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Notes on S-PLUS: A Programming Environment for Data Analysis

1. The data is given in Table 20 and may be read as a data frame from file pottery data C 19 The Beaujolais Quality Data Source Quoted in Weekes A Genstat Primer Category Multivariate analsysis Description Quality measurements for some identified samples of young Beaujolais Extracted from Table 1 in M G Jackson et al Red wine quality correlations between colour aroma and flavour and pigment and other parameters of young Beaujolais Journal of Science of Food and Agriculture 29 715 727 1978 Data The data is given in Table 21 and may be read as a data frame from file beaujolais data C 20 The Painter s Data of de Piles Source Weekes A Genstat Primer Category Multivariate Analysis Discriminant Analysis C 20 The Painter s Data of de Piles 81 OQ AC pH TSO Label OQ AC pH TSO 3 86 13 8 3 38 1 4 3 15 5 2 3 61 4 5 3 80 10 6 3 38 0 4 3 41 2 2 3 55 1 9 3 44 2 3 3 34 6 3 3 81 10 5 3 19 40 4 3 49 2 5 3 31 3 1 3 56 16 7 Table 21 Quality Measurements on Young Beaujolais Wine Samples Description The data shows the subjective assessment on a 0 20 integer scale of 54 classical painters The painters were assessed on four characteristics composition drawing colour and expression The data is due to the Eighteenth century art critic de Piles The School to which a painter belongs is indicated by a letter code as follows A Renaissance Lombard B Mannerist Sixteenth Century C Seicen
2. y lt in 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 in 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 lt in id x in x y in y or the list may be attached at position 2 of the search list see 86 4 4 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 gt X lt matrix scan light dat 0 ncol 5 byrow T There are more elaborate input facilities available and these are detailed in the manual 7 2 The read table function To read an entire data frame directly the external file must have a special form e The first line of the file should have have a name for each variable in the data frame e Each additional line of the file has 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 in Figure 2 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 functi
3. nents of the first argument here incomes defined by the levels of the second component here statref as if they were separate vector structures The result is a structure of the same length as the levels attribute of the factor containing the results The reader should consult the help document for more details Suppose further we needed to calculate the standard errors of the state income means To do this we need to write an S PLUS function to calculate the standard error for any given vector We discuss functions more fully later in these notes but since there is an in built 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 9 After this assignment the standard errors are calculated by gt incster lt 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 95 confidence limits for the state mean incomes To do this you could use tapply once more with the length function to find the sample sizes and the qt function to find the percentage points of the appropriate distributions The function tapply can be used to handle more complicated indexing of a vector by multiple categories For example we
4. 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 Al though still useful in some contexts it would now generally be replaced by the statistical models features as will be discussed in 10 6 4 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 and factors are included as 1s and non numeric 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 all have the same row size 26 Lists data frames and their uses Data frames may in many ways 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 4 1 Making data frames Objects satisfying the restrictions placed on the columns components of a data frame may be used to form one using
5. 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 8 2 Control statements The language has available a conditional construction of the form gt if expri expre else exprs where expr must evaluate to a logical value and the result of the entire expression 1s then evident There is also a for loop construction which has the form gt for name in expri expre where name is the loop variable expr is a vector expression often a sequence like 1 20 and expr is often a grouped expression with its sub expressions written in terms of the dummy name expre is repeatedly evaluated as name ranges through the values in the vector result of expr 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 to be discussed later 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 yc lt split y ind xc lt split x ind gt for i in 1 length yc plot xc ill yc i abline lsfit xc ill yc i Note the function split which produces a list of vectors got by splitting a larger vector according to the classes specified by a category This 1s a useful function mostly used in connection with boxplots S
6. however there are other more flexible sorting facilities available see order or sort list which produce a permutation to do the sorting rnorm x is a function which generates a vector or more generally an array of pseudo random standard normal deviates of the same size as x 2 3 Generating regular sequences S PLUS 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 highest priority within an expression so for example 2 1 15 is the vector c 2 4 6 28 30 Put n lt 10 and compare the sequences 1 n 1 and 1 n 1 The construction 30 1 may be used to generate a sequence backwards The function seq 1s a more general facility for generating sequences It has five argu ments only some of which may be specified in any one call The first two arguments if given specify the beginning and end of the sequence and if these are the only two arguments given the result is the same as the colon operator That is seq 2 10 is the same vector as 2 10 Parameters to seq and to many other S PLUS functions can also be given in named form in which case the order in which they appear is irrelevant The first two parameters may be named from value and to value thus seq 1 30 seq from 1 to 30 and seq to 30 from 1 are all the same as 1 30 The next two parameters to seq may be named by value and length value which specify a step
7. 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 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 func tions which are taken as an intercept and slope in that order 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 Adds a legend to the current plot at the specified position Plot ting characters line styles colours etc are identified with the la bels in the character vector legend At least one other argument v a vector the same length as legend with the corresponding values of the plotting unit must also be given as follows Shading angles Shading densities Colours for filled boxes Colours in which points or lines will be drawn 52 Graphical procedures legend lty v Line styles legend pch v Plotting characters character vector legend marks v Plotting symbols as obtained when using a numeric argument to pch numeric vector title main sub
8. 5 4 1 below 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 1n 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 1s an array formed by element by element operations on the data vector The dim attributes of operands generally need to be the same and this becomes the dimension vector of the result So if A B and C are all similar arrays then gt D lt 2 A B C 1 makes D a similar array with data vector the result of the evident 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 I 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 previous or subsequent operands e As long as short vectors and arrays only are encountered the arrays must all have the same dim attribute or an error results e Any vecto
9. ALK xX is a quadratic form The function crossprod forms crossproducts meaning that gt 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 Other important matrix functions include solve A b for solving equations solve A for the matrix inverse svd for the singular value decomposition qr for QR decom position and eigen for eigenvalues and eigenvectors of symmetric matrices The meaning of diag depends on its argument diag vector gives a diagonal matrix with elements of the vector as the diagonal entries On the other hand diag matrix gives the vector of main diagonal entries of matrix This is the same convention as that used for diag in MATLAB Also somewhat confusingly if k is a single numeric value then diag k is the k x k identity matrix A surprising omission from the suite of matrix facilities is a function for the determinant of a square matrix however the absolute value of the determinant is easy to calculate for example as the product of the singular values See later Note that x x is ambiguous as it could mean either X X or XX 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 rbind x since the
10. The Iowa Historical Wheat Yield Data 5 83 3 83 5 17 1 64 3 49 7 00 2 85 3 80 4 67 5 32 3 56 6 20 5 93 6 16 5 88 4 70 6 41 10 39 3 42 5 51 5 70 6 11 5 40 5 31 6 36 3 07 2 56 4 84 3 80 4 11 4 43 3 36 3 12 69 0 75 0 72 0 77 8 77 2 65 9 70 1 69 0 69 2 71 4 71 3 70 0 69 7 71 6 71 7 64 1 69 8 66 3 68 6 72 4 68 4 65 2 74 2 73 2 72 9 67 2 74 7 68 9 66 4 71 5 67 4 69 4 69 1 1 49 2 72 3 12 3 45 3 85 3 35 0 51 2 63 4 24 3 15 4 57 2 24 4 89 4 56 3 73 2 96 2 45 1 72 4 14 3 47 4 65 4 45 3 84 3 28 1 79 3 29 4 51 3 54 7 55 2 29 2 76 5 51 6 27 quantitative variate to account for linear or polynomial changes in the measurement over the course of a single experiment Data The data is given in Table 9 and may be read as a data frame from file morley data in a form suitable for constructing a data frame C 8 The Rat Genotype Data 71 x9 x3 L4 y Sample X1 2 x3 L4 y 6 6 6 T T T 8 8 8 9 9 9 9 e FP e c c c Table 8 The gasoline recovery data C 8 The Rat Genotype Data Source Quoted in Scheffe H The Analysis of Variance Category Unbalanced double classification Description Data from a foster feeding experiment with rat mothers and litters of four different genotypes A F I and J The measurement is the litter weight gain after a trial feeding period Data The data is given in Table 10 and may be read as a data fra
11. 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 11 1 3 Display graphics Other high level graphics functions produce different types of plots Some examples are tsplot xi x2 Plots any number of time series on the same scale This auto matic simultaneous scaling feature 1s also useful when the x s are ordinary numeric vectors in which case they are plotted against the numbers 1 2 3 50 qqnorm x qqplot x y hist x hist x nclass n hist x breaks dotchart x pie slices names explode Graphical procedures Distribution comparison plots The first form plots the numeric vector x against the expected Normal order scores a normal scores plot The second form plots the quantiles of x against those of y to compare their respective distributions 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 If the probabil ity T argument is given the bars represent relative frequencies instead of counts Construct a dotchart of the data in x In a dotchart the y axis gives a labelling of the data in x and the r
12. X This is particularly useful for large integer arrays where patterns are the real interest rather than the values 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 list The example in Figure 7 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 furtherest from linear There is however a heavy over head and the function is only competitive with other algorithms when the integrand is both smooth and very difficult to evaluate 9 5 Customizing the environment First and Last 35 The example is also given partly as a little puzzle in S PLUS programming area lt function f a b eps 1 0e 06 lim 10 funi lt function f a b fa fb a0 eps lim fun 1 d a b 2 h b a 4 fd lt f d ail lt h fa fd a2 lt h fd fb if abs a0 al a2 lt eps lim 0 return ai a
13. a percent of crude oil say y and four independent variables which may influence yield These are x The crude oil gravity in API xo The crude oil vapour pressure x3 The crude oil 1096 point ASTM 4 The gasoline end point The data comes as 10 separate samples and within each sample the values for 1 2 and z3 are constant Data The data is shown in Table 8 and is available as the file oil data in a form suitable for constructing a data frame C 7 The Michaelson Morley Speed of Light Data Source Weekes A Genstat Primer Category Analysis of Variance Description The classical data of Michaelson and Morley on the speed of light The data consists of five experiments each consisting of 20 consecutive runs The response is the speed of light measurement suitably coded The data is here viewed as a randomized block experiment with erperiment and run as the factors run may also be considered a 10 Exercises Year Rain0 Templi Raini Temp2 Rain2 Temp3 Rain3 Temp4 17 75 14 76 27 99 16 76 11 36 22 71 17 91 23 31 18 53 18 56 12 45 16 05 27 10 19 05 20 79 21 88 20 02 23 17 19 15 18 28 18 45 22 00 19 05 15 67 15 92 16 75 12 34 15 82 15 24 21 72 25 08 17 79 26 61 60 2 57 5 62 3 60 5 69 5 55 0 66 2 61 8 59 5 66 4 58 4 66 0 59 3 57 5 64 6 55 1 56 5 55 6 59 2 63 5 59 8 62 2 59 6 60 0 55 6 63 6 62 4 59 0 62 5 62 8 59 7 57 4 66 6 Table 7
14. access the current device or temporarily affecting only a single graphics function call 11 4 1 Permanent changes the par function The par function is used to access and modify the list of graphics parameters for the current graphics device par Without arguments returns a list of all graphics parameters and their values for the current device par c col With a character vector argument returns only the named graph 1ty ics parameters again as a list par col 4 1ty 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 1s given 54 Graphical procedures Note that calls to par always affect the global values of graphics parameters even when par is called from within a function This is often undesirable behaviour 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 S session You can restore the initial values by saving the result
15. also be named and in this case the component may be referred to either by giving the component name as a character string in place of the number in double square brackets or more conveniently by giving an expression of the form gt 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 if the components of St above had been named and the names were x y coefficients and covariance they could be referred to as St y St covariance and so on or indeed as St y 11 St covariance but this form is rarely if ever needed It is very important to distinguish St 1 from St 1 CE 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 St and if it is a named list the name is not included The latter is a sublist of the list St consisting of the first entry only If it is a named list the name is transferred to the sublist The names of components may be abbreviated down to the minimum number of letters needed to identify them uniquely Thus St coefficients may be minimally specified as St coe and St covariance as St cov The vector of names is in fact simply an attribute of the list like any other and may be handled as such Other structures besides lists may of course similarly be given a names attribute also 6 2 Constru
16. axis gives its value For example it allows easy visual selection of all data entries with values lying in specified ranges Make a pie diagram including the possibility of some pieces dis placed or exploded out from the centre Pie diagrams are es pecially good for showing to administrators and bosses but not much else in my opinion 11 1 4 Arguments to high level plotting functions There are a number of arguments which may be passed to high level graphics functions as follows add T axes F log x log y log xy type type p type 1 type b type o type h type s type S type n Forces the function to act as a low level graphics function super imposing the plot on the current plot some functions only Suppresses generation of axes useful for adding your own cus tom axes with the axis function The default axes T means include axes Causes the x y or both axes to be logarithmic Only works for scatterplots and variants The type argument controls the type of plot produced as follows Plot individual points the default Plot lines Plot points connected by lines both Plot points overlaid by lines Plot vertical lines from points to the zero axis high density Step function plots In the first form the top of the vertical defines the point in the second the bottom No plotting at all However axes are still drawn by default and the coordinate system is set
17. by perm j becoming the new jth dimension The easiest way to 20 Arrays and matrices think of this operation is as a generalization of transposition for matrices Indeed if A is a matrix i e 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 t A 5 7 Matrix facilities Multiplication inversion and solving linear equations As noted above a matrix is Just an array with two subscripts However it is such an important special case it needs a separate discussion S PLUS 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 The operator is used for matrix multiplication An n x 1 or 1 x 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 4 and B are square matrices of the same size then gt A B is the matrix of element by element products and gt A AB is the matrix product If x is a vector then gt x WA
18. defaults may be arbitrary expressions even involving other arguments to the same function they are not restricted to be constants as in our simple example here Another frequent requirement 1s 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 This can be done by including an extra argument literally of the function which may then be passed on An outline example is given in Figure 4 Note here that the ellipses are literal S PLUS not a typographical device 9 3 Assignments within functions are local Frames Note that any ordinary assignments done within the function are local and temporary and 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 T This ellipsis is used as a customary typographical device to mean an abridgement This is not the case in the latter part of section 9 4 More advanced examples 33 funi lt function data data frame graph T limit 20 omitted statements if graph par pch more omissions Figure 4 Use of the ellipsis argument To understand completely the rules governing the scope of S PLUS assignments the reader needs to be familiar with the notion of an evaluation frame This is a
19. for example for the poisson distribution the Var y n 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 de pends 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 incidently For example consider fitting the non linear regression 0124 Ll 1 nua t 1 this may be written alternatively as 1 e bizi Boxe where z 22 21 2 1 a1 B 1 01 and f 02 01 Supposing a suitable data frame to be set up we could fit this non linear regression as gt nlfit lt glm y xitx2 1 family quasi link inverse variance constant data biochem The reader is referred to the manual and the help document for further information as needed 10 7 Nonlinear regression models parametrized data frames S PLUS provides two functions to fit nonlinear models that do not conform even to the partially linear paradigm of generalized linear models These are ms for arbitrary minimization problems where the objective functions is a sum of similar terms and nls for conventional nonlinear least squares estimation of normal nonlinear regression models In this brief introduction we only consider the nonlinear regression f
20. function eigen Sm calculates the eigenvalues and eigenvectors of a symmetric ma trix 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 at the terminal 6 3 2 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 W 4 t V D is actually returned as a vector of the diagonal elements The result of svd M 1s actually a list of three components named d u and v with evident meanings If M is in fact square then it is not hard to see that gt absdetM lt prod svd M d 6 4 Data frames 25 calculates the absolute value of the determinant of M If this calculation were needed often with a variety of m
21. model the LD50 that is the age at which the chance of blindness for a male inhabitant is 50 44 Statistical models in S PLUS Age 20 35 45 55 70 No tested 50 50 50 50 50 No blind 6 17 26 37 44 Table 4 The Kalythos blindness data If y is the number of blind at age x and n the number tested both models have the form y B n F Bo 12 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 1 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 two possibilities for the response e If the response is a vector it 1s assumed to hold binary data and so must be a 0 1 vector e If the response is a two column matrix it is assumed that the first column holds the number of successes for the trial and the second holds the number of failures 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 lt glm Ymat x family binomial link probit data kalythos gt fml glm Ymat x family binomial data kalythos Since the logit link 1s the default the
22. of par when making changes and restoring the initial values when plotting is complete gt oldpar lt par col 4 1ty 2 plotting commands gt par oldpar 11 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 11 5 Graphics parameters list The following sections detail many of the commonly used graphical parameters The S help documentation for the par function provides a more concise summary this is provided as a somewhat more detailed alternative Graphics parameters will be presented in the following form name value A description of the parameter s effect name is the name of the pa rameter 1 e 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 11 5 1 Graphical elements S plots are made up of points lines text and polygons filled regions Graphical pa rameters exist which control how these graphical elements are drawn as follows peh Character to be used for plotting points The default varies with graphi
23. of the same length as the vector from which elements are to be selected Values corresponding to T in the index vector are selected and those corresponding to F 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 xti is na x amp x 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 vector of positive integral quantities In this case the values in the index vec tor must lie in the 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 1s 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 gt 10 Also c x y rep c 1 2 2 1 times 4 an admittedly unlikely thing to do produces a character vector of length 16 consisting of x y y x repeated four times vector of negative integral quantities Such an index vector specifies the val ues to be excluded rather than included Thus y lt x 1 5 gives y all but the first five elements of x vector of character strings This possibility only applies where an
24. parameter may be omitted on the second call To see the results of each fit we could use gt summary fmp gt summary fml Both models fit all too well To find the LD50 estimate we can use a simple function gt ld50 lt function b b 1 b 2 gt ldp lt 1d50 coef fmp 1dl lt 1d50 coef fmp c ldp ldl The actual estimates from this data are 43 663 years and 43 601 years respectively Poisson models With the poisson family the default link is the log and in practice the major use of this family is to fit surrogate poisson log linear models to frequency data whose actual distribution is often multinomial This is a large and important subject we will not 10 7 Nonlinear regression models parametrized data frames 45 discuss further here It even forms a major part of the use of non gaussian generalized models overall Occasionally genuinely poisson data arises in practice and in the past it was often anal ysed 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 gt fmod lt gim 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
25. result of rbind or cbind is always a matrix 5 8 Forming partitioned matrices cbind and rbind 21 5 8 Forming partitioned matrices cbind and rbind Matrices can be built up from given 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 arg arg3 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 argi args forming the columns If some of the arguments to cbind are vectors they may be shorter than the column size of any matrices present in which case they are cyclically extended to match the matrix column size or the length of the longest vector if no matrices are given The function rbind does the corresponding operation for rows In this case any vector argument possibly cyclically extended are of course taken as row vectors Suppose X1 and X2 have the same number of rows To combine these by columns into a matrix X together with an initial column of 1s we can use 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 resp
26. thus including it in a document electronically as opposed to physical cut and paste can be rather problematic For this type of application a better solution is to use the fig driver available from statlib and use a conversion program such as fig2dev to convert the resultant fig code to Encapsulated PostScript 60 Graphical procedures The main commands used for operating with multiple devices and their meanings are as follows motif postscript dev dev dev dev def dev dev list next prev set which k off k copy device which k print device which k graphics off 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 returns the number and name of all active devices The device at position 1 on the list is always the null device which does not accept graphics commands at all returns the number and name of the graphics device next to or previous to the current device respectively 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 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
27. x y The identify functions performs no plotting itself but simply allows the user to move the mouse pointer and click the left mouse button near a point The point nearest the mouse pointer will be and highlighted with its index number i e 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 highlighting altogether with the plot F argument When the middle button is pressed identify returns the indices of the selected points you can use these indices to extract the selected points from the original vectors x and y 11 4 Using graphics parameters When creating graphics particularly for presentation or publication purposes S does not always produce exactly that which 1s required You can however customise almost every aspect of the display using graphics parameters S maintains a list of a large number of graphics parameters which control things such as line style colours figure arrangement and text justification among many others Every graphics parameter has a name such as col which controls colours and a value a colour 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 initialised Graphics parameters can be set in two ways either permanently affecting all graphics functions which
28. 1 Logical vectors may be used in ordinary arithmetic in which case they are coerced into numeric vectors F becoming 0 and T becoming 1 However there are situations where logical vectors and their coerced numeric counterparts are not equivalent for example see the next subsection 2 5 Missing values In some cases the components of a vector may not be completely known When an element or value is not available or a missing value in the statistical sense a place within a vector may be reserved for it by assigning it the special value NA In general any operation on an NA becomes an NA The motivation for this rule is simply that if the specification of an operation 1s incomplete the result cannot be known and hence 1s not available The function is na x gives a logical vector of the same size as x with value T if and only if the corresponding element in x is NA ind is na z Notice that the logical expression x NA is quite different from is na x since NA 1s not really a value but a marker for a quantity that is not available Thus x NA isa vector of the same length as x all of whose values are NA as the logical expression itself 1s incomplete and hence undecidable 2 6 Character vectors Character quantities and character vectors are used frequently in S PLUS 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 iteratio
29. 1 9 6 3 gt x i lt 0 Replace those elements by zeros gt xX 1 2 3 4 5 1 1 5 O 13 17 2 2 O 10 14 18 3 0 7 11 15 19 4 4 8 12 16 20 gt Figure 1 Using an index array 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 lt matrix 0 n b gt Xv lt matrix 0O n v gt ib lt cbind 1 n blocks gt iv lt cbind i n varieties gt Xb ib lt 1 gt Xv iv lt 1 gt X lt cbind Xb Xv Further to construct the incidence matrix N say we could use gt N lt crossprod Xb Xv However a simpler direct way of producing this matrix is to use table N table blocks varieties 18 Arrays and matrices 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 lt array h dim c 3 4 2 would use h to set up 3 x 4 x 2 array in Z If the size of h is exactly 24 the result is the same as gt dim Z lt c 3 4 2 However if h is shorter than 24 its values recycled from the beginning again to make it up to size 24 See
30. 2 else return fun f a d fa fd ai 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 funi f a b fa fb a0 eps lim funi Figure 7 A recursive function within a function 9 5 Customizing the environment First and Last Any function named First in the Data directory has a special status It is automat ically performed at the beginning of an S PLUS session and may be used to initialize the environment For example the definition in Figure 8 alters the prompt to and sets up various other useful things that can then be taken for granted in the rest of the session Similarly a function Last if defined is executed at the very end of the session An gt First lt function options prompt continue t is the prompt options digits 5 length 999 custom numbers and printout options gui motif default graphics user interface tek4014 for terminal work par pch plotting character attach paste unix echo HOME Data sep Home of my personal library library examples attach also the system examples Figure 8 An example of a First function example is given in Figure 9 36 Writing your own functions gt Last lt function graphics off a small safety measure cat paste unix date nAdios n Is it time for lunch Figure 9 An example of a Last funct
31. 4 9 1 4 10 oma c 2 0 3 0 omi c 0 0 0 8 0 Figure 10 Anatomy of an S figure Set size of multiple figure array The first value is the number of rows the second is the number of columns The only difference between these two parameters is that setting mfcol causes figures to be filled by column mfrow fills by rows The arrangement in Figure 11 would have been created by setting mfrow c 3 2 the figure shows the page after four plots have been drawn Position of 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 Position of the current figure on the page Values are the po sitions 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 Size of outer margins Like mar and mai the first measures in text lines and the second in inches starting with the bottom margin and working clockwise 58 Graphical procedures mfrow c 3 2 mfg c 2 2 3 2 Figure 11 Page layout in multiple figure mode O
32. 73 Litter s Mother s em seer 61 5 55 0 68 2 42 0 64 0 60 2 65 0 59 7 Table 10 The Rat Genotype Data Data The data is given in Table 12 and may be read as a data frame from file barley data C 11 The Snail Mortality Data Source Zoology Department The University of Adelaide Category Generalized Linear Modelling Description Groups of 20 snails were held for periods of 1 2 3 or 4 weeks exposure in carefully controlled conditions of temperature 3 levels and relative humidity 4 levels There were two species of snail A and B and the experiment was designed as a4 x 3 x 4 x 2 completely randomized design At the end of the exposure time the snails were tested to see if they had survived this process itself is fatal for the animals The object of the exercise was to model the probability of survival in terms of the stimulus variables and in particular to test for differences between species 74 Exercises Block 1 Block 2 Block 3 Block 4 V N P KI No Wj No Wt No Wt No Wt a 1 124 162 able 11 Fisher s sugar beet data The data is unusual in that in most cases fatalities during the experiment were fairly small Data The data is given in Table 13 and may be read as a data frame from file snails data C 12 The Kalythos Blindness Data Source S D Silvey Statistical Inference Fictitious Category Generalized linear modelling C 12 The Kalythos Blindness Data 75 M Ns Ns Maf Bl
33. 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 F argument Low level plotting functions usually require some positioning information e g x and y coordinates to determine where to place the new plot elements Coordinates are given in terms of user coordinates which are defined by the previous high level graphics command and are chosen based on the supplied data Where x and y arguments are required it 1s also sufficient to supply a single argument being a list with elements named x and y Similarly a matrix with two columns is also valid input In this way functions such as locator see below may be used to specify positions on a plot interactively 11 3 Interactive graphics functions S also provides functions which allow users to extract or add information to a plot using a mouse The simplest of these 1s the 1ocator function locator n type Waits for the user to select locations on the current plot using the left mouse button This continues until n default 500 points have been selected or the middle mouse button is pressed The type argument a
34. Data Time Temp Time Temp min F min CF Table 18 The Rumford Friction Cooling Data The data is given in Table 18 and may be read as a data frame from file rumford data C7 The Jellyfish Data 79 C 17 The Jellyfish Data Source Interactive Statistics Ed Don McNeil Category Bivariate two sample data Description Two samples of jellyfish from Danger Island and Salamander Bay respectively were measured for length and width Data Danger Island Salamander Bay Width Length Width Length Width Length Width Length Table 19 The Jellyfish data Danger Island and Salamander Bay The data is given in Table 19 and may be read as a data frame from file jellyfish data C 18 Archaelogical Pottery Data Source Tubb A et al Archaeometry 22 153 171 1980 Category Multivariate analysis Description The data arises from a chemical analysis of 26 samples of pottery found at Romano British kiln sites in Wales Gwent and the New Forest The variables describe the composition in terms of various metals and are expressed as percentages of the oxides of the metals The metals are aluminium iron magnesium calcium and sodium and the sites are L Llanederyn C Caldicot I Island Thorns A Ashley Rails 80 Exercises Data Site Al Mg Site Al Mg Sie Ale Me Ce Na Ste al me Me ca Ne e e et ee ee L L L L L L L L L L L L L L Table 20 The Pottery Composition Data
35. Esc Esc Esc Esc k j text i text Esc a text Esc shift x x dw shift d shift y xp Return NOTE With vi style commands the Esc need only be issued before the first recall command and to terminate insert and append commands as is usual in vi The final Return terminates the command line editing sequence for commands of either style 66 Exercises C Exercises C 1 The Cloud Point Data Source Draper amp Smith Applied Regression Analysis p 162 Category Polynomial regression Simple plots Description The cloud point of a liquid is a measure of the degree of crystalization in a stock that can be measured by the refractive index It has been suggested that the percentage of Is in the base stock 1s an excellent predictor of cloud point using the second or third order model Y Bo Bit Bow b3 E E N 0 0 Data The following data was collected on stocks with known percentage of Ig Is Cloud Point Is Cloud Point Is Cloud Point Is Cloud Point 0 5 The data may be read from file cloud data in a form suitable to construct a data frame C 2 The Janka Hardness Data Source E J Williams Regression Analysis Wiley 1959 Category Polynomial regression Transformations Description The Janka hardness is an important structural property of Australian timbers which is difficult to measure It is however related to the density of the timber which is relatively easy
36. Least squares fitting and the QR decomposition Data frames 2 e os ons 6 4 4 Making data frames e 6 4 2 attach anddetach 22e 6 4 5 Working with data frames 6 4 4 Attaching arbitrary lists 7 Reading data from files 7 1 7 2 7 3 The scan function a 20000000000 0000 a The read table function ooa a a a a a a a Other facilities editing data 8 More language features Loops and conditional execution 8 1 8 2 Grouped expressions ooo e ss Control statements ooo a a 16 16 16 16 18 18 19 19 19 20 21 21 21 23 23 23 24 24 24 24 25 25 26 26 26 27 28 28 28 29 9 10 11 Writing your own functions 9 1 Defining new binary operators aa 9 2 Named arguments and defaults 5 2 2 les 9 3 Assignments within functions are local Frames 0 00004 9 4 More advanced examples 0 0 0 20000 002 eee 9 5 Customizing the environment First and Last 9 6 Classes generic functions and object orientation Statistical models in S PLUS 10 1 Defining statistical models formule leen 10 2 Regression models fitted model objects a 10 3 Generic functions for extracting information 10 4 Analysis of variance comparing models 10 4 1 ANOVA tables ee 10 5 Updating fitted models The ditto name lll 10 6 Generalized li
37. Notes on S PLUS A Programming Environment for Data Analysis and Graphics Bill Venables and Dave Smith Department of Statistics The University of Adelaide Email venablesOstats adelaide edu au W Venables 1990 1992 Version 2 1 July 1992 Preface These notes were originally intended only for local consumption at the University of Adelaide South Australia After some encouraging comments from students the author decided to release them to a larger readership in the hope that in some small way they promote good data analysis S or S PLUS is no panacea of course but in offering simply a coherent suite of general and flexible tools to devise precisely the right kind of analysis rather than a collection of packaged standard analyses in my view it represents the single complete environment most conducive to good data analysis so far available Some evidence of the local origins of these notes is still awkwardly apparent For example they use the Tektronics graphics emulations on terminals and workstations which at the time of writing is still the only one available to the author The X11 windowing system which allows separate windows for characters and graphics simultaneously to be displayed is much to be preferred and it will be used in later versions Also the local audience would be very familiar with MINITAB MATLAB Glim and Genstat and various echoes of these persist At one point some passing acquaintance with the Au
38. The Gasoline Yield Data ee C 7 The Michaelson Morley Speed of Light Data css C 8 The Rat Genotype Data ll se C 9 Fisher s Sugar Beet Data en C 10 A Barley Split Plot Field Trial 5 2222 ens C 11 The Snail Mortality Data s C 12 The Kalythos Blindness Data 2s C 13 The Stormer Viscometer Data een C 14 The Chlorine availability data 2 2 ss C 15 The Saturated Steam Pressure Data eee C 16 Count Rumford s Friction Data oaa aaa ee C 17 The Jellyfish Data 4 0020000000 2000048 C 18 Archaelogical Pottery Data 2 0 a C 19 The Beaujolais Quality Data 2s C 20 The Painter s Data of de Piles een 61 64 64 64 65 Introduction and Preliminaries 1 1 Introduction and Preliminaries S PLUS 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 graphical facilities for data analysis and display either at a workstation or on hard copy and a well developed simple and effective programming language which includes con ditionals loops user defined recursive functions and input and output facilities Indeed most of the system supplied functions are themselves written in the S PLUS l
39. The second component 1s 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 56 Graphical procedures tck 0 01 Length of tick marks as a fraction of the size of the plotting region When tck is small less than 0 5 the tick marks on the x and y axes are forced to be the same size A value of 1 gives grid lines Negative values give tick marks outside the plotting region Use tck 0 01 and mgp c 1 1 5 0 for internal tick marks xaxs s Axis styles for the x and y axes respectively With styles s stan yaxs d dard and e extended the smallest and largest tick marks always lie outside the range of the data Extended axes may be widened slightly if any points are very near the edge This style of axis can sometimes leave large blank gaps near the edges With styles i in ternal 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 Setting this parameter to d direct axis locks in the current axis and uses it for all future plots or until the parameter is set to one of the other values above at least Useful for generating series of fixed scale plots 11 5 3 Figure Margins A single plot in S is known as a figure and comprises a plot region surrounded by margins possibly containing ax
40. a different S PLUS prompt if you wish In these notes we will assume that the shell prompt is In using S PLUS 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 S PLUS for this problem This will be the working directory whenever you use S PLUS for this particular problem mkdir work cd work 2 Place any data files you wish to use with S PLUS in work 3 Create a sub directory of work called Data for use by S PLUS mkdir Data 4 Start the S PLUS program with the command Splus e 5 At this point S PLUS commands may be issued see later 6 To quit the S PLUS program the command is gt qO The procedure is simpler using S PLUS after the first time Make work the working directory and start the program as before cd work Splus e Use the S PLUS program terminating with the q command at the end of the session 1 4 An introductory session Readers wishing to get a feel for S PLUS at a workstation or terminal before pro ceeding are strongly advised to work through the model introductory session given in Appendix starting on page 61 1 5 S PLUS and UNIX 3 1 5 S PLUS and UNIX S PLUS allows escape to the operating system at any time in the session If a command on a new line begins with an exclamation mark then the rest of the line is interpreted as a UNIX command So for example to lo
41. a frames to be attached to the search list but other classes of object as well In particular any object of mode list may be attached in the same way gt attach any old list It is also possible to attach objects of class pframe to so called parametrized data frames needed for nonlinear regression and elsewhere Being a generic function it 1s also possible to add methods for attaching yet more classes of object should the need arise 28 Reading data from files 7 Reading data from files Large data objects will usually be read as values from external files rather than entered during an S PLUS session at the keyboard This is done most conveniently with the scan function for simple data items and the read table function for reading entire data frames directly 7 1 The scan function Suppose the data vectors are of equal length and are to be read in 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 in 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 in 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 in 1 x lt in 2
42. ach has 20 runs col Run and s1 is the recorded speed of hght suitably coded Place mm on the top of the search list position 1 Change Expt and Run into factors Save the changes and make the data frame visible at position 2 the default Compare the five experiments with simple boxplots Analyse as a randomized block with runs and ex periments as factors Fit the sub model omitting runs and compare using a formal analysis of variance Cleanup before moving on S PLUS An Introductory Session x lt seq pi pi len 50 y lt x f lt outer x y function x y cos y 1 x 2 oldpar lt par par pty s contour x y f contour x y f nint 15 add T fa lt f t f 2 contour x y fa nint 15 par oldpar persp x y f persp x y fa image x y f image x y fa objects rm x y f fa th lt seq pi pi len 100 z lt exp 1i th par pty s plot z type 1 w lt rnorm 100 rnorm i100 1i w lt ifelse Mod w gt 1 1 w w plot w xlim c 1 1 ylim c 1 1 pch xlab x ylab y lines z w lt sqrt runif 100 exp 2 pixrunif 100 1i plot w xlim c 1 1 ylim c 1 1 pch xlab x ylab y lines z rm th w z par oldpar butterfly rm oldpar qO gt 63 We now look at some more graphical features con tour and 3 dimensional perspective plots x is a vector of 50 equally spaced values
43. actionally until they match the length of the longest vector In particular a constant is simply repeated So with the above assignments the command gt v lt 2 x y td generates a new vector v of length 11 constructed by adding together element by element 2 x repeated 2 2 times y repeated just once and 1 repeated 11 times The elementary arithmetic operators are the usual and for raising to a power In addition all of the common arithmetic functions are available log exp sin cos tan sqrt and so on all have their usual meaning max and min select the largest and smallest elements of an vector respectively range is a function whose value is a vector 1With other than vector types of argument such as list mode arguments the action of c is at first sight rather different See 86 2 1 2 3 Generating regular sequences 7 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 1length x 1 or sample variance If the argument to var is an n x p matrix the value is a p x 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
44. anguage The term environment is intended to characterize it as a fully planned and coherent system rather than an incremental accretion of very specific and inflexible tools as is frequently the case with other data analysis software S PLUS is very much a vehicle for newly developing methods of interactive data analysis As such it is very dynamic and new releases have not always been fully upwardly com patible with previous releases Some users welcome the changes because of the bonus of new technology and new methods that come with new releases others seem to be more worried by the fact that old code no longer works Although S PLUS is intended as a programming language in my view one should regard programmes written in S PLUS as essentially ephemeral The name S or S PLUS as with many names within the UNIX world is not explained but left as a cryptic puzzle and probably a weak pun However its authors insist it does not stand for Statistics These notes will be mainly concerned with S PLUS an enhanced version of S distributed by Statistical Sciences Inc Seattle Washington 1 1 Reference manuals 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 August 1991 release of S are covered in Statistical Models in S Edited by John M Chambers and Trevor J Hastie In addition there are specif
45. arch list are displayed by the function gt search The names of the objects held in any place in the search list can be displayed by giving the objects function an argument For example gt objects 2 lists the contents of the entity at position 2 of the search list The search list can contain either data frames and allies which are themselves internal S PLUS objects as well as directories of files which are UNIX objects Extra entities can be added to this list with the attach function and removed with the detach function details of which can be found in the manual or the help facility To remove objects permanently the function rm is available gt rm x y Z ink junk temp foo bar The function remove can be used to remove objects with non standard names Also the ordinary UNIX facility rm may be used to remove the appropriate files in the Data directory as mentioned above 6 Simple manipulations numbers and vectors 2 Simple manipulations numbers and vectors 2 1 Vectors S PLUS operates on named data structures The simplest such structure is the 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 S PLUS 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 vect
46. as atomic structures since their components are all of the same type or mode namely numeric complex logical and character respectively Vectors must have their values all of the same mode Thus any given vector must be unambiguously either logical numeric complex or character The only mild exception to this rule is the special value listed as NA for quantities not available 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 S PLUS also operates on objects called lists which are of mode list These are ordered sequences of objects which individually can be of any mode lists are known as recursive rather than atomic structures since their components can themselves be lists in their own right The other recursive structures are those of mode function and expression Functions are the functions that form part of the S PLUS system along with similar user written functions which we discuss in some detail later in these notes Expressions as objects form an advanced part of S PLUS which will not be discussed in these notes except indirectly when we discuss formulae with we discuss modelling in S PLUS By the mode of an object we mean the basic type of its fundamental constituents This is a special case of an attribute of an object The attributes of an object provide specific information about the object its
47. as the class of the object has been introduced in the Au gust 1991 release of S and S PLUS to allow for an object oriented style of programming in S PLUS For example if an object has class data frame it will be printed in a certain way the plot function will display 1t graphically in a certain way and other 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 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 9 6 but only briefly Categories and factors 13 4 Categories and factors A category is a vector object used to specify a discrete classification of the components of other vectors of the same length A factor is similar but has the class factor which means that it 1s adapted to the generic function mechanism Whereas a category can also be used as a plain numeric vector for example a factor generally cannot 4 1 A specific example Suppose for example we have a sample of 30 tax accountants from the all states and territories and their i
48. atrices it could be defined as an S PLUS function gt absdet lt function M prod svd M d after which we could use absdet as just another S PLUS 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 Functions will be discussed formally later in these notes 6 3 3 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 for among other things regression diagnostics Note that a grand mean term is automatically included and need not be included explicitly as a column of X Another closely related function is qr and its allies Consider the following assignments gt Xplus lt qr X gt b lt 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
49. aying multivariate data S provides two very useful functions for representing multivariate data If X is a numeric matrix or data frame the command gt pairs X produces a pairwise scatterplot matrix of the variables defined by the columns of X i e every column of X is plotted against every other column of X and the resulting n n 1 plots are arranged 1n 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 given values of c If c is a factor this simply means that a 1s plotted against b for every level of c When c 1s 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 and d The coplot and pairs function both take an argument panel which can be used to customise the type of plot which appears in each panel
50. ble s is an operator either or implying the inclusion or exclusion of a term in the model the first is optional term is either e a 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 used by such programs a Glim and Genstat One inevitable change is that the operator becomes since the period is a valid name character in S PLUS The notation is summarised as in the Table 1 based on Chambers amp Hastie p 29 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 only to allow terms in model formulz to be defined using arithmetic operators Note particularly that the model formula specify the columns of the model matriz spec ification of the parameters is implicit This is not the case in other contexts for example in fitting nonlinear models 10 2 Regression models fitted model objects The basic function for fitting ordinary multiple models is 1m and a streamlined version of the call is a
51. ce Hence only for orthogonal experiments will the order of inclusion be inconsequential 10 5 Updating fitted models The ditto name 41 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 and Hastie 5 A more flexible alternative to the default full ANOVA table is to compare two or more models directly using the anova function gt 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 hierar chical sequence of course This does not give different information to the default but rather makes it easier to comprehend and control 10 5 Updating fitted models The ditto name The update function is largely a convenience function that allows a model to be fitted that differs from one previously fitted usually by just a few additional or removed terms Its form is 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 lm y x1 x2 x3 x4 xb data production gt fm6 lt update fm05 x6 gt smf6 update fm6 sqrt wou
52. cs drivers but it is usually for terminals or window devices and e for PostScript devices Plotted points tend to appear slightly above or below the appropriate position unless you use as the plotting character which produces centred points peh 4 When pch is given as an integer between 0 and 18 inclusive a spe cialised plotting symbol is produced To see what the symbols are use the command gt legend locator 1 as character 0 18 marks 0 18 11 5 Graphics parameters list 55 lty 2 lwd 2 col 2 font 2 adj 0 1 cex 1 5 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 and line types 2 and onwards are dotted or dashed lines or some combination of both Line widths Desired width of lines in multiples of the standard line width Affects axis lines as well as lines drawn with lines etc Colours to be used for points lines text filled regions and 1mages Each of these graphic elements has a list of possible colours and the value of this parameter 1s an index to that list Obviously this pa rameter applies only to a limited range of devices Font to use for text The appropriate value of this parameter is depen dent on the graphics device being used for the postscript device this 1s an index to the system dataset ps fonts Justification of text relative to the plotting position 0 means
53. cting and modifying lists New lists may be formed from existing objects by the function list An assignment of the form gt St lt list name object names objects nameg 70object sets up a list St of m components using comp compm for the components and giving them names as specified by the argument names which can be freely chosen If these names are omitted the components are numbered only The components used to form the list are copied when forming the new list and the originals are not affected 24 Lists data frames and their uses Lists like any subscripted object can be extended by specifying additional components For example gt St 5 lt 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 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 Some functions returning a list result Functions and expressions in S PLUS must return a single object as their result in cases where the result has several component parts the usual form is that of a list with named components 6 3 1 Eigenvalues and eigenvectors The
54. ctors Selecting and modifying subsets of a dataset 9 3 Objects their modes and attributes 11 3 1 Intrinsic attributes mode and length len 11 3 2 Changing the length of an object ee 12 3 3 attributes and attr a 12 3 4 The class of an object 2s 12 4 Categories and factors 13 4 1 A specific example ll 00000000004 13 4 2 The function tapply and ragged arrays llle 14 5 Arrays and matrices 5 1 5 2 5 3 5 4 5 5 5 6 5 7 5 8 5 9 Array indexing Subsections of an array oss Index arrays 2 2 2 a The array function 22s 5 4 1 Mixed vector and array arithmetic The recycling rule The outer product of two arrays 2n 5 5 1 An example Determinants of 2x 2 digit matrices Generalized transpose of an array 2 e Matrix facilities Multiplication inversion and solving linear equations Forming partitioned matrices cbind and rbind The concatenation function c with arrays 0 0 5 10 Frequency tables from factors The table function 6 Lists data frames and their uses 6 1 6 2 6 3 6 4 Lists ee m hh s s Constructing and modifying lists ess 6 2 1 Concatenating lists en Some functions returning alist result 2 een 6 3 1 Eigenvalues and eigenvectors 2 0 0 0 0 0 000004 6 3 2 Singular value decomposition and determinants 6 3 8
55. d for 1m glm and gam fitted objects It must be used for example in cases where orthogonal polyno mials are used as the original basis functions and the addition of new data implies different basis functions to the original print object Print a concise version of the object Most often used implicitly residuals object Extract the matrix of residuals weighted as appropri ate Short e arn summary object Print a comprehensive summary of the results of the re gression analysis Table 2 Commonly used generic functions on class 1m objects response mean formula Error strata formula Specifies a multi stratum experiment with error strata defined by the strata formula In the simplest case strata formula is simply a factor when it defines a two strata experiment namely between and within the levels of the factor For example with all determining variables factors a model formula such as that in gt fm lt 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 10 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 1n the residual sums of squares resulting from an inclusion of that term 1n the model at that place in the sequen
56. depending on how the device was initiated Make a copy of the device k device is a device function such as postscript with extra arguments 1f 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 See also printgraph Terminate all graphics devices on the list except the null device S PLUS An Introductory Session 61 A S PLUS An Introductory Session The following session is intended to introduce to you some features of the S PLUS envi ronment by using them Many features of the system will be unfamiliar and puzzling at first but this will soon disappear login gt ls a gt ls a Data gt Splus e help findsum Data help start motif x lt rnorm 50 y lt rnorm x hull lt chull x y plot x y polygon x hul1 y hu11 dens 15 objects rm x y hull x lt 1 20 w lt 1 sqrt x 2 dummy lt data frame x x y x rnorm x w dummy fm lt 1m y x data dummy summary fm fmi lt lm y x data dummy weight 1 w 2 summary fmi Login start your windowing system ask a demon strator if you need help and check that your work ing directory has a subdirectory Data which in turn contains the files First Last and possibly Audit You should also have the file morley data in your working directory If not see a demonstrator If you
57. design efficiencies It is numerically slightly better to work with the singular value decomposition on this occasion rather than the eigenvalue routines 34 Writing your own functions The result of the function is a list giving not only the efficiency factors as the first component but also the block and variety cononical contrasts since sometimes these give additional useful qualitative information 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 in Figure 6 as a wrap around to achieve the same result It also illustrates how some effective and useful user functions can be quite short With this function defined an no dimnames lt function a 1 Remove all dimension names from an array for compact printing d lt list 1 lt 0 for i in dim a d L 1 lt 1 1 lt rep i dimnames a lt d a Figure 6 A function for printing arrays in compact form array may be printed in close format using gt no dimnames
58. e 6 and may be read as a data frame from file sawheat data C 5 The Iowa Wheat Yield Data Source CAED Report 1964 Quoted in Draper amp Smith Category Multiple regression diagnostics 68 Exercises Price 000s Floor m Block m Age years 5 6 2 no 5 5 6 5 6 6 6 6 T 5 T 6 T 6 6 6 6 T T Table 5 The Tuggeranong house price data Description The data gives the pre season and three growing months precipitation the mean tem peratures for the three growing months and harvest month the year and the yield of wheat for the USA state of Iowa for the years 1930 1962 Data The data is given in Table 7 and may be read as a data frame from file iowheat data C 6 The Gasoline Yield Data Source Estimate gasoline ytelds from crudes by Nilon H Prater Petroleum Refiner 35 5 Category Analysis of variance covariance and multiple regression Modern regression C 7 The Michaelson Morley Speed of Light Data 69 Year Rain0 Raini Rain2 Yield Year RainO Rainl Rain Yield 1 61 3 52 91 2 11 4 60 60 3 46 00 1 55 3D 4 94 3 00 5 47 1 20 00 11 26 4 91 16 73 1 80 00 10 95 4 23 10 54 3 55 2 98 4 96 1l 5 89 93 11 89 67 2 17 03 2 5 6 56 8 49 11 95 16 03 4 02 1 30 3 60 2 8 6 57 4 93 03 2 69 1 97 8 43 1 97 00 6 88 92 8 68 1 65 3 30 3 97 2 49 3 31 1 93 1 16 98 Table 6 Yorke Penninsula Wheat Yield Data Description The data gives the gasoline yield as
59. e matrix X as well as polynomial terms in to degree 2 y A Single classification analysis of variance model of y with classes deter mined by A y At xX Single classification analysis of covariance model of y with classes de termined by A and with covariate x 38 Statistical models in S PLUS y A B Two factor non additive model of y on A and B The first two specify y A B A B the same crossed classification and the second two specify the same y B in A nested classification In abstract terms all four specify the same model y A B subspace y A B 0 2 Three factor experiment but with a model containing main effects and y A B C A B C two factor interactions only Both formulz specify the same model y A x Separate simple linear regression models of y on z within the levels of y A x A with different codings The last form produces explicit estimates of y A 1 x 1 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 S PLUS The form for an ordinary linear model is response term terms terms response is a vector or matrix or expression evaluating to a vector or matrix defining the response varia
60. ectively 5 9 The concatenation function c with arrays It should be noted that whereas cbind and rbind are concatenation functions that respect dim attributes the basic c function does not but rather clears numeric objects of all dim and dimnames attributes This is occasionally useful in its own right The official way to coerce an array back to a simple vector object is to use the function as vector gt vec lt as vector X However a similar result can be achieved by using c with just one argument simply for this side effect gt vec c X There are slight differences between the two but ultimately the choice between them is largely a matter of style with the former being preferable 5 10 Frequency tables from factors The table function Recall that a factor defines a partition into groups Similarly a pair of factors defines a two way cross classification and so on The function table allows frequency tables to be calculated from equal length factors If there are amp category 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 22 Arrays and matrices 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 category This simple case is equivalent to but m
61. ed by typing a further Esc Pressing the Return command at any time causes the command to be re submitted Other editing actions are summarised in the following table Unfortunately it does not seem to be possible to bind the motion keys for example to the arrow keys which is something of a nuisance B 3 Command Line Editor Summary B 3 Command Line Editor Summary 1 Command recall and vertical motion emacs style Go to the previous command backwards in the history P Go to the next command forwards in the history N Find the last command with the text string in it R text 2 Horizontal motion of the cursor Go to the beginning of the command A Go to the end of the line E Go back one word Esc b Go forward one word Esc f Go back one character B Go forward one character F 3 Editing and re submission Insert text at the cursor text Append text after cursor Ftext Delete the previous character left of the cursor Delete Delete the character under the cursor D Delete rest of the word under the cursor and save it Esc d Delete from cursor to end of command and save it K Insert yank the last saved text here Y Iranspose the character under the cursor with the next T Change the rest of the word to lower case Esc 1 Change the rest of the word to capitals upper case Esc c Re submit the command to S PLUS Return 65 vi style Esc Esc Esc Esc Esc Esc Esc Esc Esc Esc Esc Esc Esc
62. ee the help facility for further details 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 abnormally and next can be used to discontinue one particular cycle Control statements are most often used in connection with functions which are discussed in 9 and where more examples will emerge Writing your own functions 31 9 Writing your own functions As we have seen informally along the way the S PLUS language allows the user to create objects of mode function These are true S PLUS 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 S PLUS comfortable and productive It should be emphasized that most of the functions supplied as part of the S PLUS system such as mean var postscript and so on are themselves written in S PLUS and thus do not differ materially from user written functions A function is defined by an assignment of the form gt name lt function argi argo expression The expression is an S PLUS expression usually a grouped expression that uses the arguments arg to calculate a value The value of the expression 1s the value returned for the function A call to the f
63. eems a little complicated at first sight 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 its use is quite simple The names of the standard supplied family generators are given under Family Name in Table 3 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 lt Ilm y x1 x2 data sales but much less efficiently Note how the gaussian family is not automatically provided with a choice of links so no parameter is allowed If a problem requires a gaussian family with a nonstandard link this can usually be achieved through the quasi family as we shall see later The binomial family Consider a small artificial example On the Greek 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 in Table 4 The problem we consider is to fit both logistic and probit models to this data and to estimate for each
64. el fit in a graphical way 48 Graphical procedures 11 Graphical procedures The graphical facilities are an important and extremely versatile component of the S PLUS environment Best results are obtained when S PLUS is used with a high quality graphics system such as X windows although even a simple ASCII terminal can be quite effective for some purposes Before the graphical facilities of S PLUS may be used it is necessary to inform S what type of device 1s being used by starting a device driver In an X windows environment the command to do this may be gt X110 which creates a separate window in which high quality graphical output will appear or for a simpler graphics terminal the command tek4014 may be appropriate Once a device driver is running S PLUS plotting commands can be used to construct and display graphical objects Plotting commands are divided into three basic groups High level plotting functions create a new plot on the graphics device possibly with axes labels titles and so on Low level plotting functions add more information to an existing plot such as extra points lines and labels Interactive graphics functions allow you interactively add information to or extract information from an existing plot using a pointing device such as a mouse Furthermore S maintains a list of graphical parameters which allow you to customise your plots 11 1 High level plotting commands High lev
65. el plotting functions are designed to generate a complete plot of the data passed as arguments to the function Where appropriate axes labels and titles are automat ically generated unless you request otherwise High level plotting commands always start a new plot erasing the current plot if necessary 11 1 1 The plot function One of the most frequently used plotting functions in S is the plot function This is a generic function the type of plot produced 1s dependent on the type or class of the first argument plot x y If x and y are vectors plot x y produces a scatterplot of x plot xy against y 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 Produces a time series plot if x is a numeric vector or time series object or an Argand diagram if x is a complex vector 11 1 High level plotting commands 49 plot f f is a factor object y is a numeric vector The first form generates plot f y a bar plot of f the second form produces boxplots of y for each level of f plot df df 1s a data frame y 1s any object expr is a list of object names plot expr separated by e g a b c The first two forms produce plot y expr 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 11 1 2 Displ
66. elf Another attribute 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 For example if z is complex vector of length 100 then in an expression mode z is the character string complex and length z is 100 S PLUS 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 O 1 2 9 A further coercion or change of mode reconstructs the numerical vector again gt d lt as numeric digits 4 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 posses The reader should consult the help file to become familiar with them numeric mode is actually an amalgam of three distinct modes namely integer single precision and double precision as explained in the manual Tn general coercion from numeric to character and back again will not be exactly reversible because of roundoff errors in the character representation 12 Objects their modes and attributes 3 2 Changing the length of an object An empty object may still have a mode For example e numeric ma
67. en here are useful and a spur to the reader to seek further enlightenment in the standard reference materials The present notes are also centred around S PLUS rather than plain vanilla S This simply reflects a change in the implementation of S made available to my students and me by my employer It should not be read as any sort of endorsement one way or another My sincere thanks to David Smith James Pearce and Ron Baxter for many useful sug gestions Bill Venables University of Adelaide 13th April 1992 Contents Preface i 1 Introduction and Preliminaries 1 1 1 Reference manuals a 1 1 2 S PLUS and X windows 2s 1 1 8 Using S PLUS interactively 2 2 00 00 0200 0 00000 000 2 1 4 An introductory session 0 2 ll s es 2 1 5 S PLUS and UNIX lll Ses 3 1 6 Getting help with functions and features 2 3 1 7 S PLUS commands Case sensitivity o a 3 1 8 Recall and correction of previous commands les 4 1 831 S PLUS ll ls sss 4 1 82 VanillaS 2 2 2 222 sl 4 1 9 Executing commands from or diverting output to a file 4 1 10 Data directories Permanency Removing objects iss 5 2 Simple manipulations numbers and vectors 6 2 1 Vectors 2 e a 6 2 2 Vector arithmetic 2 2l lll se 6 2 3 Generating regular sequences en 7 2 4 Logical vectors 2 e S e 8 2 5 Missing values eS e 8 2 6 Character vectors 2 2 22222 ll ese 8 2 7 Index ve
68. f y through a single linear func tion only This linear function is called the linear predictor and is usually written n p t Baza Pptp hence z has no influence on the distribution of y if and only if 8 0 e The distribution of y is of the form fv Y H p exp yA m y Alu T v where y is a scale parameter possibly known and is constant for all observa tions A represents a prior weight assumed known but possibly varying with the observations and x 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 and this inverse function is called the link function These assumptions are loose enough to encompass a wide class of models useful in sta tistical 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 Generalized linear models by Peter McCullagh and John A Nelder 2nd edition Chapman and Hall 1989 or An introduction to generalized linear models by Annette J Dobson Chapman and Hall 1990 10 6 1 Families The class of generalized linear models handled by facilities supplied in S PLUS includes gaussian binomial poisson inverse gaussian and gamma response distrib
69. gain 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 list the entity currently at posi tion 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 6 4 3 Working with data frames A useful convention that allows you to work with many different problems comfortably together in the same working directory 1s 6 4 Data frames 27 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 a 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 4 4 Attaching arbitrary lists attach is a generic function that allows not only directories and dat
70. h dimnames lt pairs signif lt aperm dim formula plot summary LL atan dimnames ordered lt print t 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 variants for classes of object data frame default glm pregam surv fit design factor lm preloess tree formula gam loess profile tree sequence and perhaps more A complete list can be got again by using the methods function gt methods plot he reader 1s referred to the official references for a complete discussion of this mecha nism Statistical models in S PLUS 37 10 Statistical models in S PLUS 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 Since the August 1991 release S PLUS provides an interlocking suite of facilities that make fitting statistical models very simple However these are not at the same high level as those in say Genstat especially in t
71. have proceed Start Splus with the inbuilt command line editor en abled The S PLUS program begins with a banner Within S PLUS the prompt on the left hand side will not be shown to avoid confusion Set things up for the help facility Need only be done once for this directory Start the help facility You should briefly explore the features of this facility with the mouse Standard X window conventions apply Iconify the help window and move on to the next part Turn on the graphics window You may need to re position and re size to make it convenient to work with both windows Generate two pseudo random normal vectors of z and y coordinates Find their convex hull in the plane Plot the points in the plane and mark in their convex hull See which S PLUS objects are now in the Data di rectory Remove objects no longer needed cleanup Make z 1 2 20 A weight vector of standard deviations Make a data frame of two columns and y and look at 1t Fit a simple linear regression of y on x and look at the analysis Since we know the standard deviations we can do a weighted regression 62 lrf lt loess y x dummy attach dummy plot x y lines x fitted lrf abline 0O 1 lty 3 abline coef fm abline coef fml1 lty 4 detach plot fitted fm resid fm xlab Fitted values ylab Residuals main Residuals vs Fitted qqnorm resid fm ma
72. he form of the output which in keeping with general S PLUS policy is rather minimal 10 1 Defining statistical models formulz The template for a statistical model is a linear regression model with independent ho moscedastic errors p yi J Bj wig tei e NID 0 o 1 2 n j 0 In matrix terms this would be written y XxB e where the y is the response vector X is the model matrix or design matrix and has columns o 1 Zp the determining variables Very often o will be a column of 1s defining an intercept term Examples Before giving a formal specification a few examples may usefully set the picture Suppose y x x0 x1 x2 are numeric variables X is a matrix and A B C are factors The following formulae on the left side below specify statistical models as described on the right y x Both imply the same simple linear regression model of y on x The first y 1 x has an implicit intercept term and the second an explicit one y 1 x Simple linear regression of y on z through the origin that is without y x 1 an intercept term log y x1 x2 Multiple regression of the transformed variable log y on z and zo with an implicit intercept term y poly x 2 Polynomial regression of y on of degree 2 The first form uses orthog y 1 x I x 2 onal polynomials and the second uses explicit powers as basis y X poly x 2 Multiple regression y with model matrix consisting of th
73. ibute that allows it to be treated as a 3 x 5 x 100 array Other functions such as matrix and array are available for simpler and more natural looking assignments as we shall see in 85 4 The values in the data vector give the values in the array in the same order as they would occur in Fortran that is column major order with the first subscript moving fastest and the last subscript slowest For example if the dimension vector for an array say a is c 3 4 2 then there are 3x4 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 21 a 3 4 2 5 2 Array indexing Subsections of an array Individual elements of an array may be referenced as above by giving the name of the array followed by the subscripts in square brackets separated by commas More generally subsections of an array may be specified by giving a sequence of index vectors in place of subscripts however if any index position is given an empty index vector then the full range of that subscript is taken Continuing the previous example a 2 is a 4 x 2 array with dimension vector c 4 2 and data vector a 2 1 1 a 2 2 1 al2 3 1 al 2 4 1 a 2 1 2 al2 2 2 a 2 3 2 a 2 4 2 in that order a stands for the entire array which is the same as omitting the subscripts entirely and using a alone For any array say Z the dimension vector may be referenced explicitly as dim Z on either side of an assignme
74. ically S PLUS reference books S PLUS User s Manual Volumes 1 amp 2 and S PLUS Reference Manual in two volumes A K and L Z It 1s not the intention of these notes to replace these manuals Rather these notes are intended as a brief introduction to the S PLUS programming language and a minor amplification of some important points Ultimately the user of S PLUS will need to consult this reference manual probably frequently 1 2 S PLUS and X windows The most convenient way to use S PLUS is at a high quality graphics workstation running a windowing system Since these are becoming more readily available these notes are aimed at users who have this facility In particular we will occasionally refer to the use of S PLUS on an X window system and even with the motif window manager although 2 Introduction and Preliminaries the vast bulk of what is said applies generally to any implementation of the S PLUS environment Setting up a workstation to take full advantage of the customizable features of S PLUS is a straightforward if somewhat tedious procedure and will not be considered further here Users in difficulty should seek local expert help 1 3 Using S PLUS interactively When you use the S PLUS program it issues a prompt when it expects input commands The default prompt is gt which is sometimes 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
75. in 7 lt z lt 7 y is the same f is asquare matrix with rows and columns indexed by x and y respectively of values of the function cos y 1 2 Save the plotting parameters and set the plotting re gion to square Make a contour map of f add in more lines for more detail fa is the asymmetric part of f t is transpose Make a contour and restore the old graphics parameters Make some pretty perspective and high density image plots of which you can get hardcopies if you wish and clean up before moving on S PLUS can do complex arithmetic also 1i is used for the complex number 7 Plotting complex arguments means plot imaginary versus real parts This should be a circle Suppose we want to sample points within the unit cir cle One method would be to take complex numbers with standard normal real and imaginary parts and to map any outside the circle onto their recipro cal All points are inside the unit circle but the distribu tion is not uniform The second method uses the uniform distribution The points should now look more evenly spaced over the disc Cleanup again Restore standard graphics parameters An old favourite Take a hardcopy if you wish Cleanup again Quit the S PLUS program and return to Unix 64 The Inbuilt Command Line Editor in S PLUS B The Inbuilt Command Line Editor in S PLUS B 1 Preliminaries The August 1991 release of S PLUS has i
76. in Residuals Rankit Plot rm fm fmi lrf x dummy more morley data mm read table morley data mm attach mm 1 objects Expt lt factor Expt Run lt factor Run detach 1 save mm attach mm plot Expt Speed main Michaelson Morley Data xlab Experiment No fm lt aov Speed Run Expt data mm summary fm fmO lt update fm Run anova fmO fm detach rm fm fm0 S PLUS An Introductory Session Make a nonparametric local regression function Make the columns in the data frame visible as vari ables Standard point plot Add in the local regression The true regression line intercept 0 slope 1 Unweighted regression line Weighted regression line At any time you can make a hardcopy of the graph ics window by clicking on the Graph section of the window and selecting the Print option Remove data frame from the search list A standard regression diagnostic plot to check for het eroscedasticity Can you see 1t A normal scores plot to check for skewness kurtosis and outliers Not very useful here Clean up again The next section will look at data from the classical experiment of Michaelson and Morley to measure the speed of light Optional Temporarily interrupt S PLUS and look at the file This is a standard way to escape to the operating system Read in the MM data as a data frame and look at it There are five experiments col Expt and e
77. ion 9 6 Classes generic functions and object orientation The class of an object determines how it will be treated by what are known as generic functions Put the other way round a generic function performs a task or action on its arguments specific to the class of the argument itself If the argument lacks any class attribute or has a class not catered for specifically by the generic function in question there 1s always a default action provided An example makes things clearer In a sense the print function has always been generic since its action is to adopt a style of output appropriate to its arguments Thus a matrix appears as a matrix a vector as a vector and so on Note that the print function can be called explicitly or implicitly by giving an expression as a complete command The August 1991 release of S PLUS increases the number of such functions alters the mechanism by which they are implemented and via the class mechanism offers the user the facility of designing and writing generic functions for special purposes Among the other new or newly generic functions are plot for displaying objects graphically summary for summarising analyses of various types and anova for comparing sta tistical 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 L lt dbdetac
78. is labels titles etc and usually bounded by the axes themselves A typical figure appears in Figure 10 Graphics parameters controlling figure layout include mai Widths of the bottom left top and right margins respectively c 1 0 5 0 5 0 measured in inches mar c 4 2 2 1 Similar to mai except the measurement unit is text lines mar and mai are equivalent in the sense that setting one changes the value of the other The default values chosen for this parameter are often too large the right hand margin is rarely needed and neither is the top margin if no title is being used The bottom and left margins must be large enough to accommodate the axis and tick labels Furthermore the default is chosen without regard to the size of the device surface for example using the postscript driver with the height 4 argument will result in a plot which is about 50 margin unless mar or mai are set explicitly When multiple figures are in use see below the margins are reduced by half however this may not be enough when many figures share the same page 11 5 4 Multiple figure environment S allows you to create an n x 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 Figure 11 The graphical parameters relating to multiple figures are as follows 11 5 Graphics parameters list 57 mfcol c 3 2 mfrow c 2 4 mfg c 2 2 3 2 fig c
79. kes e an empty vector structure of mode numeric Similarly character is a empty character vector and so on Once an object of any size has been created new components may be added to it simply by giving it an index value outside its previous range Thus e 3 17 now makes e a vector of length 3 the first two components of which are at this point both NA This applies to any structure at all provided the mode of the additional component s agrees with the mode of the object in the first place This automatic adjustment of lengths of an object is used often for example in the scan function for input Conversely to truncate the size of an object requires only an assignment to do so Hence if alpha is a structure of length 10 then gt alpha lt alpha 2 1 5 makes 1t an object of length 5 consisting of Just the former components with even index The old indices are not retained of course 3 3 attributes and attr The function attributes object gives 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 S PLUS object The concept however is very important 3 4 The class of an object A special attribute known
80. ld 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 fit ting function this information is passed on through the fitted model object to update and its allies The name For example can also be used in other contexts but with slightly different meaning 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 addi dropi1 step and stepwise The names of these give a good clue to their purpose but for full details see the help document 10 6 Generalized linear models families Generalized linear modelling is a development of linear models to accommodate both non normal response distributions and transformations to linearity in a clean and straight forward 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 1 2 whose values influence the distribution of the response 42 Statistical models in S PLUS e The stimulus variables influence the distribution o
81. left justify 1 means right justify and 0 5 means to centre horizontally about the plotting position The actual value is the percentage 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 11 5 2 Axes and Tick marks Many of S s high level plots have axes and you can axes yourself with the low level axis graphics function Axes have three main components the azis 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 customised with the following graphics parameters lab c 5 7 12 las 1 mgp c 3 1 0 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 Orientation of axis labels 0 means always parallel to axis 1 means always horizontal and 2 mean always perpendicular to the axis Positions of axis components The first component is the distance from the axis label to the axis position in text lines
82. llows for plotting at the selected points and has the same effect as for high level graphics commands the default 1s 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 1t 1s 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 still work if the current device does not support a mouse in this case the user will be prompted for x and y coordinates identify x y Allow the user to highlight any of the points defined by x and labels 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 the middle button is pressed 11 4 Using graphics parameters 53 Sometimes we want to identify particular points on a plot rather than their positions For example we may wish the user to select some observation of interest from a graphical display and then manipulate that observation in some way Given a number of x y coordinates in two numeric vectors x and y we could use the identify function as follows gt plot x y gt identify
83. lp information This may be either scanned at the screen and dismissed or sent to a printer for hardcopy or both 1 7 S PLUS commands Case sensitivity Technically S PLUS is a function language with a very simple syntax It is case sensitive as are most UNIX based packages so A and a are different variables Elementary commands consist of either expressions or assignments If an expression 1s given as a command it is evaluated printed and the value is lost An assignment also evaluates an expression and passes the value to a variable but the result is not printed automatically Commands are separated either by a semi colon or by a newline If a command is not complete at the end of a line S PLUS will give a different prompt for example 4 Introduction and Preliminaries on second and subsequent lines and continue to read input until the command is syn tactically complete This prompt may also be changed if the user wishes In these notes we will generally omit the continuation prompt and indicate continuation by simple indenting 1 8 Recall and correction of previous commands 1 8 1 S PLUS S PLUS but not plain S provides a mechanism for recall and correction of previous commands For interactive use this is a vital facility and greatly increases the productive output of most people To invoke S PLUS with the command recall facility enabled use the e flag gt Splus e Within the session command recall is available usi
84. ly labels print summary anova dropi formula plot proj A brief description of the most commonly used ones is given in Table 2 10 4 Analysis of variance comparing models The model fitting function aov formula data data frame operates at the simplest level in a very similar way to the function 1m and most of the generic functions listed in Table 2 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 envisaged Model formula 40 Statistical models in S PLUS Value or Effect anova object Compare a submodel with an outer model and product objects an analysis of variance table coefficients object Extract the regression coefficient matrix Short form coef object deviance object Residual sum of squares weighted if appropriate formula object Extract the model formula plot object Product two plots one of the observations against the fitted values the other of the absolute residuals against the fitted values predict object The data frame supplied must have variables specified newdata data frame with the same labels as the original The value is a vec predict gam object tor or matrix of predicted values corresponding to the newdata data frame determining variable values in data frame predict gam is a safe alternative to predict that can be use
85. m list ti th 1 t2 th 2 Note that the function parameters may either be used as an expression in which case it extracts the list of parameters from a data frame or it may be used as the target for an assignment in which case it accepts a parameter list for a specified data frame In this respect it is very similar to the attributes function There is also a function param analogous to attr which handles one parameter at a time under a character string name 10 8 Some non standard models We conclude this section with just a brief mention of some of the other facilities available in S PLUS for special regression and data analysis problems Local approximating regressions The loess function fits a nonparametric re gression 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 Robust regression There are several functions available for fitting regression models in a way resistant to the influence of extreme outliers in the data The most sophisticated of these is rreg but others include lmsfit for least median squares regression and 11fit for regression using the norm However these do not as yet have the facility of using formule to specify the model function for example and conform to an older protocol which makes them sometimes rather tedious to use There is also a robust facility to cha
86. me from file genotype data C 9 Fisher s Sugar Beet Data Source R A Fisher Design of Experiments Category Analysis of variance and covariance 72 Exercises Runs 1 10 Runs 11 20 E E gt E3 E4 Es E Ez E3 E4 Es Table 9 The Michaelson Morley speed of light data Description A classical 3 x 23 randomized block experiment in four blocks of size 24 The response is the total weight of sugarbeet roots off the plot but this is accompanied by the number of roots measured The suggestion is that number of roots should be a covariate to allow for varying plot size The factors are Variety 3 levels a b and c and N P and K each at 2 levels present or absent Data The data is given in Table 11 and may be read from the file sugar data in a form suitable to construct a data frame C 10 A Barley Split Plot Field Trial Source Unknown Traditional data Category Multistratum analysis of variance Description An experiment involving barley varieties and manure nitrogen was conducted in 6 blocks of 3 whole plots Each whole plot was divided into 4 subplots Three varieties of barley were used in the experiment with one variety being sown in each whole plot while the four levels of manure 0 0 01 0 02 and 0 04 tons per acre were used one level in each of the four subplots of each whole plot In the above table V denotes the ith variety and N denotes the jth level of nitrogen C 11 The Snail Mortality Data
87. might wish to split the tax accountants by both state and sex However 1n this simple instance what happens can be thought of as follows The values in the vector are collected into groups corresponding to the distinct entries in the category The function is then applied to each of these groups individually The value 1s a vector of function results labelled by the levels attribute of the category 4 2 The function tapply and ragged arrays 15 The combination of a vector and a labelling factor or category is an example of what is 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 16 Arrays and matrices 5 Arrays and matrices 5 1 Arrays An array can be considered as a multiply subscripted collection of data entries for example numeric S PLUS allows simple facilities for creating and handling arrays and in particular the special case of matrices A dimension vector is a vector of positive integers If its length is k then the array is k dimensional The values in the dimension vector give the upper limits for each of the k subscripts The lower limits are always 1 A vector can be used by S PLUS 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 lt c 3 5 100 gives it the dim attr
88. n results Character vectors may be concatenated into a vector by the c function examples of their use will emerge frequently The paste function takes an arbitrary number of arguments and concatenates them into a single character string Any numbers given among the arguments are coerced into character strings in the evident way that is in the same way they would be if they were printed The arguments are by default separated in the result by a single blank character but this can be changed by the named parameter sep string which changes it to string possibly empty For example gt labs lt paste c X Y 1 10 sep 2 1 Index vectors Selecting and modifying subsets of a data set 9 makes labs into the character vector X1 Y2 X3 yg X5 y6 XT YS XO Y10 Note particularly that recycling of short lists takes place here too thus c X Y is repeated 5 times to match the sequence 1 10 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 be appending an index vector in square brackets immediately after the expression Such index vectors can be any of four distinct types 1 A logical vector In this case the index vector must be
89. nbuilt command line editor that allows recall editing and re submission of prior commands To use it start the S PLUS programme with Splus e Inside the editor either emacs or vi conventions are available according to the shell environment variable S_CLEDITOR To get the emacs conventions use in csh and variants setenv S CLEDITOR emacs and for the vi conventions to apply put vi instead of emacs This statement would normally be included in your login file or equivalent and would then be done au tomatically at login time To avoid forgetting to include the e a handy alias for your cshrc file is say alias S Splus e after which S is the command to start Splus with command line editor The usual typographical conventions apply M means Hold the Control down while you press the m key but Esc m means First press the Esc key and then the m key Note that after Esc case is significant B 2 Editing Actions The S PLUS programme keeps a history of the commands you type including the error 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 Esc ior Esc a characters are typed and insertion mode is finish
90. ndividual state of origin is specified by a character vector of state mnemonics as gt state c tas sa qld nsw nsw nt Wa Wa qld yic nsw vic qld qld sa tas sa nt Wa vic qld nsw nsw Wa sa act nsw vic vic act For some purposes it is convenient to represent such information by a numeric vector with the distinct values in the original in this case the state labels represented by a small integer Such a numeric vector is called a category However at the same time it is important to preserve the correspondence of these new integer labels with the originals This is done via the levels attribute of the category More formally when a category is formed from such a vector the sorted unique values in the vector form the levels attribute of the category and the values in the category are in the set 1 2 k where k is the number of unique values The value at position j in the factor is i if the ith sorted unique value occurred at position j of the original vector Hence the assignment gt stcode lt category state creates a category with values and attributes as follows gt stcode 111654223884727445653874228512771 attr levels 1 act nsw nt qld sa tas vic wa 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 sta
91. near models families 22s 10 6 1 Families ns 10 6 2 The gimO function 2 0 en 10 7 Nonlinear regression models parametrized data frames 10 7 1 Changes to the form of the model formula i n 10 7 2 Specifying the parameters 22s 10 8 Some non standard models a Graphical procedures 11 1 High level plotting commands 2s 11 1 1 The plot function aaa aaa 11 1 2 Displaying multivariate data ss 11 1 8 Display graphics 11 1 4 Arguments to high level plotting functions iss 11 2 Low level plotting commands s 11 3 Interactive graphics functions 11 4 Using graphics parameters rh 11 4 1 Permanent changes the par function 11 4 2 Temporary changes arguments to graphics functions 11 5 Graphics parameters list aa ee 11 5 1 Graphical elements 2 22e 11 5 2 Axes and Tick marks 2 2 020 000 0200 0 000 0 31 31 32 32 33 35 36 3T 3T 38 39 39 40 41 41 42 42 45 45 46 4T A S PLUS An Introductory Session B The Inbuilt Command Line Editor in S PLUS B 1 Preliminaries s B 2 Editing Actions 2l ss B 3 Command Line Editor Summary ls C Exercises C 1 The Cloud Point Data ees C 2 The Janka Hardness Data en C 3 The Tuggeranong House Price Data a C 4 Yorke Penninsula Wheat Data 2 2 ee C 5 The Iowa Wheat Yield Data 0 000 200 000 0000 C 6
92. ng either emacs style or vi style com mands The former is very similar to command recall with an interactive shell such as tcsh Details are given in Appendix B of these notes or they may be found in the reference manual or the online help documents 1 8 2 Vanilla S With S no built in mechanism is available but there are two common ways of obtaining command recall for interactive sessions e Run the S session under emacs using S mode a major mode designed to support S This is probably more convenient than even the inbuilt editor of S PLUS in the long term however it does require a good deal of preliminary effort for persons not familiar with the emacs editor It also often requires a dedicated workstation with a good deal of memory and other resources e Run the S session under some front end processor such as the public domain fep program available from the public sources archives This provides essentially the same service as the inbuilt S PLUS editor but with somewhat more overhead but a great deal less overhead than emacs requires 1 9 Executing commands from or diverting output to a file If commands are stored on an external file say commands S in the working directory work they may be executed at any time in an S PLUS session with the command gt source commands S Similarly gt sink record lis will divert all subsequent output from the terminal to an external file record lis The command gt sink
93. nge a glm family object into a robust version for use with the glm model fitting function Generalized additive models This technique aims to construct an regression func tion from smooth additive functions of the determining variables usually one for each determining variable The function gam is in many ways similar to the other model fitting functions outlined above In addition there are other model fitting functions that do a similar job These include avas and ace On the other hand ppreg is available for projection pursuit regression but this technique is still very much in need of a complete theoretical treatment and further practical experience These latter functions are again conforming to an older protocol for model fitting functions and lack the convenience of the newer functions Tree based models Rather than seek an explicit global linear model for prediction or interpretation tree based models seek to bifurcate the data recursively at critical points of the determining variables in order to partition the data ultimately into groups that are as homogeneous as possible within and as heterogeneous as possible between The results often lead to insights that other data analysis methods tend not to yield Models are again specified in the ordinary linear model form The model fitting function is tree but many other generic functions such as plot and text are well adapted to displaying the results of a tree based mod
94. nk at most 1 Notice that the outer product operator is of course non commutative 5 5 1 An example Determinants of 2 x 2 digit matrices As an artificial but cute example consider the determinants of 2 x 2 matrices f 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 1s chosen independently and uniformly at 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 88 2 is so inefficient as to be impractical It is also perhaps surprising that about 1 1n 20 such matrices 1s 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 2 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
95. nt Also if an array name is given with just one subscript or index vector then the corre sponding values of the data vector only are used in this case the dimension vector is ignored This is not the case however if the single index is not a vector but itself an array as we next discuss 5 3 Index arrays As well as an index vector in any subscript position an array may be used with a single index array 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 5 3 Index arrays 17 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 x 5 array X and we wish to do the following 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 0s In this case we need a 3 x 2 subscript array as in the example given in Figure 1 gt x lt array 1 20 dim c 4 5 Generate a 4 x 5 array gt x 1 2 3 4 5 1 1 5 9 13 17 2 2 6 10 14 18 3 3 7 11 15 19 4 4 8 12 16 20 gt i lt array c 1 3 3 1 dim c 3 2 gt i iis a 3 x 2 index array 1 2 1 1 3 2 2 2 3 3 1 gt x i Extract those elements
96. object has a names attribute to identify its components In this case a subvector of the names vector may be used in the same way as the positive integral labels in 2 above gt lunch lt fruit c apple orange This option is particularly useful in connection with data frames as we shall see later An indexed expression can also appear on the receiving end of an assignment in which case the assignment operation is performed only on those elements of the vector The 10 Simple manipulations numbers and vectors expression must be of the form vector index_vector as having an arbitrary expression in place of the vector name does not make much sense here The vector assigned must match the length of the index vector and in the case of a logical index vector it must again be the same length as the vector it is indexing For example gt x is na x lt 0 replaces any missing values in x by zeros and gt y y lt 0 lt yLy 0 has the same effect as gt y lt abs y 2 Note that abs does not work as expected with complex arguments The appropriate function for the complex modulus is Mod Objects their modes and attributes 11 3 Objects their modes and attributes 3 1 Intrinsic attributes mode and length The entities S PLUS 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
97. ock Variety M_M Ns Ma V 74 89 801 12 1 2 3 1 2 3 1 2 V3 Table 12 A split plot Barley field trial Relative Humidity P 60 0 65 8 70 5 75 8 Temperature Temperature Temperature Temperature 10 15 20 10 15 20 10 1 2 1 1 5 0 10 15 20 A 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 0 1 2 3 3 5 2 3 3 Table 13 The snail mortality data V V V V V V V Description On the Greek 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 Data Age 20 35 45 55 70 No tested 50 50 50 50 50 No blind 6 17 26 37 44 Table 14 The Kalythos blindness data The data is given in Table 14 and may be read as a data frame from file kalythos data 76 Exercises Problems C 13 The Stormer Viscometer Data Source E J Williams Regression Analysis Wiley 1959 Category Nonlinear regression special regression Description The stormer viscometer measures the viscosity of a fluid by measuring the time taken for an inner cylinder in the mechanism to perform a fixed number of revolutions in response to an actuating weight The viscometer is calibrated by measuring the time taken with varying weights while the mechanism is suspended in fluids of accurately known viscosity The data comes from such a calibra
98. ok through a data file without leaving S PLUS you could use gt more curious dat When you finish paging the file the S PLUS session is resumed In fact the integration of S PLUS into UNIX is very complete For example there is a command unix that executes any unix command specified as a character string argument and passes on any output from the command as a character string to the program Essentially the full power of the operating system remains easily available to the user of the S PLUS program during any session There are non UNIX implementations of S PLUS for example for DOS Users should consult the appropriate user guides for more information 1 6 Getting help with functions and features S PLUS 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 1s gt solve For a feature specified by special characters the argument must be enclosed in double quotes making it a character string gt help CL A much more comprehensive help facility is available with the X windows version of S PLUS The command gt help start gui motif causes a help window to appear with the motif graphical user interface It is at this point possible to select items interactively from a series of menus and the selection process again causes other windows to appear with the he
99. on read table can then be used to read the data frame directly gt HousePrice lt read table houses data 7 3 Other facilities editing data 29 Price Floor Area Rooms Age Cent heat 01 52 00 111 0 830 5 6 2 no 02 54 75 128 0 710 5 7 5 no 03 57 50 101 0 1000 5 4 2 no 04 57 50 131 0 690 6 8 8 no 05 59 75 93 0 900 5 1 9 yes Figure 2 Input file form with names and row labels 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 Figure 3 The data frame may 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 Figure 3 Input file form without row labels then be read as gt HousePrice lt read table houses data header T where the heading T 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 3 Other facilities editing data S PLUS input facilities are simple and their requirements are fairly strict There is a clear presumption that rather than use S PLUS to accommodate a subtle variety of input and more so output protocols you will be able to modify your input files using other tools such file editors and the UNIX utilities sed and awk to fit in with the requirements of S PLUS Generally this very simple There i
100. or 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 assignment operator is not the usual operator which is reserved for another purpose It consists of the two characters lt less than and minus occurring strictly side by side and it points to the structure receiving the value of the expression Assignments can also be made in the other direction using the obvious change in the assignment operator So the same assignment could be made using 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 Shorter vectors in the expression are recycled as often as need be perhaps fr
101. ore convenient than gt statefr lt tapply statef statef length Further suppose that incomef is a category giving a suitably defined income class for each entry in the data vector for example with the cut 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 act nsw nt qld sa tas vic wa 35 thru 45 1 1 0 10 0 1 0 45 thru 55 1 1 1 12 0 1 3 55 thru 65 0 3 1 3 2 2 2 1 65 thru 75 0 1 0 0 0 0 1 0 Extension to higher way frequency tables is immediate Lists data frames and their uses 23 6 Lists data frames and their uses 6 1 Lists An S PLUS list is an object consisting of an ordered collection of objects known as its components There is no particular need for the components to be of the same mode or type and for example a list could consist of a numeric vector a logical value a matrix a complex vector a character array a function and so on Components are always numbered and may always be referred to as such Thus if St is the name of a list with four components these may be individually referred to as St 111 St 2 SELL31 and St 411 If further St 3 is a triply subscripted array then St 3 1 1 1 is its first entry and dim St 3 1 is its dimension vector and so on If St is a list then the function length St gives the number of top level components it has Components of lists may
102. own functions Note the use of quote marks The function could then be used as X y The backslash symbol itself is not a convenient choice as it presents special problems in this context The matrix multiplication operator 4 4 and the outer product matrix operator 0 are other examples of binary operators defined in this way 9 2 Named arguments and defaults As first noted in 82 3 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 f 4 Then the function may be invoked in several ways for example gt ans funi d df 20 T gt ans lt funi d df graph T limit 20 gt ans lt funi data d limit 20 graph T 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 function data data frame graph T limit 20 0 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
103. r operand longer than some previous array immediately converts the calculation to one in which all operands are coerced to vectors A diagnostic message is issued if the size of the long vector is not a multiple of the common size of all previous arrays e If array structures are present and no error or coercion to vector has been precipi tated the result is an array structure with the common dim attribute of its array operands 5 5 The outer product of two arrays 19 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 1s an array whose dimension vector 1s got by concatenating their two dimension vectors order is important and whose data vector is got by forming all possible products of elements of the data vector of a with those of b The outer product is formed by the special operator 404 gt ab lt a 4o b An alternative 1s gt ab lt 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 cos y f y TF over a regular grid of values with z and y coordinates defined by the S PLUS vectors x and y respectively we could proceed as follows gt f lt function x y cos y 1 x72 gt z lt outer x y f In particular the outer product of two ordinary vectors is a doubly subscripted array i e a matrix of ra
104. rce Quoted in Draper amp Smith Applied Regression Analysis Category Nonlinear regression Description The data gives the temperature C and pressure Pascals in a saturated steam driven experimental device The relationship between pressure Y and temperature in sat urated steam can be written as t y ocp oh ee y tt However a more realistic model may have the experimental errors multiplicative rather than additive in which case an analysis in the log scale using the model t logY loge f2 p ob E y t may be more appropriate Data The data is given in Table 17 and may be read as a data frame from file steam data 78 Exercises Table 17 Temperature and Pressure in Saturated Steam C 16 Count Rumford s Friction Data Source Bates amp Watts Nonlinear Regression Analysis Category Nonlinear regression Description Data on the amount of heat generated by friction was obtained by Lord Rumford in 1798 A bore was fitted into a stationary cylinder and pressed against the bottom by a screw The bore was turned by a team of horses for 30 minutes after which Lord Rumford suffered the thermometer to remain in its place nearly three quarters of an hour observing and noting down at small intervals of time the temperature indicated by it Newton s law of cooling suggests a nonlinear regression model of the form Y Bo Bi exp t where Y is the temperature and is the time in minutes
105. restores it to the terminal once again 1 10 Data directories Permanency Removing objects 5 1 10 Data directories Permanency Removing objects All objects created during your S PLUS sessions are stored in a special form in the Data sub directory of your working directory work say Each object is held as a separate file of the same name and so may be manipulated by the usual UNIX commands such as rm cp and mv This means that if you resume your S PLUS session at a later time objects created in previous sessions are still available which is a highly convenient feature This also explains why it is recommended that you should use separate working directo ries for different jobs Common names for objects are single letter names like x y and so on and if two problems share the same Data sub directory the objects will become mixed up and you may overwrite one with another There is however another method of partitioning variables within the same Data di rectory using data frames These are discussed further in 6 4 In S PLUS to get a list of names of the objects currently defined use the command gt objects whose result is a vector of character strings giving the names When S PLUS looks for an object it searches in turn through a sequence of places known as the search list Usually the first entry in the search list is the Data sub directory of the current working directory The names of the places currently on the se
106. s however a function make fields that can be used to convert a file with fixed width non separated input fields into a file with separated fields There is also a facility count fields that will count the number of fields on each line of such a file Occasionally for very simple conversion and checking problems these may be adequate to the task but in most cases it is better to do the preliminary spade work before the S PLUS session begins Once a data set has been read there is an X window based facility in S PLUS for making small changes The command gt xnew lt data ed xold will allow you to edit your data set xold using a spreadsheet like environment in a separate editing window and on completion the changed object is assigned to xnew xold and hence xnew can be any matrix vector data frame or atomic data object 30 More language features Loops and conditional execution 8 More language features Loops and conditional execution 8 1 Grouped expressions S PLUS 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 expri exprs exprm in which case the value of the group is the result of the last expression in the group evaluated
107. s follows gt fitted model lt 1m formula data data frame 10 3 Generic functions for extracting information 39 Y is modelled as M Include M and M5 Include M leaving out terms of M The tensor product of M and M If both terms factors then the subclasses factor in M Similar to Mj M but with a different coding Ma Mi Ms M M M3 Mi Ms in Mi All terms in M together with interactions up to order n Insulate M Inside M all operators have their normal arithmetic meaning and that term appears in the model matrix Table 1 Summary of model operator semantics For example gt fm2 lt 1m y x1 x2 data production would fit a multiple regression model of y on a and x2 with implicit intercept term The important but technically optional parameter data production specifies that any variables needed to construct the model should come first from the production data frame This is the case regardless of whether data frame production has been attached to the search list or not 10 3 Generic functions for extracting information The value of 1m is fitted model object technically a list of results of class 1m Infor mation 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 A full list of these at the present time is addi coef effects kappa predict residuals alias deviance fami
108. 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 gt s4 lt seq length 51 from 5 by 2 generates the same vector in s4 The fifth parameter may be named along vector which if used must be the only pa rameter and creates a sequence 1 2 length vector or the empty sequence if the vector is empty as it can be A related function is rep which can be used for replicating a structure 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 8 Simple manipulations numbers and vectors 2 4 Logical vectors As well as numerical vectors S PLUS allows manipulation of logical quantities The elements of a logical vectors have just two possible values represented formally as F for false and T for true 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 F corresponding to elements of x where the condition is not met and T 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 c1 is their intersection c1 c2 is their union and c1 1s the negation of c
109. somewhat advanced though hardly difficult topic and is not covered further in these notes If global and permanent assignments are intended within a function then the superas signment operator lt lt can be used See the help document for details and also see the synchronize function 9 4 More advanced examples Efficiency factors in block designs As a more complete if a little pedestrian example of a function consider finding the efficiency factors for a block design Some aspects of this problem have already been discussed in 5 3 A block design is defined by two factor say blocks b levels and varieties v levels If R and K are the replications and block size matrices and N is the incidence matrix then the efficiency factors are defined as the eigenvalues of the matrix E 2I R V N K NR V 7 AA where A K 1 N R One way to write the function is as in Figure 5 gt bdeff lt function blocks varieties blocks lt as factor blocks minor safety move b lt length levels blocks varieties lt as factor varieties minor safety move v lt length levels varieties K lt as vector table blocks remove dim attr R lt as vector table varieties remove dim attr N lt table blocks varieties A lt 1 sqrt K N rep 1 sqrt R rep b v sv lt svd A list eff 1 sv d 2 blockcv sv u varietycv sv v Figure 5 A function for block
110. stralian States is assumed but the elementary facts are given in a footnote for foreign readers Comments and corrections are always welcome Please address email correspondence to the author at wvenable stats adelaide edu au The author is indebted to many people for useful contributions but in particular Lucien W Van Elsen who did the basic TEX to IATpX conversion and Rick Becker who offered an authoritative and extended critique on an earlier version Responsibility for this version however remains entirely with the author and the notes continue to enjoy a fully unofficial and unencumbered status These notes may be freely copied and redistributed for any educational purpose provided the copyright notice remains intact Where appropriate a small charge to cover the costs of production and distribution only may be made Bill Venables University of Adelaide 16th December 1990 Addendum Version 2 0 April 1992 As foreshadowed above the present version of the notes contains references to the use of S in a workstation environment although I hope they remain useful to the user on an ordinary graphics terminal Of much greater importance however are the language developments that have taken place in S itself in the August 1991 release These are only partially addressed in this version of the notes as a complete coverage would require a document of much greater length than was ever intended I trust however that the sketches giv
111. tef lt factor state The print function now handles the factor object slightly differently 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 If we remove the factor class however using the function unclass it becomes virtually identical to the category 5Foreign 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 14 Categories and factors gt unclass statef 11 654223884727445653874228512771 attr levels 1 act nsw nt qld sa tas vic wa 4 2 The function tapply and ragged arrays To continue the previous example suppose we have the incomes of the same tax accoun tants 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 gt incmeans act nsw nt qld sa tas vic wa 44 5 57 333 55 5 53 6 55 60 5 56 52 25 The function tapply is used to apply a function here mean to each group of compo
112. the command dev off This ensures that the device finishes cleanly for example in the case of hardcopy devices this ensures that every page 1s completed and has been sent to the printer 11 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 portrait orientation unless the horizontal T 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 height 4 will produce a file containing PostScript code for a figure 4 inches high perhaps for inclusion in a document It is important to note that if the file named in the command already exists it will be overwritten This is the case even if the file was only created earlier in the same S session 11 6 2 Multiple graphics devices In advanced use of S it is often useful to have several graphics devices in use at the one 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 Warning The PostScript code produced by the postscript device driver is not Encapsulated PostScript and
113. the function data frame gt accountants lt data frame home statef loot income 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 T 6 4 2 attach and detach The notation such as accountants statef for list components is not always very convenient A useful facility would be somehow to make the components of a list or data frame temporarily visible as variables under their component name without the need to quote the list name explicitly each time The attach function as well as having a directory name as its argument may also have a data frame Thus suppose lentils is a data frame with three variables lentils u lentils v lentils w The attach gt attach lentils places the data frame in the search list 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 list To make a permanent change to the data frame itself the simplest way is to resort once a
114. tial approximation for each with which to start the iterative estimation procedure There are two ways of specifying this information e If the call to nls has a start parameter specified its value must be a list of named components The names of the list specify the names of the parameters and the values specify the starting values e If the data is held in a data frame the parameters may similarly be defined as a parameters attribute of the data frame Since our policy is generally to work with data frames as much as follows we show the second possibility in the next example Example Consider again a nonlinear regression of the form 1 An easy way to find initial estimates for the parameters is to regress roy on x and zs gt fmO lt Ilm x2 y x1 x2 1 data biochem gt th lt coef fm0 To name the parameters and associate them with the biochem is done as follows gt parameters biochem lt list ti th 1 t2 th 2 Now to fit the nonlinear regression model gt fm lt nls y ti xi x2 t2 data biochem At this point we could use the summary function and most of the other generics to investigate the model and display information To extract the coefficients we could now use for example gt th lt coef fm and to make these least squares estimates the new values of the parameters associated with biochem we could simply repeat the step 10 8 Some non standard models 47 gt parameters bioche
115. tion and theoretical considerations suggest a nonlinear relationship between time weight and viscosity of the form ro p w 60 where 8 and 0 are unknown parameters to be estimated Data Weight Viscosity 20 50 ur mepus Pars ns Pana eo a0 Paap r ee ar s 897 isos 58 3 30 0 146 6 229 0 85 6 an aa o ma Pons oma Poe Pana 298 3 187 2 89 0 86 5 Table 15 The Stormer viscometer calibration data The data is given in Table 15 and may be read as a data frame from file stormer data C 14 The Chlorine availability data Source Draper amp Smith Applied Regression Analysis adapted Category Nonlinear regression C 15 The Saturated Steam Pressure Data 77 Description The following set of industrial chemical data shows the amount of chlorine available in a certain product at various times of testing after manufacture A nonlinear regression model for the chlorine decay of the form Y fo B exp 0t has been suggested on theoretical grounds with Y the amount remaining at time t Data Percent available pues Percent available pue Percent available 49 49 42 43 42 47 47 48 48 40 41 41 43 45 46 46 40 40 42 43 43 45 40 41 41 43 43 44 40 41 45 46 38 40 40 Table 16 The Chlorine availability data The data is given in Table 16 and may be read as a data frame from file chlorine data C 15 The Saturated Steam Pressure Data Sou
116. to Seventeenth Century D Venetian French Data The data is given in Table 22 and may be read as a data frame from file painters data 82 Exercises Po Composition Drawing Colour Expression Da Udine Da Vinci Del Piombo Del Sarto Fr Penni Guilio Romano Michelangelo Perino del Vaga Perugino Raphael F Zucarro Fr Salviata Parmigiano Primaticcio T Zucarro Volterra Barocci Cortona Josepin L Jordaens Testa Vanius Bassano Bellini Giorgione Murillo Palma Giovane Palma Vecchio Pordenone Tintoretto Titian Veronese Albani Caravaggio Corregio Domenichino Guercino Lanfranco The Carraci Durer Holbein Pourbus Van Leyden Diepenbeck J Jordaens Otho Venius Rembrandt Rubens Teniers Van Dyck Bourdon Le Brun Le Suer Poussin Table 22 The Subjective Assessment Data of de Piles
117. to measure A low degree polynomial regression is suggested as appropriate Y bo Piz Goa E where Y is the hardness and x the density Data The following data comes from samples of 36 Australian Eucalypt hardwoods C 3 The Tuggeranong House Price Data 67 The data may be read as a data frame from file janka data C 3 The Tuggeranong House Price Data Source Dr Ray Correll Personal communication Category Multiple regression coplots Description Before buying a house in Tuggeranong in February 1987 a cautious potential house holder collected some data on houses on the market The data for 20 such houses is shown in the table and is available as the file house dat The variables collected are in order price total floor area block area number of main rooms age of house and whether or not the house was centrally heated Data The data is given in Table 5 and is available as the file house data C 4 Yorke Penninsula Wheat Data Source K W Morris private communication Category Multiple regression Description The annual yield of wheat in a marginal wheat growing district on the Yorke Penninsula South Australia together with the rainfall for the three growing months for the years 1931 1955 The year itself is potentially a surrogate predictor to allow for improvements in varieties and farm practice Yield is in bushels per acre and rainfall is in inches Data The data is given in Tabl
118. unction n1s and leave ms for the reader to pursue as needed 10 7 1 Changes to the form of the model formula In specifying a linear or generalized linear model we could allow the regression parame ters to be defined implicitly and to be given names by transference from the column of the model matrix that they multiply 46 Statistical models in S PLUS In arbitrary nonlinear models no such simplicity applies and we have to specify the model as an ordinary expression that includes both determining variables and parameters together For example to specify a model for a nonlinear regression such as 1 above we would use y ti xi x2 t2 where y is the response variable x1 and x2 are determining variables and t1 and t2 are scalar parameters In such model formula all operators have their usual arithmetic expression meaning and the useful facility of expanding factors and forming cross and nested structures 1s no longer available All parameters must be explicitly defined in the formula even if they come from a linear part of the model 10 7 2 Specifying the parameters Since the model formula now contains both determining variables and parameters there has to be some mechanism for specifying which are which But of course once the parameters have been specified the remaining variates in the model formula must be variables As well as specifying which are the parameters it is also necessary in this case to spec ifying an ini
119. unction then usually takes the form name expr expro and may occur anywhere a function call is legitimate For example consider a function to emulate directly the MATLAB backslash command which returns the coefficients of the orthogonal projection of the vector y onto the column space of the matrix X Thus given a vector y and a matrix X then XN def X X X y where X 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 is permanent like all objects and may be used in statements such as gt regcoeff lt bslash Design yvar and so on The classical S PLUS 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 9 1 Defining new binary operators Had we given the bslash function a different name namely one of the form fanything it could have been used as a binary operator in expressions rather than in function form Suppose for example we choose for the internal character The function definition would then start as gt AIY lt function X y 0 32 Writing your
120. up according to the data Ideal for creating plots with subsequent low level graphics functions 11 2 Low level plotting commands 51 xlab string ylab string main string sub string Axis labels for the z 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 Figure title placed at the top of the plot in a large font Sub title placed just below the z axis in a smaller font 11 2 Low level plotting commands Sometimes the high level plotting functions don t produce exactly the kind of plot you desire In this case low level plotting commands can be used to add extra information such as points lines or text to the current plot Some of the more useful low level plotting functions are points x y lines x y text x y labels abline a b abline h y abline v x abline Im obj polygon x y e legend x y legend legend angle v legend density v legend fill v legend col v Adds points or connected lines to the current plot plot stype argument can also be passed to these functions and defaults to p for points and 1 for lines 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 yLil The default is 1 1ength x Note This function is often used in the sequence gt
121. uter margins are particularly useful for page wise titles etc Text can be added to the outer margins with the mtext function with argument outer T There are no outer margins by default however so you must create them explicitly using oma or omi 11 6 Device drivers S can generate graphics of varying levels of quality on almost any type of display or printing device Before this can begin however S 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 S draw a line for example into 11 6 Device drivers 59 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 motif For use with the X11 or Open Windows window systems openlook X110 suntools For use with the SunView windowing system postscript For printing on PostScript printers or creating PostScript graphics files printer For terminals with little or no graphics capabilities ASCII based crt graphics are generated When you have finished with a device be sure to terminate the device driver by issuing
122. utions and also quasi Iikelihood 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 as in Table 3 The combination of a response distribution a link function and various other pieces of information that are needed to carry out the modelling exercise is called the family of the generalized linear model 10 6 2 The gim function Since the distribution of the response depends on the stimulus variables through a single linear function only the same mechanism as was used for linear models can still be used to specify the linear part of a generalized model The family has to be specified in a different way The S PLUS function to fit a generalized linear model is glm which uses the form gt fitted model glm formula family family generator data data frame 10 6 Generalized linear models families 43 Family Name Link Function binomial gaussian Gamma inverse gaussian poisson quasi logit probit cloglog identity inverse log 1 mu 2 sqrt Table 3 Families and the link functions available to them The only new feature is the family generator which is the way the family is described Although it s

Notes on S-PLUS: A Programming Environment for Data Analysis

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