sequence respectively If neither of these is given the default by1 is assumed For example gt seq5 5 by2 gt SS generates in 3 the vector c 50 48 46 46 48 50 Similarly gt 54 lt seqlength51 from5 by2 generates the same vector in s4 Chapter 2 Simple manipulations numbers and vectors 9 The fth parameter may be named alongvector which if used must be the only parameter and creates a sequence 1 2 lengthvector or the empty sequence if the vector is empty as it can be A related function is repO which can be used for replicating an object in various complicated ways The simplest form is gt 55 lt repx times5 which will put ve copies of x end to end in 55 Another useful version is gt 56 lt repx each5 which repeats each element of x ve times before moving on to the next 24 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 rst 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 gt temp lt x gt 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 lt lt gt gt for exact equality and for inequality In addition if c1 and c2 are logical expressions then c1 amp c2 is their intersection and cl 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 O 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 25 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 speci cation of an operation is incomplete the result cannot be known and hence is not available The function isnax gives a logical vector of the same size as x with value TRUE if and only if the corresponding element in x is NA gt z lt c13NA ind lt isnaz Notice that the logical expression x NA is quite different from isnax 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 gt 00 Chapter 2 Simple manipulations numbers and vectors 10 gt Inf Inf which both give NaN since the result cannot be de ned sensibly In summary isnaxx is TRUE both for NA and NaN values To differentiate these is nanxx is only TRUE for NaNs Missing values are sometimes printed as ltNAgt when character vectors are printed without quotes 26 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 eg quotxvaluesquot quotNew iteration resultsquot Character strings are entered using either double quot 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 quot is entered as quot Other useful escape sequences are n newline t tab and b backspace 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 sepstring which changes it to string possibly empty For example gt labs lt pastecquotXquot quotYquot 110 sepquotquot makes labs into the character vector CquotXlquot quotY2quot quotX3quot quotY4quot quotX5quot quotY6quot quotX7quot quotY8quot quotX9quot quotY10quot Note particularly that recycling of short lists takes place here too thus CquotXquot quotYquot is repeated 5 times to match the sequence 1103 27 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 gt y lt xisnax creates or recreates 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 gt x1 isnax amp xgt0 gt 2 creates an object 2 and places in it the values of the vector x1 for which the corresponding value in x was both non missing and positive t collapsess joins the arguments into a single Character string putting ss in between There are pas s 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 lengthx 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 gt x 1 10 selects the rst 10 elements of x assuming lengthx is not less than 10 Also gt cquotxquotquotyquot repc1221 times4 an admittedly unlikely thing to do produces a character vector of length 16 consisting of quotxquot quotyquot quotyquot quotxquot repeated four times OJ A vector of negative integral quantities Such an index vector speci es the values to be excluded rather than included Thus gt y lt x 15 gives y all but the rst ve elements of x q 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 gt fruit lt 65 10 1 20 gt namesfruit lt Cquotorangequot quotbananaquot quotapplequot quotpeachquot gt lunch lt fruit Cquotapplequot quotorangequot 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 indexvect0r 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 gt xisnax lt 0 replaces any missing values in x by zeros and gt yy lt O lt yy lt O has the same effect as gt y lt absy 28 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 0 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 0 factors provide compact ways to handle categorical data See Chapter 4 Factors page 16 0 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 61 Lists page 26 Chapter 2 Simple manipulations numbers and vectors 12 0 data frames are matrix like structures in which the columns can be of different types Think of data frames as data matrices7 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 63 Data frames page 27 0 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 31 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 numericl 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 character0 and the empty numeric vector as numeri c O 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 modeobject and lengthobject can be used to nd out the mode and length of any de ned structure2 Further properties of an object are usually provided by attributesobject see Section 33 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 modez is the character string complex and lengthz 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 gt z lt 09 we could put gt digits lt ascharacterz after which digits is the character vector cquot0quot quot1quot quot2quot quot9quot A further coercion or change of mode reconstructs the numerical vector again gt d lt asintegerdigits Now d and z are the same3 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 les 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 lengthobject does not always contain intrinsic useful information eg when object is a function 3 In general coercion from numeric to character and back again will not be exactly reversible because of roundoif errors in the character representation Chapter 3 Objects their modes and attributes 14 32 Changing the length of an object An empty object may still have a mode For example gt e lt 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 gt e 3 lt 17 now makes e a vector of length 3 the rst two components of which are at this point both NA This applies to any structure at all provided the mode of the additional components agrees with the mode of the object in the rst place This automatic adjustment of lengths of an object is used often for example in the scan function for input see Section 72 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 gt alpha lt 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 rst three values by gt lengthalpha lt 3 and vectors can be extended by missing values in the same way 33 Getting and setting attributes The function attributesobject returns a list of all the non intrinsic attributes currently de ned for that object The function attrobject name can be used to select a speci c 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 gt attrz quotdim lt c 10 10 allows R to treat 2 as if it were a 10 by 10 matrix 34 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 quotlistquot but matrix array factor and dataframe 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 O For example if winter has the class data frame then 4 A diHerent style using formal or S4 classes is provided in package methods Chapter 3 Objects their modes and attributes 15 gt winter will print it in data frame form which is rather like a matrix whereas gt unclasswinter 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 109 Object orientation page 48 but only brie y Chapter 4 Ordered and unordered factors 16 4 Ordered and unordered factors A factor is a vector object used to specify a discrete classi cation 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 1111 Contrasts page 52 we here look at a speci c example 41 A speci c 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 speci ed by a character vector of state mnemonics as gt state lt Cntasn quotsaw nqldn vvnswn vvnswn nntn quotwan quotwan nqldn quotVicquot nnsw quotVicquot nqldn nqldn quotsaw quottasquot 5 5 5 5 5 5 5 5 saw nntn quotwan quotVicquot nqldn vvnswn vvnswn quotwan saw quotactquot vvnswn quotVicquot quotVicquot nactn 5 Notice that in the case of a character vector sorted means sorted in alphabetical order A factor is similarly created using the factor function gt statef lt factorstate The print function handles factors slightly differently from other objects gt 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 nd out the levels of a factor the function levels can be used gt levelsstatef quotactquot quot11st quotmtquot qudn quotsaw quottasquot quotVicquot quotwan 42 The function tapplyO 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 gt incomes lt c60 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 tapplyO gt incmeans lt tapplyincomes statef mean giving a means vector with the components labelled by the levels act nsw nt 1d 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 rst argument here incomes de ned 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 tapplyO also works in this case when its second argument is not a factor eg tapplyincomes state 7 and this is true for quite a few other functions since arguments are coerced to factors when necessary using asfactor 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 varO to calculate the sample variance such a function is a very simple one liner speci ed by the assignment gt stderr lt functionx sqrt var x lengthx 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 gt incster lt tapplyincomes statef stderr and the values calculated are then gt incster act nsw nt 1d sa tas Vic wa 15 43102 45 41061 27386 05 5244 26575 As an exercise you may care to nd the usual 95 con dence limits for the state mean incomes To do this you could use tapply once more with the length function to nd the sample sizes and the qt function to nd the percentage points of the appropriate 25 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 rugged army since the subclass sizes are possibly irregular When the subclass sizes are all the same the indexing may be done implicitly and much more ef ciently as we see in the next section 43 Ordered factors The levels of factors are stored in alphabetical order or in the order they were speci ed to factor if they were speci ed explicitly Sometimes the levels will have a natural ordering that we want to record and want our statistical analysis to make use of The orderedO 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 tting linear models are different Chapter 5 Arrays and matrices 18 5 Arrays and matrices 51 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 eg 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 2 is a vector of 1500 elements The assignment gt dimz lt c35100 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 54 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 order77 with the rst subscript moving fastest and the last subscript slowest For example if the dimension vector for an array say a is 6 342 then there are 3 x 4 x 2 24 entries in a and the data vector holds them in the order a1 1 1 a21 1 a242 a342 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 52 Array indexing Subsections of an array lndividual 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 speci ed 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 a2 is a 4 x 2 array with dimension vector 6 42 and data vector containing the values ca211 a221 a231 a241 a212 a222 a232 a242 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 dimZ 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 53 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 0 Extract elements X 1 3 X 2 2 and X 3 1 as a vector structure and 0 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 gt x lt array120 dimc 45 Generate a 4 by 5 array gt 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 gt i lt arrayc1331 dimc32 gt i i is a 3 by 2 index array 1 2 1 1 3 2 2 2 3 3 1 gt xi Extract those elements 1 9 6 3 gt xi lt O Replace those elements by zeros gt x 1 2 3 4 5 1 1 5 o 13 17 2 2 o 10 14 18 3 o 7 11 15 19 4 4 8 12 16 20 gt 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 de ned by factors blocks b levels and varieties v levels Further suppose there are n plots in the experiment We could proceed as follows gt Xb lt matrix0 n b Xv lt matrix0 n v ib lt cbind1n blocks iv lt cbind1n varieties Xb ib lt 1 Xv iv lt 1 gt X lt cbindXb Xv VVVV V To construct the incidence matrix N say we could use gt N lt crossprodXb Xv Chapter 5 Arrays and matrices 20 However a simpler direct way of producing this matrix is to use table gt N lt tableblocks varieties Index matrices must be numerical any other form of matrix eg a logical or character matrix supplied as a matrix is treated as an indexing vector 54 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 gt Z lt arraydatavector dimvect0r For example if the vector h contains 24 or fewer numbers then the command gt Z lt arrayh dimc342 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 gt dimZ lt c342 However if h is shorter than 24 its values are recycled from the beginning again to make it up to size 24 see Section 541 The recycling rule page 20 As an extreme but common example gt Z lt array0 c342 makes Z an array of all zeros At this point dimZ stands for the dimension vector 6 342 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 gtDlt 2ABC1 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 541 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 nd in the references From experience we have found the following to be a reliable guide 0 The expression is scanned from left to right 0 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 55 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 700 gt ab lt a 700 b An alternative is gt ab lt outera b quotquot The multiplication function can be replaced by an arbitrary function of two variables For example if we wished to evaluate the function m y cosy1 2 over a regular grid of values with z and y coordinates de ned by the R vectors x and y respectively we could proceed as follows gt f lt functionx y cosy1 xquot2 gt z lt outerx 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 De ning your own R functions will be considered further in Chapter 10 Writing your own functions page 42 An example Determinants of 2 by 2 singledigit matrices As an arti cial but cute example consider the determinants of 2 by 2 matrices ab cd where each entry is a non negative integer in the range 0 1 9 that is a digit The problem is to nd the determinants ad 7 be of all possible matrices of this form and represent the frequency with which each value occurs as a high density plot This amounts to nding 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 gt d lt outer09 09 gt fr lt tableouterd d quotquot gt plot as numeric names fr fr typequothquot xlabquotDeterminantquot ylabquotFrequencyquot 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 inef cient as to be impractical It is also perhaps surprising that about 1 in 20 such matrices is singular 56 Generalized transpose of an array The function aperma 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 gt B lt apermA c21 is just the transpose of A For this special case a simpler function 00 is available so we could have used B lt tA Chapter 5 Arrays and matrices 22 57 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 tX is the matrix transpose function as noted above The functions nrowA and ncolA give the number of rows and columns in the matrix A respectively 571 Matrix multiplication The operator o o is used for matrix multiplication An 71 by 1 or 1 by 71 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 gt A B is the matrix of element by element products and gt A 796 B is the matrix product If x is a vector then gt x A x is a quadratic form1 The function crossprodO forms crossproducts meaning that crossprodX y is the same as tX o o y but the operation is more e icient If the second argument to crossprodO is omitted it is taken to be the same as the rst The meaning of diagO depends on its argument diagv where v is a vector gives a diagonal matrix with elements of the vector as the diagonal entries On the other hand di agM where M is a matrix gives the vector of main diagonal entries of M This is the same convention as that used for diagO in MATLAB Also somewhat confusingly if k is a single numeric value then diag k is the k by k identity matrix 572 Linear equations and inversion Solving linear equations is the inverse of matrix multiplication When after gt b lt A x only A and b are given the vector x is the solution of that linear equation system In R gt solveAb solves the system returning x up to some accuracy loss Note that in linear algebra formally x A lb where A 1 denotes the inverse of A which can be computed by solve A but rarely is needed Numerically it is both ine icient and potentially unstable to compute x lt solve A o o b instead of solveAb The quadratic form x A lx which is used in multivariate computations should be computed by something like x o o solve Ax rather than computing the inverse of A 1 Note that x 3939 x is ambiguous as it could mean either xx 0r xx where x is the column form In such cases the smaller matrix seems implicitly to be the interpretation adopted so the scalar xx is in this case the result The matrix xx may be calculated either by cbindx 3939 x or x 3939 rbindx since the result of rbindo 0r cbindo is always a matrix However the best way to compute xx 0r xx is crossprodx or x 39o39 K respectively 0 EVen better would be to form a matrix square root B with A BB and nd the squared length of the solution of By x perhaps using the Cholesky 0r eigendecomposition of A Chapter 5 Arrays and matrices 23 573 Eigenvalues and eigenvectors The function eigenSm t the and s of a symmetric matrix Sm The result of this function is a list of two components named values and vectors The assignment gt ev lt eigenSm will assign this list to ev Then evval is the vector of eigenvalues of Sm and evvec is the matrix of corresponding eigenvectors Had we only needed the eigenvalues we could have used the assignment gt evals lt eigenSmvalues evals now holds the vector of eigenvalues and the second component is discarded If the expression gt eigenSm 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 gt evals lt eigenSm onlyvalues TRUEvalues 574 Singular value decomposition and determinants The function svdM 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 tV D is actually returned as a vector of the diagonal elements The result of svdM 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 gt absdetM lt prodsvdMd calculates the absolute value of the determinant of M If this calculation were needed often with a variety of matrices it could be de ned as an R function gt absdet lt functionM prodsvdMd 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 diagO 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 575 Least squares tting and the QR decomposition The function lsfit returns a list giving results of a least squares tting procedure An assignment such as gt ans lt lsfitX y gives the results of a least squares t 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 lsdiag 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 112 Linear models page 53 to lsfit for regression modelling Another closely related function is qu and its allies Consider the following assignments Chapter 5 Arrays and matrices 24 gt Xplus lt qrX gt b lt qrcoef Xplus y gt fit lt qrfittedXplus y gt res lt qrresidXplus 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 58 Forming partitioned matrices CbindO and rbind As we have already seen informally matrices can be built up from other vectors and matrices by the functions cbindO and rbind Roughly cbindO forms matrices by binding together matrices horizontally or column wise and rbind vertically or row wise In the assignment gt X lt cbindarg1 arg2 arg3 the arguments to cbindO 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 argJ arg2 forming the columns If some of the arguments to cbindO 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 is we can use gt X lt cbind1 X1 X2 The result of rbind or cbindO always has matrix status Hence cbindx and rbindx are possibly the simplest ways explicitly to allow the vector x to be treated as a column or row matrix respectively 59 The concatenation function 60 with arrays It should be noted that whereas cbindO and rbind are concatenation functions that respect dim attributes the basic 6 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 asvector gt vec lt asvectorX However a similar result can be achieved by using c with just one argument simply for this side effect gt vec lt CX 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 25 510 Frequency tables from factors Recall that a factor de nes a partition into groups Similarly a pair of factors de nes a two way cross classi cation 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 gt statefr lt tablestatef 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 gt statefr lt tapplystatef statef length Further suppose that incomef is a factor giving a suitably de ned income class77 for each entry in the data vector for example with the cut function gt factorcutincomes breaks 351007 gt incomef Then to calculate a two way table of frequencies gt tableincomef 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 61 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 gt Lst lt list namequotFredquot wifequotMaryquot nochildren3 child agesc 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 Lst1 Lst 2 L51 3 and L51 4 lf further L51 4 is a vector subscripted array then Lst4 1 is its rst entry If Lst is a list then the function lengthLst 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 gt namecomponentname 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 Lstname is the same as L51 1 and is the string quotFredquot Lstwife is the same as L51 2 and is the string quotMaryquot Lstchildages 1 is the same as Lst4 1 and is the number 4 Additionally one can also use the names of the list components in double square brackets ie Lst quotnamequot is the same as Lstname This is especially useful when the name of the component to be extracted is stored in another variable as in gt x lt quotnamequot Lstx It is very important to distinguish L51 1 from L51 1 is the operator used to select a single element whereas is a general subscripting operator Thus the former is the rst 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 rst 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 Lstcoefficients may be minimally speci ed as Lstcoe and Lstcovariance as Lstcov 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 62 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 gt Lst lt listname1object1 namemobjectm sets up a list Lst of m components using objectJ objectm for the components and giving them names as speci ed 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 gt Lst 5 lt listmatrixMat 621 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 gt listABC lt ClistA listB listC 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 63 Data frames A data frame is a list with class quotdata frame There are restrictions on lists that may be made into data frames namely 0 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 63 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 gt accountants lt dataframe homestatef lootincomes shotincomef A list whose components conform to the restrictions of a data frame may be coerced into a data frame using the function asdata frame The simplest way to construct a data frame from scratch is to use the readtable function to read an entire data frame from an external le This is discussed further in Chapter 7 Reading data from les page 30 632 attach and detach The notation such as accountantsstatef 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 attachO function takes a database such as a list or data frame as its argument Thus suppose lentils is a data frame with three variables lentilsu lentilsv lentilsw The attach gt attachlentils places the data frame in the search path at position 2 and provided there are no variables 11 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 gt u lt vw does not replace the component 11 of the data frame but rather masks it with another variable 11 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 gt lentilsu lt vw However the new value of component 11 is not visible until the data frame is detached and attached again To detach a data frame use the function gt detachO More precisely this statement detaches from the search path the entity currently at position 2 Thus in the present context the variables 11 v and w would be no longer visible except under the list notation as lentilsu 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 detachlentils or detachquotlentilsquot 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 633 Working With data frames A useful convention that allows you to work with many different problems comfortably together in the same working directory is o gather together all variables for any well de ned and separate problem in a data frame under a suitably informative name 0 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 detachO nally 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 634 Attaching arbitrary lists attachO 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 quotlistquot may be attached in the same way gt attachany 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 29 635 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 gt search 1 quot GlobalEnvquot quotAutoloadsquot quotpackagezbasequot where GlobalEnv is the workspace1 After lentils is attached we have gt search 1 quot GlobalEnvquot lentils quotAutoloadsquot quotpackage basequot gt ls2 quotuquot quotVquot quotwquot and as we see ls or obj ects can be used to examine the contents of any position on the search path Finally7 we detach the data frame and con rm it has been removed from the search path gt detachquotlentilsquot gt search 1 quot GlobalEnvquot quotAutoloadsquot quotpackagezbasequot 1 See the online help for autoload for the meaning of the second term Chapter 7 Reading data from les 30 7 Reading data from les Large data objects will usually be read as values from external les rather than entered during an R session at the keyboard R input facilities are simple and their requirements are fairly strict and even rather in exible There is a clear presumption by the designers of R that you will be able to modify your input les using other tools such as le editors or Perll to t 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 readtable 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 B Data Im portExport manual 71 The readtab1e function To read an entire data frame directly the external le will normally have a special form 0 The rst line of the le should have a name for each variable in the data frame 0 Each additional line of the le has as its rst item a row label and the values for each variable If the le has one fewer item in its rst line than in its second this arrangement is presumed to be in force So the rst few lines of a le to be read as a data frame might look as follows lnput le form with names and row labels Price Floor Area Rooms Age Centheat 01 5200 1110 830 5 62 no 02 5475 1280 710 5 75 no 03 5750 1010 1000 5 42 no 04 5750 1310 690 6 88 no 05 5975 930 900 5 19 yes By default numeric items except row labels are read as numeric variables and non numeric variables such as Centheat in the example as factors This can be changed if necessary The function readtable can then be used to read the data frame directly gt HousePrice lt read table quothouses dataquot Often you will want to omit including the row labels directly and use the default labels In this case the le may omit the row label column as in the following lnput le form without row labels Price Floor Area Rooms Age Cent heat 5200 1110 830 5 62 no 5475 1280 710 5 75 no 5750 1010 1000 5 42 no 5750 1310 690 6 88 no 5 1 9 yes 59 75 93 0 900 1 Under UNIX the utilities Sed or Awk can be used Chapter 7 Reading data from les 31 The data frame may then be read as gt HousePrice lt readtablequothouses dataquot headerTRUE where the headerTRUE option speci es that the rst line is a line of headings and hence by implication from the form of the le that no explicit row labels are given 72 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 rst of mode character and the remaining two of mode numeric and the le is inputdat The rst step is to use scan to read in the three vectors as a list as follows gt inp lt scanquotinput datquot listquot quot 0 O 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 gt label lt inp1 x lt inp2 y lt inp3ll 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 gt inp lt scanquotinput datquot listidquot quot x0 y0 If you wish to access the variables separately they may either be re assigned to variables in the working frame gt label lt inpid x lt inpx y lt inpy or the list may be attached at position 2 of the search path see Section 634 Attaching arbitrary lists page 28 lfthe 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 gt X lt matrixscanquotlightdatquot O ncol5 byrowTRUE There are more elaborate input facilities available and these are detailed in the manuals 73 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 200 all the datasets suppied 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 datainfert 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 731 Loading data from other R packages To access data from a particular package use the package argument for example datapackagequotrpartquot dataPuromycin packagequotdatasetsquot 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 les 32 74 Editing data When invoked on a data frame or matrix7 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 gt xnew lt editxold will allow you to edit your data set xold7 and on completion the changed object is assigned to xnew If you want to alter the original dataset xold7 the simplest way is to use fixxold7 which is equivalent to xold lt editxold Use gt xnew lt edit data frame to enter new data via the spreadsheet interface Chapter 8 Probability distributions 33 8 Probability distributions 81 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 PX x the probability density function and the quantile function given q the smallest x such that PX z gt q and to simulate from the distribution Distribution R name additional arguments beta beta shapel shape2 ncp binomial binom size prob Cauchy cauchy location scale chi squared chisq df ncp exponential exp rate f df 1 df 2 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 poi s lambda Student s t t df ncp uniform unif min max Weibull weibull shape scale Wilcoxon wilcox m n Pre x the name given here by d for the density p for the CDF q for the quantile function and r for simulation random deviates The rst argument is x for dxxx q for pxxx p for qxxx and n for rxxx except for rhyper and rwilcox for which it is ln 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 lowertail and logp and the dxxx ones have log This allows eg getting the cumulative or integrated hazard function 11175 10g1 Ft7 by pXXXt lowertail FALSE logp 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 gt 2 tailed p value for t distribution gt 2pt243 df 13 gt upper 1 point for an F2 7 distribution gt qf O 01 2 7 lower tail FALSE 82 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 leaf77 plot Chapter 8 Probability distributions 34 gt attachfaithful gt summaryeruptions Min lst Qu Median Mean 3rd Qu Max 1600 2163 4000 3488 4454 5100 gt fivenumeruptions 1 16000 21585 40000 44585 51000 gt stemeruptions The decimal point is 1 digits to the left of the I 16 I 070355555588 18 lllI l888 20 I 00002223378800035778 22 I 0002335578023578 24 I 00228 26 I 23 28 I 080 30 I 7 32 I 2337 34 I 250077 36 I 0000823577 38 I 2333335582225577 40 I 0000003357788888002233555577778 42 I 03335555778800233333555577778 44 I l8888 46 I 0000233357700000023578 48 I 00000022335800333 50 I 0370 A stem and leaf plot is like a histogram7 and R has a function hist to plot histograms gt histeruptions make the bins smaller7 make a plot of density gt hist eruptions seq1 6 5 2 O 2 probTRUE gt linesdensityeruptions bw0 1 gt rugeruptions 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 available7 and in this example bw quotSJquot gives a good result Histogram of eruptions Relawe Frequency m n l iii 539 HlllllllllllllHllllllllllllllllll ll lll l llllllll W 15 2D 25 an 35 An 45 an empuuns We can plot the empirical cumulative distribution function by using the function ecdf gt plot ecdf eruptions do pointsFALSE verticalsTRUE This distribution is obviously far from any standard distribution How about the right hand mode7 say eruptions of longer than 3 minutes Let us t a normal distribution and overlay the tted CDF gt long lt eruptions eruptions gt 3 gt plot ecdf long do pointsFALSE verticalsTRUE gt x lt seq3 54 001 gt linesx pnormx meanmeanlong sdsqrtvarlong lty3 ec dflo ng Quantile quantile Q Q plots can help us examine this more carefully parptyquotsquot arrange for a square figure region qqnormlong qqlinelong Chapter 8 Probability distributions 36 which shows a reasonable t but a shorter right tail than one would expect from a normal distribution Let us compare this with some simulated data from a 25 distribution Normal Q Q Plot Sample Ouanhles Theuveucal Ouarmles x lt rt250 df 5 qqnorm x qqlinex 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 ppoints250 df 5 x xlab quotQQ plot for t dsnquot qqline x Finally7 we might want a more formal test of agreement with normality or not R provides the Shapiro Wilk test gt shapiro test long ShapiroWilk normality test data long W 09793 pvalue 001052 and the Kolmogorov Smirnov test gt ks testlong quotpnormquot mean meanlong sd sqrtvarlong Onesample KolmogorovSmirnov test data long D 00661 pvalue 04284 alternative hypothesis twosided Note that the distribution theory is not valid here as we have estimated the parameters of the normal distribution from the same sample 83 One and twosample 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 R7 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 calgm from Rice 19957 p490 Chapter 8 Probability distributions 37 Method A 7998 8004 8002 8004 8003 8003 8004 7997 8005 8003 8002 8000 8002 Method B 8002 7994 7998 7997 7997 8003 7995 7997 Boxplots provide a simple graphical comparison of the two samples A lt scan 7998 8004 8002 8004 8003 8003 8004 7997 8005 8003 8002 8000 8002 B lt scan 8002 7994 7998 7997 7997 8003 7995 7997 boxplot A B which indicates that the rst group tends to give higher results than the second an 02 an m l l an an 79 99 l a 79 99 l 79 94 l To test for the equality of the means of the two examples7 we can use an unpaired t test by gt ttestA B Welch Two Sample ttest data A and B t 32499 df 12027 pvalue 000694 alternative hypothesis true difference in means is not equal to 0 95 percent confidence interval 001385526 007018320 sample estimates mean of x mean of y 8002077 7997875 which does indicate a signi cant difference7 assuming normality By default the R function does not assume equality of variances in the two samples in contrast to the similar S PLUS ttest function We can use the F test to test for equality in the variances7 provided that the two samples are from normal populations gt vartestA B F test to compare two variances Chapter 8 Probability distributions 38 data A and B F 05837 num df 12 denom df 7 pvalue 03938 alternative hypothesis true ratio of variances is not equal to 1 95 percent confidence interval 01251097 21052687 sample estimates ratio of variances 05837405 which shows no evidence of a signi cant difference7 and so we can use the classical t test that assumes equality of the variances gt ttestA B varequalTRUE Two Sample ttest data A and B t 34722 df 19 pvalue 0002551 alternative hypothesis true difference in means is not equal to 0 95 percent confidence interval 001669058 006734788 sample estimates mean of x mean of y 8002077 7997875 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 gt wilcoxtestA B Wilcoxon rank sum test with continuity correction data A and B W 89 pvalue 0007497 alternative hypothesis true location shift is not equal to 0 Warning message Cannot compute exact pvalue with ties in wilcoxtestA B Note the warning there are several ties in each sample7 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 gt plot ecdf A do pointsFALSE verticalsTRUE xlimrange A B gt plot ecdf B do pointsFALSE verticalsTRUE addTRUE will show the two empirical CDFs7 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 s7 assuming a common continuous distribution gt kstestA B Twosample KolmogorovSmirnov test data A and B D 05962 p value 005919 Chapter 8 Probability distributions alternative hypothesis twosided Warning message cannot compute correct pvalues with ties in kstestA B Chapter 9 Grouping loops and conditional execution 40 9 Grouping loops and conditional execution 91 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 expr1 exprm 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 92 Control statements 921 Conditional execution if statements The language has available a conditional construction of the form gt if expr1 expr2 else expr3 where exprJ must evaluate to a single logical value and the result of the entire expression is then evident The short circuit operators ampamp and I are often used as part of the condition in an if statement Whereas amp and apply element wise to vectors ampamp and apply to vectors of length one and only evaluate their second argument if necessary There is a vectorized version of the ifelse construct the ifelse function This has the form ifelsecondition a b and returns a vector of the length of its longest argument with elements ai if condition i is true otherwise bi 922 Repetitive execution for loops repeat and while There is also a for loop construction which has the form gt for name in expr1 expr2 where name is the loop variable expril is a vector expression often a sequence like 1 20 and expr2 is often a grouped expression with its sub expressions written in terms of the dummy name expri2 is repeatedly evaluated as name ranges through the values in the vector result of expril 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 coplot1 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 gt xc lt splitx ind gt yc lt splity ind gt for i in 1lengthyc plotxc il yC ill 5 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 speci ed 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 Grouping7 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 object7 View is likely to be both clearer and faster in R Other looping facilities include the gt repeat expr statement and the gt while condition expr statement The break statement can be used to terminate any loop7 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 nex 77 Control statements are most often used in connection with functions which are discussed in Chapter 10 Writing your own functions7 page 427 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 varO postscriptO and so on are themselves written in R and thus do not differ materially from user written functions A function is de ned by an assignment of the form gt name lt functionarg1 arg2 expression The expression is an R expression usually a grouped expression that uses the arguments argj 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 expr1 expr2 and may occur anywhere a function call is legitimate 101 Simple examples As a rst example consider a function to calculate the two sample t statistic showing all the steps This is an arti cial example of course since there are other simpler ways of achieving the same end The function is de ned as follows gt twosam lt functiony1 y n1 lt lengthy1 n2 lt lengthy2 ybl lt meany1 yb2 lt meany2 51 lt vary1 52 lt vary2 s lt n11sl n2152n1n22 tst lt ybl yb2sqrts1n1 1n2 tst With this function de ned you could perform two sample t tests using a call such as gt tstat lt twosamdatamale datafemale tstat As a second example consider a function to emulate directly the MATLAB backslash com mand which returns the coef cients 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 coef cients This would ordinarily be done with the qu 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 de ned as X X X y where X X is a generalized inverse of X X gt bslash lt functionX y f X lt qrX qr coef X y After this object is created it may be used in statements such as gt regcoeff lt bslashXmat yvar and so on Chapter 10 Writing your own functions 43 The classical R function lsfitO does this job quite well and more It in turn uses the functions qu 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 102 De ning new binary operators Had we given the bslash function a different name namely one of the form ooanyt11i11g o 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 de nition would then start as gt quotVOWoquot lt functionX y Note the use of quote marks The function could then be used as X AVo 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 700 are other examples of binary operators de ned in this way 103 Named arguments and defaults As rst noted in Section 23 Generating regular sequences page 8 if arguments to called functions are given in the nameobject 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 funl de ned by gt funl lt functiondata dataframe graph limit function body omitted then the function may be invoked in several ways for example gt ans lt funl d df TRUE 20 gt ans lt funl d df graphTRUE limit20 gt ans lt fun1datad limit20 graphTRUE dataframedf 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 funl were de ned as gt funl lt functiondata dataframe graphTRUE limit20 it could be called as gt ans lt fun1d df which is now equivalent to the three cases above or as gt ans lt funl d df limit10 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 44 104 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 1241 The par function page 67 for more details on the par function This can be done by including an extra argument literally 7 of the function which may then be passed on An outline example is given below funl lt functiondata dataframe graphTRUE limit20 omitted statements if graph parpchquotquot more omissions 105 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 lt qrX 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 dif cult topic and is not covered further here If global and permanent assignments are intended within a function then either the su perassignment77 operator ltlt or the function assign can be used See the help document for details S PLUS users should be aware that ltlt has different semantics in R These are discussed further in Section 107 Scope page 46 106 More advanced examples 1061 Ef ciency factors in block designs As a more complete if a little pedestrian example of a function consider nding the ef ciency factors for a block design Some aspects of this problem have already been discussed in Sec tion 53 Index matrices page 19 A block design is de ned 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 12 incidence matrix then the ef ciency factors are de ned as the eigenvalues of the matrix E I 7 R lQN K lNR lQ I 7 AA where A K 12NR 12 One way to write the function is given below gt bdeff lt functionblocks varieties blocks lt asfactorblocks minor safety move b lt lengthlevelsblocks varieties lt asfactorvarieties minor safety move v lt lengthlevelsvarieties K lt asvectortableblocks remove dim attr R lt asvectortablevarieties remove dim attr N lt tableblocks varieties A lt 1sqrtK N rep1sqrtR repb v sv lt svdA listeff1 svdquot2 blockcvsvu varietycvsvv 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 ef ciency factors as the rst component but also the block and variety canonical contrasts since sometimes these give additional useful qualitative information 1062 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 gt temp lt X gt dimnamestemp lt listrepquotquot nrowX repquot quot ncolX gt temp rmtemp This can be much more conveniently done using a function nodimnames shown below as a wrap around77 to achieve the same result It also illustrates how some effective and useful user functions can be quite short nodimnames lt functiona Remove all dimension names from an array for compact printing d lt list 1 lt O fori in dima d1 lt 1 1 lt Iepquotquot i dimnamesa lt d a With this function de ned an array may be printed in close format using gt no dimnames X This is particularly useful for large integer arrays where patterns are the real interest rather than the values 1063 Recursive numerical integration Functions may be recursive and may themselves de ne 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 onepanel 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 dif cult to evaluate The example is also given partly as a little puzzle in R programming area lt functionf a b eps 1 Oe06 lim 10 funl lt functionf a b fa fb a0 eps lim fun function funl is only visible inside area d lt a b2 Chapter 10 Writing your own functions h lt b a4 fd lt fd a1 lt h fa fd 212 lt h fd fb ifabsa0 a1 a2 lt eps II lim 0 returna1 a2 else returnfunf a d fa fd a1 eps lim 1 fun funf d b fd fb a2 eps lim 1 fun fa lt fa fb lt fb 210 lt fa fb b a2 fun1f a b fa fb a0 eps lim funl 107 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 de nition f lt functionx y lt 2x printx printy printz 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 rst looking in the environment in which the function was created This is called lexical scope First we de ne a function called cube cube lt functionn sq lt functionO nn nsq The variable 11 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 11 Under lexical scope R it is the parameter to the function cube since that is the active binding for the variable 11 at the time the function sq was de ned The difference between evaluation in R and evaluation in S PLUS is that S PLUS looks for a global variable called 11 while R rst looks for a variable called 11 in the environment created when cube was invoked rst evaluation in S Sgt cube2 Error in sq Object quotnquot not found Chapter 10 Writing your own functions 47 Dumped Sgt n lt 3 Sgt cube2 1 18 then the same function evaluated in R Rgt cube2 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 total7 a function for making withdrawals7 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 de ned in an environment which contains total7 they will have access to its value The special assignment operator7 ltlt 7 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 nds such an environment it replaces the value7 in that environment7 with the value of right hand side If the global or toplevel environment is reached without nding the symbol total then that variable is created and assigned to there For most users ltlt creates a global variable and assigns the value of the right hand side to it2 Only when ltlt has been used in a function that was returned as the value of another function will the special behavior described here occur openaccount lt functiontotal list deposit functionamount ifamount lt 0 stopquotDeposits must be positivenquot total ltlt total amount catamount quotdeposited Your balance isquot total quotnnquot withdraw functionamount ifamount gt total stopquotYou don t have that much moneynquot total ltlt total amount catamount quotwithdrawn Your balance isquot total quotnnquot balance function catquotYour balance isquot total quotnnquot ross lt openaccount100 robert lt openaccount200 rosswithdraw30 rossbalance robertbalance 2 In some sense this mimics the behavior in SePLUS since in SePLUS this operator always creates or assigns to a global Variable Chapter 10 Writing your own functions 48 rossdeposit 50 rossbalance0 rosswithdraw500 108 Customizing the environment Users can customize their environment in several different ways There is a site initialization le and every directory can have its own special initialization le Finally the special functions First and Last can be used The location ofthe site initialization le is taken from the value of the RPROFILE environment variable If that variable is unset the le Rprofilesite in the R home subdirectory etc is used This le should contain the commands that you want to execute every time R is started under your system A second personal pro le le named Rprofile73 can be placed in any directory If R is invoked in that directory then that le will be sourced This le gives individual users control over their workspace and allows for different startup procedures in different working directories If no Rprofile7 le is found in the startup directory then R looks for a Rprofile7 le in the user s home directory and uses that if it exists Any function named First in either of the two pro le les or in the RData7 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 de nition 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 les are executed is Rprofilesite Rprofile RData7 and then First A de nition in later les will mask de nitions in earlier les gt First lt functionO optionspromptquot quot continuequottquot is the prompt optionsdigits5 length999 custom numbers and printout x110 for graphics parpch quotquot plotting character source file pathSys getenvquotHOMEquot quotRquot quotmystuff Rquot my personal functions libraryMASS attach a package Similarly a function Last if de ned is normally executed at the very end of the session An example is given below gt Last lt functionO graphics off a small safety measure cat pastedate quotnAdiosnquot Is it time for lunch 109 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 speci c to the class of the argument itself If the argument lacks any class attribute or has a class not catered for speci cally 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 speci c way can be quite large For example the functions that can accommodate in some fashion objects of class quotdata frame include lt any asmatrix lt mean plot summary A currently complete list can be got by using the methods function gt 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 dataframe density factor and more A complete list can be got again by using the methods function gt methods plot For many generic functions the function body is quite short for example gt 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 gt methods coef 1 coefaov coefArima coefdefault coeflistof 5 coefnls coefsummarynls Nonvisible 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 gt getAnywhere coef aov A single object matching coefaov was found It was found in the following places registered SS method for coef from namespace stats namespacezstats with value function object z lt objectcoef zisnaz gt getSBmethod coef aov function object z lt objectcoef z isnaz The reader is referred to the R Language De nition 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 tting statistical models are suf ciently well de ned 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 tting 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 111 De ning statistical models formulae The template for a statistical model is a linear regression model with independent homoscedastic errors 7 yiZ jmljei ei NlD002 i1n j0 ln matrix terms this would be written yX e where the y is the response vector X is the model matrix or design matrix and has columns m0m1mp the determining variables Very often mo will be a column of ones de ning an intercept term Examples Before giving a formal speci cation 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 quot x y quot 1 x Both imply the same simple linear regression model of y on m The rst has an implicit intercept term and the second an explicit one y quot O x y quot 1 x y quot x 1 Simple linear regression of y on x through the origin that is without an intercept term logy quot x1 x2 Multiple regression of the transformed variable logy on 1 and 2 with an implicit intercept term y quot polyx2 y 1xIxquot2 Polynomial regression of y on x of degree 2 The rst form uses orthogonal polyno mials and the second uses explicit powers as basis y quot X polyx2 Multiple regression y with model matrix consisting of the matrix X as well as polynomial terms in z to degree 2 Chapter 11 Statistical models in R 51 y quot A Single classi cation analysis of variance model of y with classes determined by A y quot A x Single classi cation analysis of covariance model of y with classes determined by A and with covariate z y AB y quot A B AzB y quot B oin o A y quot AB Two factor non additive model of y on A and B The rst two specify the same crossed classi cation and the second two specify the same nested classi cation In abstract terms all four specify the same model subspace y quot A B C quot2 y quot ABC ABC Three factor experiment but with a model containing main effects and two factor interactions only Both formulae specify the same model y quot 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 quot AB 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 de ne a model formula in R The form for an ordinary linear model is response quot op1 tem1 op2 tem2 op3 term3 where response is a vector or matrix or expression evaluating to a vector or matrix de ning the response variables opi is an operator either or implying the inclusion or exclusion of a term in the model the rst is optional termj is either 0 a vector or matrix expression or 1 o a factor or o a formula expression consisting of factors vectors or matrices connected by formula operators In all cases each term de nes 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 z since the period is a valid name character in R The notation is summarized below based on Chambers amp Hastie 1992 p29 Y quot M Y is modeled as M M1 M2 lnclude MJ and M2 Chapter 11 Statistical models in R 52 M1 M2 Include MJ leaving out terms of M2 M1 M2 The tensor product of M1 and M2 If both terms are factors then the subclasses factor M1 oin o M2 Similar to M1 M2 but with a different coding M1 M2 M1 M2 M1M2 M1 M2 M1 M2 oin o M1 Mquotn All terms in M together with interactions up to order n I M lnsulate M lnside 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 de ned using arithmetic operators Note particularly that the model formulae specify the columns of the model matrix the speci cation of the parameters being implicit This is not the case in other contexts for example in specifying nonlinear models 1111 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 7 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 rst For ordered factors the k 7 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 rst 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 optionscontrasts c quotcontr treatment quot quotcontr polyquot 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 optionscontrasts c quotcontr helmertquot quotcontr polyquot This is a deliberate difference as treatment contrasts R s default are thought easier for new comers to interpret We have still not nished 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 53 112 Linear models The basic function for tting ordinary multiple models is lm and a streamlined version of the call is as follows gt fittedmode1 lt lmformu1a data dataframe For example gt fm2 lt lmy quot x1 x2 data production would t a multiple regression model of y on 1 and 2 with implicit intercept term The important but technically optional parameter data production speci es that any variables needed to construct the model should come rst from the production data frame This is the case regardless of whether data frame production has been attached on the search path or not 113 Generic functions for extracting model information The value of lm is a tted model object technically a list of results of class quot1mquot Information about the tted model can then be displayed extracted plotted and so on by using generic functions that orient themselves to objects of class quot1mquot These include addl deviance formula predict step alias dropl kappa print summary anova effects labels proj vcov coef family plot residuals A brief description of the most commonly used ones is given below anovaobject1 object2 Compare a submodel with an outer model and produce an analysis of variance table coef object Extract the regression coef cient matrix Long form coefficientsobject devianceobject Residual sum of squares weighted if appropriate formulaobject Extract the model formula plotobject Produce four plots showing residuals tted values and some diagnostics predict object newdatadata frame The data frame supplied must have variables speci ed with the same labels as the original The value is a vector or matrix of predicted values corresponding to the determining variable values in dataframe printobject Print a concise version of the object Most often used implicitly residualsobject Extract the matrix of residuals weighted as appropriate Short form residobject stepobject Select a suitable model by adding or dropping terms and preserving hierarchies The model with the smallest value of A10 Akaike s An Information Criterion discovered in the stepwise search is returned Chapter 11 Statistical models in R 54 summaryobject Print a comprehensive summary of the results of the regression analysis vcovobject Returns the variancecovariance matrix of the main parameters of a tted model object 114 Analysis of variance and model comparison The model tting function aovformula datadataframe 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 113 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 quot mean formula Error strata formula speci es a multi stratum experiment with error strata de ned by the strataformula In the simplest case strataformula is simply a factor when it de nes 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 gt fm lt aovyield quot v npk Errorfarmsblocks datafarmdata would typically be used to describe an experiment with mean model v npk and three error strata namely between farms within farms between blocks77 and within blocks 1141 ANOVA tables Note also that the analysis of variance table or tables are for a sequence of tted 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 rst to project the response onto the error strata again in sequence and to t the mean model to each projection For further details see Chambers amp Hastie 1992 A more exible alternative to the default full ANOVA table is to compare two or more models directly using the anovaO function gt anovafittedmode1 1 fittedmode1 2 The display is then an ANOVA table showing the differences between the tted models when tted in sequence The tted 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 115 Updating tted models The update function is largely a convenience function that allows a model to be tted that differs from one previously tted usually by just a few additional or removed terms lts form is gt newmode1 lt updateoldmode1 newformu1a In the newformula the special name consisting of a period 3 only can be used to stand for the corresponding part of the old model formula For example gt meS lt lmy quot x1 x2 x3 x4 x5 data production gt fm6 lt updatefm05 quot x6 gt smf6 lt updatefm6 sqrt quot Chapter 11 Statistical models in R 55 would t a ve variate multiple regression with variables presumably from the data frame production t an additional model including a sixth regressor variable and t a variant on the model where the response had a square root transform applied Note especially that if the data argument is speci ed on the original call to the model tting function this information is passed on through the tted model object to update and its allies The name can also be used in other contexts but with slightly different meaning For example gt fmfull lt lmy quot data production would t 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 116 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 0 There is a response y of interest and stimulus variables m1 m2 whose values in uence the distribution of the response The stimulus variables in uence the distribution of y through a single linear function only This linear function is called the linear predictor and is usually written 7 511 22 p7 hence x has no in uence on the distribution of y if and only if B O The distribution of y is of the form A My 7 w exp MM v WW W w where 4p 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 a 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 0 The mean ILL is a smooth invertible function of the linear predictor and this inverse function to 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 uni ed 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 amp Nelder 1989 or Dobson 1990 Chapter 11 Statistical models in R 56 1161 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 speci ed In the latter case the variance function must be speci ed 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 inversegaussian 1muquot2 identity inverse log poisson identity log sqrt quasi logit probit cloglog identity inverse log 1muquot2 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 1162 The glmO 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 speci ed in a different way The R function to t a generalized linear model is glmO which uses the form gt fittedmode1 lt glmfomu1a familyfami1ygenerator datadataframe The only new feature is the familygenerator 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 de ne and control the model and estimation process Although this may seem a little complicated at rst sight its use is quite simple The names of the standard supplied family generators are given under Family Name77 in the table in Section 1161 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 speci ed in this way Some examples make the process clear The gaussian family A call such as gt fm lt glmy quot x1 x2 family gaussian data sales achieves the same result as gt fm lt lmy quot x1x2 datasales but much less ef ciently 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 arti cial example from Silvey 1970 Chapter 11 Statistical models in R 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 t 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 z and n the number tested both models have the form y N 13717 F o 5190 ltIgtz is the standard normal distribution function and in the 1 5 In both cases the LD50 is LD50 76061 that is the point at which the argument of the distribution function is zero The rst step is to set the data up as a data frame where for the probit case logit case the default 5 gt kalythos lt dataframex C20 35 45 5570 n rep50 5 y C617263744 To t a binomial model using glmO there are three possibilities for the response o If the response is a vector it is assumed to hold binary data and so must be a 01 vector o If the response is a two column matrix it is assumed that the rst column holds the number of successes for the trial and the second holds the number of failures o If the response is a factor its rst level is taken as failure 0 and all other levels as success Here we need the second of these conventions so we add a matrix to our data frame gt kalythosYmat lt cbindkalythosy kalythosn kalythosy To t the models we use gt fmp lt glmYmat quot x family binomiallinkprobit data kalythos gt fml lt glmYmat quot 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 t we could use gt summaryfmp gt summaryfml Both models t all too well To nd the LD50 estimate we can use a simple function gt ld50 lt functionb b1 b 2 gt ldp lt ld50coef fmp ldl lt ld50coeffml Cldp ldl The actual estimates from this data are 43663 years and 43601 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 t 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 squareroot transformation As a graceful alternative to the latter a Poisson generalized linear model may be tted as in the following example Chapter 11 Statistical models in R 58 gt fmod lt glmy quot A B x family poisson1inksqrt data wormcounts Quasilikelihood 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 Vary 1 For quasi likelihood estimation and inference the precise response distribution is not speci ed 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 tting gaussian models with non standard link functions or variance functions incidentally For example consider tting the non linear regression t9121 y2202e which may be written alternatively as 1 5 y B i 52 where ml 2221 2 7121 31 101 and 82 0201 Supposing a suitable data frame to be set up we could t this non linear regression as gt nlfit lt glmy quot x1 x2 1 family quasi1inkinverse varianceconstant data biochem The reader is referred to the manual and the help document for further information as needed 117 Nonlinear least squares and maximum likelihood models Certain forms of nonlinear model can be tted by Generalized Linear Models glmO But in the majority of cases we have to approach the nonlinear curve tting problem as one of nonlinear optimization R s nonlinear optimization routines are optim n1m and from R 220 n1minb which provide the functionality and more of S PLUS s ms and n1minb We seek the parameter values that minimize some index of lack of t 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 1171 Least squares One way to t 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 amp Watts 1988 page 51 The data are gt x lt C002 002 006 006 011 011 022 022 056 056 1 10 1 10 gt y lt C76 47 97 107 123 139 159 152 191 201 207 200 The model to be tted is Chapter 11 Statistical models in R 59 gt fn lt functionp sumy p1 xp2 xquot2 In order to do the t we need initial estimates of the parameters One way to nd sensible starting values is to plot the data guess some parameter values and superimpose the model curve using those values gt plot x y gt xfit lt seq02 11 05 gt yfit lt 200 xfit01 xfit gt linessplinexfit yfit We could do better but these starting values of 200 and 01 seem adequate Now do the t gt out lt nlmfn p C200 01 hessian TRUE After the tting outminimum is the SSE and outestimate are the least squares estimates of the parameters To obtain the approximate standard errors SE of the estimates we do gt sqrt diag2outminimumlengthy 2 solveouthessian The 2 in the line above represents the number of parameters A 95 con dence interval would be the parameter estimate i 196 SE We can superimpose the least squares t on a new plot gt plot x y gt xfit lt seq02 11 05 gt yfit lt 21268384222 xfit006412146 xfit gt linessplinexfit yfit The standard package stats provides much more extensive facilities for tting non linear models by least squares The model we have just tted is the Michaelis Menten model so we can use gt df lt dataframexx yy gt fit lt nlsy quot SSmicmenx Vm K df gt fit Nonlinear regression model model y quot SSmicmenx Vm K data df K 21268370711 006412123 residual sumofsquares 1195449 gt summaryfit Formula y quot SSmicmenx Vm K Parameters Estimate Std Error t value Prgtt Vm 2127e02 6947e00 30615 324e11 K 6412e02 8281e03 7743 157e05 Residual standard error 1093 on 10 degrees of freedom Correlation of Parameter Estimates Vm K 07651 1172 Maximum likelihood Maximum likelihood is a method of nonlinear model tting that applies even if the errors are not normal The method nds 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 1087111 This example ts a logistic model to dose response data which clearly could also be t by glmO The data are gt x lt c16907 17242 17552 17842 18113 1 8369 1 8610 1 8839 gt y lt c 6 13 18 28 52 53 61 60 gt n lt C59 60 62 56 63 59 62 60 The negative log likelihood to minimize is gt fn lt functionp sum yp1p2 x nlog1expp1p 2 x logchoosen y We pick sensible starting values and do the t gt out lt nlmfn p c5020 hessian TRUE After the tting outminimum is the negative log likelihood and outestimate are the maxi mum likelihood estimates of the parameters To obtain the approximate SEs of the estimates we do gt sqrt diagsolve outhessian A 95 con dence interval would be the parameter estimate i 196 SE 118 Some nonstandard 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 0 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 coef cients correspond to random effects These functions make heavy use of formulae to specify the models Local approximating regressions The loess function ts 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 tting regression models in a way resistant to the in uence of extreme outliers in the data Function lqs in the rec ommended package MASS provides stateof art algorithms for highly resistant ts Less resistant but statistically more ef cient 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 Treebased 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 speci ed in the ordinary linear model form The model tting function is tree but many other generic functions such as plot and text are well adapted to displaying the results of a treebased model t in a graphical way Chapter 11 Statistical models in R Tree models are available in R via the user contributed packages rpart and tree Chapter 12 Graphical procedures 62 1 2 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 lnteractive use is also easy because at startup time R initiates a graphics device diiuer 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 and windows under Windows 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 0 Highlevel plotting functions create a new plot on the graphics device possibly with axes labels titles and so on Lowlevel 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 7 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 121 Highlevel 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 1211 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 rst argument plotx y plotxy lfx and y are vectors plotx y produces a scatterplot ofy 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 plotx 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 rst 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 quot expr ploty quot expr df is a data frame y is any object expr is a list of object names separated by eg a b c The rst two forms produce distributional plots of the variables in a data frame rst form or of a number of named objects second form The third form plots y against every object named in expr 1212 Displaying multivariate data R provides two very useful functions for representing multivariate data If X is a numeric matrix or data frame the command gt pairsX produces a pairwise scatterplot matrix of the variables de ned by the columns of X that is every column of X is plotted against every other column of X and the resulting nn 7 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 gt coplota quot b I 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 6 is numeric it is divided into a number of conditioning intervals and for each interval a is plotted against b for values of 6 within the interval The number and position of intervals can be controlled with givenva1ues argument to coplotOithe function co intervals is useful for selecting intervals You can also use two given variables with a command like gt coplota b I C d which produces scatterplots of a against b for every joint conditioning interval of c and d The coplotO and pairsO 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 1213 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 rst 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 hi st x hist x nclass11 histx breaksb 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 speci ed exactly with the breaks argument Chapter 12 Graphical procedures 64 If the probabilityTRUE argument is given7 the bars represent relative frequencies instead of counts dotchartx Constructs a dotchart of the data in x In a dotchart the y axis gives a labelling of the data in x and the m axis gives its value For example it allows easy visual selection of all data entries with values lying in speci ed ranges imagex y z contourx y z perspx y z Plots of three variables The image plot draws a grid of rectangles using different colours to represent the value of 27 the contour plot draws contour lines to represent the value of 27 and the persp plot draws a 3D surface 1214 Arguments t0 highlevel plotting functions There are a number of arguments which may be passed to high level graphics functions7 as follows addTRUE Forces the function to act as a low level graphics function7 superimposing the plot on the current plot some functions only axesFALSE Suppresses generation of axesiuseful for adding your own custom axes with the axis function The default7 axesTRUE7 means include axes lognxn lognyn logquotxyquot Causes the z y or both axes to be logarithmic This will work for many7 but not all7 types of plot type The type argument controls the type of plot produced7 as follows typequotpquot Plot individual points the default typequotlquot Plot lines typequotbquot Plot points connected by lines both typequotoquot Plot points overlaid by lines typequothquot Plot vertical lines from points to the zero axis high density typensn typequotSquot Step function plots In the rst form7 the top of the vertical de nes the point in the second7 the bottom typequotnquot 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 xlabstring ylabstring Axis labels for the z and y axes Use these arguments to change the default labels7 usually the names of the objects used in the call to the high level plotting function mainstring Figure title7 placed at the top of the plot in a large font substring Sub title7 placed just below the m axis in a smaller font Chapter 12 Graphical procedures 65 122 Lowlevel 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 pointsx y linesx y Adds points or connected lines to the current plot plotO s type argument can also be passed to these functions and defaults to quotpquot for points and quotlquot for lines textx 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 yi The default is 11engthx Note This function is often used in the sequence gt plotx y typequotnquot textx y names The graphics parameter typequotnquot suppresses the points but sets up the axes and the text function supplies special characters as speci ed by the character vector names for the points abline a b abline hy abline vX abline 1m obj Adds a line of slope b and intercept a to the current plot hy may be used to specify y coordinates for the heights of horizontal lines to go across a plot and vx similarly for the m coordinates for vertical lines Also lmobj may be list with a coefficients component of length 2 such as the result of model tting functions which are taken as an intercept and slope in that order polygonx y Draws a polygon de ned by the ordered vertices in x y and optionally shade it in with hatch lines or ll it if the graphics device allows the lling of gures legendx y legend Adds a legend to the current plot at the speci ed position Plotting characters line styles colors etc are identi ed 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 fillV Colors for lled boxes legend colV Colors in which points or lines will be drawn legend ltyV Line styles legend lwdV Line widths legend pchV Plotting characters character vector Chapter 12 Graphical procedures 66 titlemain 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 axisside Adds an axis to the current plot on the side given by the rst 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 axesFALSE argument Low level plotting functions usually require some positioning information eg z and y co ordinates to determine where to place the new plot elements Coordinates are given in terms of user coordinates which are de ned 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 locatorO see below may be used to specify positions on a plot interactively 1221 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 gt textx y expressionpastebgroupquotquot atopn x quotquot pquotx qquotnx More information including a full listing of the features available can obtained from within R using the commands gt helpplotmath gt exampleplotmath gt demoplotmath 1222 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 andor 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 gt helpHershey gt demoHershey gt helpJapanese gt demoJapanese 123 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 locatorO function Chapter 12 Graphical procedures 67 locatorn type Waits for the user to select locations on the current plot using the left mouse button This continues until 11 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 locatorO returns the locations of the points selected as a list with two components x and y locatorO 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 gt textlocator1 quotOutlierquot adj0 may be useful locatorO will be ignored if the current device such as postscript does not support interactive pointing identifyx y labels Allow the user to highlight any of the points de ned 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 z y coordinates in two numeric vectors x and y we could use the identifyO function as follows gt plot x y gt identifyx y The identifyO 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 xy 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 identifyO returns the indices of the selected points you can use these indices to extract the selected points from the original vectors x and y 124 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 gure arrangement and text justi ca 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 1241 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 68 par Without arguments returns a list of all graphics parameters and their values for the current device Parcncoln nltyn With a character vector argument returns only the named graphics parameters again as a list par col4 lty2 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 parO is called from within a function This is often undesirable behavioriusually 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 gt oldpar lt parcol4 lty2 plotting commands gt paroldpar To save and restore all settable l graphical parameters use gt oldpar lt parnoreadonlyTRUE plotting commands gt paroldpar 1242 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 gt plot x y pchquotquot 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 parO 125 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 nameva1ue A description of the parameters 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 1251 Graphical elements R plots are made up of points lines text and polygons lled regions Graphical parameters exist which control how these graphical elements are drawn as follows Pchnn pch4 lty2 lwd2 col2 colaxis collab colmain colsub font2 font axis font lab font main font sub adj O 1 cex15 Character to be used for plotting points The default varies with graphics drivers but it is usually 0 Plotted points tend to appear slightly above or below the appropriate position unless you use quot quot as the plotting character which produces centered points 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 gt legendlocator1 ascharacter025 pch 025 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 32255 representing a character in the current font Line types Alternative line styles are not supported on all graphics devices and vary on those that do but line type 1 is always a solid line line type 0 is always invis ible and line types 2 and onwards are dotted or dashed lines or some combination of both Line widths Desired width of lines in multiples of the standard line width 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 Colors to be used for points lines text lled regions and images A number from the current palette see palette or a named colour The color to be used for axis annotation z and y labels main and sub titles re spectively An integer which speci es 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 The font to be used for axis annotation z and y labels main and sub titles respec tively Justi cation of text relative to the plotting position 0 means left justify 1 means right justify and 05 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 O 1 leaves a gap of 10 of the text width between the text and the plotting position Character expansion The value is the desired size of text characters including plotting characters relative to the default text size Chapter 12 Graphical procedures 70 G ex axis cexlab G ex main cexsub The character expansion to be used for axis annotation z and y labels main and sub titles respectively 1252 Axes and tick marks Many of Rs high level plots have axes and you can construct axes yourself with the low level axisO 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 labc5 7 12 The rst two numbers are the desired number of tick intervals on the z 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 las1 Orientation of axis labels 0 means always parallel to axis 1 means always horizon tal and 2 means always perpendicular to the axis mgpc3 1 0 Positions of axis components The rst 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 nal component is the distance from the axis position to the axis line usually zero Positive numbers measure outside the plot region negative numbers inside tck001 Length of tick marks as a fraction of the size of the plotting region When tck is small less than 05 the tick marks on the z 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 tck0 01 and mgpc 1 15 0 for internal tick marks xaxsquotrquot axsquotiquot Axis st les for the z and axes res ectivel With st les quotiquot internal and quotrquot Y y y 7 p y y the default tick marks always fall within the range of the data however style quotIquot leaves a small amount of space at the edges S has other styles not implemented in 1253 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 71 A typical gure is Plot region mailll Margin Graphics parameters controlling gure layout include maic1 05 05 O Widths of the bottom7 left7 top and right margins7 respectively7 measured in inches marc4 2 2 1 Similar to mai7 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 needed7 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 Furthermore7 the default is chosen without regard to the size of the device surface for example7 using the postscriptO driver with the height4 argument will result in a plot which is about 50 margin unless mar or mai are set explicitly When multiple gures are in use see below the margins are reduced7 however this may not be enough when many gures share the same page Chapter 12 Graphical procedures 72 1254 Multiple gure environment R allows you to create an n by m array of gures on a single page Each gure has its own margins and the array of gures is optionally surrounded by an outer margin as shown in the following gure oma3 omi4 ngwmm omi1 mfrowc32 The graphical parameters relating to multiple gures are as follows mfcolc3 2 mfrowc 2 4 Set the size of a multiple gure array The rst 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 gures to be lled by column mfrow lls by rows The layout in the Figure could have been created by setting mfrowc32 the gure shows the page after four plots have been drawn Setting either of these can reduce the base size of symbols and text controlled by parquotcexquot and the pointsize of the device In a layout with exactly two rows and columns the base size is reduced by a factor of 083 if there are three or more of either rows or columns the reduction factor is 066 mfgc2 2 S 2 Position of the current gure in a multiple gure environment The rst two numbers are the row and column of the current gure the last two are the number of rows and columns in the multiple gure array Set this parameter to jump between gures in the array You can even use different values for the last two numbers than the true values for unequally sized gures on the same page figc4 9 1 410 Position of the current gure 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 gure in the bottom right of the page Set this parameter for arbitrary positioning of gures within a page If you want to add a gure to a current page use newTRUE as well unlike S Chapter 12 Graphical procedures 73 omac2 O 3 O omic0 O 08 0 Size of outer margins Like mar and mai the rst measures in text lines and the second in inches starting with the bottom margin and working clockwise Outer margins are particularly useful for page wise titles etc Text can be added to the outer margins with the mtextO function with argument outerTRUE There are no outer margins by default however so you must create them explicitly using oma or omi More complicated arrangements of multiple gures can be produced by the split screen and layout functions as well as by the grid and lattice packages 126 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 line77 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 helpDevices for a list of them all For example issuing the command gt postscriptO 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 0 For use on Windows quartzO For use on MacOS X postscript O For printing on PostScript printers or creating PostScript graphics les pdf Produces a PDF le which can also be included into PDF les pngO Produces a bitmap PNG le Not always available see its help page jpegO Produces a bitmap JPEG le best used for image plots Not always available see its help page When you have nished with a device be sure to terminate the device driver by issuing the command gt dev of f O This ensures that the device nishes 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 1261 PostScript diagrams for typeset documents By passing the file argument to the postscriptO device driver function you may store the graphics in PostScript format in a le of your choice The plot will be in landscape orientation unless the horizontalFALSE 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 t these dimensions For example the command Chapter 12 Graphical procedures 74 gt postscriptquotfile psquot horizontalFALSE height5 pointsize10 will produce a le containing PostScript code for a gure ve inches high perhaps for inclusion in a document It is important to note that if the le named in the command already exists it will be overwritten This is the case even if the le was only created earlier in the same R session Many usages of PostScript output will be to incorporate the gure 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 onefileFALSE 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 speci cation Thus to produce a plot for inclusion use something like gt postscript quotplotl epsquot horizontalFALSE onefileFALSE height8 width6 pointsize10 1262 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 deulce 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 x110 UNIX windows win printer 0 win metafile 0 Windows quartz MacOS X postscript 0 pdf 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 Some platforms may have further devices available dev li st 0 Returns the number and name of all active devices The device at position 1 on the list is always the null deulce which does not accept graphics commands at all devnext devprev Returns the number and name of the graphics device next to or previous to the current device respectively dev set whichk 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 of f k Terminate the graphics device at point k of the device list For some devices such as postscript devices this will either print the le immediately or correctly complete the le for later printing depending on how the device was initiated Chapter 12 Graphical procedures 75 devcopydevice whichk devprintdevice whichk Make a copy of the device k Here device is a device function such as postscript with extra arguments if needed speci ed by 7 devprint is similar but the copied device is immediately closed so that end actions such as printing hardcopies are immediately performed graphics off 0 Terminate all graphics devices on the list except the null device 127 Dynamic graphics R does not have builtin capabilities for dynamic or interactive graphics eg 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 httpwwwggobiarg and these can be accessed from R via the package rggobi described at htLpwa ggobi orgggobi Also package rgl provides ways to interact with 3D plots for example of surfaces Chapter 13 Packages 76 1 3 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 gt library with no arguments To load a particular package eg the boot package containing functions from Davison amp Hinkley 1997 use a command like gt libraryboot Users connected to the Internet can use the install packages and update packagesO functions available through the Packages menu in the Windows and RAqua GUls see section Installing packages in R Installation and Adminstration to install and update packages To see which packages are currently loaded use gt search to display the search list Some packages may be loaded but not available on the search list see Section 133 Namespaces page 76 these will be included in the list given by gt loadedNamespacesO To see a list of all available help topics in an installed package use gt help start to start the HTML help system and then navigate to the package listing in the Reference section 131 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 132 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 ORAN hLLp HCRAIl Rproject0g and its mirrors and other repositories such as Bioconductor htcpz 39rnbioccnductor org The R FAQ contains a list that was current at the time of release but the collection of available packages changes frequently 133 Namespaces Packages can have namespaces and currently all of the base and recommended packages do expect 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 00 is the transpose function in R but users might de ne their own function named t Namespaces prevent the user s de nition 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 de nitions from a particular namespace In the example above the transpose function will always be available as base t because it is de ned in the base package Only functions that are exported from the package can be retrieved in this way The triplecolon operator z 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 getAnywhereO 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 rst 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 0 Start the HTML interface to on line help using a web browser available at your machine You should brie y explore the features of this facility with the mouse lconify the help window and move on to the next part x lt rnorm50 y lt rnormx Generate two pseudo random normal vectors of z and y coordinates plot x y Plot the points in the plane A graphics window will appear automatically 150 See which R objects are now in the R workspace rmx y Remove objects no longer needed Clean up x lt 120 Make z 1220 w lt 1 sqrtx2 A weight vector of standard deviations dummy lt dataframexx y x rnormxw dummy Make a data frame of two columns z and y and look at it fm lt lmy quot x datadummy summaryfm Fit a simple linear regression of y on x and look at the analysis fml lt lmy quot x datadummy weight1wquot2 summaryfm1 Since we know the standard deviations we can do a weighted regression attachdummy Make the columns in the data frame visible as variables lrf lt lowessx y Make a nonparametric local regression function plot x y Standard point plot linesx lrfy Add in the local regression abline O 1 lty3 The true regression line intercept O slope 1 abline coef fm Unweighted regression line Appendix A A sample session 79 ablinecoeffm1 col quotredquot Weighted regression line detachO Remove data frame from the search path plotfittedfm residfm xlabquotFitted valuesquot ylabquotResidualsquot mainquotResiduals vs Fittedquot A standard regression diagnostic plot to check for heteroscedasticity Can you see it qqnormresidfm mainquotResiduals Rankit Plotquot normal scores plot to check for skewness7 kurtosis and outliers Not very useful here rmfm fml 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 object7 but we will read it to illustrate the readtable function filepath lt systemfilequotdataquot quotmorleyta quot packagequotdatasetsquot filepath Get the path to the data le file showfilepath Optional Look at the le m lt readtablefilepath mm Read in the Michaelson and Morley data as a data frame7 and look at it There are ve experiments column Expt and each has 20 runs column Run and 1 is the recorded speed of light7 suitably coded mmExpt lt factor mmExpt mmRun lt factor mmRun Change Expt and Run into factors attachmm Make the data frame visible at position 3 the default plotExpt Speed mainquotSpeed of Light Dataquot xlabquotExperiment No quot Compare the ve experiments with simple boxplots fm lt aovSpeed quot Run Expt datamm summaryfm Analyze as a randomized block7 with runs and experiments as factors me lt updatefm quot Run anovafm0 fm Fit the submodel omitting runs and compare using a formal analysis of variance detachO rmfm me Clean up before moving on We now look at some more graphical features contour and image plots x lt seqpi pi len50 y lt x z is a vector of 50 equally spaced values in 77139 g z 3 7139 y is the same Appendix A A sample session 80 f lt outerx y functionx y cosy1 xquot2 f is a square matrix with rows and columns indexed by z and y respectively of values of the function cosy1 m2 oldpar lt par no readonly TRUE par ptyquotsquot Save the plotting parameters and set the plotting region to square contourx y f contourx y f nlevels15 addTRUE Make a contour map of 1 add in more lines for more detail fa lt f tf2 fa is the asymmetric part77 of f t is transpose contourx y fa nlevels15 Make a contour plot par oldpar and restore the old graphics parameters imagex y f imagex y fa Make some high density image plots of which you can get hardcopies if you wish objectsO rmx y f fa and clean up before moving on R can do complex arithmetic also th lt seqpi pi len100 z lt exp1ith ii is used for the complex number 2 par ptyquotsquot plot 2 typequotlquot Plotting complex arguments means plot imaginary versus real parts This should be a circle w lt rnorm100 rnorm1001i 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 lt ifelseModw gt 1 1w w and to map any outside the circle onto their reciprocal plot w xlimc 1 1 ylimc 1 1 pChquotquot Xlabquotxquot ylabquotyquot lines 2 All points are inside the unit circle but the distribution is not uniform w lt sqrtrunif 100exp2pirunif 1001i plot w xlimc 1 1 ylimc 1 1 pChquotquot Xlabquotxquot ylabquotyquot lines 2 The second method uses the uniform distribution The points should now look more evenly spaced over the disc rmth w 2 Clean up again 10 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 B1 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 ltin le gtout le or via the R CMD interface as a wrapper to various R tools eg for processing les 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 les 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 speci c details 0 Unless no environ was given R searches for user and site les to process for setting environment variables The name of the site le is the one pointed to by the environment variable RENVIRON if this is unset RHOMEetcRenvironsite is used if it exists The user le searched for is Renviron7 in the current or in the user s home directory in that order These les should contain lines of the form name Va1ue 7 See helpStartup for a precise description Variables you might want to set include RPAPERSIZE the default paper size RPRINTCMD the default print command and RLIBS speci es the list of R library trees searched for add on packages Then R searches for the site wide startup pro le unless the command line option no site file was given The name of this le is taken from the value of the RPROFILE environment variable If that variable is unset the default RHOMEetcRprofilesite is used if this exists Then unless no init file was given R searches for a le called Rprofile7 in the current directory or in the user s home directory in that order and sources it norestore7 or It also loads a saved image from RData7 if there is one unless no restore data was speci ed 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 de ned in the appropriate startup pro les 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 version7 Print version information to standard output and exit successfully encodingenc 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 RHOME Print the path to the R home directory77 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 Appendix B Invoking R 82 save nosave 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 speci ed or implied by some other option see below noenviron Do not read any user le to set environment variables nositefile Do not read the site wide pro le at startup noinitfile Do not read the user s pro le at startup restore norestore norestoredata Control whether saved images le RData7 in the directory where R was started should be restored at startup or not The default is to restore no restore7 implies all the speci c no restore options norestorehistory Control whether the history le normally le Rhistory7 in the directory where R was started but can be set by the environment variable RHISTFILE should be restored at startup or not The default is to restore noRconsole Windows only Prevent loading the Rconsole le at startup vanilla Combine nosave noenviron nositefile noinitfile and no restore Under Windows this also includes no Rconsole f file7 filefi1e7 Take input from le means stdin lmplies no save unless save has been set e expression7 Use expression as an input line One or more e7 options can be used but not together with f or file lmplies no save unless save has been set There is a limit of 10000 bytes on the total length of expressions used in this way 4 noreadline 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 ess Windows only Set Rterm up for use by R inferior mode in ESS minvsizeN maxvsizeN Specify the minimum or maximum amount of memory used for variable size objects by setting the vector heap77 size to N bytes Here N must either be an integer or an integer ending with G M K or k meaning Giga 2quot30 Mega 2quot20 computer Kilo 2quot10 or regular kilo 1000 Appendix B Invoking R 83 minnsizeN maxnsizeN Specify the amount of memory used for xed size objects by setting the number of cons cells77 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 maxppsizeN 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 maxmemsizeN 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 15Gb1 and the amount of physical RAM in the machine and must be between 32Mb and 3Gb quiet silent7 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 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 debuggername d name7 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 guit pe7 g type7 UNIX only Use type as graphical user interface note that this also includes in teractive graphics Currently possible values for type are Xll the default pro vided that TclTk support is available Tk and gnome provided that package gnomeGUI is installed For back compatibility xll tk and GNOME are ac cepted args This ag does nothing except cause the rest of the command line to be skipped this can be useful to retrieve values from it with commandArgsTRUE Note that input and output can be redirected in the usual way using lt7 and gt7 but the line length limit of 1024 bytes still applies Warning and error messages are sent to the error channel stderr except on Windows 9XME 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 les for use with R 1 25Gb on Versions of Windows that support 3Gb per process and have the support enabled see the rwFAQ Q29 Appendix B Invoking R 84 SHLIB Build shared library for dynamic loading INSTALL Install add on packages REMOVE Remove add on packages build Build that is7 package add on packages check Check add on packages LINK UNIX only Mont end for creating executable programs Rprof Post process R pro ling les Rdconv Convert Rd format to various other formats7 including HTML7 Nroff7 ETEX plain text7 and S documentation format Rd2dvi Convert Rd format to DVIPDF Rd2txt Convert Rd format to text Sd2Rd Convert S documentation to Rd format config UNIX only Obtain con guration information Use R CMD command help to obtain usage information for each of the tools accessible via the R CMD interface B2 Invoking R under Windows There are two ways to run R under Windows Within a terminal window eg cmdexe or commandcom or a more capable shell7 the methods described in the previous section may be used7 invoking by Rexe or more directly by Rtermexe These are principally intended for batch use For interactive use7 there is a consolebased GUI Rgui exe The startup procedure under Windows is very similar to that under UNIX7 but references to the home directory7 need to be clari ed7 as this is not always de ned on Windows If the environment variable RUSER is de ned7 that gives the home directory Next7 if the environment variable HOME is de ned7 that gives the home directory After those two user controllable settings7 R tries to nd system de ned home directories It rst tries to use the Windows quotpersonalquot directory typically CDocuments and SettingsusernameMy Documents in Windows XP If that fails7 and environment variables HOMEDRIVE and HOMEPATH are de ned and they normally are under Windows NT2000XP these de ne the home directory Failing all those7 the home directory is taken to be the starting directory You need to ensure that either the environment variables TMPDIR7 TMP and TEMP are either unset or one of them points to a valid place to create temporary les and directories Environment variables can be supplied as name Va1ue7 pairs at the end of the command line The following additional command line options are available when invoking RGui exe mdi7 sdi nomdi 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 console7 graphics and pager The command line setting overrides the setting in the user s Rconsole le Appendix B Invoking R 85 debug Enable the Break to debugger77 menu item in Rgui7 and trigger a break to the debugger during command line processing In Windows with R CMD you may also specify your own bat or exe7 le instead of one of the built in commands It will be run with the following environment variables set appropriately RHOMERVERSIONRCMDROSTYPEPATH7PERL5LIB7and TEXINPUTS Fbrexanipb7ifyou already have latexexe on your path7 then R CMD latex exe mydoc will run ETEX on mydoc tex 7 with the path to Rs sharetexmf macros added to TEXINPUTS B3 Invoking R under Mac OS X There are two ways to run R under Mac OS X Within a Terminalapp window by invoking R7 the methods described in the previous sections apply There is also consolebased 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 directory7 is the one inside the Rframework7 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 B4 Scripting with R If you just want to run a le fooR of R commands7 the recommended way is to use R CMD BATCH fooR If you want to run this in the background or as a batch job use OS speci c facilities to do so for example in most shells R CMD BATCH fooR amp 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 fooR amp will pass arguments to a script which can be retrieved as a character vector by args lt commandArgsTRUE This is made simpler by the alternative front end Rscript7 which can be invoked by Rscript fooR arg1 arg2 and this can also be used to write executable script les like at least on Unix alikes7 and in some Windows shells pathtoRscript args lt commandArgsTRUE qstatusltexit status codegt If this is entered into a text le runfoo and this is made executable by chmod 755 runfoo7 it can be invoked for different arguments by runfoo arg1 arg2 For further options see helpquotRscriptquot 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 R7 use usrbinenv Rscript At least in Bourne and bash shells7 the mechanism does not allow extra arguments like usrbinenv Rscript vanilla One thing to consider is what stdin refers to It is commonplace to write R scripts with segments like Appendix B Invoking R chem lt scann24 290 310 340 340 370 370 280 250 240 240 270 220 528 337 303 303 2895 377 340 220 350 360 370 370 and stdinO refers to the script le to allow such traditional usage If you want to refer to the process s stdin use quotstdinquot as a file connection7 eg scanquotstdinquot Another way to write executable script les suggested by F ranois Pinard is to use a here document like binsh environment variables can be set here R slave other options ltltEOF R program goes here EOF but here stdin refers to the program source and quotstdinquot will not be usable Appendix C The command line editor 87 Appendix C The commandline editor C 1 Preliminaries When the GNU readline library is available at the time R is con gured for compilation un der UNIX an inbuilt command line editor allowing recall editing and re submission of prior commands is used It can be disabled useful for usage with ESSl 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 le README R cerm7 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 down while you press the Q key and are written as C m below Meta characters such as Meta b are typed by holding down 2 and pressing E and written as M b in the following If your terminal does not have a key you can still type Meta characters using two character sequences starting with ESC Thus to enter M b you could type E The ESC character sequences are also allowed on terminals with real Meta keys Note that case is signi cant for Meta characters 02 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 ln 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 11239 mode character insertion mode is started by M i or M a characters are typed and insertion mode is nished by typing a further Pressing the command at any time causes the command to be re submitted Other editing actions are summarized in the following table 03 Commandline 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 0 11 respectively Horizontal motion of the cursor C a Go to the beginning of the command C e Go to the end of the line M b Go back one word M f Go forward one word 1 The Emacs Speaks Statistics7 package see the URL hrxp lz39ESS hproject org 2 On a PC keyboard this is usually the Alt key occasionally the Windows key Appendix C The command line editor 88 017 Cf Go back one character Go forward one character On most terminals7 you can also use the left and right arrow keys instead of 0 12 and C f7 respectively Editing and re submission text Cf text DEL C d M d C k CV C t M l MC RET Insert text at the cursor Append text after the cursor Delete the previous character left of the cursor Delete the character under the cursor Delete the rest of the word under the cursor7 and save it Delete from cursor to end of command7 and save it lnsert yank the last saved text here Transpose the character under the cursor with the next Change the rest of the word to lower case Change the rest of the word to upper case Re submit the command to R The nal terminates the command line editing sequence Appendix D Function and variable index Appendix D Function and variable index coef coefficients 53 contour contrasts devlist 74 devnext devoff Appendix D Function and variable index 90 dim dotchart dropl 55 mea n methods formula 53 function 42 G getAnywhere 49 getSSmethod 49 glm 56 H help help search levels 16 quartz H 73 Appendix D Function and variable index 91 range 8 rbind 24 readtable 30 S U um lass 14 update 54 Appendix E Concept index Appendix E Concept index A Accessing builtin datasets 31 Additive models 60 Analysis of variance 54 Arithmetic functions and operators 7 Arrays 18 Assignment 7 Attributes 13 B Binary operators 43 Box plots 37 C Character vectors 10 Classes 147 48 Concatenating lists 27 Contrasts 52 Control statements 4O 76 Customizing the environment 48 D Data frames 27 Default values 43 Density estimation 34 Determinants 23 Diverting input and output 5 Dynamic graphics 75 E Eigenvalues and eigenvectors 23 Empirical CDFs 35 F Factors 167 52 amilies 56 Formulae 50 G Generalized linear models 55 Generalized transpose of an array 21 Generic functions 48 Graphics dev1ce drivers 73 Graphics parameters 67 Grouped expressions 40 I Indexing of and by arrays 18 Indexing vectors 1O 92 KolmogorovSmirnov test 36 Least squares tting 23 Linear equations 22 Linear models 53 Lists 26 Local approximating regressions Loops and conditional execution 40 M Matrices 18 Matrix multiplication 22 Maximum likelihood 59 Missing values 9 Mixed models 60 N Named arguments Ordered factors Outer products of arrays 21 P Probability distributions Q R decomposition 23 Quantile quantile plots 35 R Reading data from les 3O Recycling rule 77 20 Regular sequences 8 Removing objects 6 Robust regression 60 S Scope 46 Search path 29 ShapiroWilk test 36 Singular value decomposition 23 Statistical models 5O Appendix E Concept index Student s t test 37 T Tabulation 25 Treebased models 60 U Updating tted models 54 V Vectors 7 W Wilcoxon test 38 Workspace 5 Writing functions 42 Appendix F References 94 Appendix F References D M Bates and D G Watts 19887 Nonlinear Regression Analysis and Its Applications John Wiley amp Sons7 New York Richard A Becker7 John M Chambers and Allan R Wilks 19887 The New S Language Chap man amp Hall7 New York This book is often called the Blue Book John M Chambers and Trevor J Hastie eds 19927 Statistical Models in S Chapman amp Hall7 New York This is also called the White Book John M Chambers 1998 Programming with Data Springer7 New York This is also called the Green Book A C Davison and D V Hinkley 19977 Bootstrap Methods and Their Applications7 Cambridge University Press Annette J Dobson 19907 An Introduction to Generalized Linear Models7 Chapman and Hall7 London Peter McCullagh and John A Nelder 19897 Generalized Linear Models Second edition7 Chap man and Hall7 London John A Rice 19957 Mathematical Statistics and Data Analysis Second edition Duxbury Press7 Belmont7 CA S D Silvey 19707 Statistical Inference Penguin7 London


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