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An Introduction to R

1. 1 Under UNIX the utilities sed orawk can be used Chapter 7 Reading data from files 31 The data frame may then be read as gt HousePrice lt read table houses data header TRUE where the header TRUE option specifies that the first line is a line of headings and hence by implication from the form of the file that no explicit row labels are given 7 2 The scan function Suppose the data vectors are of equal length and are to be read in parallel Further suppose that there are three vectors the first of mode character and the remaining two of mode numeric and the file is input dat The first step is to use scan to read in the three vectors as a list as follows gt inp lt scan input dat list 0 0 The second argument is a dummy list structure that establishes the mode of the three vectors to be read The result held in inp is a list whose components are the three vectors read in To separate the data items into three separate vectors use assignments like gt label lt inp 1 x lt inp 21 y lt inp 31 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 scan input dat list id x 0 y 0 If you wish to access the variables separately they may either be re assigned to variables in the working frame gt label inp id x inp x y inp y or the list may be attached at posit
2. To understand completely the rules governing the scope of R assignments the reader needs to be familiar with the notion of an evaluation frame This is a somewhat advanced though hardly difficult topic and is not covered further here If global and permanent assignments are intended within a function then either the su perassignment operator or the function assign can be used See the help document for details S PLus users should be aware that lt lt has different semantics in R These are discussed further in Section 10 7 Scope page 46 10 6 More advanced examples 10 6 1 Efficiency factors in block designs As a more complete if a little pedestrian example of a function consider finding the effi ciency factors for a block design Some aspects of this problem have already been discussed in Section 5 3 Index matrices page 19 A block design is defined by two factors say blocks b levels and varieties v levels If R and K are the v by v and b by b replications and block size matrices respectively and N is the b by v incidence matrix then the efficiency factors are defined as the eigenvalues of the matrix E I ROPNTKONR I I ATA where A K N R17 One way to write the function is given below gt bdeff lt function blocks varieties Y blocks lt as factor blocks minor safety move b lt length levels blocks varieties lt as factor varieties minor safety move v lt length
3. e Extract elements X 1 3 X 2 2 and X 3 1 as a vector structure and e 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 array 1 20 dim c 4 5 st 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 i 15 19 4 4 8 12 16 20 gt i lt array c 1 3 3 1 dim c 3 2 i iisa 3 by 2 index array 1 2 1 1 3 2 2 2 3 3 1 gt x i Extract those elements 1 963 gt x i lt 0 Replace those elements by zeros gt x 1 2 3 4 5 1 1 5 0 13 17 2 2 0 10 14 18 3 0 7 11 15 19 4 4 8 12 16 20 Negative indices are not allowed in index matrices NA and zero values are allowed rows in the index matrix containing a zero are ignored and rows containing an NA produce an NA in the result As a less trivial example suppose we wish to generate an unreduced design matrix for a block design defined by factors blocks b levels and varieties v levels Further suppose there are n plots in the experiment We could proceed as follows gt Xb matrix 0 n b gt Xv matrix 0 n v gt ib lt cbind 1 n blocks gt iv lt cbind 1 n varieties gt Xb ib 1 gt Xvliv lt 1 gt X lt cbind Xb Xv To construct the incidence matrix N say we could use gt N lt crossprod Xb Xv Chapter 5 Arrays and m
4. ross lt open account 100 2 n some sense this mimics the behavior in S PLUS since in S PLUS this operator always creates or assigns to a global variable Chapter 10 Writing your own functions 48 robert lt open account 200 rossfwithdraw 30 rossfbalance robert balance ross deposit 50 ross balance ross withdraw 500 10 8 Customizing the environment Users can customize their environment in several different ways There is a site initialization file and every directory can have its own special initialization file Finally the special functions First and Last can be used The location of the site initialization file is taken from the value of the R PROFILE environment variable If that variable is unset the file Rprofile site in the R home subdirectory etc is used This file should contain the commands that you want to execute every time R is started under your system A second personal profile file named Rprofile can be placed in any directory If R is invoked in that directory then that file will be sourced This file gives individual users control over their workspace and allows for different startup procedures in different working directories If no Rprofile file is found in the startup directory then R looks for a Rprofile file in the user s home directory and uses that if it exists If the environment variable RB PROFILE USER is set the file it points to is used instead of the Rprofile files
5. 7 4 Editing data When invoked on a data frame or matrix edit brings up a separate spreadsheet like environment for editing This is useful for making small changes once a data set has been read The command gt xnew lt edit xold will allow you to edit your data set xold and on completion the changed object is assigned to xnew If you want to alter the original dataset xold the simplest way is to use fix xold which is equivalent to xold edit xold Use xnew edit data frame to enter new data via the spreadsheet interface Chapter 8 Probability distributions 33 8 Probability distributions 8 1 R as a set of statistical tables One convenient use of R is to provide a comprehensive set of statistical tables Functions are provided to evaluate the cumulative distribution function P X lt x the probability density function and the quantile function given q the smallest x such that P X lt x 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 f dfi df2 ncp gamma gamma shape scale geometric geom prob hypergeometric hyper m n k log normal lnorm meanlog sdlog logistic logis location scale negative binomial nbinom size prob normal norm mean sd Poisson pois lambda signed rank signrank n Student s t t df ncp uniform
6. Any function named First in either of the two profile files or in the RData image has a special status It is automatically performed at the beginning of an R session and may be used to initialize the environment For example the definition in the example below alters the prompt to and sets up various other useful things that can then be taken for granted in the rest of the session Thus the sequence in which files are executed is Rprofile site the user profile RData and then First A definition in later files will mask definitions in earlier files gt First lt function 4 options prompt continue t is the prompt options digits 5 length 999 custom numbers and printout x110 for graphics par pch plotting character source file path Sys getenv HOME R mystuff R my personal functions library MASS attach a package Similarly a function Last if defined is normally executed at the very end of the session An example is given below gt Last lt function 4 graphics off a small safety measure cat paste date OO nAdios n Is it time for lunch 3 So it is hidden under UNIX Chapter 10 Writing your own functions 49 10 9 Classes generic functions and object orientation The class of an object determines how it will be treated by what are known as generic functions Put the other way round a generic function performs a task or action on i
7. R 3 At this point R commands may be issued see later 4 To quit the R program the command is gt qO 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 O command at the end of the session To use R under Windows the procedure to follow is basically the same Create a folder as the working directory and set that in the Start In field in your R shortcut Then launch R by double clicking on the icon Chapter 1 Introduction and preliminaries 4 1 6 An introductory session Readers wishing to get a feel for R at a computer before proceeding are strongly advised to work through the introductory session given in Appendix A A sample session page 82 1 7 Getting help with functions and features R has an inbuilt help facility similar to the man facility of UNIX To get more information on any specific named function for example solve the command is gt help solve An alternative is gt solve For a feature specified by special c
8. points x y lines x y Adds points or connected lines to the current plot plot s type argument can also be passed to these functions and defaults to p for points and 1 for lines text x y labels Add text to a plot at points given by x y Normally labels is an integer or character vector in which case labels i is plotted at point x il y i The default is 1 length x Note This function is often used in the sequence plot x y type n text x y names The graphics parameter type n suppresses the points but sets up the axes and the text function supplies special characters as specified by the character vector names for the points abline a b abline h y abline v x abline 1m obj Adds a line of slope b and intercept a to the current plot h y may be used to specify y coordinates for the heights of horizontal lines to go across a plot and v x similarly for the x coordinates for vertical lines Also Im obj may be list with a coefficients component of length 2 such as the result of model fitting functions which are taken as an intercept and slope in that order polygon x y Draws a polygon defined by the ordered vertices in x y and optionally shade it in with hatch lines or fill it if the graphics device allows the filling of figures legend x y legend Adds a legend to the current plot at the specified position Plotting characters line styles colors etc are
9. Farnilies ia HU ee EUN n rer inset 57 11 6 2 Th gim functione ee tere dee eee pr eed beer ener ERES 57 11 7 Nonlinear least squares and maximum likelihood model 59 TIL beast Square A RIAM RN UNES aeger et arie 59 11 7 2 Maximum Dkelbhood es ee 60 11 8 Some non standard model 61 12 Graphical porocedures esses 63 12 1 High level plotting commande 63 Schi Fhe plot O function 5 1 betas RR A be 63 12 1 2 Displaying multivariate data 64 12 1 3 Display graphics cce era fei RES Musis UU Va ERU wate nines 64 12 1 4 Arguments to high level plotting function 65 12 2 Low level plotting commande 66 12 2 1 Mathematical annotation e 67 12 2 2 Hershey vector fonts ccce ceca E aa 67 12 3 Interacting with erapbies ee mn 67 12 4 Using graphics parameters rr 68 12 4 1 Permanent changes The par Dumnction eee 68 12 4 2 Temporary changes Arguments to graphics functions ssssesesusnss 69 12 5 Graphics parameters jet 69 12 5 1 Graphical elements ee he men 70 12 5 2 Axes and tick marks ci n das aaa ii S Dem Der Ed 71 12 5 3 Figure REEL ive dont i See tea aot ede eta 71 12 5 4 Multiple figure environment 0 cece eee men 73 12 6 Device drivers ox iere tbe Decr As 74 12 6 1 PostScript diagrams for typeset documents 74 12 6 2 Multiple graphics deviecen 0 00 cece cece een eect ene eneas 75 127 Dynamic graphics i geste waist Ate CREER ot een te A e Kae hele 76 13 EE TT I3 1 Standard pac
10. In R the free variable bindings are resolved by first looking in the environment in which the function was created This is called lexical scope First we define a function called cube cube lt function n sq lt function n n n sq The variable n in the function sq is not an argument to that function Therefore it is a free variable and the scoping rules must be used to ascertain the value that is to be associated with it Under static scope S PLUS the value is that associated with a global variable named n Under lexical scope R it is the parameter to the function cube since that is the active binding for the variable n at the time the function sq was defined The difference between evaluation Chapter 10 Writing your own functions 47 in R and evaluation in S PLUS is that S PLus looks for a global variable called n while R first looks for a variable called n in the environment created when cube was invoked first evaluation in S S cube 2 Error in sq Object n not found Dumped S gt n lt 3 S gt cube 2 1 18 then the same function evaluated in R R gt cube 2 1 8 Lexical scope can also be used to give functions mutable state In the following example we show how R can be used to mimic a bank account A functioning bank account needs to have a balance or total a function for making withdrawals a function for making deposits and a function for stating the current balance We achieve this by c
11. Note particularly that recycling of short lists takes place here too thus c X Y is repeated 5 times to match the sequence 1 10 2 7 Index vectors selecting and modifying subsets of a data set Subsets of the elements of a vector may be selected by appending to the name of the vector an index vector in square brackets More generally any expression that evaluates to a vector may have subsets of its elements similarly selected by appending an index vector in square brackets immediately after the expression Such index vectors can be any of four distinct types 1 A logical vector In this case the index vector is recycled to the same length as the vector from which elements are to be selected Values corresponding to TRUE in the index vector are selected and those corresponding to FALSE are omitted For example gt y lt x is na x creates or re creates an object y which will contain the non missing values of x in the same order Note that if x has missing values y will be shorter than x Also gt x 1 is na x amp x gt 0 gt z creates an object z and places in it the values of the vector x 1 for which the corresponding value in x was both non missing and positive P paste collapse ss joins the arguments into a single character string putting ss in between e g ss There are more tools for character manipulation see the help for sub and substring Chapter 2 Simple manipulations numbers and ve
12. Using e f or file asserts non interactive use even if interactive is given Note that this does not turn on command line editing ess Windows only Set Rterm up for use by R inferior mode in ESS including assert ing interactive use without the command line editor and no buffering of stdout verbose Print more information about progress and in particular set R s option verbose to TRUE R code uses this option to control the printing of diagnostic messages debugger name d name UNIX only Run R through debugger name For most debuggers the exceptions are valgrind and recent versions of gdb further command line options are disregarded and should instead be given when starting the R executable from inside the debugger gui type g type UNIX only Use type as graphical user interface note that this also includes in teractive graphics Currently possible values for type are X11 the default and 1 2 5Gb on versions of Windows that support 3Gb per process and have the support enabled see the rw FAQ Q2 9 3 5Gb on most 64 bit versions of Windows Appendix B Invoking R 88 provided that Tcl Tk support is available Tk For back compatibility x11 and tk are accepted arch name UNIX only Run the specified sub architecture args This flag does nothing except cause the rest of the command line to be skipped this can be useful to retrieve values from it with commandAr
13. and its allies Consider the following assignments Chapter 5 Arrays and matrices 24 gt Xplus lt qr X gt b qr coef Xplus y gt fit lt qr fitted Xplus y gt res lt qr resid Xplus y These compute the orthogonal projection of y onto the range of X in fit the projection onto the orthogonal complement in res and the coefficient vector for the projection in b that is b is essentially the result of the MATLAB backslash operator It is not assumed that X has full column rank Redundancies will be discovered and removed as they are found This alternative is the older low level way to perform least squares calculations Although still useful in some contexts it would now generally be replaced by the statistical models features as will be discussed in Chapter 11 Statistical models in R page 51 5 8 Forming partitioned matrices cbind and rbind As we have already seen informally matrices can be built up from other vectors and matrices by the functions cbind and rbind Roughly cbind forms matrices by binding together matrices horizontally or column wise and rbind vertically or row wise In the assignment gt X lt cbind arg 1 arg 2 arg 3 the arguments to cbind must be either vectors of any length or matrices with the same column size that is the same number of rows The result is a matrix with the concatenated arguments arg 1 arg_2 forming the columns If some of
14. and so must be a 0 1 vector e If the response is a two column matriz it is assumed that the first column holds the number of successes for the trial and the second holds the number of failures e If the response is a factor its first level is taken as failure 0 and all other levels as success 1 Here we need the second of these conventions so we add a matrix to our data frame gt kalythos Ymat lt cbind kalythos y kalythos n kalythos y To fit the models we use gt fmp glm Ymat x family binomial link probit data kalythos gt fml lt glm Ymat x family binomial data kalythos Since the logit link is the default the parameter may be omitted on the second call To see the results of each fit we could use gt summary fmp gt summary fm1 Both models fit all too well To find the LD50 estimate we can use a simple function gt 1d50 lt function b b 1 b 2 gt ldp lt 1d50 coef fmp ldl lt 1d50 coef fm1 c ldp 1d1 The actual estimates from this data are 43 663 years and 43 601 years respectively Poisson models With the Poisson family the default link is the log and in practice the major use of this family is to fit surrogate Poisson log linear models to frequency data whose actual distribution is often multinomial This is a large and important subject we will not discuss further here It even forms a major part of the use of non gaussian generalized models overall Occasiona
15. df gt fit Nonlinear regression model model y SSmicmen x Vm K data df Vm K 212 68370711 0 06412123 residual sum of squares 1195 449 gt summary fit Formula y SSmicmen x Vm K Parameters Estimate Std Error t value Pr gt tl Vm 2 127e 02 6 947e 00 30 615 3 24e 11 K 6 412e 02 8 281e 03 7 743 1 57e 05 Residual standard error 10 93 on 10 degrees of freedom Correlation of Parameter Estimates Vm K 0 7651 11 7 2 Maximum likelihood Maximum likelihood is a method of nonlinear model fitting that applies even if the errors are not normal The method finds the parameter values which maximize the log likelihood or Chapter 11 Statistical models in R 61 equivalently which minimize the negative log likelihood Here is an example from Dobson 1990 pp 108 111 This example fits a logistic model to dose response data which clearly could also be fit by glm The data are gt x lt c 1 6907 1 7242 1 7552 1 7842 1 8113 1 8369 1 8610 1 8839 gt y lt c 6 13 18 28 52 53 61 60 gt n lt c 59 60 62 56 63 59 62 60 The negative log likelihood to minimize is gt fn function p sum y p 1 p 2 x n log 1 exp p 1 p 2 x log choose n y We pick sensible starting values and do the fit gt out lt nlm fn p c 50 20 hessian TRUE After the fitting out minimum is the negative log likelihood and out estimate are the maxi mum likelihood estimat
16. gt methods coef 1 coef aov coef Arimax coef default coef listof 5 coef nls coef summary nls Non visible functions are asterisked In this example there are six methods none of which can be seen by typing its name We can read these by either of gt getAnywhere coef aov A single object matching coef aov was found It was found in the following places registered S3 method for coef from namespace stats namespace stats with value function object 1 z lt object coef z is na z Chapter 10 Writing your own functions 50 gt getS3method coef aov function object 1 z lt object coef z is na z A function named gen cl will be invoked by the generic gen for class cl so do not name functions in this style unless they are intended to be methods The reader is referred to the R Language Definition for a more complete discussion of this mechanism Chapter 11 Statistical models in R 51 11 Statistical models in R This section presumes the reader has some familiarity with statistical methodology in particular with regression analysis and the analysis of variance Later we make some rather more ambitious presumptions namely that something is known about generalized linear models and nonlinear regression The requirements for fitting statistical models are sufficiently well defined to make it possible to construct general tools that apply in a broad spectrum of problems R prov
17. implies all the specific no restore options no restore history Control whether the history file normally file Rhistory in the directory where R was started but can be set by the environment variable R_HISTFILE should be restored at startup or not The default is to restore no Rconsole Windows only Prevent loading the Rconsole file at startup vanilla Combine no save no environ no site file no init file and no restore Under Windows this also includes no Rconsole f file file file not Rgui exe Take input from file means stdin Implies no save unless save has been set On a Unix alike shell metacharacters should be avoided in file but spaces are allowed e expression not Rgui exe Use expression as an input line One or more e options can be used but not together with f or file Implies no save unless save has been set There is a limit of 10 000 bytes on the total length of expressions used in this way Expressions containing spaces or shell metacharacters will need to be quoted Appendix B Invoking R 87 no readline UNIX only Turn off command line editing via readline This is useful when run ning R from within Emacs using the ESS Emacs Speaks Statistics package See Appendix C The command line editor page 92 for more information Command line editing is enabled for default interactive use see interactive This option also affects tilde expansion
18. key On a Mac keyboard normally no meta key is available Appendix C The command line editor 93 Horizontal motion of the cursor C a C e M b M f C b C f Go to the beginning of the command Go to the end of the line Go back one word Go forward one word Go back one character Go forward one character On most terminals you can also use the left and right arrow keys instead of C b and C f respectively Editing and re submission text C f text DEL C d M d C k C y C t M 1 M c 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 cursor and save it Delete from cursor to end of command and save it Insert 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 final RET terminates the command line editing sequence The readline key bindings can be customized in the usual way via a inputrc file These customizations can be conditioned on application R that is by including a section like if R NC xd ot no NMn endif Appendix D Function and variable index Appendix D Function and variable index D ioa da tese men ei ee a 9 EHE 9 06 KEE 22 MO da Sanka A ue bi a 21 a
19. log x log y log xy Causes the z y or both axes to be logarithmic This will work for many but not all types of plot type The type argument controls the type of plot produced as follows type p Plot individual points the default type 1 Plot lines type b Plot points connected by lines both type o Plot points overlaid by lines type h Plot vertical lines from points to the zero axis high density type s type S Step function plots In the first form the top of the vertical defines the point in the second the bottom type n No plotting at all However axes are still drawn by default and the coordinate system is set up according to the data Ideal for creating plots with subsequent low level graphics functions xlab string ylab string Axis labels for the x and y axes Use these arguments to change the default labels usually the names of the objects used in the call to the high level plotting function main string Figure title placed at the top of the plot in a large font sub string Sub title placed just below the x axis in a smaller font Chapter 12 Graphical procedures 66 12 2 Low level plotting commands Sometimes the high level plotting functions don t produce exactly the kind of plot you desire In this case low level plotting commands can be used to add extra information such as points lines or text to the current plot Some of the more useful low level plotting functions are
20. mation from an existing plot using a pointing device such as a mouse In addition R maintains a list of graphical parameters which can be manipulated to customize your plots This manual only describes what are known as base graphics A separate graphics sub system in package grid coexists with base it is more powerful but harder to use There is a recommended package lattice https CRAN R project org package lattice which builds on grid and provides ways to produce multi panel plots akin to those in the Trellis system in S 12 1 High level plotting commands High level plotting functions are designed to generate a complete plot of the data passed as ar guments to the function Where appropriate axes labels and titles are automatically generated unless you request otherwise High level plotting commands always start a new plot erasing the current plot if necessary 12 1 1 The plot function One of the most frequently used plotting functions in R is the plot function This is a generic function the type of plot produced is dependent on the type or class of the first argument plot x y plot xy If x and y are vectors plot x y produces a scatterplot of y against x The same effect can be produced by supplying one argument second form as either a list containing two elements x and y or a two column matrix plot x If x is a time series this produces a time series plot If x is a numeric vector it produces
21. see the help for path expand min vsize N min nsize N For expert use only set the initial trigger sizes for garbage collection of vector heap in bytes and cons cells number respectively Suffix M specifies megabytes or millions of cells respectively The defaults are 6Mb and 350k respectively and can also be set by environment variables R_NSIZE and R_VSIZE max ppsize N Specify the maximum size of the pointer protection stack as N locations This defaults to 10000 but can be increased to allow large and complicated calculations to be done Currently the maximum value accepted is 100000 max mem size N Windows only Specify a limit for the amount of memory to be used both for R objects and working areas This is set by default to the smaller of the amount of physical RAM in the machine and for 32 bit R 1 5Gb and must be between 32Mb and the maximum allowed on that version of Windows quiet silent q Do not print out the initial copyright and welcome messages slave Make R run as quietly as possible This option is intended to support programs which use R to compute results for them It implies quiet and no save interactive UNIX only Assert that R really is being run interactively even if input has been redirected use if input is from a FIFO or pipe and fed from an interactive program The default is to deduce that R is being run interactively if and only if stdin is connected to a terminal or pty
22. the same and this becomes the dimension vector of the result So if A B and C are all similar arrays then gt D lt 2xAx B C d1 makes D a similar array with its data vector being the result of the given element by element operations However the precise rule concerning mixed array and vector calculations has to be considered a little more carefully 5 4 1 Mixed vector and array arithmetic The recycling rule The precise rule affecting element by element mixed calculations with vectors and arrays is somewhat quirky and hard to find in the references From experience we have found the following to be a reliable guide e The expression is scanned from left to right e Any short vector operands are extended by recycling their values until they match the size of any other operands e Aslong as short vectors and arrays only are encountered the arrays must all have the same dim attribute or an error results e Any vector operand longer than a matrix or array operand generates an error e Ifarray structures are present and no error or coercion to vector has been precipitated the result is an array structure with the common dim attribute of its array operands Chapter 5 Arrays and matrices 21 5 5 The outer product of two arrays An important operation on arrays is the outer product If a and b are two numeric arrays their outer product is an array whose dimension vector is obtained by concatenating their two dimension vectors o
23. 10 matrix 3 4 The class of an object All objects in R have a class reported by the function class For simple vectors this is just the mode for example numeric logical character or list but matrix array factor and data frame are other possible values A special attribute known as the class of the object is used to allow for an object oriented style of programming in R For example if an object has class data frame it will be printed in a certain way the plot function will display it graphically in a certain way and other so called generic functions such as summary will react to it as an argument in a way sensitive to its class To remove temporarily the effects of class use the function unclass For example if winter has the class data frame then A different 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 unclass winter will print it as an ordinary list Only in rather special situations do you need to use this facility but one is when you are learning to come to terms with the idea of class and generic functions Generic functions and classes will be discussed further in Section 10 9 Object orientation page 49 but only briefly Chapter 4 Ordered and unordered factors 16 4 Ordered and unordered factors A factor is
24. 22 DM A ed ade Yee we ae TRS 18 Got chart ENEE Se EH NEE TER 65 o EE 56 E cdi airada adidas ada 35 UA A ae 32 eigen E 23 li A A O Reha egit 40 EXrOI4 EE 55 XML EEN 4 EE e EE E 8 F Pacto ea we RE en od wae 16 FALSE E ta ea Wide quU Ie Moa del 9 LAA A AA AE 34 DO ti AA EE E E ATA 40 formula 2551 24 A AAA 54 EU 42 E EE 9 EE 49 getS3method rr 49 EE 57 H liue PE 4 help searcha nii En eMe EE eie eus 4 Belg Stage Ae e abhi div baht AE Ze LER 4 GSE EE 34 64 I identify Via ir does 68 DE o ste px tems EEN 40 El a A ata 40 E A 65 DS ATA RR ANO 9 O EE E n D ER M Sa 10 DEE Geet Y pee PIDE E Y pene ebe Ze 74 K ESOS ida da ds a a 36 L l gend 2 aa Sas bv X Wu T ncs 66 95 levels do xs AES XAR EPI RESPICE ES 16 lineSi si6 yv e lE RRERRMERERPSMUSIFEQRe quee 66 listo ibo RA Or NER Ede WCENE NE SN Fred ed E 26 Amst ege LIONS Ue eie E eats 54 lm oou e e EE DUET uve des 61 LOCATOR e BIER RM IBIDEM 68 TESS e ee eege xev taedet aus 61 La AAA AAA 8 LAS Metals Sd el 61 TEA a T nt ede 23 M MACS hya nes A erre erue wwe Red ege We 61 MAC A ecrire e ANE We N yak quw C HE DS 8 MOAN ranas EE 8 Mt iia ita ad 49 MI ti Aa RA 8 Moda tt EN Pied E desde EUR Ms 13 N NaN erue p Sark cede a Ene ru Ue ERR C CER 9 NA ee td dat ates 9 nCOl tvbsssntbt da A ege Ho 22 EE 41 Ee cas 59 60 61 Le ia A E EEN A 61 DUO iris a A A A e a 59 LrOWilsenilLehhrQx pr dida 22 O e E EE E NE abe De eo Lena 59 ee EE 8 ordered EE ELAD e i
25. 35 too much smoothing it usually does for interesting densities Better automated methods of bandwidth choice are available and in this example bw SJ gives a good result Histogram of eruptions tov 2 S i o Relative Frequency e f LI INN M Lf 1 Lil JIDD 15 2 0 25 3 0 3 5 4 0 45 5 0 eruptions We can plot the empirical cumulative distribution function by using the function ecdf gt plot ecdf eruptions do points FALSE verticals TRUE This distribution is obviously far from any standard distribution How about the right hand mode say eruptions of longer than 3 minutes Let us fit a normal distribution and overlay the fitted CDF gt long lt eruptions eruptions gt 3 gt plot ecdf long do points FALSE verticals TRUE gt x lt seq 3 5 4 0 01 gt lines x pnorm x mean mean long sd sqrt var long lty 3 ecdf long Quantile quantile Q Q plots can help us examine this more carefully par pty s arrange for a square figure region qqnorm long qqline long Chapter 8 Probability distributions 36 which shows a reasonable fit but a shorter right tail than one would expect from a normal distribution Let us compare this with some simulated data from a t distribution Normal Q Q Plot Sample Quantiles Theoretical Quan
26. Arrays and matrices 23 5 7 3 Eigenvalues and eigenvectors The function eigen Sm calculates the eigenvalues and eigenvectors of a symmetric matrix Sm The result of this function is a list of two components named values and vectors The assignment gt ev lt eigen Sm will assign this list to ev Then ev val is the vector of eigenvalues of Sm and ev vec is the matrix of corresponding eigenvectors Had we only needed the eigenvalues we could have used the assignment gt evals lt eigen Sm values evals now holds the vector of eigenvalues and the second component is discarded If the expression gt eigen Sm is used by itself as a command the two components are printed with their names For large matrices it is better to avoid computing the eigenvectors if they are not needed by using the expression gt evals lt eigen Sm only values TRUE values 5 7 4 Singular value decomposition and determinants The function svd M takes an arbitrary matrix argument M and calculates the singular value decomposition of M This consists of a matrix of orthonormal columns U with the same column space as M a second matrix of orthonormal columns V whose column space is the row space of M and a diagonal matrix of positive entries D such that M U D t V D is actually returned as a vector of the diagonal elements The result of svd M is actually a list of three components named d u and v with evident meanings If M is in fa
27. R from the command line When working at a command line on UNIX or Windows the command R can be used both for starting the main R program in the form R options lt infile gt outfile or via the R CMD interface as a wrapper to various R tools e g for processing files in R documentation format or manipulating add on packages which are not intended to be called directly At the Windows command line Rterm exe is preferred to R You need to ensure that either the environment variable TMPDIR is unset or it points to a valid place to create temporary files and directories Most options control what happens at the beginning and at the end of an R session The startup mechanism is as follows see also the on line help for topic Startup for more informa tion and the section below for some Windows specific details e Unless no environ was given H searches for user and site files to process for setting environment variables The name of the site file is the one pointed to by the environ ment variable R ENVIRON if this is unset R_HOME etc Renviron site is used if it exists The user file is the one pointed to by the environment variable R ENVIRON USER if this is set otherwise files Renviron in the current or in the user s home directory in that order are searched for These files should contain lines of the form name value See help Startup for a precise description Variables you might want to set inclu
28. SESSION EE 4 1 7 Getting help with functions and features 4 1 8 R commands case sensitivity etc 4 1 9 Recall and correction of previous Commande 5 1 10 Executing commands from or diverting output toa le eee eee eee 5 1 11 Data permanency and removing objects 0 cece cece eee eee eet eens 5 2 Simple manipulations numbers and vectors 7 2 1 Vectors and assignment 0 cece ccc eee cede eben mr 7 2 2 Vector arithmeticice iis ced See EE ea ees Sa Rad yee ae ea HEAT EN UNS 7 2 3 Generating regular sequences 0 cece ccc ented ene e ene eenneees 8 2A L pical Veetorss i fannie a a let ena eal 9 2 5 Missing values NEE sek Oey nee EENS 9 20 Character Vectors tie bee ed ea ter bee at ae eet aed 10 2 7 Index vectors selecting and modifying subsets of a data set 10 2 8 Other types of objects o ee ia Ment ee e Ete EE ce ate 11 3 Objects their modes and attributes 13 3 1 Intrinsic attributes mode and length 0 0 c cece eee e 13 3 2 Changing the length of an object 0 cnet n 14 3 3 Getting and setting attributes sisse me 14 KEN of am object EEN 14 4 Ordered and unordered foctors suasanana 16 GE lege 16 4 2 The function tapply and ragged aas 16 4 3 Ordered tacto ii A e d EEN 17 5 AAPrays and Matrices sonia 18 b T ATINA Ses A ER Eee EID DEIN E cec UI Memes a 18 5 2 Array indexing Subsections of an array sssssssssses
29. The names of these give a good clue to their purpose but for full details see the on line help 11 6 Generalized linear models Generalized linear modeling is a development of linear models to accommodate both non normal response distributions and transformations to linearity in a clean and straightforward way A generalized linear model may be described in terms of the following sequence of assumptions e There is a response y of interest and stimulus variables x1 2 whose values influence the distribution of the response e The stimulus variables influence the distribution of y through a single linear function only This linear function is called the linear predictor and is usually written n Biz Baza Bp hence z has no influence on the distribution of y if and only if 6 0 e The distribution of y is of the form fv Y u p exp f Juin y Alu Ty p where y 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 u 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 e The mean u is a smooth invertible function of the linear predictor p m np mcm u n and this inverse function is called the link function These assumptions are loose enough to encompass a wide class of models useful in statistica
30. also appear on the receiving end of an assignment in which case the assignment operation is performed only on those elements of the vector The expression must be of the form vector index vector as having an arbitrary expression in place of the vector name does not make much sense here For example gt x is na x lt 0 replaces any missing values in x by zeros and gt yly lt 0 yly 0 has the same effect as gt y abs y 2 8 Other types of objects Vectors are the most important type of object in R but there are several others which we will meet more formally in later sections e 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 e factors provide compact ways to handle categorical data See Chapter 4 Factors page 16 e lists are a general form of vector in which the various elements need not be of the same type and are often themselves vectors or lists Lists provide a convenient way to return the results of a statistical computation See Section 6 1 Lists page 26 e data frames are matrix like structures in which the columns can be of different types Think of data frames as data matrices with one row per observational unit but with possibly Chapter 2 Simple manipulations numbers and vectors 12 both numerical and categor
31. attach dummy Make the columns in the data frame visible as variables lrf lowess x y Make a nonparametric local regression function plot x y Standard point plot lines x lrf y Add in the local regression abline 0 1 lty 3 The true regression line intercept 0 slope 1 abline coef fm Unweighted regression line Appendix A A sample session 83 abline coef fm1 col red Weighted regression line detach Remove data frame from the search path plot fitted fm resid fm xlab Fitted values ylab Residuals main Residuals vs Fitted A standard regression diagnostic plot to check for heteroscedasticity Can you see it qqnorm resid fm main Residuals Rankit Plot A normal scores plot to check for skewness kurtosis and outliers Not very useful here rm fm fmi lrf x dummy Clean up again The next section will look at data from the classical experiment of Michelson to measure the speed of light This dataset is available in the morley object but we will read it to illustrate the read table function filepath system file data morley tab package datasets filepath Get the path to the data file file show filepath Optional Look at the file mm lt read table filepath mm Read in the Michelson data as a data frame and look at it There are five exper iments column Expt and each has 20 runs column Run and s1 is the recorded speed of light suitably cod
32. chmod 755 runfoo it can be invoked for different arguments by runfoo argl arg2 For further options see help Rscript This writes R output to stdout and stderr and this can be redirected in the usual way for the shell running the command If you do not wish to hardcode the path to Rscript but have it in your path which is normally the case for an installed R except on Windows but e g OS X users may need to add usr local bin to their path use usr bin env Rscript At least in Bourne and bash shells the mechanism does not allow extra arguments like st usr bin env 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 91 chem lt scan n 24 2 90 3 10 3 40 3 40 3 70 3 70 2 80 2 50 2 40 2 40 2 70 2 20 5 28 3 37 3 03 3 03 28 95 3 77 3 40 2 20 3 50 3 60 3 70 3 70 and stdin refers to the script file to allow such traditional usage If you want to refer to the process s stdin use stdin as a file connection e g scan stdin Another way to write executable script files suggested by Francois Pinard is to use a here document like bin sh environment variables can be set here R slave other options EOF R program goes here EOF but here stdin refers to the program source and stdin will not be usable Short scripts can be passed to Rscript on the command line v a the e flag Empty scripts a
33. could type ESCb The ESC character sequences are also allowed on terminals with real Meta keys Note that case is significant for Meta characters C 2 Editing actions The R program keeps a history of the command lines you type including the erroneous lines and commands in your history may be recalled changed if necessary and re submitted as new commands In Emacs style command line editing any straight typing you do while in this editing phase causes the characters to be inserted in the command you are editing displacing any characters to the right of the cursor In vi mode character insertion mode is started by M i or M a characters are typed and insertion mode is finished by typing a further ESC The default is Emacs style and only that is described here for vi mode see the readline documentation Pressing the RET command at any time causes the command to be re submitted Other editing actions are summarized in the following table C 3 Command line editor summary Command recall and vertical motion C p Go to the previous command backwards in the history C n Go to the next command forwards in the history C r text Find the last command with the text string in it On most terminals you can also use the up and down arrow keys instead of C p and C n respectively The Emacs Speaks Statistics package see the URL http ESS R project org 2 Ona PC keyboard this is usually the Alt key occasionally the Windows
34. f a b fa fb a0 eps lim fun Y function funl is only visible inside area d a b 2 h lt b a 4 fd lt f d al lt h fa fd a2 lt h fd fb if abs a0 al a2 lt eps lim 0 return al a2 else return fun f a d fa fd al eps lim 1 fun fun f d b fd fb a2 eps lim 1 fun fa lt f a fb lt f b a0 lt fa fb b a 2 fun1 f a b fa fb a0 eps lim fun1 10 7 Scope The discussion in this section is somewhat more technical than in other parts of this document However it details one of the major differences between S PLUS and R The symbols which occur in the body of a function can be divided into three classes formal parameters local variables and free variables The formal parameters of a function are those occurring in the argument list of the function Their values are determined by the process of binding the actual function arguments to the formal parameters Local variables are those whose values are determined by the evaluation of expressions in the body of the functions Variables which are not formal parameters or local variables are called free variables Free variables become local variables if they are assigned to Consider the following function definition f function x y lt 2 x print x print y print z In this function x is a formal parameter y is a local variable and z is a free variable
35. first place This automatic adjustment of lengths of an object is used often for example in the scan O function for input see Section 7 2 The scan function page 31 Conversely to truncate the size of an object requires only an assignment to do so Hence if alpha is an object of length 10 then gt alpha alpha 2 1 5 makes it an object of length 5 consisting of just the former components with even index The old indices are not retained of course We can then retain just the first three values by gt length alpha 3 and vectors can be extended by missing values in the same way 3 3 Getting and setting attributes The function attributes object returns a list of all the non intrinsic attributes currently defined for that object The function attr object name can be used to select a specific attribute These functions are rarely used except in rather special circumstances when some new attribute is being created for some particular purpose for example to associate a creation date or an operator with an R object The concept however is very important Some care should be exercised when assigning or deleting attributes since they are an integral part of the object system used in R When it is used on the left hand side of an assignment it can be used either to associate a new attribute with object or to change an existing one For example gt attr z dim c 10 10 allows R to treat z as if it were a 10 by
36. graphical comparison of the two samples A lt scan 79 98 80 04 80 02 80 04 80 03 80 03 80 04 79 97 80 05 80 03 80 02 80 00 80 02 B scan 80 02 79 94 79 98 79 97 79 97 80 03 79 95 79 97 boxplot A B which indicates that the first group tends to give higher results than the second 79 96 79 98 80 00 80 02 80 04 79 94 To test for the equality of the means of the two examples we can use an unpaired t test by t test A B Welch Two Sample t test data A and B t 3 2499 df 12 027 p value 0 00694 alternative hypothesis true difference in means is not equal to O 95 percent confidence interval 0 01385526 0 07018320 sample estimates mean of x mean of y 80 02077 79 97875 which does indicate a significant difference assuming normality By default the R function does not assume equality of variances in the two samples in contrast to the similar S PLUS t test function We can use the F test to test for equality in the variances provided that the two samples are from normal populations gt var test A B F test to compare two variances Chapter 8 Probability distributions 38 data A and B F 0 5837 num df 12 denom d 7 p value 0 3938 alternative hypothesis true ratio of variances is not equal to 1 95 percent confidence interval 0 1251097 2 1052687 sample estimates ratio of variances 0 5837405 which shows no evidence of a signific
37. identified with the labels in the character vector legend At least one other argument v a vector the same length as legend with the corre sponding values of the plotting unit must also be given as follows legend fill v Colors for filled boxes legend col v Colors in which points or lines will be drawn legend lty v Line styles legend lwd v Line widths legend pch v Plotting characters character vector Chapter 12 Graphical procedures 67 title main sub Adds a title main to the top of the current plot in a large font and optionally a sub title sub at the bottom in a smaller font axis side Adds an axis to the current plot on the side given by the first argument 1 to 4 counting clockwise from the bottom Other arguments control the positioning of the axis within or beside the plot and tick positions and labels Useful for adding custom axes after calling plot with the axes FALSE argument Low level plotting functions usually require some positioning information e g r and y co ordinates to determine where to place the new plot elements Coordinates are given in terms of user coordinates which are defined by the previous high level graphics command and are chosen based on the supplied data Where x and y arguments are required it is also sufficient to supply a single argument being a list with elements named x and y Similarly a matrix with two columns is also valid input In this way function
38. in the same R session Many usages of PostScript output will be to incorporate the figure in another document This works best when encapsulated PostScript is produced R always produces conformant output but only marks the output as such when the onefile FALSE argument is supplied This unusual notation stems from S compatibility it really means that the output will be a single page which is part of the EPSF specification Thus to produce a plot for inclusion use something like gt postscript ploti eps horizontal FALSE onefile FALSE height 8 width 6 pointsize 10 12 6 2 Multiple graphics devices In advanced use of R it is often useful to have several graphics devices in use at the same time Of course only one graphics device can accept graphics commands at any one time and this is known as the current device When multiple devices are open they form a numbered sequence with names giving the kind of device at any position The main commands used for operating with multiple devices and their meanings are as follows X110 UNIX windows win printer win metafile Windows quartzO OS X postscript pdf O pngO jpegO tiffO bitmapO Each new call to a device driver function opens a new graphics device thus extending by one the device list This device becomes the current device to which graphics output will be sent dev list Returns the number and name of all active devices The device at position 1 o
39. is the distance from the axis label to the axis position in text lines The second component is the distance to the tick labels and the final component is the distance from the axis position to the axis line usually zero Positive numbers measure outside the plot region negative numbers inside tck 0 01 Length of tick marks as a fraction of the size of the plotting region When tck is small less than 0 5 the tick marks on the zr and y axes are forced to be the same size A value of 1 gives grid lines Negative values give tick marks outside the plotting region Use tck 0 01 and mgp c 1 1 5 0 for internal tick marks xaxs r yaxs i Axis styles for the x and y axes respectively With styles i internal and r the default tick marks always fall within the range of the data however style r leaves a small amount of space at the edges S has other styles not implemented in R 12 5 3 Figure margins A single plot in R is known as a figure and comprises a plot region surrounded by margins possibly containing axis labels titles etc and usually bounded by the axes themselves Chapter 12 Graphical procedures 72 A typical figure is ot a a A A A EE Plot region mai 2 mail 1 Margin Graphics parameters controlling figure layout include mai c 1 0 5 0 5 0 Widths of the bottom left top and right margins respectively measured in inches mar c 4 2 2 1 Similar to mai except the measu
40. levels varieties K lt as vector table blocks remove dim attr R lt as vector table varieties remove dim attr Chapter 10 Writing your own functions 45 N lt table blocks varieties A lt 1 sqrt K N rep 1 sqrt R rep b v sv svd A list eff 1 sv d 2 blockcv sv u varietycv sv v It is numerically slightly better to work with the singular value decomposition on this occasion rather than the eigenvalue routines The result of the function is a list giving not only the efficiency factors as the first component but also the block and variety canonical contrasts since sometimes these give additional useful qualitative information 10 6 2 Dropping all names in a printed array For printing purposes with large matrices or arrays it is often useful to print them in close block form without the array names or numbers Removing the dimnames attribute will not achieve this effect but rather the array must be given a dimnames attribute consisting of empty strings For example to print a matrix X gt temp lt X gt dimnames temp lt list rep nrow X rep ncol X gt temp rm temp This can be much more conveniently done using a function no dimnames shown below as a wrap around to achieve the same result It also illustrates how some effective and useful user functions can be quite short no dimnames function a Remove all dimension names from an array for com
41. random A neat way of doing this uses the outer function twice gt d lt outer 0 9 0 9 fr table outer d d gt plot as numeric names fr fr type h xlab Determinant ylab Frequency Notice the coercion of the names attribute of the frequency table to numeric in order to recover the range of the determinant values The obvious way of doing this problem with for loops to be discussed in Chapter 9 Loops and conditional execution page 40 is so inefficient as to be impractical It is also perhaps surprising that about 1 in 20 such matrices is singular 5 6 Generalized transpose of an array The function aperm a perm may be used to permute an array a The argument perm must be a permutation of the integers 1 k where k is the number of subscripts in a The result of the function is an array of the same size as a but with old dimension given by perm j becoming the new j th dimension The easiest way to think of this operation is as a generalization of transposition for matrices Indeed if A is a matrix that is a doubly subscripted array then B given by gt B aperm A c 2 1 is just the transpose of A For this special case a simpler function t is available so we could have used B lt t A Chapter 5 Arrays and matrices 22 5 7 Matrix facilities As noted above a matrix is just an array with two subscripts However it is such an important special case it needs a separate disc
42. set of data Given a univariate set of data we can examine its distribution in a large number of ways The simplest is to examine the numbers Two slightly different summaries are given by summary and fivenum and a display of the numbers by stem a stem and leaf plot gt attach faithful gt summary eruptions Min ist Qu Median Mean 3rd Qu Max 1 600 2 163 4 000 3 488 4 454 5 100 gt fivenum eruptions 1 1 6000 2 1585 4 0000 4 4585 5 1000 gt stem eruptions The decimal point is 1 digit s to the left of the 16 070355555588 18 000022233333335577777777888822335777888 20 00002223378800035778 22 0002335578023578 24 00228 26 23 28 080 30 7 32 2337 34 250077 36 0000823577 38 2333335582225577 40 0000003357788888002233555577778 42 03335555778800233333555577778 44 02222335557780000000023333357778888 46 0000233357700000023578 48 00000022335800333 50 0370 A stem and leaf plot is like a histogram and R has a function hist to plot histograms gt hist eruptions make the bins smaller make a plot of density gt hist eruptions seq 1 6 5 2 0 2 prob TRUE gt lines density eruptions bw 0 1 gt rug eruptions show the actual data points More elegant density plots can be made by density and we added a line produced by density in this example The bandwidth bw was chosen by trial and error as the default gives Chapter 8 Probability distributions
43. set of symbols which can be used in R names depends on the operating system and country within which R is being run technically on the locale in use Normally all alphanumeric symbols are allowed and in some countries this includes accented letters plus and _ with the restriction that a name must start with or a letter and if it starts with the second character must not be a digit Names are effectively unlimited in length Elementary commands consist of either expressions or assignments If an expression is given as a command it is evaluated printed unless specifically made invisible and the value is lost An assignment also evaluates an expression and passes the value to a variable but the result is not automatically printed 1 For portable R code including that to be used in R packages only A Za z0 9 should be used Chapter 1 Introduction and preliminaries 5 Commands are separated either by a semi colon or by a newline Elementary commands can be grouped together into one compound expression by braces C and Y Comments can 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 th
44. squares estimate of the regression coefficients This would ordinarily be done with the qr 3 function however this is sometimes a bit tricky to use directly and it pays to have a simple function such as the following to use it safely Thus given a n by 1 vector y and an n by p matrix X then X y is defined as X7 X XTy where X7 X is a generalized inverse of X X gt bslash lt function X y 1 X qr X qr coef X y After this object is created it may be used in statements such as gt regcoeff lt bslash Xmat yvar and so on Chapter 10 Writing your own functions 43 The classical R function 1sfit does this job quite well and more It in turn uses the functions qr and qr coef in the slightly counterintuitive way above to do this part of the calculation Hence there is probably some value in having just this part isolated in a simple to use function if it is going to be in frequent use If so we may wish to make it a matrix binary operator for even more convenient use 10 2 Defining new binary operators Had we given the bslash function a different name namely one of the form anything it could have been used as a binary operator in expressions rather than in function form Suppose for example we choose for the internal character T he function definition would then start as gt h 4 lt function X y 1 Note the use of quote marks The function could then be used as X 4 y The backs
45. term 2 op 3 term 3 where response is a vector or matrix or expression evaluating to a vector or matrix defining the response variable s op i is an operator either or implying the inclusion or exclusion of a term in the model the first is optional term i is either e vector or matrix expression or 1 e a factor or e a formula expression consisting of factors vectors or matrices connected by formula operators In all cases each term defines a collection of columns either to be added to or removed from the model matrix A 1 stands for an intercept column and is by default included in the model matrix unless explicitly removed The formula operators are similar in effect to the Wilkinson and Rogers notation used by such programs as Glim and Genstat One inevitable change is that the operator becomes since the period is a valid name character in R The notation is summarized below based on Chambers amp Hastie 1992 p 29 Y M Y is modeled as M M_1 M_2 Include M_1 and M_2 Chapter 11 Statistical models in R 53 M_1 M_2 Include M_1 leaving out terms of M_2 M1 9M2 The tensor product of M_1 and M_2 If both terms are factors then the subclasses factor M 1 hinh M 2 Similar to M 1 M 2 but with a different coding M1 M2 M1 M2 M_1 M_2 M1 M2 M1 M2 in M1 Mn All terms in M together with interactions up to order n ICM Insulate M Inside M all operators have their normal arithme
46. the proportion of text that appears to the left of the plotting position so a value of 0 1 leaves a gap of 1096 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 71 cex axis cex lab cex main cex sub The character expansion to be used for axis annotation x and y labels main and sub titles respectively 12 5 2 Axes and tick marks Many of R s high level plots have axes and you can construct axes yourself with the low level axis graphics function Axes have three main components the axis line line style controlled by the 1ty graphics parameter the tick marks which mark off unit divisions along the axis line and the tick labels which mark the units These components can be customized with the following graphics parameters lab c 5 7 12 The first two numbers are the desired number of tick intervals on the x and y axes respectively The third number is the desired length of axis labels in characters including the decimal point Choosing a too small value for this parameter may result in all tick labels being rounded to the same number las 1 Orientation of axis labels O means always parallel to axis 1 means always horizon tal and 2 means always perpendicular to the axis mgp c 3 1 0 Positions of axis components The first component
47. to an external file record Lis The command gt sink restores it to the console once again 1 11 Data permanency and removing objects The entities that R creates and manipulates are known as objects These may be variables arrays of numbers character strings functions or more general structures built from such components During an R session objects are created and stored by name we discuss this process in the next session The R command gt objects alternatively 1s 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 definition 3 some of the consoles will not allow you to enter more and amongst those which do some will silently discard the excess and some will use it as the start of the next line of unlimited length Chapter 1 Introduction and preliminaries 6 gt rm x y z ink junk temp foo bar All objects created during an R session can be stored permanently in a file for use in future R sessions At the end of each R session you are given the opportunity to save all the currently available objects If you indicate that you want to do this the objects are written to a file called RData in the current directory and the command lines used in the session are saved to a file
48. via the package rggobi https CRAN R project org package rggobi described at http www ggobi org rggobi Also package rgl https CRAN R project org package rgl provides ways to interact with 3D plots for example of surfaces Chapter 13 Packages 77 13 Packages All R functions and datasets are stored in packages Only when a package is loaded are its contents available This is done both for efficiency the full list would take more memory and would take longer to search than a subset and to aid package developers who are protected from name clashes with other code The process of developing packages is described in Section Creating R packages in Writing R Extensions Here we will describe them from a user s point of view To see which packages are installed at your site issue the command gt library with no arguments To load a particular package e g the boot https CRAN R project org package boot package containing functions from Davison amp Hinkley 1997 use a com mand like gt library boot Users connected to the Internet can use the install packages and update packages functions available through the Packages menu in the Windows and OS X GUIs see Section Installing packages in R Installation and Administration 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 avai
49. y f image x y fa Make some high density image plots of which you can get hardcopies if you wish objects O rm x y f fa and clean up before moving on R can do complex arithmetic also th seq pi pi len 100 z lt exp 1i th 1i is used for the complex number 7 par pty s plot z type 1 Plotting complex arguments means plot imaginary versus real parts This should be a circle w lt rnorm 100 rnorm 100 1i Suppose we want to sample points within the unit circle One method would be to take complex numbers with standard normal real and imaginary parts w ifelse Mod w gt 1 1 w w and to map any outside the circle onto their reciprocal plot w xlim c 1 1 ylim c 1 1 pch xlab x ylab y lines z All points are inside the unit circle but the distribution is not uniform w lt sqrt runif 100 exp 2 pi runif 100 1i plot w xlim c 1 1 ylim c 1 1 pch xlab x ylab y lines z The second method uses the uniform distribution The points should now look more evenly spaced over the disc rm th w z Clean up again qO 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 85 Appendix B Invoking R Users of R on Windows or OS X should read the OS specific section first but command line use is also supported B 1 Invoking
50. 2 dimensional array The dimensions are indexed from one up to the values given in the dimension vector A vector can be used by R as an array only if it has a dimension vector as its dim attribute Suppose for example z is a vector of 1500 elements The assignment gt dim z c 3 5 100 gives it the dim attribute that allows it to be treated as a 3 by 5 by 100 array Other functions such as matrix and array O are available for simpler and more natural looking assignments as we shall see in Section 5 4 The array function page 20 The values in the data vector give the values in the array in the same order as they would occur in FORTRAN that is column major order with the first subscript moving fastest and the last subscript slowest For example if the dimension vector for an array say a is c 3 4 2 then there are3 x 4 x 2 24 entries in a and the data vector holds them in the order a 1 1 1 a 2 1 1 a 2 4 2 a 3 4 2 Arrays can be one dimensional such arrays are usually treated in the same way as vectors including when printing but the exceptions can cause confusion 5 2 Array indexing Subsections of an array Individual elements of an array may be referenced by giving the name of the array followed by the subscripts in square brackets separated by commas More generally subsections of an array may be specified by giving a sequence of indez vectors in place of subscripts however if any index posi
51. 4 OS facilities 79 14 OS facilities R has quite extensive facilities to access the OS under which it is running this allows it to be used as a scripting language and that ability is much used by R itself for example to install packages Because R s own scripts need to work across all platforms considerable effort has gone into make the scripting facilities as platform independent as is feasible 14 1 Files and directories There are many functions to manipulate files and directories Here are pointers to some of the more commonly used ones To create an empty file or directory use file create or create dir These are the analogues of the POSIX utilities touch and mkdir For temporary files and directories in the R session directory see tempfile Files can be removed by either file remove or unlink the latter can remove directory trees For directory listings use list files also available as dir or list dirs These can select files using a regular expression to select by wildcards use Sys glob Many types of information on a filepath including for example if it is a file or directory can be found by file info There are several ways to find out if a file exists a file can exist on the filesystem and not be visible to the current user There are functions file exists file access and file test with various versions of this test file test is a version of the POSIX test command for those familiar with shell scripti
52. A B Two sample Kolmogorov Smirnov test data A and B D 0 5962 p value 0 05919 alternative hypothesis two sided Chapter 8 Probability distributions Warning message cannot compute correct p values with ties in ks test A B 39 Chapter 9 Grouping loops and conditional execution 40 9 Grouping loops and conditional execution 9 1 Grouped expressions R is an expression language in the sense that its only command type is a function or expression which returns a result Even an assignment is an expression whose result is the value assigned and it may be used wherever any expression may be used in particular multiple assignments are possible Commands may be grouped together in braces expr 1 expr_m in which case the value of the group is the result of the last expression in the group evaluated Since such a group is also an expression it may for example be itself included in parentheses and used a part of an even larger expression and so on 9 2 Control statements 9 2 1 Conditional execution if statements The language has available a conditional construction of the form gt if expr 1 expr 2 else expr 3 where expr 1 must evaluate to a single logical value and the result of the entire expression is then evident The short circuit operators amp amp and are often used as part of the condition in an if statement Whereas amp and apply element wise to vectors amp amp and apply to vec
53. An Introduction to R Notes on R A Programming Environment for Data Analysis and Graphics Version 3 2 3 2015 12 10 W N Venables D M Smith and the R Core Team This manual is for R version 3 2 3 2015 12 10 Copyright c 1990 W N Venables Copyright 1992 W N Venables amp D M Smith Copyright 1997 R Gentleman amp R Ihaka Copyright 1997 1998 M Maechler Copyright c 1999 2015 R Core Team Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one Permission is granted to copy and distribute translations of this manual into an other language under the above conditions for modified versions except that this permission notice may be stated in a translation approved by the R Core Team Table of Contents IEN 1 1 Introduction and preliminaries 00005 2 Ts Aber environ Menten e voee t Anka been xe eb ERES 2 1 2 Related software and documentation 2 1 32 and ostatlsUCs ues vafer A SUUS iN Dieta was eter notar Ho A 2 1 4 Rand the window system 0 ccc e n 3 1 5 Using R interactively iii ARR KEE Re ENARRARE 3 1 6 Adr introductory
54. Script graphics files pdf Produces a PDF file which can also be included into PDF files pngO Produces a bitmap PNG file Not always available see its help page jpegO Produces a bitmap JPEG file best used for image plots Not always available see its help page When you have finished with a device be sure to terminate the device driver by issuing the command gt dev off This ensures that the device finishes cleanly for example in the case of hardcopy devices this ensures that every page is completed and has been sent to the printer This will happen automatically at the normal end of a session 12 6 1 PostScript diagrams for typeset documents By passing the file argument to the postscript device driver function you may store the graphics in PostScript format in a file of your choice The plot will be in landscape orientation unless the horizontal FALSE argument is given and you can control the size of the graphic with the width and height arguments the plot will be scaled as appropriate to fit these dimensions For example the command gt postscript file ps horizontal FALSE height 5 pointsize 10 will produce a file containing PostScript code for a figure five inches high perhaps for inclusion in a document It is important to note that if the file named in the command already exists Chapter 12 Graphical procedures 75 it will be overwritten This is the case even if the file was only created earlier
55. Working with data frames A useful convention that allows you to work with many different problems comfortably together in the same working directory is e gather together all variables for any well defined and separate problem in a data frame under a suitably informative name e 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 e before leaving a problem add any variables you wish to keep for future reference to the data frame using the form of assignment and then detach e finally remove all unwanted variables from the working directory and keep it as clean of left over temporary variables as possible In this way it is quite simple to work with many problems in the same directory all of which have variables named x y and z for example 6 3 4 Attaching arbitrary lists attach is a generic function that allows not only directories and data frames to be attached to the search path but other classes of object as well In particular any object of mode list may be attached in the same way Chapter 6 Lists and data frames 29 gt attach any old list Anything that has been attached can be detached by detach by position number or prefer ably by name 6 3 5 Managing the search path The function search shows the current search path and so is a very useful way to keep track of which data frames and lists and packag
56. a plot of the values in the vector against their index in the vector If x is a complex vector it produces a plot of imaginary versus real parts of the vector elements plot f plot f y f is a factor object y is a numeric vector The first form generates a bar plot of f the second form produces boxplots of y for each level of f Chapter 12 Graphical procedures 64 plot df plot expr plot y expr df is a data frame y is any object expr is a list of object names separated by e g a b c The first two forms produce distributional plots of the variables in a data frame first form or of a number of named objects second form The third form plots y against every object named in expr 12 1 2 Displaying multivariate data R provides two very useful functions for representing multivariate data If X is a numeric matrix or data frame the command pairs X produces a pairwise scatterplot matrix of the variables defined by the columns of X that is every column of X is plotted against every other column of X and the resulting n n 1 plots are arranged in a matrix with plot scales constant over the rows and columns of the matrix When three or four variables are involved a coplot may be more enlightening If a and b are numeric vectors and c is a numeric vector or factor object all of the same length then the command gt coplot a b c produces a number of scatterplots of a against b for give
57. a vector object used to specify a discrete classification grouping of the components of other vectors of the same length R provides both ordered and unordered factors While the real application of factors is with model formulae see Section 11 1 1 Contrasts page 53 we here look at a specific example 4 1 A specific example Suppose for example we have a sample of 30 tax accountants from all the states and territories of Australia and their individual state of origin is specified by a character vector of state mnemonics as gt state lt c tas sa qld nsw nsw nt wa wa qld vic nsw vic qld qld sa tas sa nt wa vic qld nsw nsw wa Sa act nsw vic vic act Notice that in the case of a character vector sorted means sorted in alphabetical order A factor is similarly created using the factor function gt statef lt factor state The print O 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 find out the levels of a factor the function levels can be used gt levels statef 1 act nsw nt qld sa tas yic wa 4 2 The function tapply and ragged arrays To continue the previous example suppose we have the incomes of the same tax acco
58. all have the same row size A data frame may for many purposes be regarded as a matrix with columns possibly of differing modes and attributes It may be displayed in matrix form and its rows and columns extracted using matrix indexing conventions 6 3 1 Making data frames Objects satisfying the restrictions placed on the columns components of a data frame may be used to form one using the function data frame gt accountants lt data frame home statef loot incomes shot incomef A list whose components conform to the restrictions of a data frame may be coerced into a data frame using the function as data frame The simplest way to construct a data frame from scratch is to use the read table function to read an entire data frame from an external file This is discussed further in Chapter 7 Reading data from files page 30 1 Conversion of character columns to factors is overridden using the stringsAsFactors argument to the data frame function Chapter 6 Lists and data frames 28 6 3 2 attach and detach The notation such as accountants home for list components is not always very convenient A useful facility would be somehow to make the components of a list or data frame temporarily visible as variables under their component name without the need to quote the list name explicitly each time The attach function takes a database such as a list or data frame as its argument Thus suppose lentils is a da
59. alues corresponding to the determining variable values in data frame print object Print a concise version of the object Most often used implicitly residuals object Extract the matrix of residuals weighted as appropriate Short form resid object step object Select a suitable model by adding or dropping terms and preserving hierarchies The model with the smallest value of AIC Akaike s An Information Criterion discovered in the stepwise search is returned Chapter 11 Statistical models in R 55 summary object Print a comprehensive summary of the results of the regression analysis vcov object Returns the variance covariance matrix of the main parameters of a fitted model object 11 4 Analysis of variance and model comparison The model fitting function aov formula data data frame operates at the simplest level in a very similar way to the function 1m O and most of the generic functions listed in the table in Section 11 3 Generic functions for extracting model information page 54 apply It should be noted that in addition aov allows an analysis of models with multiple error strata such as split plot experiments or balanced incomplete block designs with recovery of inter block information The model formula response mean formula Error strata formula specifies a multi stratum experiment with error strata defined by the strata formula In the simplest case strata formula is simply a factor when it
60. ant difference and so we can use the classical t test that assumes equality of the variances gt t test A B var equal TRUE Two Sample t test data A and B t 3 4722 df 19 p value 0 002551 alternative hypothesis true difference in means is not equal to O 95 percent confidence interval 0 01669058 0 06734788 sample estimates mean of x mean of y 80 02077 79 97875 All these tests assume normality of the two samples The two sample Wilcoxon or Mann Whitney test only assumes a common continuous distribution under the null hypothesis gt wilcox test A B Wilcoxon rank sum test with continuity correction data A and B W 89 p value 0 007497 alternative hypothesis true location shift is not equal to 0 Warning message Cannot compute exact p value with ties in wilcox test A B Note the warning there are several ties in each sample which suggests strongly that these data are from a discrete distribution probably due to rounding There are several ways to compare graphically the two samples We have already seen a pair of boxplots The following gt plot ecdf A do points FALSE verticals TRUE xlim range A B gt plot ecdf B do points FALSE verticals TRUE add TRUE will show the two empirical CDFs and qqplot will perform a Q Q plot of the two samples The Kolmogorov Smirnov test is of the maximal vertical distance between the two ecdf s assuming a common continuous distribution gt ks test
61. as recursive rather than atomic structures since their components can themselves be lists in their own right The other recursive structures are those of mode function and expression Functions are the objects that form part of the R system along with similar user written functions which we discuss in some detail later Expressions as objects form an advanced part of R which will not be discussed in this guide except indirectly when we discuss formulae used with modeling in R By the mode of an object we mean the basic type of its fundamental constituents This is a special case of a property of an object Another property of every object is its length The functions mode object and length object can be used to find out the mode and length of any defined structure Further properties of an object are usually provided by attributes object see Section 3 3 Getting and setting attributes page 14 Because of this mode and length are also called intrinsic attributes of an object For example if z is a complex vector of length 100 then in an expression mode z is the character string complex and length z is 100 R caters for changes of mode almost anywhere it could be considered sensible to do so and a few where it might not be For example with gt z lt 0 9 we could put gt digits lt as character z after which digits is the character vector c 0 1 2 9 A further coercion or change of mo
62. ata Im port Export manual 7 1 The read table function To read an entire data frame directly the external file will normally have a special form e The first line of the file should have a name for each variable in the data frame e Each additional line of the file has as its first item a row label and the values for each variable If the file has one fewer item in its first line than in its second this arrangement is presumed to be in force So the first few lines of a file to be read as a data frame might look as follows E Input file form with names and row labels Price Floor Area Rooms Age Cent besat 01 52 00 111 0 830 5 6 2 no 02 54 75 128 0 710 5 7 5 no 03 57 50 101 0 1000 5 4 2 no 04 57 50 131 0 690 6 8 8 no 05 59 75 93 0 900 5 1 9 yes J By default numeric items except row labels are read as numeric variables and non numeric variables such as Cent heat in the example as factors This can be changed if necessary The function read table can then be used to read the data frame directly gt HousePrice lt read table houses data Often you will want to omit including the row labels directly and use the default labels In this case the file may omit the row label column as in the following S Input file form without row labels Price Floor Area Rooms Age Cent heat 52 00 111 0 830 5 6 2 no 54 75 128 0 710 5 7 5 no 57 50 101 0 1000 5 4 2 no 57 50 131 0 690 6 8 8 no 59 75 93 0 900 5 1 9 yes
63. ation exists for R in The R statistical system FAQ 1 3 R and statistics Our introduction to the R environment did not mention statistics yet many people use R as a statistics system We prefer to think of it of an environment within which many classical and modern statistical techniques have been implemented A few of these are built into the base R environment but many are supplied as packages There are about 25 packages supplied with R called standard and recommended packages and many more are available through the CRAN family of Internet sites via https CRAN R project org and elsewhere More details on packages are given later see Chapter 13 Packages page 77 Most classical statistics and much of the latest methodology is available for use with R but users may need to be prepared to do a little work to find it Chapter 1 Introduction and preliminaries 3 There is an important difference in philosophy between S and hence R and the other main statistical systems In S a statistical analysis is normally done as a series of steps with intermediate results being stored in objects Thus whereas SAS and SPSS will give copious output from a regression or discriminant analysis R will give minimal output and store the results in a fit object for subsequent interrogation by further R functions 1 4 R and the window system The most convenient way to use R is at a graphics workstation running a windowing system Thi
64. atrices 20 However a simpler direct way of producing this matrix is to use table gt N lt table blocks varieties Index matrices must be numerical any other form of matrix e g a logical or character matrix supplied as a matrix is treated as an indexing vector 5 4 The array function As well as giving a vector structure a dim attribute arrays can be constructed from vectors by the array function which has the form gt Z lt array data_vector dim_vector For example if the vector h contains 24 or fewer numbers then the command gt Z array h dim c 3 4 2 would use h to set up 3 by 4 by 2 array in Z If the size of h is exactly 24 the result is the same as gt Z lt h dim Z lt c 3 4 2 However if h is shorter than 24 its values are recycled from the beginning again to make it up to size 24 see Section 5 4 1 The recycling rule page 20 but dim h lt c 3 4 2 would signal an error about mismatching length As an extreme but common example gt Z lt array 0 c 3 4 2 makes Z an array of all zeros At this point dim Z stands for the dimension vector c 3 4 2 and Z 1 24 stands for the data vector as it was in h and Z with an empty subscript or Z with no subscript stands for the entire array as an array Arrays may be used in arithmetic expressions and the result is an array formed by element by element operations on the data vector The dim attributes of operands generally need to be
65. ble 33 8 2 Examining the distribution of a set of data 34 8 3 One and two sample tests 36 Grouping loops and conditional execution 40 9 1 Grouped expressions ee ehh he 40 9 2 Control statements rin erdiei A UL A pO e Hd e D pte e 40 9 2 1 Conditional execution if atatements nee 40 9 2 2 Repetitive execution for loops repeat and wbile cc 40 10 Writing your own Dunctions ra arannana 42 10 Smple examples ccce Cot hed heed Ai XE euet a RE eR 42 10 2 Defining new binary operatorg ee mm 43 10 3 Named arguments and defaults n 43 10 4 Ehe argumelt 22er ette a RIA 44 10 5 Assignments within function 44 10 6 More advanced examples e m 44 10 6 1 Efficiency factors in block designs eee 44 10 6 2 Dropping all names in a printed aan 45 10 6 3 Recursive numerical integration e 45 LORE SCOPE eck sedge tie eege See EES AEN 46 10 8 Customizing the environment nc 48 10 9 Classes generic functions and object orientation 49 11 Statistical models in EK ses reir Eh EEN 51 11 1 Defining statistical models formulae e 51 TELEL XCOonbr sts l2exesdeve IR eue E erar ee ERE 53 I1 2 Linearamodels bue RE A EES 54 11 3 Generic functions for extracting model information 0oooocccoocccccnoncccno 54 11 4 Analysis of variance and model comparison sssssssessssssse rro 55 HAAT ANOVA tables Iv A n bubo qc fie tes 55 11 5 Updating fitted models e hen 55 11 6 Generalized linear model 56 TE
66. called Rhistory When R is started at later time from the same directory it reloads the workspace from this file At the same time the associated commands history is reloaded It is recommended that you should use separate working directories for analyses conducted with R It is quite common for objects with names x and y to be created during an analysis Names like this are often meaningful in the context of a single analysis but it can be quite hard to decide what they might be when the several analyses have been conducted in the same directory 5 The leading dot in this file name makes it invisible in normal file listings in UNIX and in default GUI file listings on OS X and Windows Chapter 2 Simple manipulations numbers and vectors 7 2 Simple manipulations numbers and vectors 2 1 Vectors and assignment R operates on named data structures The simplest such structure is the numeric vector which is a single entity consisting of an ordered collection of numbers To set up a vector named x say consisting of five numbers namely 10 4 5 6 3 1 6 4 and 21 7 use the R command gt x lt c 10 4 5 6 3 1 6 4 21 7 This is an assignment statement using the function c which in this context can take an arbitrary number of vector arguments and whose value is a vector got by concatenating its arguments end to end A number occurring by itself in an expression is taken as a vector of length one Notice that the assignmen
67. cessfully version Print version information to standard output and exit successfully Appendix B Invoking R 86 encoding enc Specify the encoding to be assumed for input from the console or stdin This needs to be an encoding known to iconv see its help page encoding enc is also accepted The input is re encoded to the locale H is running in and needs to be representable in the latter s encoding so e g you cannot re encode Greek text in a French locale unless that locale uses the UTF 8 encoding RHOME Print the path to the R home directory to standard output and exit success fully Apart from the front end shell script and the man page R installation puts everything executables packages etc into this directory save no save Control whether data sets should be saved or not at the end of the R session If neither is given in an interactive session the user is asked for the desired behavior when ending the session with q in non interactive use one of these must be specified or implied by some other option see below no environ Do not read any user file to set environment variables no site file Do not read the site wide profile at startup no init file Do not read the user s profile at startup restore no restore no restore data Control whether saved images file RData in the directory where R was started should be restored at startup or not The default is to restore no restore
68. ct in various complicated ways The simplest form is gt sb lt rep x times 5 which will put five copies of x end to end in s5 Another useful version is gt s6 lt rep x each 5 which repeats each element of x five times before moving on to the next 2 4 Logical vectors As well as numerical vectors R allows manipulation of logical quantities The elements of a logical vector can have the values TRUE FALSE and NA for not available see below The first two are often abbreviated as T and F respectively Note however that T and F are just variables which are set to TRUE and FALSE by default but are not reserved words and hence can be overwritten by the user Hence you should always use TRUE and FALSE Logical vectors are generated by conditions For example 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 ci c2 is their union or and c1 is the negation of c1 Logical vectors may be used in ordinary arithmetic in which case they are coerced into numeric vectors FALSE becoming 0 and TRUE becoming 1 However there are situations where logical vectors and their coerced numeric counterparts are
69. ct square then it is not hard to see that gt absdetM prod svd M d calculates the absolute value of the determinant of M If this calculation were needed often with a variety of matrices it could be defined as an R function gt absdet lt function M prod svd M d after which we could use absdet as just another R function As a further trivial but potentially useful example you might like to consider writing a function say tr to calculate the trace of a square matrix Hint You will not need to use an explicit loop Look again at the diag function R has a builtin function det to calculate a determinant including the sign and another determinant to give the sign and modulus optionally on log scale 5 7 5 Least squares fitting and the QR decomposition The function 1sfit returns a list giving results of a least squares fitting procedure An assignment such as gt ans lt lsfit X y gives the results of a least squares fit where y is the vector of observations and X is the design matrix See the help facility for more details and also for the follow up function 1s diag O 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 1m see Section 11 2 Linear models page 54 to 1sfit for regression modelling Another closely related function is qr
70. ctors 11 2 A vector of positive integral quantities In this case the values in the index vector must lie in the set 1 2 length x The corresponding elements of the vector are selected and concatenated in that order in the result The index vector can be of any length and the result is of the same length as the index vector For example x 6 is the sixth component of x and x 1 10 selects the first 10 elements of x assuming length x is not less than 10 Also gt c x y rep c 1 2 2 1 times 4 an admittedly unlikely thing to do produces a character vector of length 16 consisting of x y y x repeated four times 3 A vector of negative integral quantities Such an index vector specifies the values to be excluded rather than included Thus gt y lt x 1 5 gives y all but the first five elements of x 4 A vector of character strings This possibility only applies where an object has a names attribute to identify its components In this case a sub vector of the names vector may be used in the same way as the positive integral labels in item 2 further above gt fruit lt c 5 10 1 20 gt names fruit lt c orange banana apple peach gt lunch lt fruit c apple orange The advantage is that alphanumeric names are often easier to remember than numeric indices This option is particularly useful in connection with data frames as we shall see later An indexed expression can
71. culate the standard errors of the state income means To do this we need to write an R function to calculate the standard error for any given vector Since there is an builtin function var to calculate the sample variance such a function is a very simple one liner specified by the assignment gt stderr lt function x sqrt var x length x Writing functions will be considered later in Chapter 10 Writing your own functions page 42 and in this case was unnecessary as R also has a builtin function sd After this assignment the standard errors are calculated by gt incster tapply incomes statef stderr and the values calculated are then gt incster act nsw nt qid sa tas vic wa 1 5 4 3102 4 5 4 1061 2 7386 0 5 5 244 2 6575 As an exercise you may care to find the usual 9596 confidence limits for the state mean incomes To do this you could use tapply once more with the length function to find the sample sizes and the qt function to find the percentage points of the appropriate t distributions You could also investigate R s facilities for t tests The function tapply can also be used to handle more complicated indexing of a vector by multiple categories For example we might wish to split the tax accountants by both state and sex However in this simple instance just one factor what happens can be thought of as follows The values in the vector are collected into groups corresponding to the distinct entries i
72. de reconstructs the numerical vector again gt d lt as integer digits Now d and z are the same There is a large collection of functions of the form as something for either coercion from one mode to another or for investing an object with some other attribute it may not already possess The reader should consult the different help files to become familiar with them 1 numeric mode is actually an amalgam of two distinct modes namely integer and double precision as explained in the manual Note however that length object does not always contain intrinsic useful information e g when object is a function 3 In general coercion from numeric to character and back again will not be exactly reversible because of roundoff errors in the character representation Chapter 3 Objects their modes and attributes 14 3 2 Changing the length of an object An empty object may still have a mode For example gt e numeric makes e an empty vector structure of mode numeric Similarly character is a empty character vector and so on Once an object of any size has been created new components may be added to it simply by giving it an index value outside its previous range Thus gt el3 lt 17 now makes e a vector of length 3 the first two components of which are at this point both NA This applies to any structure at all provided the mode of the additional component s agrees with the mode of the object in the
73. de R_ PAPERSIZE the default paper size R PRINTCMD the default print command and R_LIBS specifies the list of R library trees searched for add on packages e Then R searches for the site wide startup profile unless the command line option no site file was given The name of this file is taken from the value of the R PROFILE environment variable If that variable is unset the default R HOME etc Rprofile site is used if this exists e Then unless no init file was given R searches for a user profile and sources it The name of this file is taken from the environment variable R PROFILE USER if unset a file called Rprofile in the current directory or in the user s home directory in that order is searched for e It also loads a saved workspace from file RData in the current directory if there is one unless no restore or no restore data was specified e Finally if a function First exists it is executed This function as well as Last O which is executed at the end of the R session can be defined in the appropriate startup profiles or reside in RData In addition there are options for controlling the memory available to the R process see the on line help for topic Memory for more information Users will not normally need to use these unless they are trying to limit the amount of memory used by R R accepts the following command line options help h Print short help message to standard output and exit suc
74. defines a two strata experiment namely between and within the levels of the factor For example with all determining variables factors a model formula such as that in gt fm aov yield v n p k Error farms blocks data farm data would typically be used to describe an experiment with mean model v n p k and three error strata namely between farms within farms between blocks and within blocks 11 4 1 ANOVA tables Note also that the analysis of variance table or tables are for a sequence of fitted models The sums of squares shown are the decrease in the residual sums of squares resulting from an inclusion of that term in the model at that place in the sequence Hence only for orthogonal experiments will the order of inclusion be inconsequential For multistratum experiments the procedure is first to project the response onto the error strata again in sequence and to fit the mean model to each projection For further details see Chambers amp Hastie 1992 A more flexible alternative to the default full ANOVA table is to compare two or more models directly using the anova function anova fitted model 1 fitted model 2 The display is then an ANOVA table showing the differences between the fitted models when fitted in sequence The fitted models being compared would usually be an hierarchical sequence of course This does not give different information to the default but rather makes it easier to c
75. e continuation prompt and indicate continuation by simple indenting Command lines entered at the console are limited to about 4095 bytes not characters 1 9 Recall and correction of previous commands Under many versions of UNIX and on Windows R provides a mechanism for recalling and re executing previous commands The vertical arrow keys on the keyboard can be used to scroll forward and backward through a command history Once a command is located in this way the cursor can be moved within the command using the horizontal arrow keys and characters can be removed with the DEL key or added with the other keys More details are provided later see Appendix C The command line editor page 92 The recall and editing capabilities under UNIX are highly customizable You can find out how to do this by reading the manual entry for the readline library Alternatively the Emacs text editor provides more general support mechanisms via ESS Emacs Speaks Statistics for working interactively with R See Section R and Emacs in The R statistical system FAQ 1 10 Executing commands from or diverting output to a file If commands are stored in an external file say commands R in the working directory work they may be executed at any time in an R session with the command gt source commands R For Windows Source is also available on the File menu The function sink gt sink record lis will divert all subsequent output from the console
76. e summary this is provided as a somewhat more detailed alternative Graphics parameters will be presented in the following form name value A description of the parameter s effect name is the name of the parameter that is the argument name to use in calls to par or a graphics function value is a typical value you might use when setting the parameter Note that axes is not a graphics parameter but an argument to a few plot methods see xaxt and yaxt Some graphics parameters such as the size of the current device are for information only Chapter 12 Graphical procedures 70 12 5 1 Graphical elements R plots are made up of points lines text and polygons filled regions Graphical parameters exist which control how these graphical elements are drawn as follows pch pch 4 lty 2 lwd 2 col 2 col axis col lab col main col sub font 2 font axis font lab font main font sub adj 0 1 cex 1 5 Character to be used for plotting points The default varies with graphics drivers but it is usually o Plotted points tend to appear slightly above or below the appropriate position unless you use 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 legend locator 1 as character 0 25 pch 0 25 Those from 21 to 25 may appear to du
77. eS 17 Uter Ee A EE bg 21 P BS leg ere 4 alee tate NE eae hens 64 Pai tate a ae taa e d 68 Paste AE EE EE e lee a one 10 POE eo i habbo eM EN best 74 E EE 65 PLENAS e gees 54 63 A EE 8 PM TOUS r 8 PUR A A 74 HELLER EE eg id 66 POLY BONS di vira dp eee EE ER 66 postseoript i ie fa eee regenti e e 74 PECOL iio da ona dica 54 BEL a ee E E a E Ee ERSA des 54 EE 8 Q Co BEE 35 64 ARNM a i 35 64 et tri a reo dal eS 64 di il eee oM ese AO 23 quartz BLA UL RR DPI Le EE 74 Appendix D Function and variable index Tange T at AN ewe Sis 8 e RE EEN 24 FOAds tabla a 30 E EE 9 reped Gerini ok CE E EEEN ENE OE Wares ph eSI 41 TOS di A A AS 54 Yesidual s v 06 cece say cw eee Gales a e 54 PLM A AS 61 E IE 6 SOMA AA ANTAS ASA NANO da 31 Sdulil A A A A DE S 17 SOLTERA ata 29 Sii ss coa tn tad 8 shapiro test 2g doe e a b b EEN 36 SIRO AE Vere d o xS e ee EN 8 Chbu gcc 5 SOLlVesitioneltensaa pic EE EE 22 SOft 4 l4 gni uL CUN SR NE ERI USER DERE e 8 SOUL Su vos EU a ES 5 EE 40 SE EE 8 EE 34 EUA do 54 56 SUN Ae E A ado Ae LRL ada Ae EIER 8 SUMMA ee SE 34 54 Ud A CO REOS 23 96 T Bi AAA AAA AAA nb tides 21 CU EE 37 table a 20 25 EN 8 tapply ut rese sU eae Se ee ee eee ih gee 16 A GER 66 ler AA Ee dd 67 Crees pee Amena id dad e oh de 62 a bn does eon rs e da det ni AT 9 TRUE e See e Gute Ze CE 9 UNCLASS eee eee wa Sea erie thee Lau xis 14 updat earann aree EE aE E A 55 MEE EE 8 17 VALES A o AAA 38 ME Vi
78. ed mm Expt lt factor mm Expt mm Run lt factor mm Run Change Expt and Run into factors attach mm Make the data frame visible at position 3 the default plot Expt Speed main Speed of Light Data xlab Experiment No Compare the five experiments with simple boxplots fm aov Speed Run Expt data mm summary fm Analyze as a randomized block with runs and experiments as factors fm0 lt update fm Run anova fmO fm Fit the sub model omitting runs and compare using a formal analysis of variance detach rm fm fmO Clean up before moving on We now look at some more graphical features contour and image plots x lt seq pi pi len 50 y lt x x is a vector of 50 equally spaced values in 7 lt x m y is the same Appendix A A sample session 84 f lt outer x y function x y cos y 1 x 2 f is a square matrix with rows and columns indexed by x and y respectively of values of the function cos y 1 z oldpar lt par no readonly TRUE par pty gn Save the plotting parameters and set the plotting region to square contour x y f contour x y f nlevels 15 add TRUE Make a contour map of f add in more lines for more detail fa lt f t f 2 fa is the asymmetric part of f tO is transpose contour x y fa nlevels 15 Make a contour plot par oldpar and restore the old graphics parameters image x
79. ees Boh 8 ULL m BE 9 Ui Ice A EVITE 40 e M AAT 8 A BE 8 a o ioe ed e DLL ELSE 8 ge A E be A ees E eee 55 FA3rSt i bi E A ere ducts 48 LASA A A AGIT EQUUS 48 EE 8 NA ii Igi 8 O EE 78 a ate acy heey acess 78 lt EE 9 E A EEEE A a 47 KS AE AA O et Si 9 Ee A m AT dd I 9 gt a a A mice eo Fut ame os N 9 KEE 9 7 e a ELE Sue ME 4 94 E quU Et Ve aeons une e d a as rude 4 RER 8 EE EE EEE E alada 9 KEEN 40 KEE 52 A abLine lian AA e ee Fue 66 ACC roth enh od hate oink Dh E 61 addl want Ge iert E aces See dee Se e SE 56 ANOVA eblebeewe e eds A eet ew ete beh TAS 54 55 BOW is Se eee Po ae EN et DER NE ee 55 Ch EE 21 ATAN is Ee 20 as data framers oi sew ns eee Lae tas esi 27 AS ECO EE 24 atada a 28 o e A 14 attributes A A Ee 14 VAS a a 61 AS AE ES 67 B boXplot extr use E 3T break ilsiledjlw uw da lS URN EOM DEDUCI WS A1 DUO it AIMQUHS E ET DUE RI ade 61 C Ct Lm 7 10 24 27 G T leg bh leg awh ses en ese do 24 Dl rc cL 54 Coefficients royi rri peti E aey Rua Tela LEE ue 54 CONTOU eiri pan a a ee 65 Contras EE 53 COPLOL EE 64 COSS Ee 8 EE e Ee 19 22 CUA A A A 25 A A 53 D data ti e ewe Dies e EE 31 data frame a A ds 27 density ds dle LEM 34 det 4 oli Aere bah ELS E e A 23 detachi il BEEN EEN Rave 28 determinant dure rdv snc aeu Ern Rn ed 23 Appendix D Function and variable index O i 5oiawetre ee wakes Bei ee eho rts 75 GeV EE 75 CGV prev geesde 75 A NN 75 deyi nGe A E 54 AA a elei
80. ents in double square brackets i e Let name is the same as Lst name This is especially useful when the name of the component to be extracted is stored in another variable as in gt x lt name Lst x It is very important to distinguish Lst 1 1 from Lst 11 is the operator used to select a single element whereas is a general subscripting operator Thus the former is the first object in the list Lst and if it is a named list the name is not included The latter is a sublist of the list Lst consisting of the first entry only If it is a named list the names are transferred to the sublist The names of components may be abbreviated down to the minimum number of letters needed to identify them uniquely Thus Lst coefficients may be minimally specified as Lst coe and Lst covariance as Lst cov The vector of names is in fact simply an attribute of the list like any other and may be handled as such Other structures besides lists may of course similarly be given a names attribute also Chapter 6 Lists and data frames 27 6 2 Constructing and modifying lists New lists may be formed from existing objects by the function list An assignment of the form gt Lst lt list name_1 object_1 name m object m sets up a list Lst of m components using object 1 object m for the components and giving them names as specified by the argument names which can be freely chosen If these names are o
81. er contributed packages rpart https CRAN R project org package rpart and tree https CRAN R project org package tree Chapter 12 Graphical procedures 63 12 Graphical procedures Graphical facilities are an important and extremely versatile component of the R environment It is possible to use the facilities to display a wide variety of statistical graphs and also to build entirely new types of graph The graphics facilities can be used in both interactive and batch modes but in most cases interactive use is more productive Interactive use is also easy because at startup time R initiates a graphics device driver which opens a special graphics window for the display of interactive graphics Although this is done automatically it may useful to know that the command used is X110 under UNIX windows under Windows and quartz under OS X A new device can always be opened by dev new Once the device driver is running R plotting commands can be used to produce a variety of graphical displays and to create entirely new kinds of display Plotting commands are divided into three basic groups e High level plotting functions create a new plot on the graphics device possibly with axes labels titles and so on e Low level plotting functions add more information to an existing plot such as extra points lines and labels e Interactive graphics functions allow you interactively add information to or extract infor
82. er 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 vy lt 2 x y 1 generates a new vector v of length 11 constructed by adding together element by element 2 x repeated 2 2 times y repeated just once and 1 repeated 11 times 1 With other than vector types of argument such as 1ist mode arguments the action of c is rather different See Section 6 2 1 Concatenating lists page 27 2 Actually it is still available as Last value before any other statements are executed Chapter 2 Simple manipulations numbers and vectors 8 The elementary arithmetic operators are the usual and for raising to a power In addition all of the common arithmetic functions are available log exp sin cos tan sqrt and so on all have their usual meaning max and min select the largest and smallest elements of a vector respectively range is a function whose value is a vector of length two namely c min x max x length x is the number of elements in x sum x gives the total of the elements in x and prod x their product Two statistical functions are mean x which calculates the sample mean which is the same as sum x length x and var x which gives sum x mean x 2 Aength x 1 or sample variance If the argument to var is an n by p mat
83. es have been attached and detached Initially it gives gt search 1 GlobalEnv Autoloads package base where GlobalEnv is the workspace After lentils is attached we have gt search 1 GlobalEnv lentils Autoloads package base gt 1s 2 1 u y y and as we see 1s or objects can be used to examine the contents of any position on the search path Finally we detach the data frame and confirm it has been removed from the search path gt detach lentils gt search 1 GlobalEnv Autoloads package base 2 See the on line help for autoload for the meaning of the second term Chapter 7 Reading data from files 30 7 Reading data from files Large data objects will usually be read as values from external files rather than entered during an R session at the keyboard R input facilities are simple and their requirements are fairly strict and even rather inflexible There is a clear presumption by the designers of R that you will be able to modify your input files using other tools such as file editors or Perl to fit in with the requirements of R Generally this is very simple If variables are to be held mainly in data frames as we strongly suggest they should be an entire data frame can be read directly with the read table function There is also a more primitive input function scan that can be called directly For more details on importing data into R and also exporting data see the R D
84. es of the parameters To obtain the approximate SEs of the estimates we do gt sqrt diag solve out hessian A 95 confidence interval would be the parameter estimate 1 96 SE 11 8 Some non standard models We conclude this chapter with just a brief mention of some of the other facilities available in R for special regression and data analysis problems Mixed models The recommended nlme https CRAN R project org package nlme package provides functions lme and nlme for linear and non linear mixed effects models that is linear and non linear regressions in which some of the coefficients correspond to random effects These functions make heavy use of formulae to specify the models Local approximating regressions The loess function fits a nonparametric regression by using a locally weighted regression Such regressions are useful for highlighting a trend in messy data or for data reduction to give some insight into a large data set Function loess is in the standard package stats together with code for projection pursuit regression Robust regression There are several functions available for fitting regression models in a way resistant to the influence of extreme outliers in the data Function Los in the recom mended package MASS https CRAN R project org package MASS provides state of art algorithms for highly resistant fits Less resistant but statistically more efficient methods are available in packages for e
85. f the response depends on the stimulus variables through a single linear function only the same mechanism as was used for linear models can still be used to specify the linear part of a generalized model The family has to be specified in a different way The R function to fit a generalized linear model is glm which uses the form gt fitted model lt glm formula family family generator data data frame The only new feature is the family generator which is the instrument by which the family is described It is the name of a function that generates a list of functions and expressions that together define and control the model and estimation process Although this may seem a little complicated at first sight its use is quite simple The names of the standard supplied family generators are given under Family Name in the table in Section 11 6 1 Families page 57 Where there is a choice of links the name of the link may also be supplied with the family name in parentheses as a parameter In the case of the quasi family the variance function may also be specified in this way Some examples make the process clear The gaussian family A call such as gt fm lt glm y x1 x2 family gaussian data sales achieves the same result as gt fm Im y x1 x2 data sales but much less efficiently Note how the gaussian family is not automatically provided with a choice of links so no parameter is allowed If a problem requ
86. g session is intended to introduce to you some features of the R environment by using them Many features of the system will be unfamiliar and puzzling at first but this puzzlement will soon disappear Start R appropriately for your platform see Appendix B Invoking R page 85 The R program begins with a banner Within R code the prompt on the left hand side will not be shown to avoid con fusion help start Start the HTML interface to on line help using a web browser available at your machine You should briefly explore the features of this facility with the mouse Iconify the help window and move on to the next part x lt rnorm 50 y lt rnorm x Generate two pseudo random normal vectors of x and y coordinates plot x y Plot the points in the plane A graphics window will appear automatically 1sQ See which R objects are now in the R workspace rm x y Remove objects no longer needed Clean up x lt 1 20 Make x 1 2 20 w lt 1 sqrt x 2 A weight vector of standard deviations dummy lt data frame x x y x rnorm x w dummy Make a data frame of two columns x and y and look at it fm lt Im y x data dummy summary fm Fit a simple linear regression and look at the analysis With y to the left of the tilde we are modelling y dependent on z fmi Im y x data dummy weight 1 w 2 summary fm1 Since we know the standard deviations we can do a weighted regression
87. gs TRUE Note that input and output can be redirected in the usual way using lt and gt but the line length limit of 4095 bytes still applies Warning and error messages are sent to the error channel stderr The command R CMD allows the invocation of various tools which are useful in conjunction with R but not intended to be called directly The general form is R CMD command args where command is the name of the tool and args the arguments passed on to it Currently the following tools are available BATCH Run R in batch mode Runs R restore save with possibly further options see BATCH COMPILE UNIX only Compile C C Fortran files for use with R SHLIB Build shared library for dynamic loading INSTALL Install add on packages REMOVE Remove add on packages build Build that is package add on packages check Check add on packages LINK UNIX only Front end for creating executable programs Rprof Post process R profiling files Rdconv Rd2txt Convert Rd format to various other formats including HTML IATEX plain text and extracting the examples Rd2txt can be used as shorthand for Rd2conv t txt Rd2pdf Convert Rd format to PDF Stangle Extract S R code from Sweave or other vignette documentation Sweave Process Sweave or other vignette documentation Rdiff Diff R output ignoring headers etc config Obtain configuration information javareconf Unix only Update the Java configura
88. haracters 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 help Either form of quote mark may be used to escape the other as in the string It s important Our convention is to use double quote marks for preference On most R installations help is available in HTML format by running 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 Keywords link in the page loaded by help start is particularly useful as it is contains a high level concept list which searches though available functions It can be a great way to get your bearings quickly and to understand the breadth of what R has to offer The help search command alternatively allows searching for help in various ways For example gt solve Try help search for details and more 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 1 8 R commands case sensitivity etc Technically R is an expression language with a very simple syntax It is case sensitive as are most UNIX based packages so A and a are different symbols and would refer to different variables The
89. hat calls to par always affect the global values of graphics parameters even when par is called from within a function This is often undesirable behavior usually we want to set some graphics parameters do some plotting and then restore the original values so as not to affect the user s R session You can restore the initial values by saving the result of par O when making changes and restoring the initial values when plotting is complete gt oldpar lt par col 4 lty 2 plotting commands gt par oldpar To save and restore all settable graphical parameters use gt oldpar lt par no readonly TRUE plotting commands gt par oldpar 12 4 2 Temporary changes Arguments to graphics functions Graphics parameters may also be passed to almost any graphics function as named arguments This has the same effect as passing the arguments to the par function except that the changes only last for the duration of the function call For example gt plot x y pch produces a scatterplot using a plus sign as the plotting character without changing the default plotting character for future plots Unfortunately this is not implemented entirely consistently and it is sometimes necessary to set and reset graphics parameters using par 12 5 Graphics parameters list The following sections detail many of the commonly used graphical parameters The R help documentation for the par function provides a more concis
90. havior can be changed by the options setting for contrasts The default setting in R is options contrasts c contr treatment contr poly The main reason for mentioning this is that R and S have different defaults for unordered factors S using Helmert contrasts So if you need to compare your results to those of a textbook or paper which used S PLUS you will need to set options contrasts c contr helmert contr poly This is a deliberate difference as treatment contrasts R s default are thought easier for new comers to interpret We have still not finished as the contrast scheme to be used can be set for each term in the model using the functions contrasts and C We have not yet considered interaction terms these generate the products of the columns introduced for their component terms Although the details are complicated model formulae in R will normally generate the models that an expert statistician would expect provided that marginality is preserved Fitting for example a model with an interaction but not the corresponding main effects will in general lead to surprising results and is for experts only Chapter 11 Statistical models in R 54 11 2 Linear models The basic function for fitting ordinary multiple models is 1m and a streamlined version of the call is as follows gt fitted model lt lm formula data data frame For example gt fm2 lt Ilm y x1 x2 data production would fit a mu
91. her than an incremental accretion of very specific and inflexible tools as is frequently the case with other data analysis software R is very much a vehicle for newly developing methods of interactive data analysis It has developed rapidly and has been extended by a large collection of packages However most programs written in R are essentially ephemeral written for a single piece of data analysis 1 2 Related software and documentation R can be regarded as an implementation of the S language which was developed at Bell Labora tories by Rick Becker John Chambers and Allan Wilks and also forms the basis of the S PLUS systems The evolution of the S language is characterized by four books by John Chambers and coauthors For R the basic reference is The New S Language A Programming Environment for Data Analysis and Graphics by Richard A Becker John M Chambers and Allan R Wilks The new features of the 1991 release of S are covered in Statistical Models in S edited by John M Chambers and Trevor J Hastie The formal methods and classes of the methods package are based on those described in Programming with Data by John M Chambers See Appendix F References page 99 for precise references There are now a number of books which describe how to use R for data analysis and statistics and documentation for S S PLUS can typically be used with R keeping the differences between the S implementations in mind See Section What document
92. hogonal polyno mials and the second uses explicit powers as basis y X poly x 2 Multiple regression y with model matrix consisting of the matrix X as well as polynomial terms in x to degree 2 Chapter 11 Statistical models in R 52 yA Single classification analysis of variance model of y with classes determined by A y A x Single classification analysis of covariance model of y with classes determined by A and with covariate x y AXB y A B A B y B in A y A B Two factor non additive model of y on A and B The first two specify the same crossed classification and the second two specify the same nested classification In abstract terms all four specify the same model subspace y A B C 72 y A B C A B C Three factor experiment but with a model containing main effects and two factor interactions only Both formulae specify the same model y A x y A x y A 1 x 1 Separate simple linear regression models of y on x within the levels of A with different codings The last form produces explicit estimates of as many different intercepts and slopes as there are levels in A y A B Error C An experiment with two treatment factors A and B and error strata determined by factor C For example a split plot experiment with whole plots and hence also subplots determined by factor C The operator is used to define a model formula in R The form for an ordinary linear model is response op 1 term 1 op 2
93. hs as needed for commands in the current OS 14 4 Compression and Archives Recent versions of R have extensive facilities to read and write compressed files often transpar ently Reading of files in R is to a vey large extent done by connections and the file function which is used to open a connection to a file or a URL and is able to identify the compression used from the magic header of the file The type of compression which has been supported for longest is gzip compression and that remains a good general compromise Files compressed by the earlier Unix compress utility can also be read but these are becoming rare Two other forms of compression those of the Chapter 14 OS facilities 81 bzip2 and xz utilities are also available These generally achieve higher rates of compression depending on the file much higher at the expense of slower decompression and much slower compression There is some confusion between xz and 1zma compression see https en wikipedia org wiki Xz and https en wikipedia org wiki LZMA R can read files compressed by most versions of either File archives are single files which contain a collection of files the most common ones being tarballs and zip files as used to distribute R packages R can list and unpack both see functions untar and unzip and create both for zip with the help of an external program Appendix A A sample session 82 Appendix A A sample session The followin
94. i ue KNIE ER ad 55 Vector ii A A ete A e qw ee ee T ENEE 41 Wilcox C 6St EE le ENEE rentas 38 MIDA Sii a adds 74 Appendix E Concept index Appendix E Concept index A Accessing builtin datasets 0 eee 31 Additive model 61 Analysis of variance no 55 Arithmetic functions and operatorg 7 ATTIVI oe bM a da ae 18 Assignment oll lel price NSA 7 Attributes 22 04 ae aed WIR HR id 13 B Binary operators 2 e eee eee eee eee 43 Box Oe EEGEN 37 C Character vectorg ono 10 Classes tiles 14 49 Concatenating jete 27 ContrastiS s 4lol l ferte ETA EN ie cea EEN EE 53 Control statements sese ee 40 GRAN scien Ponsa Eder AAA 77 Customizing the environment 48 D Data frames bu e AAA es 27 Default v lues c odie Bugs dg ebe ae da Ad Ap meets 43 Density estimation eee 34 IR EE prx Rae 23 Diverting input and output oo 5 Dynamic graphies 5225 be Ae RP GP nade 76 E Eigenvalues and eigenvectors ooooocoococooo 23 Empirical CDE3 o 35 PACO iii A cer dE deed 16 53 Familiegusi s dro prada ioe we dees 57 Eormulas dui a died 51 G Generalized linear model 56 Generalized transpose of an array 21 Generic function 49 Graphics device drivers noc 74 Graphics parameters eee 68 Grouped expressions 00 cece sence eee oo 40 I Indexing of and by arrays ooooooocccocccccccoo 18 Indexing EEN 10 97 K Kolmogorov Smir
95. ical variables Many experiments are best described by data frames the treatments are categorical but the response is numeric See Section 6 3 Data frames page 27 e functions are themselves objects in R which can be stored in the project s workspace This provides a simple and convenient way to extend R See Chapter 10 Writing your own functions page 42 Chapter 3 Objects their modes and attributes 13 3 Objects their modes and attributes 3 1 Intrinsic attributes mode and length The entities R operates on are technically known as objects Examples are vectors of numeric real or complex values vectors of logical values and vectors of character strings These are known as atomic structures since their components are all of the same type or mode namely numeric complex logical character and raw Vectors must have their values all of the same mode Thus any given vector must be un ambiguously either logical numeric complex character or raw The only apparent exception to this rule is the special value listed as NA for quantities not available but in fact there are several types of NA Note that a vector can be empty and still have a mode For example the empty character string vector is listed as character 0 and the empty numeric vector as numeric 0 R also operates on objects called lists which are of mode list These are ordered sequences of objects which individually can be of any mode lists are known
96. ides an interlocking suite of facilities that make fitting statistical models very simple As we mention in the introduction the basic output is minimal and one needs to ask for the details by calling extractor functions 11 1 Defining statistical models formulae The template for a statistical model is a linear regression model with independent homoscedastic errors p yi A Bau ei e NID 0 o i 1 n j 0 In matrix terms this would be written y XP e where the y is the response vector X is the model matrix or design matriz and has columns Zo T1 Tp the determining variables Very often xy will be a column of ones defining an intercept term Examples Before giving a formal specification a few examples may usefully set the picture Suppose y x x0 x1 x2 are numeric variables X is a matrix and A B C are factors The following formulae on the left side below specify statistical models as described on the right yx y 1 x Both imply the same simple linear regression model of y on x The first has an implicit intercept term and the second an explicit one y 0 x y 1 x y x 1 Simple linear regression of y on x through the origin that is without an intercept term log y x1 x2 Multiple regression of the transformed variable log y on zl and z2 with an implicit intercept term y poly x 2 y 1 x I x 2 Polynomial regression of y on x of degree 2 The first form uses ort
97. ink inverse variance constant data biochem The reader is referred to the manual and the help document for further information as needed 11 7 Nonlinear least squares and maximum likelihood models Certain forms of nonlinear model can be fitted by Generalized Linear Models g1m OQ But in the majority of cases we have to approach the nonlinear curve fitting problem as one of nonlinear optimization R s nonlinear optimization routines are optim nlm and nlminb which provide the functionality and more of S PLUS s ms and nlminb We seek the parameter values that minimize some index of lack of fit and they do this by trying out various parameter values iteratively Unlike linear regression for example there is no guarantee that the procedure will converge on satisfactory estimates All the methods require initial guesses about what parameter values to try and convergence may depend critically upon the quality of the starting values 11 7 1 Least squares One way to fit a nonlinear model is by minimizing the sum of the squared errors SSE or residuals This method makes sense if the observed errors could have plausibly arisen from a normal distribution Here is an example from Bates amp Watts 1988 page 51 The data are gt x lt c 0 02 0 02 0 06 0 06 0 11 0 11 0 22 0 22 0 56 0 56 1 10 1 10 gt y lt c 76 47 97 107 123 139 159 152 191 201 207 200 The fit criterion to be minimized is Chap
98. ion 2 of the search path see Section 6 3 4 Attaching arbitrary lists page 28 If the second argument is a single value and not a list a single vector is read in all components of which must be of the same mode as the dummy value X matrix scan light dat 0 ncol 5 byrow TRUE There are more elaborate input facilities available and these are detailed in the manuals 7 3 Accessing builtin datasets Around 100 datasets are supplied with R in package datasets and others are available in packages including the recommended packages supplied with R To see the list of datasets currently available use data All the datasets supplied with R are available directly by name However many packages still use the obsolete convention in which data was also used to load datasets into R for example data infert and this can still be used with the standard packages as in this example In most cases this will load an R object of the same name However in a few cases it loads several objects so see the on line help for the object to see what to expect 7 3 1 Loading data from other R packages To access data from a particular package use the package argument for example data package rpart data Puromycin package datasets If a package has been attached by library its datasets are automatically included in the search User contributed packages can be a rich source of datasets Chapter 7 Reading data from files 32
99. ires a gaussian family with a nonstandard link this can usually be achieved through the quasi family as we shall see later The binomial family Consider a small artificial example from Silvey 1970 Chapter 11 Statistical models in R 58 On the Aegean island of Kalythos the male inhabitants suffer from a congenital eye disease the effects of which become more marked with increasing age Samples of islander males of various ages were tested for blindness and the results recorded The data is shown below Age 20 35 45 55 70 No tested 50 50 50 50 50 No blind 6 17 26 37 44 The problem we consider is to fit both logistic and probit models to this data and to estimate for each model the LD50 that is the age at which the chance of blindness for a male inhabitant is 50 If y is the number of blind at age x and n the number tested both models have the form y Bin F Bo Gah where for the probit case F z z is the standard normal distribution function and in the logit case the default F z e 1 e In both cases the LD50 is LD50 Bo A that is the point at which the argument of the distribution function is zero The first step is to set the data up as a data frame gt kalythos lt data frame x c 20 35 45 55 70 n rep 50 5 y c 6 17 26 37 44 To fit a binomial model using glm there are three possibilities for the response e If the response is a vector it is assumed to hold binary data
100. kages a EAE AA eR E AE E A TT 13 2 Contributed packages and CHANNEL 77 13 37 Namespace td A A A EH 78 I4 OS facilities ere da idad ENEE 79 14 1 Files and directories iio e SNE S 79 14 2 A tee EE tre ede 79 TJ System commands o a da i Asa etra 80 14 4 Compression and Archives 80 Appendix A A sample sesaion 000 e cece eee eee 82 Appendix B Invoking Rai A A tt 85 B 1 Invoking R from the command ne 85 B 2 Invoking R under Windows 89 B 3 Invoking R under OS X asser eienn tr ars 4 REDRPP EFC RENE 90 B4 Serip ng witbzIusza a zoe een A AA EE di 90 Appendix C The command line editor 92 Cl Brehminares ito dee a a ad ada d A 92 C2 Editing actions ect a e elas 92 C 3 Command line editor summar rr 92 Appendix D Function and variable index 94 Appendix E Concept mmpdex 00 cece eee 97 Appendix F References 0 cee 99 Preface 1 Preface This introduction to R is derived from an original set of notes describing the S and S PLUS environments written in 1990 2 by Bill Venables and David M Smith when at the University of Adelaide We have made a number of small changes to reflect differences between the R and S programs and expanded some of the material We would like to extend warm thanks to Bill Venables and David Smith for granting permission to distribute this modified version of the notes in this way and for being a supporter of R from way back Comme
101. kages have namespaces and have since R 2 14 0 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 For example tQ is the transpose function in R but users might define their own function named t Namespaces prevent the user s definition from taking precedence and breaking every function that tries to transpose a matrix There are two operators that work with namespaces The double colon operator selects definitions from a particular namespace In the example above the transpose function will always be available as base t because it is defined in the base package Only functions that are exported from the package can be retrieved in this way The triple colon operator may be seen in a few places in R code it acts like the double colon operator but also allows access to hidden objects Users are more likely to use the getAnywhere function which searches multiple packages Packages are often inter dependent and loading one may cause others to be automatically loaded T he 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 Chapter 1
102. l practice but tight enough to allow the development of a unified methodology of estimation and inference at least approximately The reader is referred to any of the current reference works on the subject for full details such as McCullagh amp Nelder 1989 or Dobson 1990 Chapter 11 Statistical models in R 57 11 6 1 Families The class of generalized linear models handled by facilities supplied in R includes gaussian binomial poisson inverse gaussian and gamma response distributions and also quasi likelihood models where the response distribution is not explicitly specified In the latter case the variance function must be specified as a function of the mean but in other cases this function is implied by the response distribution Each response distribution admits a variety of link functions to connect the mean with the linear predictor Those automatically available are shown in the following table Family name Link functions binomial logit probit log cloglog gaussian identity log inverse Gamma identity inverse log inverse gaussian 1 mu 2 identity inverse log poisson identity log sqrt quasi logit probit cloglog identity inverse log 1 mu 2 sqrt The combination of a response distribution a link function and various other pieces of infor mation that are needed to carry out the modeling exercise is called the family of the generalized linear model 11 6 2 The glm function Since the distribution o
103. lable on the search list see Section 13 3 Namespaces page 78 these will be included in the list given by gt loadedNamespaces 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 13 1 Standard packages The standard or base packages are considered part of the R source code They contain the basic functions that allow R to work and the datasets and standard statistical and graphical functions that are described in this manual They should be automatically available in any R installation See Section R packages in R FAQ for a complete list 13 2 Contributed packages and CRAN There are thousands of contributed packages for R written by many different authors Some of these packages implement specialized statistical methods others give access to data or hard ware and others are designed to complement textbooks Some the recommended packages are distributed with every binary distribution of R Most are available for download from CRAN https CRAN R project org and its mirrors and other repositories such as Bioconductor https www bioconductor org and Omegahat http www omegahat org The R FAQ contains a list of CRAN packages current at the time of release but the collection of available packages changes very frequently Chapter 13 Packages 78 13 3 Namespaces All pac
104. lash 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 40 are other examples of binary operators defined in this way 10 3 Named arguments and defaults As first noted in Section 2 3 Generating regular sequences page 8 if arguments to called functions are given in the name object form they may be given in any order Furthermore the argument sequence may begin in the unnamed positional form and specify named arguments after the positional arguments Thus if there is a function funi defined by gt funi lt function data data frame graph limit 1 function body omitted D then the function may be invoked in several ways for example gt ans funi d df TRUE 20 gt ans lt funi d df graph TRUE limit 20 gt ans lt funi data d limit 20 graph TRUE data frame df are all equivalent In many cases arguments can be given commonly appropriate default values in which case they may be omitted altogether from the call when the defaults are appropriate For example if funi were defined as gt funi lt function data data frame graph TRUE limit 20 it could be called as gt ans funi d df which is now equivalent to the three cases above or as gt ans funi d df limit 10 which changes one of the defaults It is important to note that defaults may be arbitrary expres
105. lly genuinely Poisson data arises in practice and in the past it was often analyzed as gaussian data after either a log or a square root transformation As a graceful alternative to the latter a Poisson generalized linear model may be fitted as in the following example Chapter 11 Statistical models in R 59 gt fmod glm y A B x family poisson link sqrt data worm counts Quasi likelihood models For all families the variance of the response will depend on the mean and will have the scale parameter as a multiplier The form of dependence of the variance on the mean is a characteristic of the response distribution for example for the poisson distribution Var y u For quasi likelihood estimation and inference the precise response distribution is not specified but rather only a link function and the form of the variance function as it depends on the mean Since quasi likelihood estimation uses formally identical techniques to those for the gaussian distribution this family provides a way of fitting gaussian models with non standard link functions or variance functions incidentally For example consider fitting the non linear regression 0121 EPUM NEL which may be written alternatively as 1 Wo eo ee Dm Boxe where 1 22 21 2 1 21 61 1 0 and f 05 0 Supposing a suitable data frame to be set up we could fit this non linear regression as gt nlfit lt glm y x1 x2 1 family quasi l
106. ltiple regression model of y on x1 and z2 with implicit intercept term The important but technically optional parameter data production specifies that any variables needed to construct the model should come first from the production data frame This is the case regardless of whether data frame production has been attached on the search path or not 11 3 Generic functions for extracting model information The value of 1m O is a fitted model object technically a list of results of class 1m Information about the fitted model can then be displayed extracted plotted and so on by using generic functions that orient themselves to objects of class 1m These include addi deviance formula predict step alias dropi kappa print summary anova effects labels proj VCOV coef family plot residuals A brief description of the most commonly used ones is given below anova object_1 object_2 Compare a submodel with an outer model and produce an analysis of variance table coef object Extract the regression coefficient matrix Long form coefficients object deviance object Residual sum of squares weighted if appropriate formula object Extract the model formula plot object Produce four plots showing residuals fitted values and some diagnostics predict object newdata data frame The data frame supplied must have variables specified with the same labels as the original The value is a vector or matrix of predicted v
107. mewhat confusingly if k is a single numeric value then diag k is the k by k identity matrix 5 7 2 Linear equations and inversion Solving linear equations is the inverse of matrix multiplication When after 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 solve A b solves the system returning x up to some accuracy loss Note that in linear algebra formally x A lb where A denotes the inverse of A which can be computed by solve A but rarely is needed Numerically it is both inefficient and potentially unstable to compute x lt solve A b instead of solve A b The quadratic form xt A x which is used in multivariate computations should be computed by something like x solve A x rather than computing the inverse of A l Note that x x is ambiguous as it could mean either xl x or xxl where x is the column form In such cases the smaller matrix seems implicitly to be the interpretation adopted so the scalar x x is in this case the result The matrix xx may be calculated either by cbind x x or x 4 4 rbind x since the result of rbind or cbind is always a matrix However the best way to compute x x or xx is crossprod x or x o x respectively Even better would be to form a matrix square root B with A BB and find the squared length of the solution of By x perhaps using the Cholesky or eigen decomposition of A N Chapter 5
108. mitted 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 list matrix Mat 6 2 1 Concatenating lists When the concatenation function c is given list arguments the result is an object of mode list also whose components are those of the argument lists joined together in sequence gt List ABC lt c list A list B list C Recall that with vector objects as arguments the concatenation function similarly joined together all arguments into a single vector structure In this case all other attributes such as dim attributes are discarded 6 3 Data frames A data frame is a list with class data frame There are restrictions on lists that may be made into data frames namely e The components must be vectors numeric character or logical factors numeric matrices lists or other data frames e Matrices lists and data frames provide as many variables to the new data frame as they have columns elements or variables respectively e Numeric vectors logicals and factors are included as is and by default character vectors are coerced to be factors whose levels are the unique values appearing in the vector e Vector structures appearing as variables of the data frame must all have the same length and matrix structures must
109. n the list is always the null device which does not accept graphics commands at all dev next dev prev Returns the number and name of the graphics device next to or previous to the current device respectively dev set which k Can be used to change the current graphics device to the one at position k of the device list Returns the number and label of the device dev off k Terminate the graphics device at point k of the device list For some devices such as postscript devices this will either print the file immediately or correctly complete the file for later printing depending on how the device was initiated Chapter 12 Graphical procedures 76 dev copy device which k dev print device which k Make a copy of the device k Here device is a device function such as postscript with extra arguments if needed specified by dev print is similar but the copied device is immediately closed so that end actions such as printing hardcopies are immediately performed graphics off Terminate all graphics devices on the list except the null device 12 7 Dynamic graphics R does not have builtin capabilities for dynamic or interactive graphics e g rotating point clouds or to brushing interactively highlighting points However extensive dynamic graphics facilities are available in the system GGobi by Swayne Cook and Buja available from http www ggobi org and these can be accessed from R
110. n the factor The function is then applied to each of these groups individually The value is a vector of function results labelled by the levels attribute of the factor The combination of a vector and a labelling factor is an example of what is sometimes called a ragged array since the subclass sizes are possibly irregular When the subclass sizes are all the same the indexing may be done implicitly and much more efficiently as we see in the next section 4 3 Ordered factors The levels of factors are stored in alphabetical order or in the order they were specified to factor if they were specified explicitly Sometimes the levels will have a natural ordering that we want to record and want our statistical analysis to make use of The ordered function creates such ordered factors but is otherwise identical to factor For most purposes the only difference between ordered and unordered factors is that the former are printed showing the ordering of the levels but the contrasts generated for them in fitting linear models are different Chapter 5 Arrays and matrices 18 5 Arrays and matrices 5 1 Arrays An array can be considered as a multiply subscripted collection of data entries for example numeric R allows simple facilities for creating and handling arrays and in particular the special case of matrices A dimension vector is a vector of non negative integers If its length is k then the array is k dimensional e g a matrix is a
111. n values of c If c is a factor this simply means that a is plotted against b for every level of c When c is numeric it is divided into a number of conditioning intervals and for each interval a is plotted against b for values of c within the interval The number and position of intervals can be controlled with given values argument to coplot the function co intervals is useful for selecting intervals You can also use two given variables with a command like gt coplot a b c d which produces scatterplots of a against b for every joint conditioning interval of c and d The coplot and pairs function both take an argument panel which can be used to customize the type of plot which appears in each panel The default is points to produce a scatterplot but by supplying some other low level graphics function of two vectors x and y as the value of panel you can produce any type of plot you wish An example panel function useful for coplots is panel smooth 12 1 3 Display graphics Other high level graphics functions produce different types of plots Some examples are qqnorm x qqline x qqplot x y Distribution comparison plots The first form plots the numeric vector x against the expected Normal order scores a normal scores plot and the second adds a straight line to such a plot by drawing a line through the distribution and data quartiles The third form plots the quantiles of x against those of y to compare their re
112. nction normalizePath will find a canonical filepath Windows has the concepts of short 8 3 and long file names normalizePath will return an absolute path using long file names and shortPathName will return a version using short names The latter does not contain spaces and uses backslash as the separator so is sometimes useful for exporting names from R File permissions are a related topic R has support for the POSIX concepts of read write execute permission for owner group all but this may be only partially supported on the filesystem so for example on Windows only read only files for the account running the R session are recognized Access Control Lists ACLs are employed on several filesystems but do not have an agreed standard and R has no facilities to control them Use Sys chmod to change permissions 14 3 System commands Functions system and system2 are used to invoke a system command and optionally collect its output system2 is a little more general but its main advantage is that it is easier to write cross platform code using it system behaves differently on Windows from other OSes because the API C call of that name does Elsewhere it invokes a shell to run the command the Windows port of R has a function shell to do that To find out if the OS includes a command use Sys which which attempts to do this in a cross platform way unfortunately it is not a standard OS service Function shQuote will quote filepat
113. nd 1 n 1 The construction 30 1 may be used to generate a sequence backwards The function seq is a more general facility for generating sequences It has five arguments only some of which may be specified in any one call The first two arguments if given specify the beginning and end of the sequence and if these are the only two arguments given the result is the same as the colon operator That is seq 2 10 is the same vector as 2 10 Arguments to seq and to many other R functions can also be given in named form in which case the order in which they appear is irrelevant The first two arguments may be named from value and to value thus seq 1 30 seq from 1 to 30 and seq to 30 from 1 are all the same as 1 30 The next two arguments to seq may be named by value and length value which specify a step size and a length for the sequence respectively If neither of these is given the default by 1 is assumed For example gt seq 5 5 by 2 gt s3 generates in s3 the vector c 5 0 4 8 4 6 4 6 4 8 5 0 Similarly Chapter 2 Simple manipulations numbers and vectors 9 gt s4 lt seq length 51 from 5 by 2 generates the same vector in s4 The fifth argument may be named along vector which is normally used as the only argu ment to create the sequence 1 2 length vector or the empty sequence if the vector is empty as it can be A related function is repO which can be used for replicating an obje
114. ng Function file copy is the R analogue of the POSIX command cp Choosing files can be done interactively by file choose the Windows port has the more versatile functions choose files and choose dir and there are similar functions in the tcltk package tk choose files and tk choose dir Functions file show and file edit will display and edit one or more files in a way appro priate to the R port using the facilities of a console such as RGui on Windows or R app on OS X if one is in use There is some support for links in the filesystem see functions file link and Sys readlink 14 2 Filepaths With a few exceptions R relies on the underlying OS functions to manipulate filepaths Some aspects of this are allowed to depend on the OS and do even down to the version of the OS There are POSIX standards for how OSes should interpret filepaths and many R users assume POSIX compliance but Windows does not claim to be compliant and other OSes may be less than completely compliant The following are some issues which have been encountered with filepaths e POSIX filesystems are case sensitive so foo png and Foo PNG are different files However the defaults on Windows and OS X are to be case insensitive and FAT filesystems com monly used on removable storage are not normally case sensitive and all filepaths may be mapped to lower case e Almost all the Windows OS services support the use of slash or backslash as the filepath sepa
115. not equivalent for example see the next subsection 2 5 Missing values In some cases the components of a vector may not be completely known When an element or value is not available or a missing value in the statistical sense a place within a vector may be reserved for it by assigning it the special value NA In general any operation on an NA becomes an NA The motivation for this rule is simply that if the specification of an operation is incomplete the result cannot be known and hence is not available The function is na x gives a logical vector of the same size as x with value TRUE if and only if the corresponding element in x is NA gt z lt c 1 3 NA ind lt is na z Notice that the logical expression x NA is quite different from is na x since NA is not really a value but a marker for a quantity that is not available Thus x NA is a vector of the same length as x all of whose values are NA as the logical expression itself is incomplete and hence undecidable Note that there is a second kind of missing values which are produced by numerical com putation the so called Not a Number NaN values Examples are 0 0 or Chapter 2 Simple manipulations numbers and vectors 10 gt Inf Inf which both give NaN since the result cannot be defined sensibly In summary is na xx is TRUE both for NA and NaN values To differentiate these is nan xx is only TRUE for NaNs Missing values are sometimes p
116. nov Let 36 L Least squares fibtid8 ooooooocccoocconcccoco 23 Linear equations 0c eee ee eee ee 22 Linear models see e eee cece eee ccoo 54 RE EEN 26 Local approximating regressions s 5 61 Loops and conditional execution s c c 40 M MatTicES evitan HEEN ANEN es 18 Matrix multiplication 0 22 Maximum likelihood 0 0 cee eee eens 60 Missing values zi Eeer MD Shad hoes koe 9 Mixed model 61 N Named arguments 00 ee eee eee 43 N mespace iilcicenie e Ic Il ENEE tees ceed 78 Nonlinear least squares 0 0 cece e eens 59 Object orientation s erse sorin enrian oein 49 Objects EE 13 One and two sample tests o ooooooooooomoc 36 Ordered factors om 16 53 Outer products of aas 21 P Packages v i vec ase Rer Md ed 2 TT Probability distributions oooooooccomccoo 33 QR decomposition ssseeeeeee eee ee 23 Quantile quantile plots 0 cece eee eee 35 R Reading data from Dies 30 Recycling me tard er Hor ZEN 7 20 Regular sequences rr 8 Removing objects 0 cee eee eee eee 6 Robust regression 0 00 e eee eee eee eee 61 ee EEN 46 Search path eene i tad 29 Shapiro Wilk test 36 Singular value decomposition sssssesrss 23 Statistical models 0 ccc ccc eens 51 Appendix E Concept index Student s t test 37 T TABA SAA Ai ag 25 Tree based models 00 ccc cece ee
117. ns 61 U Updating fitted model 55 98 V Vector aisen ee Ze EES RIA ie Wilcoxon test 38 e EE 5 Writing functions sees 42 Appendix F References 99 Appendix F References D M Bates and D G Watts 1988 Nonlinear Regression Analysis and Its Applications John Wiley amp Sons New York Richard A Becker John M Chambers and Allan R Wilks 1988 The New S Language Chap man amp Hall New York This book is often called the Blue Book John M Chambers and Trevor J Hastie eds 1992 Statistical Models in S Chapman amp Hall New York This is also called the White Book John M Chambers 1998 Programming with Data Springer New York This is also called the Green Book A C Davison and D V Hinkley 1997 Bootstrap Methods and Their Applications Cambridge University Press Annette J Dobson 1990 An Introduction to Generalized Linear Models Chapman and Hall London Peter McCullagh and John A Nelder 1989 Generalized Linear Models Second edition Chap man and Hall London John A Rice 1995 Mathematical Statistics and Data Analysis Second edition Duxbury Press Belmont CA S D Silvey 1970 Statistical Inference Penguin London
118. nts and corrections are always welcome Please address email correspondence to R core R project org Suggestions to the reader Most R novices will start with the introductory session in Appendix A This should give some familiarity with the style of R sessions and more importantly some instant feedback on what actually happens Many users will come to R mainly for its graphical facilities See Chapter 12 Graphics page 63 which can be read at almost any time and need not wait until all the preceding sections have been digested Chapter 1 Introduction and preliminaries 2 1 Introduction and preliminaries 1 1 The R environment R is an integrated suite of software facilities for data manipulation calculation and graphical display Among other things it has e an effective data handling and storage facility e a suite of operators for calculations on arrays in particular matrices e a large coherent integrated collection of intermediate tools for data analysis e graphical facilities for data analysis and display either directly at the computer or on hard copy and e a well developed simple and effective programming language called S which includes conditionals loops user defined recursive functions and input and output facilities Indeed most of the system supplied functions are themselves written in the S language The term environment is intended to characterize it as a fully planned and coherent system rat
119. omprehend and control 11 5 Updating fitted models The update function is largely convenience function that allows a model to be fitted that differs from one previously fitted usually by just a few additional or removed terms Its form is gt new model lt update old model new formula In the new formula the special name consisting of a period only can be used to stand for the corresponding part of the old model formula For example gt fm05 lt Im y x1 x2 x3 x4 xb data production gt fm6 lt update fm05 x6 gt sm 6 update fm6 sqrt Chapter 11 Statistical models in R 56 would fit a five variate multiple regression with variables presumably from the data frame production fit an additional model including a sixth regressor variable and fit a variant on the model where the response had a square root transform applied Note especially that if the data argument is specified on the original call to the model fitting function this information is passed on through the fitted model object to update and its allies The name can also be used in other contexts but with slightly different meaning For example gt fmfull lt lm y data production would fit a model with response y and regressor variables all other variables in the data frame production Other functions for exploring incremental sequences of models are add1 drop1 and stepO
120. onents 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 name Fred wife Mary no children 3 child ages c 4 7 9 Components are always numbered and may always be referred to as such Thus if Lst is the name of a list with four components these may be individually referred to as Lst 1 Lst 2 Lst 31 and Lst 4 If further Lst 4 is a vector subscripted array then Lst 4 1 is its first entry If Lst is a list then the function length Lst gives the number of top level components it has Components of lists may also be named and in this case the component may be referred to either by giving the component name as a character string in place of the number in double square brackets or more conveniently by giving an expression of the form name component name for the same thing This is a very useful convention as it makes it easier to get the right component if you forget the number So in the simple example given above Lst name is the same as Lst 1 and is the string Fred Lst wife is the same as Lst 2 and is the string Mary Lst child ages 1 is the same as Lst 4 1 and is the number 4 Additionally one can also use the names of the list compon
121. or further details 1 to be discussed later or use xyplot from package lattice https CRAN R project org package lattice Chapter 9 Grouping loops and conditional execution 41 Warning for loops are used in R code much less often than in compiled languages Code that takes a whole object view is likely to be both clearer and faster in R Other looping facilities include the gt repeat expr statement and the gt while condition expr statement The break statement can be used to terminate any loop possibly abnormally This is the only way to terminate repeat loops The next statement can be used to discontinue one particular cycle and skip to the next Control statements are most often used in connection with functions which are discussed in Chapter 10 Writing your own functions page 42 and where more examples will emerge Chapter 10 Writing your own functions 42 10 Writing your own functions As we have seen informally along the way the R language allows the user to create objects of mode function These are true R functions that are stored in a special internal form and may be used in further expressions and so on In the process the language gains enormously in power convenience and elegance and learning to write useful functions is one of the main ways to make your use of R comfortable and productive It should be emphasized that most of the functions supplied as part of the R system such a
122. pact printing d listO 1 lt 0 for i in dim a 4 apri lt 1 1 lt rep i dimnames a lt d a With this function defined 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 10 6 3 Recursive numerical integration Functions may be recursive and may themselves define functions within themselves Note however that such functions or indeed variables are not inherited by called functions in higher evaluation frames as they would be if they were on the search path The example below shows a naive way of performing one dimensional numerical integration The integrand is evaluated at the end points of the range and in the middle If the one panel trapezium rule answer is close enough to the two panel then the latter is returned as the value Otherwise the same process is recursively applied to each panel The result is an adaptive integration process that concentrates function evaluations in regions where the integrand is farthest from linear There is however a heavy overhead and the function is only competitive with other algorithms when the integrand is both smooth and very difficult to evaluate The example is also given partly as a little puzzle in H programming Chapter 10 Writing your own functions 46 area lt function f a b eps 1 0e 06 lim 10 4 funi lt function
123. plicate earlier symbols but can be coloured in different ways see the help on points and its examples In addition pch can be a character or a number in the range 32 255 representing a character in the current font 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 OO 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 filled regions and images A number from the current palette see palette or a named colour The color to be used for axis annotation x and y labels main and sub titles re spectively An integer which specifies which font to use for text If possible device drivers arrange so that 1 corresponds to plain text 2 to bold face 3 to italic 4 to bold italic and 5 to a symbol font which include Greek letters The font to be used for axis annotation x and y labels main and sub titles respec tively Justification of text relative to the plotting position O means left justify 1 means right justify and 0 5 means to center horizontally about the plotting position The actual value is
124. r being preferable Chapter 5 Arrays and matrices 25 5 10 Frequency tables from factors Recall that a factor defines a partition into groups Similarly a pair of factors defines a two way cross classification and so on The function table allows frequency tables to be calcu lated from equal length factors If there are k factor arguments the result is a k way array of frequencies Suppose for example that statef is a factor giving the state code for each entry in a data vector The assignment gt statefr lt table statef gives in statefr a table of frequencies of each state in the sample The frequencies are ordered and labelled by the levels attribute of the factor This simple case is equivalent to but more convenient than gt statefr lt tapply statef statef length Further suppose that incomef is a factor giving a suitably defined income class for each entry in the data vector for example with the cut O function gt factor cut incomes breaks 35 10 0 7 gt incomef Then to calculate a two way table of frequencies gt table incomef statef statef incomef act nsw nt qld sa tas vic wa 35 45 1 1 0 1 0 0 1 0 45 55 1 1 1 1 2 0 1 3 55 65 0 3 1 3 2 2 2 1 65 75 0 1 0 0 0 0 1 0 Extension to higher way frequency tables is immediate Chapter 6 Lists and data frames 26 6 Lists and data frames 6 1 Lists An R list is an object consisting of an ordered collection of objects known as its comp
125. rator and R converts the known exceptions to the form required by Windows Chapter 14 OS facilities 80 e The behaviour of filepaths with a trailing slash is OS dependent Such paths are not valid on Windows and should not be expected to work POSIX 2008 requires such paths to match only directories but earlier versions allowed them to also match files So they are best avoided e Multiple slashes in filepaths such as abc def are valid on POSIX filesystems and treated as if there was only one slash They are usually accepted by Windows OS functions However leading double slashes may have a different meaning e Windows UNC filepaths such as server dir1 dir2 file and UNC server diri dir2 file are not supported but they may work in some R functions POSIX filesystems are allowed to treat a leading double slash specially e Windows allows filepaths containing drives and relative to the current directory on a drive e g d foo bar refers to d a b c foo bar if the current directory on drive d is a b c It is intended that these work but the use of absolute paths is safer Functions basename and dirname select parts of a file path the recommended way to as semble a file path from components is file path Function pathexpand does tilde expansion substituting values for home directories the current user s and perhaps those of other users On filesystems with links a single file can be referred to by many filepaths Fu
126. rder 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 4o gt ab lt a fof b An alternative is gt ab outer a b The multiplication function can be replaced by an arbitrary function of two variables For example if we wished to evaluate the function f x y cos y 1 z over a regular grid of values with x and y coordinates defined by the R vectors x and y respectively we could proceed as follows gt f lt function x y cos y 1 x 2 gt z lt outer x y f In particular the outer product of two ordinary vectors is a doubly subscripted array that is a matrix of rank at most 1 Notice that the outer product operator is of course non commutative Defining your own R functions will be considered further in Chapter 10 Writing your own functions page 42 An example Determinants of 2 by 2 single digit matrices As an artificial but cute example consider the determinants of 2 by 2 matrices a b c d where each entry is a non negative integer in the range 0 1 9 that is a digit The problem is to find the determinants ad bc of all possible matrices of this form and represent the frequency with which each value occurs as a high density plot This amounts to finding the probability distribution of the determinant if each digit is chosen independently and uniformly at
127. re not accepted Note that on a Unix alike the input filename such as foo R should not contain spaces nor shell metacharacters Appendix C The command line editor 92 Appendix C The command line editor C 1 Preliminaries When the GNU readline library is available at the time R is configured for compilation un der UNIX an inbuilt command line editor allowing recall editing and re submission of prior commands is used Note that other versions of readline exist and may be used by the inbuilt command line editor this used to happen on OS X It can be disabled useful for usage with ESS using the startup option no readline Windows versions of R have somewhat simpler command line editing see Console under the Help menu of the GUI and the file README Rterm for command line editing under Rterm exe When using R with readline capabilities the functions described below are available as well as others probably documented in man readline or info readline on your system Many of these use either Control or Meta characters Control characters such as Control m are obtained by holding the CTRL down while you press the m key and are written as C m below Meta characters such as Meta b are typed by holding down META and pressing b and written as M b in the following If your terminal does not have a META key enabled you can still type Meta characters using two character sequences starting with ESC Thus to enter M b you
128. reating the three functions within account and then returning a list containing them When account is invoked it takes a numerical argument total and returns a list containing the three functions Because these functions are defined in an environment which contains total they will have access to its value The special assignment operator is used to change the value associated with total This operator looks back in enclosing environments for an environment that contains the symbol total and when it finds such an environment it replaces the value in that environment with the value of right hand side If the global or top level environment is reached without finding the symbol total then that variable is created and assigned to there For most users creates a global variable and assigns the value of the right hand side to it Only when has been used in a function that was returned as the value of another function will the special behavior described here occur open account lt function total 1 list deposit function amount 1 if amount lt 0 stop Deposits must be positive Wn total lt lt total amount cat amount deposited Your balance is total n n Ts withdraw function amount 4 if amount gt total stop You don t have that much money Wn total total amount cat amount withdrawn Your balance is total n n LE balance function cat Your balance is total n n
129. rement unit is text lines mar and mai are equivalent in the sense that setting one changes the value of the other The default values chosen for this parameter are often too large the right hand margin is rarely needed and neither is the top margin if no title is being used The bottom and left margins must be large enough to accommodate the axis and tick labels Furthermore the default is chosen without regard to the size of the device surface for example using the postscript driver with the height 4 argument will result in a plot which is about 50 margin unless mar or mai are set explicitly When multiple figures are in use see below the margins are reduced however this may not be enough when many figures share the same page Chapter 12 Graphical procedures 12 5 4 Multiple figure environment 73 R allows you to create an n by m array of figures on a single page Each figure has its own margins and the array of figures is optionally surrounded by an outer margin as shown in the following figure oma 3 omi 4 mfg c 3 2 3 2 mfrow c 3 2 omi 1 The graphical parameters relating to multiple figures are as follows mfcol c 3 2 mfrow c 2 4 Set the size of a multiple figure array The first value is the number of rows the second is the number of columns The only difference between these two parameters is that setting mfcol causes figures to be filled by column mfrow fills by
130. rinted as lt NA gt when character vectors are printed without quotes 2 6 Character vectors Character quantities and character vectors are used frequently in R for example as plot labels Where needed they are denoted by a sequence of characters delimited by the double quote character e g x values New iteration results Character strings are entered using either matching double or single quotes but are printed using double quotes or sometimes without quotes They use C style escape sequences using as the escape character so NN is entered and printed as NN and inside double quotes is entered as V Other useful escape sequences are Nn newline Nt tab and Nb backspace see Quotes for a full list Character vectors may be concatenated into a vector by the c function examples of their use will emerge frequently The paste O function takes an arbitrary number of arguments and concatenates them one by one into character strings Any numbers given among the arguments are coerced into character strings in the evident way that is in the same way they would be if they were printed The arguments are by default separated in the result by a single blank character but this can be changed by the named argument sep string which changes it to string possibly empty For example gt labs lt paste c X Y 1 10 sep makes labs into the character vector GIU Y2 X3 Y4 X5 Y6 X7 Y8 X9 Y10
131. rix the value is a p by p sample covariance matrix got by regarding the rows as independent p variate sample vectors sort x returns a vector of the same size as x with the elements arranged in increasing order however there are other more flexible sorting facilities available see order or sort list which produce a permutation to do the sorting Note that max and min select the largest and smallest values in their arguments even if they are given several vectors The parallel maximum and minimum functions pmax and pmin return a vector of length equal to their longest argument that contains in each element the largest smallest element in that position in any of the input vectors For most purposes the user will not be concerned if the numbers in a numeric vector are integers reals or even complex Internally calculations are done as double precision real numbers or double precision complex numbers if the input data are complex To work with complex numbers supply an explicit complex part Thus sqrt 17 will give NaN and a warning but sqrt 17 0i will do the computations as complex numbers 2 3 Generating regular sequences R has a number of facilities for generating commonly used sequences of numbers For example 1 30 is the vector c 1 2 29 30 The colon operator has high priority within an ex pression so for example 2 1 15 is the vector c 2 4 28 30 Put n lt 10 and compare the sequences 1 n 1 a
132. rows The layout in the Figure could have been created by setting mfrow c 3 2 the figure shows the page after four plots have been drawn Setting either of these can reduce the base size of symbols and text controlled by par cex and the pointsize of the device In a layout with exactly two rows and columns the base size is reduced by a factor of 0 83 if there are three or more of either rows or columns the reduction factor is 0 66 mfg c 2 2 3 2 Position of the current figure in a multiple figure environment The first two numbers are the row and column of the current figure the last two are the number of rows and columns in the multiple figure array Set this parameter to jump between figures in the array You can even use different values for the last two numbers than the true values for unequally sized figures on the same page fig c 4 9 1 4 10 Position of the current figure on the page Values are the positions of the left right bottom and top edges respectively as a percentage of the page measured from the bottom left corner The example value would be for a figure in the bottom right of the page Set this parameter for arbitrary positioning of figures within a page If you want to add a figure to a current page use new TRUE as well unlike S oma c 2 0 3 0 omi c 0 0 0 8 0 Size of outer margins Like mar and mai the first measures in text lines and the second in inches starting with the bottom margin and
133. s guide is aimed at users who have this facility In particular we will occasionally refer to the use of R on an X window system although the vast bulk of what is said applies generally to any implementation of the R environment Most users will find it necessary to interact directly with the operating system on their computer from time to time In this guide we mainly discuss interaction with the operating system on UNIX machines If you are running R under Windows or OS X you will need to make some small adjustments Setting up a workstation to take full advantage of the customizable features of R is a straight forward if somewhat tedious procedure and will not be considered further here Users in diffi culty should seek local expert help 1 5 Using R interactively When you use the R program it issues a prompt when it expects input commands The default prompt is 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 first occasion is as follows 1 Create a separate sub directory say work to hold data files on which you will use R for this problem This will be the working directory whenever you use R for this particular problem mkdir work cd work 2 Start the R program with the command
134. s mean var postscript and so on are themselves written in R and thus do not differ materially from user written functions A function is defined by an assignment of the form gt name lt function arg 1 arg 2 expression The expression is an R expression usually a grouped expression that uses the arguments arg i to calculate a value The value of the expression is the value returned for the function A call to the function then usually takes the form name expr 1 expr 2 and may occur anywhere a function call is legitimate 10 1 Simple examples As a first example consider a function to calculate the two sample t statistic showing all the steps This is an artificial example of course since there are other simpler ways of achieving the same end The function is defined as follows gt twosam lt function yl y2 ni lt length yl n2 lt length y2 ybi lt mean yl yb2 lt mean y2 si lt var y1 s2 lt var y2 s lt n1 1 s1 n2 1 s2 n1 n2 2 tst lt ybi yb2 sgrt s 1 n1 1 n2 tst D With this function defined you could perform two sample t tests using a call such as gt tstat twosam data male data female tstat As a second example consider a function to emulate directly the MATLAB backslash com mand which returns the coefficients of the orthogonal projection of the vector y onto the column space of the matrix X This is ordinarily called the least
135. s 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 identify function as follows plot x y gt identify x y The identify O functions performs no plotting itself but simply allows the user to move the mouse pointer and click the left mouse button near a point If there is a point near the mouse pointer it will be marked with its index number that is its position in the x y vectors plotted nearby Alternatively you could use some informative string such as a case name as a highlight by using the labels argument to identify or disable marking altogether with the plot FALSE argument When the process is terminated see above identify O returns the indices of the selected points you can use these indices to extract the selected points from the original vectors x and y 12 4 Using graphics parameters When creating graphics particularly for presentation or publication purposes R s defaults do not always produce exactly that which is required You can however customize almost every aspect of the display using graphics parameters R maintains a list of a large number of graphics parameters which control things such as line style colors figure arrangement and text justifica tion among many others Every graphic
136. s parameter has a name such as col which controls colors and a value a color number for example A separate list of graphics parameters is maintained for each active device and each device has a default set of parameters when initialized Graphics parameters can be set in two ways either permanently affecting all graphics functions which access the current device or temporarily affecting only a single graphics function call 12 4 1 Permanent changes The par function The par O function is used to access and modify the list of graphics parameters for the current graphics device Chapter 12 Graphical procedures 69 par Without arguments returns a list of all graphics parameters and their values for the current device par c col 1ty With a character vector argument returns only the named graphics parameters again as a list par col 4 lty 2 With named arguments or a single list argument sets the values of the named graphics parameters and returns the original values of the parameters as a list Setting graphics parameters with the par function changes the value of the parameters permanently in the sense that all future calls to graphics functions on the current device will be affected by the new value You can think of setting graphics parameters in this way as setting default values for the parameters which will be used by all graphics functions unless an alternative value is given Note t
137. s such as locator see below may be used to specify positions on a plot interactively 12 2 1 Mathematical annotation In some cases it is useful to add mathematical symbols and formulae to a plot This can be achieved in R by specifying an expression rather than a character string in any one of text mtext axis or title For example the following code draws the formula for the Binomial probability function gt text x y expression paste bgroup atop n x p x q n xJ More information including a full listing of the features available can obtained from within R using the commands gt help plotmath gt example plotmath gt demo plotmath 12 2 2 Hershey vector fonts It is possible to specify Hershey vector fonts for rendering text when using the text and contour functions There are three reasons for using the Hershey fonts e Hershey fonts can produce better output especially on computer screen for rotated and or small text e 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 e 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 help Hershey gt demo Hershey gt help Japanese gt demo Japanese 12 3 Interacting with graphics R al
138. sions even involving other arguments to the same function they are not restricted to be constants as in our simple example here 1 See also the methods described in Chapter 11 Statistical models in R page 51 Chapter 10 Writing your own functions 44 10 4 The argument Another frequent requirement is to allow one function to pass on argument settings to another For example many graphics functions use the function par and functions like plot allow the user to pass on graphical parameters to par to control the graphical output See Section 12 4 1 The par function page 68 for more details on the par O function This can be done by including an extra argument literally of the function which may then be passed on An outline example is given below funi lt function data data frame graph TRUE limit 20 omitted statements if graph par pch more omissions Less frequently a function will need to refer to components of The expression list evaluates all such arguments and returns them in a named list while 1 2 etc evaluate them one at a time with n returning the n th unmatched argument 10 5 Assignments within functions Note that any ordinary assignments done within the function are local and temporary and are lost after exit from the function Thus the assignment X lt qr X does not affect the value of the argument in the calling program
139. so provides functions which allow users to extract or add information to a plot using a mouse The simplest of these is the locator function Chapter 12 Graphical procedures 68 locator n type Waits for the user to select locations on the current plot using the left mouse button This continues until n default 512 points have been selected or another mouse button is pressed The type argument allows for plotting at the selected points and has the same effect as for high level graphics commands the default is no plotting locator returns the locations of the points selected as a list with two components x and y locator is usually called with no arguments It is particularly useful for interactively selecting positions for graphic elements such as legends or labels when it is difficult to calculate in advance where the graphic should be placed For example to place some informative text near an outlying point the command gt text locator 1 Outlier adj 0 may be useful locator will be ignored if the current device such as postscript does not support interactive pointing identify x y labels Allow the user to highlight any of the points defined by x and y using the left mouse button by plotting the corresponding component of labels nearby or the index number of the point if labels is absent Returns the indices of the selected points when another button is pressed Sometimes we want to identify particular point
140. spective distributions hist x hist x nclass n hist x breaks b Produces a histogram of the numeric vector x A sensible number of classes is usually chosen but a recommendation can be given with the nclass argument Alternatively the breakpoints can be specified exactly with the breaks argument Chapter 12 Graphical procedures 65 If the probability TRUE argument is given the bars represent relative frequencies divided by bin width instead of counts dotchart x Constructs a dotchart of the data in x In a dotchart the y axis gives a labelling of the data in x and the z axis gives its value For example it allows easy visual selection of all data entries with values lying in specified ranges image x y Z contour x y Z persp x y Z Plots of three variables The image plot draws a grid of rectangles using different colours to represent the value of z the contour plot draws contour lines to represent the value of z and the persp plot draws a 3D surface 12 1 4 Arguments to high level plotting functions There are number of arguments which may be passed to high level graphics functions as follows add TRUE Forces the function to act as a low level graphics function superimposing the plot on the current plot some functions only axes FALSE Suppresses generation of axes useful for adding your own custom axes with the axis function The default axes TRUE means include axes
141. sse see 18 5 9 Md matrices c 19 pd The array function voee DER ROS UDEREPWOPUN A ai 20 5 4 1 Mixed vector and array arithmetic The recycling nie 20 5 5 The outer product of two amgang 21 5 6 Generalized transpose of an array 1 6 en 21 ee EECH 22 5 1 Matrix multiplication 0 id ete daa les 22 5 7 2 Linear equations and mveraion rr 22 5 7 3 Eigenvalues and eigenvectors rr 23 5 7 4 Singular value decomposition and determunants sees 23 5 7 5 Least squares fitting and the QR decomposition 0000 e eee eee ee 23 5 8 Forming partitioned matrices cbind and rbindc eee eee 24 5 9 The concatenation function c with amgang 24 5 10 Frequency tables from factors 0 2 c ccc een 25 Lists and data frames il 26 UBNBEL HC 26 6 2 Constructing and modifying botze 27 6 2 1 Concatenating lists ic eck ber ee deed RR e rer eR NNN FIG 27 6 3 Data EE 27 6 3 1 Making data frames titi A Sy peta AAA 27 6 3 2 attach and detach iii ida 28 6 3 3 Working with data Dramen 28 6 3 4 Attaching arbitrary ste 28 6 3 5 Managing the search path 29 Reading data from Des 30 TA The read table Q Fonction ed aaa e b a e eae A a 30 2 Th scan E Te rn dd AR 31 7 3 Accessing builtin datasets isssssssesssesssssssss e m ee ennes 31 7 3 1 Loading data from other R packages sssseesesssees en 31 TA oBditing dat A VrItak DEC A ey FI 32 Probability distributions sess eese 33 8 1 Rasa set of statistical ta
142. t operator lt which consists of the two characters lt less than and minus occurring strictly side by side and it points to the object receiving the value of the expression In most contexts the operator can be used as an alternative Assignment can also be made using the function assign An equivalent way of making the same assignment as above is with gt assign x c 10 4 5 6 3 1 6 4 21 7 The usual operator can be thought of as a syntactic short cut to this Assignments can also be made in the other direction using the obvious change in the assign ment operator So the same assignment could be made using gt c 10 4 5 6 3 1 6 4 21 7 gt x If an expression is used as a complete command the value is printed and lost So now if we were to use the command gt 1 x the reciprocals of the five values would be printed at the terminal and the value of x of course unchanged The further assignment gt y lt c x 0 x would create a vector y with 11 entries consisting of two copies of x with a zero in the middle place 2 2 Vector arithmetic Vectors can be used in arithmetic expressions in which case the operations are performed element by element Vectors occurring in the same expression need not all be of the same length If they are not the value of the expression is a vector with the same length as the longest vector which occurs in the expression Short
143. ta frame with three variables lentils u lentils v lentils w The attach gt attach lentils places the data frame in the search path at position 2 and provided there are no variables u v or w in position 1 u v and w are available as variables from the data frame in their own right At this point an assignment such as gt u lt vtw does not replace the component u of the data frame but rather masks it with another variable u in the working directory at position 1 on the search path To make a permanent change to the data frame itself the simplest way is to resort once again to the notation gt lentils u lt v w However the new value of component u is not visible until the data frame is detached and attached again To detach a data frame use the function gt detach More precisely this statement detaches from the search path the entity currently at position 2 Thus in the present context the variables u v and w would be no longer visible except under the list notation as lentils u and so on Entities at positions greater than 2 on the search path can be detached by giving their number to detach but it is much safer to always use a name for example by detach lentils or detach lentils Note In R lists and data frames can only be attached at position 2 or above and what is attached is a copy of the original object You can alter the attached values via assign but the original list or data frame is unchanged 6 3 3
144. ter 11 Statistical models in R gt fn function p sum y p 1 x In order to do the fit we need initial estimates of t 60 pE2 x 9 2 he parameters One way to find sensible starting values is to plot the data guess some parameter values and superimpose the model curve using those values plot x y gt xfit lt seq 02 1 1 05 gt yfit 200 xfit 0 1 xfit gt lines spline xfit yfit We could do better but these starting values of 200 and 0 1 seem adequate Now do the fit gt out lt nlm fn p c 200 0 1 hessian TRUE After the fitting out minimum is the SSE and out estimate are the least squares estimates of the parameters To obtain the approximate standard errors SE of the estimates we do gt sqrt diag 2 out minimum length y 2 solve out hessian The 2 which is subtracted in the line above represents the number of parameters A 9596 confidence interval would be the parameter estimate squares fit on a new plot gt plot x y gt xfit lt seq 02 1 1 05 1 96 SE We can superimpose the least gt yfit 212 68384222 xfit 0 06412146 xfit gt lines spline xfit yfit The standard package stats provides much more extensive facilities for fitting non linear models by least squares The model we have just fitted is the Michaelis Menten model so we can use gt df data frame x x y y gt fit nls y SSmicmen x Vm K
145. the Applications folder on your system It is a standard double clickable OS X application The startup procedure under OS X is very similar to that under UNIX but R app does not make use of command line arguments The home directory is the one inside the R framework but the startup and current working directory are set as the user s home directory unless a different startup directory is given in the Preferences window accessible from within the GUI B 4 Scripting with R If you just want to run a file foo R of R commands the recommended way is to use R CMD BATCH foo R If you want to run this in the background or as a batch job use OS specific facilities to do so for example in most shells on Unix alike OSes R CMD BATCH foo R runs a background job You can pass parameters to scripts via additional arguments on the command line for example where the exact quoting needed will depend on the shell in use R CMD BATCH args argl arg2 foo R E will pass arguments to a script which can be retrieved as a character vector by args lt commandArgs TRUE This is made simpler by the alternative front end Rscript which can be invoked by Rscript foo R argi arg2 and this can also be used to write executable script files like at least on Unix alikes and in some Windows shells path to Rscript args lt commandArgs TRUE q status lt exit status code If this is entered into a text file runfoo and this is made executable by
146. the arguments to cbind are vectors they may be shorter than the column size of any matrices present in which case they are cyclically extended to match the matrix column size or the length of the longest vector if no matrices are given The function rbind does the corresponding operation for rows In this case any vector argument possibly cyclically extended are of course taken as row vectors Suppose X1 and X2 have the same number of rows To combine these by columns into a matrix X together with an initial column of 1s we can use gt X lt cbind 1 X1 X2 The result of rbind or cbind always has matrix status Hence cbind x and rbind x are possibly the simplest ways explicitly to allow the vector x to be treated as a column or row matrix respectively 5 9 The concatenation function cO with arrays It should be noted that whereas cbind and rbind are concatenation functions that respect dim attributes the basic c function does not but rather clears numeric objects of all dim and dimnames attributes This is occasionally useful in its own right The official way to coerce an array back to a simple vector object is to use as vector gt vec lt as vector X However a similar result can be achieved by using cO with just one argument simply for this side effect vec c X There are slight differences between the two but ultimately the choice between them is largely a matter of style with the forme
147. the methods described in the previous section may be used invoking by R exe or more directly by Rterm exe For interactive use there is a console based GUI Rgui exe The startup procedure under Windows is very similar to that under UNIX but references to the home directory need to be clarified as this is not always defined on Windows If the environment variable R USER is defined that gives the home directory Next if the environment variable HOME is defined that gives the home directory After those two user controllable settings R tries to find system defined home directories It first tries to use the Windows personal directory typically C Documents and Settings username My Documents in Windows XP If that fails and environment variables HOMEDRIVE and HOMEPATH are defined and they normally are these define the home directory Failing all those the home directory is taken to be the starting directory You need to ensure that either the environment variables TMPDIR TMP and TEMP are either unset or one of them points to a valid place to create temporary files and directories Environment variables can be supplied as name value pairs on the command line If there is an argument ending RData in any case it is interpreted as the path to the workspace to be restored it implies restore and sets the working directory to the parent of the named file This mechanism is used for drag and drop and file association with RGui e
148. tic meaning and that term appears in the model matrix Note that inside the parentheses that usually enclose function arguments all operators have their normal arithmetic meaning The function I is an identity function used to allow terms in model formulae to be defined using arithmetic operators Note particularly that the model formulae specify the columns of the model matrix the specification of the parameters being implicit This is not the case in other contexts for example in specifying nonlinear models 11 1 1 Contrasts We need at least some idea how the model formulae specify the columns of the model matrix This is easy if we have continuous variables as each provides one column of the model matrix and the intercept will provide a column of ones if included in the model What about a k level factor A The answer differs for unordered and ordered factors For unordered factors k 1 columns are generated for the indicators of the second kth levels of the factor Thus the implicit parameterization is to contrast the response at each level with that at the first For ordered factors the k 1 columns are the orthogonal polynomials on 1 k omitting the constant term Although the answer is already complicated it is not the whole story First if the intercept is omitted in a model that contains a factor term the first such term is encoded into k columns giving the indicators for all the levels Second the whole be
149. tiles x lt rt 250 df 5 qqnorm x qqline x which will usually if it is a random sample show longer tails than expected for a normal We can make a Q Q plot against the generating distribution by aqplot qt ppoints 250 df 5 x xlab Q Q plot for t dsn aqline x Finally we might want a more formal test of agreement with normality or not R provides the Shapiro Wilk test gt shapiro test long Shapiro Wilk normality test data long N 0 9793 p value 0 01052 and the Kolmogorov Smirnov test gt ks test long pnorm mean mean long sd sqrt var long One sample Kolmogorov Smirnov test data long D 0 0661 p value 0 4284 alternative hypothesis two sided Note that the distribution theory is not valid here as we have estimated the parameters of the normal distribution from the same sample 8 3 One and two sample tests So far we have compared a single sample to a normal distribution A much more common operation is to compare aspects of two samples Note that in R all classical tests including the ones used below are in package stats which is normally loaded Consider the following sets of data on the latent heat of the fusion of ice cal gm from Rice 1995 p 490 Chapter 8 Probability distributions 37 Method A 79 98 80 04 80 02 80 04 80 03 80 03 80 04 79 97 80 05 80 03 80 02 80 00 80 02 Method B 80 02 79 94 79 98 79 97 79 97 80 03 79 95 79 97 Boxplots provide a simple
150. tion is given an empty indez vector then the full range of that subscript is taken Continuing the previous example a 2 isa 4 x 2 array with dimension vector c 4 2 and data vector containing the values 668 211 411 8 2 2 1 a 2 3 1 12 41 a 2 1 2 a 2 2 2 a 2 3 2 12 421 in that order a stands for the entire array which is the same as omitting the subscripts entirely and using a alone For any array say Z the dimension vector may be referenced explicitly as dim Z on either side of an assignment Also if an array name is given with just one subscript or index vector then the corresponding values of the data vector only are used in this case the dimension vector is ignored This is not the case however if the single index is not a vector but itself an array as we next discuss Chapter 5 Arrays and matrices 19 5 3 Index matrices As well as an index vector in any subscript position a matrix may be used with a single index matrix in order either to assign a vector of quantities to an irregular collection of elements in the array or to extract an irregular collection as a vector A matrix example makes the process clear In the case of a doubly indexed array an index matrix may be given consisting of two columns and as many rows as desired The entries in the index matrix are the row and column indices for the doubly indexed array Suppose for example we have a 4 by 5 array X and we wish to do the following
151. tion variables rtags Unix only Create Emacs style tag files from C R and Rd files open Windows only Open a file via Windows file associations texify Windows only Process La TeX files with R s style files Use Appendix B Invoking R 89 R CMD command help to obtain usage information for each of the tools accessible via the R CMD interface In addition you can use options arch no environ no init file no site file and vanilla between R and CMD these affect any R processes run by the tools Here vanilla is equivalent to no environ no site file no init file However note that R CMD does not of itself use any R startup files in particular neither user nor site Renviron files and all of the R processes run by these tools except BATCH use no restore Most use vanilla and so invoke no R startup files the current exceptions are INSTALL REMOVE Sweave and SHLIB which uses no site file no init file R CMD cmd args for any other executable cmd on the path or given by an absolute filepath this is useful to have the same environment as R or the specific commands run under for example to run 1dd or pdflatex Under Windows cmd can be an executable or a batch file or if it has extension sh or pl the appropriate interpreter if available is called to run it B 2 Invoking R under Windows There are two ways to run R under Windows Within a terminal window e g cmd exe or a more capable shell
152. tors of length one and only evaluate their second argument if necessary There is a vectorized version of the if else construct the ifelse function This has the form ifelse condition a b and returns a vector of the length of its longest argument with elements a i if condition i is true otherwise b i 9 2 2 Repetitive execution for loops repeat and while There is also a for loop construction which has the form gt for name in expr 1 expr 2 where name is the loop variable expr 1 is a vector expression often a sequence like 1 20 and expr 2 is often a grouped expression with its sub expressions written in terms of the dummy name expr 2 is repeatedly evaluated as name ranges through the values in the vector result of expr 1 As an example suppose ind is a vector of class indicators and we wish to produce separate plots of y versus x within classes One possibility here is to use coplot O 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 split x and gt yc lt split y ind gt for i in 1 length yc i plot xc i yc i1 abline lsfit xc i yc i Note the function split which produces a list of vectors obtained by splitting a larger vector according to the classes specified by a factor This is a useful function mostly used in connection with boxplots See the help facility f
153. ts arguments specific to the class of the argument itself If the argument lacks any class attribute or has a class not catered for specifically by the generic function in question there is always a default action provided An example makes things clearer The class mechanism offers the user the facility of designing and writing generic functions for special purposes Among the other generic functions are plot for displaying objects graphically summary O for summarizing analyses of various types and anova for comparing statistical models The number of generic functions that can treat a class in a specific way can be quite large For example the functions that can accommodate in some fashion objects of class data frame include lt any as matrix 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 data frame density factor and more A complete list can be got again by using the methods O 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
154. unif min max Weibull weibull shape scale Wilcoxon wilcox m n Prefix the name given here by d for the density p for the CDF q for the quantile function and r for simulation random deviates The first argument is x for dxxx q for pxxx p for qxxx and n for rxxx except for rhyper rsignrank and rwilcox for which it is nn In not quite all cases is the non centrality parameter ncp currently available see the on line help for details The pxxx and qxxx functions all have logical arguments lower tail and log p and the dxxx ones have log This allows e g getting the cumulative or integrated hazard function H t log 1 F t by pxxx t lower tail FALSE log p TRUE or more accurate log likelihoods by dxxx log TRUE directly In addition there are functions ptukey and qtukey for the distribution of the studentized range of samples from a normal distribution and dmultinom and rmultinom for the multinomial distribution Further distributions are available in contributed packages notably SuppDists https CRAN R project org package SuppDists Here are some examples gt 2 tailed p value for t distribution 2 pt 2 43 df 13 upper 1 point for an F 2 7 distribution qf 0 01 2 7 lower tail FALSE See the on line help on RNG for how random number generation is done in R NW WM WM Chapter 8 Probability distributions 34 8 2 Examining the distribution of a
155. untants in another vector in suitably large units of money gt incomes lt c 60 49 40 61 64 60 59 54 62 69 70 42 56 61 61 61 58 51 48 65 49 49 41 48 52 46 59 46 58 43 To calculate the sample mean income for each state we can now use the special function tapply gt incmeans lt tapply incomes statef mean giving a means vector with the components labelled by the levels act nsw nt qid sa tas vic wa 44 500 57 333 55 500 53 600 55 000 60 500 56 000 52 250 The function tapply O is used to apply a function here mean to each group of components of the first argument here incomes defined by the levels of the second component here statef Readers should note that there are eight states and territories in Australia namely the Australian Capital Territory New South Wales the Northern Territory Queensland South Australia Tasmania Victoria and Western Australia 2 Note that tapply also works in this case when its second argument is not a factor e g tapply incomes state and this is true for quite a few other functions since arguments are coerced to factors when necessary using as factor Chapter 4 Ordered and unordered factors 17 as if they were separate vector structures The result is a structure of the same length as the levels attribute of the factor containing the results The reader should consult the help document for more details Suppose further we needed to cal
156. ussion R contains many operators and functions that are available only for matrices For example t X is the matrix transpose function as noted above The functions nrow A and ncol A give the number of rows and columns in the matrix A respectively 5 7 1 Matrix multiplication The operator x is used for matrix multiplication An n by 1 or 1 by n matrix may of course be used as an n vector if in the context such is appropriate Conversely vectors which occur in matrix multiplication expressions are automatically promoted either to row or column vectors whichever is multiplicatively coherent if possible although this is not always unambiguously possible as we see later If for example A and B are square matrices of the same size then gt AxB is the matrix of element by element products and gt A 4 4 B is the matrix product If x is a vector then gt x Leh ACD X is a quadratic form The function crossprod forms crossproducts meaning that crossprod X y is the same as t X y but the operation is more efficient If the second argument to crossprod is omitted it is taken to be the same as the first The meaning of diagO depends on its argument diag v where v is a vector gives a diagonal matrix with elements of the vector as the diagonal entries On the other hand diag M where M is a matrix gives the vector of main diagonal entries of M This is the same convention as that used for diag in MATLAB Also so
157. working clockwise Chapter 12 Graphical procedures 74 Outer margins are particularly useful for page wise titles etc Text can be added to the outer margins with the mtext function with argument outer TRUE There are no outer margins by default however so you must create them explicitly using oma or omi More complicated arrangements of multiple figures can be produced by the split screen and layout functions as well as by the grid and lattice https CRAN R project org package lattice packages 12 6 Device drivers R can generate graphics of varying levels of quality on almost any type of display or printing device Before this can begin however R needs to be informed what type of device it is dealing with This is done by starting a device driver The purpose of a device driver is to convert graphical instructions from R draw a line for example into a form that the particular device can understand Device drivers are started by calling a device driver function There is one such function for every device driver type help Devices for a list of them all For example issuing the command gt postscript causes all future graphics output to be sent to the printer in PostScript format Some commonly used device drivers are x110 For use with the X11 window system on Unix alikes windows For use on Windows quartz For use on OS X postscript For printing on PostScript printers or creating Post
158. xample function rlm in package MASS https CRAN R project org package MASS Additive models This technique aims to construct a regression function from smooth addi tive functions of the determining variables usually one for each determining variable Func tions avas and ace in package acepack https CRAN R project org package acepack and functions bruto and mars in package mda https CRAN R project org package mda provide some examples of these techniques in user contributed packages to R An extension is Generalized Additive Models implemented in user contributed pack ages gam https CRAN R project org package gam and mgcv https CRAN R project org package mgcv Tree based models Rather than seek an explicit global linear model for prediction or interpretation tree based models seek to bifurcate the data recursively at critical points of the determining variables in order to partition the data ultimately into groups that are Chapter 11 Statistical models in R 62 as homogeneous as possible within and as heterogeneous as possible between The results often lead to insights that other data analysis methods tend not to yield Models are again specified in the ordinary linear model form The model fitting function is tree but many other generic functions such as plot and text are well adapted to displaying the results of a tree based model fit in a graphical way Tree models are available in R via the us
159. xe but also works for Rterm exe If the named file does not exist it sets the working directory if the parent directory exists The following additional command line options are available when invoking RGui exe mdi sdi no mdi Control whether Rgui will operate as an MDI program with multiple child windows within one main window or an SDI application with multiple top level windows for the console graphics and pager The command line setting overrides the setting in the user s Rconsole file debug Enable the Break to debugger menu item in Rgui and trigger a break to the debugger during command line processing Under Windows with R CMD you may also specify your own bat exe sh or pl file It will be run under the appropriate interpreter Perl for p1 with several environment variables set appropriately including BR HOME R_OSTYPE PATH BSTINPUTS and TEXINPUTS For example if you already have latex exe on your path then Appendix B Invoking R 90 R CMD latex exe mydoc will run IXTEX on mydoc tex with the path to R s share texmf macros appended to TEXINPUTS Unfortunately this does not help with the MiKTeX build of IATEX but R CMD texify mydoc will work in that case B 3 Invoking R under OS X There are two ways to run R under OS X Within a Terminal app window by invoking R the methods described in the first subsection apply There is also console based GUI R app that by default is installed in

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