The Mann-Whitney U test is a test for differences in location between two samples. The data for the samples can either be stored in two separate variates, and by the parameters Y1 and Y2. Alternatively, they can be stored in a single variate, supplied by Y1, with the GROUPS option set to a factor to identify which unit belongs to each sample. The GROUPS option is ignored when the Y2 parameter is set. MANNWHITNEY calculates the test statistic U, along with its Normal approximation if both sample sizes are larger than 5. These statistics can be saved using the STATISTIC and NORMAL parameters respectively, and are displayed by the test setting of the option PRINT. The ranks setting of PRINT produces vectors of ranks (with respect to the whole data set) for each variate in DATA. Parameter SIGN holds an indicator which takes the value 1 if the ranks in the first sample are higher on average than those in the second sample, and takes the value 0 otherwise. The ranks (with respect to the combined data set) for each sample can be saved using the R1 and R2 parameters.

Option: PRINT, GROUPS. Parameters: Y1, Y2, R1, R2, STATISTIC, NORMAL, SIGN.

where T_k = ( t_k³-t_k )/12 and t_k is the number of observations with rank k. (See for example Siegel 1956, pages 116-127.) Otherwise, MANNWHITNEY looks up the probability from a stored table.

The variates in DATA can be restricted, and in different ways. MANNWHITNEY uses only those units of each variate that are not excluded by their respective restrictions.

Procedure MANOVA performs multivariate analysis of variance or covariance. The data variates are specified by the Y parameter.

The model for the design is specified by options of the procedure. TREATMENTSTRUCTURE specifies a model formula to define the treatment terms in the analysis; if this is unset, MANOVA will use the model already defined by the TREATMENTSTRUCTURE directive, or will fail if that too has not been set. BLOCKSTRUCTURE defines the underlying structure of the design, and MANOVA will use the model (if any) previously defined by the BLOCKSTRUCTURE directive if this is not set; these can both be omitted if there is only one error term (i.e. if the design is unstratified). The COVARIATES option specifies any covariates; by default MANOVA will take those already listed (if any) by the COVARIATE directive. The FACTORIAL option can be used to set a limit on the number of factors in the terms generated from the treatment formula.

The LRV option allows a pointer to be saved containing an LRV structure for each treatment term. When covariates have been specified, the pointer will also contain a final LRV structure for the covariate term. If a term is estimated in more than one stratum, the LRV is taken from the stratum that occurs last in the BLOCKTERMS pointer. The structures in the LRV hold the canonical variate loadings, roots and trace for the respective term.

The other options control printed output. PRINT indicates the output required from the multivariate analysis of covariance, with settings ssp to print the sums of squares and products matrices, and tests to print the various test statistics (Wilks Lambda, with Chi square and F approximations, the Pillai-Bartlett trace, Roy's maximum root test and the Lawley-Hotelling trace). APRINT, UPRINT and CPRINT control output from the univariate analyses of each of the y-variates, corresponding to ANOVA options PRINT, UPRINT and CPRINT, respectively.

Parameter: Y.

The relevant theory, with formulae and references for the test statistics, can be found in Chatfield & Collins (1986, Chapter 9). The procedure analyses the data variates by ANOVA first as y-variates, and then as covariates in order to obtain the SSP matrices. The SSP matrices are then adjusted for the covariates, using matrix manipulation in CALCULATE, and LRV decompositions are done, before the test statistics are calculated (again using CALCULATE).

The MENU procedure enters a menu system. By default, the standard Genstat Menu System is started, but the option FILENAME can be set to the name of an alternative file of Genstat commands. The Option INCHANNEL specifies the input channel on which the alternative file is to be opened, and OUTCHANNEL specifies the output channel which may be used within the alternative file for keeping a record of commands. The procedure only operates interactively: in batch mode, immediate exit occurs without a diagnostic.

Options: FILENAME, INCHANNEL, OUTCHANNEL. Parameters: none.

The input channel INCHANNEL is closed, without a warning being printed if it was not actually open. Then the base file is opened on INCHANNEL and control is passed to the commands in the file, with echoing of commands switched off. Automatic logging (via the COPY directive) to the channel OUTCHANNEL is switched off.

MPOWER forms powers of a square matrix, using as few matrix operations as possible in order to save time and decrease rounding errors. The square matrix is specified using the MATRIX parameter, and can be either an ordinary matrix structure (with an equal number of rows and columns), a symmetric matrix or a diagonal matrix. The required power, which must be a positive integer, is specified using the POWER parameter. The RESULT parameter supplies the identifier of the structure to save the results; this will be declared automatically to be of the same type as the input structure.

Options: none. Parameters: MATRIX, POWER, RESULT.

For general matrices, successive powers of two of the matrix are formed by matrix products, and the result formed by taking the product of those that are needed to achieve the specified power. Diagonal matrices are dealt with using simple exponentiation of the diagonal values. Symmetric matrices are spectrally decomposed, and the result formed as a product of the matrix containing the latent vectors (V) with the simple power of the diagonal matrix containing the latent roots (R):

MULTMISS estimates missing values for units in a multivariate data set, using an iterative regression technique. The input for the procedure is a set of variates contained in a pointer specified by the DATA parameter. The output can be saved in a different set of variates by supplying a similar pointer with the parameter OUT; if this is absent, the output values will overwrite the values of the variates given by DATA. The maximum number of iterations is set by the option MAXCYCLE, with a default of 10. If MAXCYCLE is set to zero, missing values will be replaced by variate means calculated from the units that have no values missing for any of the variates.

Option: MAXCYCLE. Parameters: DATA, OUT.

Initial estimates of the missing values in each variate are formed from the variate means using the values for units that have no missing values for any variate. Estimates of the missing values for each variate are then recalculated as the fitted values from the multiple regression of that variate on all the other variates. When all the missing values have been estimated the variate means are recalculated. If any of the means differs from the previous mean by more than a tolerance (the initial standard error divided by 1000) the process is repeated, subject to a maximum number of repetitions defined by the MAXCYCLE option.

All the variates must be unrestricted, or they must all be restricted to the same set of units; otherwise a fault will occur in a CALCULATE statement within MULTMISS.

Procedure MVARIOGRAM uses the directives FIT, FITCURVE and FITNONLINEAR to fit various models to the experimental variogram. Models must be authorized in the sense that they cannot give rise to negative variances when data are combined. Technically they are conditionally negative semi-definite (CNSD); see Webster & Oliver (1990) or Journel & Huijbregts (1978) for an explanation.

The MODEL option specifies the model that is to be fitted. There are bounded isotropic models with finite ranges. These all take the value c + c₀ for h ³  a, and the following values for h < a

circular c₀ + c {1 - (2/P)arccos(h/a)

+ (2h/(Pa))Ö (1-h²/a²)}

spherical c₀ + c {1.5h/a - 0.5(h/a)³ }

doublespherical c₀ + c₁ {1.5h/a₁ - 0.5(h/a₁)³ }

and unbounded models

(power function with exponent a strictly between 0 and 2)

Finally, the affinepower function can be fitted to an experimental variogram that appears unbounded and geometrically anisotropic, i.e. one that might be made isotropic by a simple linear transformation of the spatial coordinates

In all these models, the intercept term (or nugget variance) c₀ can be omitted by setting the CONSTANT option to omit; the default is estimate.

The data for the procedure can be taken directly from the FVARIOGRAM directive, with parameters DISTANCES, VARIOGRAMS and COUNTS corresponding to those with the same names in FVARIOGRAM. The data will be in variates if the variogram was calculated in only one direction. If it is in several, they can either be in matrices (as generated by FVARIOGRAM) or in variates. For MODEL=affinepower directions must be supplied, using the DIRECTIONS parameter. These should be in a variate with one value for each column if the other data are in matrices; alternatively, they should be in a variate of the same length as the other variates.

The WEIGHTING option controls the weights that are used when fitting the model. The default setting counts uses the values supplied by the COUNTS parameter, cbyvar uses the COUNTS divided by the values in VARIOGRAM, and equal uses equal weights (of one).

The procedure generates rough starting values for the parameters before calling FITNONLINEAR to convergence. If the solution does not converge there are two likely reasons. The model may be unsuited for the particular experimental variogram. For example, a bounded model is specified when the variogram is clearly unbounded, or vice versa. You should choose only models that have approximately the right shape. Alternatively, the starting values are too far from a sensible solution. Here you should estimate starting values by inspection and insert them into MVARIOGRAM.

Printed output is controlled by the PRINT option, and includes all the usual settings as in FIT, FITCURVE or FITNONLINEAR. You can also produce a high-resolution graph of the data and the fitted model, by setting the WINDOW option to the number of a suitable window. By default WINDOW is zero, and no graph is produced. The TITLE option can supply a title for the plot. Option XUPPER can define an upper value for the x-axis (i.e. distance), and PENDATA and PENMODEL can supply the numbers of the pens to be used to plot the experimental variogram and the fitted model respectively (by default 1 and 2).

The model is fitted using directives FIT, FITCURVE or FITNONLINEAR as appropriate.

Fits nonlinear contrasts to quantitative factors in ANOVA

The ANOVA directive allows linear contrasts to be fitted and incorporated into the analysis-of-variance table. NLCONTRASTS extends this to enable nonlinear contrasts to be fitted to the effects of a quantitative factor and its interaction with another factor. The analysis should include both main effects and the interaction between the factors. The procedure will work for any block structure providing each treatment term is estimated entirely within one stratum. The result is similar to ANOVA with a polynomial contrast, but with slightly different partitions of the treatment sums of squares. The main effect is partitioned into the sum of squares for the "Curve" and the remainder or "Deviations". The interaction sum of squares is partitioned into the sum of squares due to curves with "Common Nonlinear" parameters for the levels of the non-quantitative factor, and the extra sum of squares due to having "Separate Curves" for each level of that factor, and the remaining sum of squares which again represents "Deviations".

The BLOCKSTRUCTURE and TREATMENTSTRUCTURE directives must be used in the normal way before the procedure is called, and any [[r[[COVARIATES]]r]] should also be defined first. The structure of the analysis-of-variance table is then accessed from inside the procedure. The Y parameter defines the variate to be analysed, and the form of nonlinear contrast is defined using the CURVE option of the procedure. The same choices of curves are available as for FITCURVE. There are four other options, PRINT, FPROBABILITY, PSE, and WEIGHT, which are exactly as for ANOVA. The XFACTOR parameter defines the factor to which the contrasts are to be fitted, and the XLEVELS parameter may be used to define x values for the regressions if the levels already defined for the factor are unsuitable. The GROUPFACTOR parameter defines the factor whose interaction with XFACTOR is to be assessed. The final three parameters CONTRASTS, SECONTRASTS and DFCONTRASTS can be used to save the parameter estimates for the contrasts, their standard errors and degrees of freedom respectively.

Options: PRINT, CURVE, FPROBABILITY, PSE, WEIGHT.

ANOVA is used to obtain the basic analysis-of-variance table and the sums of squares for the treatment terms. FITCURVE is then used with the treatment means to fit three sets of curves: a single curve, curves with common nonlinear parameters, and entirely separate curves. The deviances and degrees of freedom obtained from these are used in conjunction with the treatment sums of squares to calculate the contrast sums of squares and degrees of freedom. Further details are given by Butler & Brain (1992). New lines for the analysis-of-variance table are then constructed using PRINT and EDIT, and these lines are then inserted into the table (saved in a text with ADISPLAY) using EDIT. The standard errors for the parameter estimates and deviances are based on the Residual Mean Square for the appropriate stratum. Standard errors for deviations are calculated using the method on page 501 of the Genstat 5 Release 3 Reference Manual.

If the Y variate is restriced, the procedure will use only the units not excluded by the restriction.

Marginal (univariate) tests - assess the normality of each variate in turn. The variates are standardized to have mean=0, variance=1 and then transformed with the NORMAL function. The test is based on the idea that, assuming normality, these transformed values should look like a sample from a uniform distribution on (0,1).

Bivariate angle tests - assess the bivariate normality of each pair of variates in turn. The variates are standardized so that they are uncorrelated and have mean=0 and variance=1. The test is based on the following idea: if x and y are the standardized values, then the angle between the x-axis and the line joining (0,0) to (x,y) should, assuming normality, be uniformly distributed on (0,2P).

For each type of test, the test statistics are empirical distribution function (EDF) statistics - i.e. they compare the empirical distribution function of the sample with the theoretical distribution expected under the null hypothesis. Three EDF statistics are provided for each type of test - the Anderson-Darling statistic, the Cramer-von Mises statistic and the Watson statistic. The idea is to provide good power against a wide range of alternatives. The test statistics are adjusted so that their null distribution is independent of the sample size; critical values can be printed by the procedure (option PRINT=critical).

The DATA parameter is used to indicate the variate(s) whose normality is to be assessed. If a single variate is supplied, its normality is tested using the marginal test. Alternatively, DATA can supply a pointer to a set of variates to be tested for multivariate normality.

The PRINT option can be used to select the type of test using the settings marginal, bivariateangle and radius. The setting critical allows tables of critical values to be printed, and stars requests that significant values of the test statistics be flagged with stars. Settings bivariateangle and radius are relevant only when testing for multivariate normality. By default PRINT=marginal,bivariateangle,radius

Option: PRINT. Parameter: DATA.

If a variate to which the DATA parameter is set is restricted, the tests will be calculated using only the units included by the restriction. Similarly, the variates in a DATA pointer can be restricted, but then must all be restricted in the same way. The procedure does not work properly with missing values. If missing values are present, RESTRICT should be used (before calling the procedure) to exclude all units for which any of the variates has a missing value.

NOTICE allows information to be printed from the Genstat 5 Notice Board. The information is stored in a backing-store file whose name is defined by Library procedure LIBFILENAME; there must be a free backing-store channel to which the file can be attached.

The PRINT option is used to specify what information is required. The possible values, with explanations in brackets, are as follows: news information about recent developments concerning Genstat, library recent developments concerning the Procedure Library, errors details about errors reported in Genstat 5 directives, and other advice and warnings, instructions instructions for authors of library procedures.

Option: PRINT. Parameters: none.

The information is held in subfile _notices of the backing-store file that holds help for the Library; the name of the file is supplied by procedure LIBFILENAME. The file is opened on the first available backing-store channel; if all the channels are in use, the procedures stops with a diagnostic. After printing the required information, the file is closed.

Polynomials of low degree can be fitted by ordinary linear regression, estimating effects of terms X, X**2, X**3, and so on for a variate X. However, it is sometimes preferable to arrange that successive polynomial terms are orthogonal to each other; certainly, there are likely to be numerical problems with polynomials of degree five or more, if they are not orthogonal. ORTHPOL calculates orthogonal polynomials up to a specified maximum degree from a given variate. The orthogonalization can be weighted by specifying a variate of weights.

Options: MAXDEGREE, WEIGHTS. Parameters: X, POLYNOMIAL.

A variate in the X parameter can be restricted: the restriction is transferred to the calculated polynomials, and to the weight variate if specified.

PAIRTEST can be used to test all pairwise differences in every situation in which a vector of estimates and a corresponding variance-covariance matrix are available. PAIRTEST is particularly useful for tests of all pairwise differences of slopes after fitting a model with an interaction between a factor and a variate. In most other situations procedure RPAIR will be more suitable.

All pairwise differences of entries in ESTIMATES with variance-covariance matrix VCOVARIANCE are calculated and tested. The results of these tests can be saved in symmetric matrices DIFFERENCES, SED, TVALUES and TPROBABILITIES. The matrices are labeled by text vector LABELS or, if LABELS is unset, by the numbers 1, 2, 3...

PRINT controls the output of PAIRTEST. The t-probabilities are based on DF degrees of freedom; by default, if DF has not been set, normal probablitities are calculated. Option SORT controls whether the estimates on the diagonal of DIFFERENCES are sorted in ascending order. The other output is sorted accordingly.

Options: PRINT, DF, SORT.

The variate ESTIMATES and the text LABELS can be restricted; the analysis is restricted according to restrictions on ESTIMATES. The lengths of the unrestricted vectors ESTIMATES and LABELS must be identical.

The configurations of points are specified using the DATA parameter. This supplies a pointer containing a matrix with the data for each configuration. The PROTATE option controls the output from the individual Procrustes rotations, and the PPCO option controls that from the principal coordinate analysis. There are M´ (M-1)/2 Procrustes rotations so, by default, PROTATE=* to suppress any output. The SCALING and STANDARDIZE options control the way in which the Procrustes rotations are carried out, using the SCALING and STANDARDIZE options of ROTATE. However, the combination of SCALING=yes and STANDARDIZE=centre should not be used, because then the results will be dependent on the order of the input matrices.

The LRV and CENTROID parameters can be used to save results from the principal coordinates analysis, and the DISTANCES parameter can be used to save the symmetric matrix of the residual sums-of-squares from the Procrustes analyses.

Options: PROTATE, PPCO, SCALING, STANDARDIZE.

Parameters: DATA, LRV, CENTROID, DISTANCES.

The pairwise Procrustes rotations are performed using the ROTATE directive, and the residual sums-of-squares are stored in a symmetric matrix of order M. This matrix is then used as input to a principal coordinate analysis, performed using the PCO directive on a suitably transformed copy of the matrix.

PDESIGN allows the treatment combinations allocated to each plot in a design to be displayed as tables, classified by the block factors.

The combinations are represented using the levels of the treatment factors. If any factor also has labels these are printed alongside the levels, as a key, after the tables. The levels are printed in formats that are determined automatically in a way that avoids wasted space or unnecessary decimal places. The block factors are obtained from the block structure of the design, which can be specified explicitly using the BLOCKSTRUCTURE option; otherwise PDESIGN will use any structure that has already been defined by a BLOCKSTRUCTURE statement earlier in the job. Similarly, the treatment factors are obtained either from the TREATMENTSTRUCTURE option of the procedure, or from an earlier TREATMENTSTRUCTURE statement.

If the display produced by the procedure is unsuitable, printing can be suppressed by setting option PRINT=* (by default PRINT=design), and the tables of treatment levels can be saved for printing outside the procedure by setting the TABLES option to a pointer. This will be returned with an element for each treatment factor, pointing to a table classified by the block factors and storing the tabulated levels of the treatment.

Options: PRINT, BLOCKSTRUCTURE, TREATMENTSTRUCTURE, TABLES. Parameters: none.

The FCLASSIFICATION directive is used to form lists of factors from the block or treatment formulae and, if the block factors do not supply a unique combination of levels for every unit of the design, procedure AFUNITS is used to form a factor to index the units with each combination. Each treatment factor is then copied into a variate and TABULATE is used to put the values into a table classified by the block factors. Numbers of decimal places for printing the factor levels are determined using the DECIMALS procedure.

PERCENT allows you to express the body of a table as percentages of the values in one of its margins. The table is specified using the OLDTABLE parameter. A table to store the new values can be specified using the NEWTABLES parameter, otherwise these replace the values of the original table. The margin is indicated by listing the factors that define it using the CLASSIFICATION option; the default is the final margin (the grand total, or grand mean etc). If the original table has no margins, option METHOD defines how these are to be calculated; the default is to form margins of totals. The values originally in the margin will be left unchanged. If you would prefer these to be replaced by values of 100%, you should set option HUNDRED=yes.

Options: CLASSIFICATION, METHOD, HUNDRED. Parameters: OLDTABLE, NEWTABLE.

If the OLDTABLE has no margins and contains no missing values, these are formed by the MARGIN directive. Alternatively, if there are missing values, margins other than variances can be formed using TABULATE. CALCULATE is then used to put the required margin into a table classified just by the factors that define the margin. The original table is divided by the marginal table and multiplied by 100 to give the required percentages. If option HUNDRED=no, the same operations are done on a dummy table that originally contains random numbers; for this table, values of 100 should occur only in the margin. Thus by using a logical test in which the values of the dummy table are compared with 100, the marginal values of the original table can be put back into the margin of the final table. The random numbers are generated using a specially written procedure URANDOM in case the Genstat random number generator is already in use in the program that called PERCENT.

PERIODTEST gives periodogram-based tests for departure from white noise in a set of time series. The series are supplied in a list of variates, using the SERIES parameter. The LENGTH option can specify that only part of each series is to be used, using either a scalar N to indicate that the first N values are to be used, or a variate of length two, holding the values of the first and last units of the required subseries. This may be used to eliminate missing values, which are otherwise not permitted.

The mean-adjusted periodogram is calculated for each series using FOURIER, and can be saved using the PERIODOGRAM parameter. The maximum periodogram ordinate test, Fisher's g-test and the Kolmogorov-Smirnov test on the cumulative periodogram are calculated using the standard formulae (Priestley 1981).

The output for each series consists of the value of the maximum periodogram ordinate (after scaling by the length of the analysed series), the frequency at which this maximum occurs (expressed as the unit number in the PERIODOGRAM variate, i.e. if the maximum occurs at w = 2Pj/N, then j is given), and the probability of exceeding this maximum; the ratio of the maximum to the total of the periodogram ordinates (Fisher's g), and the probability of exceeding this; and the Kolmogorov-Smirnov D statistic based on the maximum deviation of the cumulative periodogram from the line y=x.

Option: LENGTH. Parameters: SERIES, PERIODOGRAM.

The SERIES may not be restricted; restriction of the input series to a contiguous set of units may be achieved by use of the LENGTH parameter.

The procedure allows the calculation of PLS1 and PLS2 models with cross-validation to assist in the determination of the correct number of dimensions to include in the model. By setting the NGROUPS option the data are randomly divided into a number of groups; samples in each group are then modelled from the remaining samples only. The sum of squares of differences between these "leave out predictions" and the observed values of Y are called PRESS. Many tests of significance for determining the correct number of dimensions are based on comparing values of PRESS for PLS models of varying rank. Values of PRESS are used in the procedure to perform Osten's (1988) test of significance and may also be plotted out in a scree diagram. In addition to the factor scores, factor loadings and residuals, the procedure also calculates a leverage measure (Martens & Naes 1989 page 276) and a single linear combination of the X variables (ESTIMATES) which summarises the entire PLS model.

X or Y variates.

To use a PLS model to make predictions from new observations on the X variables, two methods are available. Either the user may do this manually by using the model as specified in the estimates matrix, or the new X data may be specified beforehand as the pointer to variates XPREDICT and the corresponding predicions obtained as YPREDICT.

Output from the PLS procedure can be selected using the following settings of the PRINT option.

The default settings are estimates, xpercent, ypercent, scores, xloadings, yloadings, ploadings.

The data for PLS are supplied using the X and Y parameters, as pointers to variates containing the columns of the X and Y matrices. Other parameters allow output to be saved in appropriate data structures.

Options: PRINT, NROOTS, YSCALING, XSCALING, NGROUPS, SEED, LABELS, PLABELS.

It is usual to centre all variables prior to a PLS analysis, the procedure will automatically do so even if the XSCALING/YSCALING options are not set. On exit from the procedure the variates pointed to by X and Y are unchanged.

The procedure will work with restricted variates, fitting a PLS model to the subset of objects indicated by the restriction. If there are different restrictions on different data variates then these restrictions will be combined and the analysis performed on the subset of samples that is common to all the restrictions. Note that the unrestricted length of all of the data variates must be the same and the number of samples in the common subset must be at least three. Any restrictions on a text supplied for the LABELS option or a factor for the SEED option will be ignored. On exit from the procedure all the data variates, and if supplied the SEED factor and LABELS text, will all be returned restricted to the common subset of samples. Output data structures that correspond to the samples (i.e. XSCORE, YSCORE, FITTED, LEVERAGE, YRESIDUAL and XRESIDUAL) will also be returned restricted to the common subset, and missing values will be used for those values that have been restricted out.

When restricted data are supplied and LABELS are also given then the appropriate subset of labels will be appear in the output; if LABELS are not defined then default labels reflecting the position of the restricted data in the unrestricted variate will be used instead.

No restrictions are allowed in the variates supplied in the XPREDICT parameter or the PLABELS option.

Procedures RPAIR and PAIRTEST produce a symmetric matrix of two-sided t-probabilities for tests of all pairwise differences of estimates. PPAIR displays this matrix at a specified level of significance in two compact schematic diagrams. This is especially useful when the number of estimates is large.

Input to PPAIR is a symmetric matrix TPROBABILITIES containing probabilities of the set of pairwise comparisons. The level of significance can be set by the PROBABILITY option. A common level is specified by a scalar, while a symmetric matrix specifies a level for each comparison separately (which may be useful for some multiple comparison methods). Output is labelled by the row labels of TPROBABILITIES. If parameter DIFFERENCES is set to a symmetric matrix the diagonal of this matrix is printed alongside these labels (with number of decimals as defined at declaration of DIFFERENCES). This is especially useful if DIFFERENCES is saved by RPAIR or PAIRTEST because it then contains the estimates on the diagonal. DIFFERENCES can also be set to a variate or table. Alternatively the output can be labelled by specifying parameter LABELS.

PRINT controls which diagram is printed. PRINT=items produces a diagram which should be read line by line. Each item (represented by a letter) is followed by those items (again represented by letters) not significantly different from that item. When there are more than 52 items, letters are repeated. PRINT=groups is only useful when the TPROBABILITIES are sorted in a sensible order, for example by specifying SORT=yes in RPAIR or PAIRTEST. This produces a diagram in which items followed by a common letter are not significantly different. Such items are said to form a homogeneous group. This is similar to common underlining of items with non-significantly different estimates. In constructing this diagram the philosophy of multistage testing is followed, see the Method section.

Options: PRINT, PROBABILITY.

Parameters: TPROBABILITIES, DIFFERENCES, LABELS.

The construction of the diagram for PRINT=groups is as follows. First the difference between the first and last item of the complete set of n items is checked for significance. Then the first and last item of all subsets of n-1 consecutive items are checked, followed by all subsets of n-2 items, and so on. If non-significance is found between the first and last item of a subset, all items of the subset are said to form a homogeneous group and they receive the same letter. This is only sensible when the TPROBABILITIES are sorted according to the estimates. The diagram only consists of homogeneous groups which are not a part of a larger group.

It is obvious that items in a homogeneous group can be significantly different. This is not displayed in the diagram, although a message is printed if this occurs. If there are no significant differences within homogenous groups, both diagrams essentially contain the same information; PRINT=groups then gives a more concise representation.

Restrictions on DIFFERENCES and LABELS are ignored.

PREWHITEN provides filtering of time series data prior to spectral analysis. Parameters SERIES and FILTERED specify the input and output series, respectively. The filtered series y is given by

where x is the input series. (Thus q = 1 would give first differencing.) The value of q is specified by the PHI option; the default value of q=0.99 is often suitable. Alternatively, an empirical approach is to use the value

Option: PHI. Parameters: SERIES, FILTERED.

The procedure uses the FILTER directive with two TSMs defined as follows:

The behaviour is as for the FILTER directive.

In probit analysis, we are interested in estimating the equation of that line. This can be done by perfoming an experiment in which there are several batches of subjects, each of which is given a different dose of the stimulus. The data then consists of a variate indicating the number of subjects that responded out of each batch, a variate to show the dose given to each batch, and a final variate for the total numbers of subjects in the batches; these are specified by parameters Y, DOSE and NBINOMIAL, respectively. The NBINOMIAL parameter can be omitted if the total numbers cannot be measured, as in some fumigation experiments ("Wadley's problem"; see for example Finney 1971, pages 202-8).

The PRINT option controls printed output: model details of the model that has been fitted,

summary summary analysis-of-variance table, estimates parameter estimates and standard errors, correlations correlations between parameter estimates, and fittedvalues fitted values and residuals. By default, PRINT=mode,summ,esti,fitt.

The TRANSFORMATION option allows other transformations to be selected. Putting TRANSFORMATION=logit requests a logit transformation:

Sometimes, subjects may respond even in the absence of any dose. For example, with some short-lived insects, some would have died simply from natural causes during the period of the experiment. By setting option MORTALITY=estimate this natural mortality can be included in the model and estimated. Similarly, there may be subjects that will not respond, no matter how high the dose. Setting option IMMUNITY=estimate will include and estimate a parameter for natural immunity.

It is also often of interest to fit study the way in which the model varies for different groups of subjects. For example, there may be groups of batches of subjects, each of which is given a different drug. The GROUPS option should then specify the group to which each batch of subjects belongs, and option SEPARATE indicates which parameters of the model (slope, mortality, and/or immunity) should have separate estimates. If SEPARATE is left at its default value, parallel lines will be fitted with identical values for any estimates of mortality and immunity.

The final option, LD, can request the estimation of one or more lethal (or effective) doses, specifying a scalar if there is just one, or a variate if there are several. The LD50 value (that is the dose at which 50% of the population would respond) is always printed as one of the parameters of the fitted line.

The model is fitted using the FITNONLINEAR directive, and the final two parameters, INITIAL and STEPLENGTHS, allow initial values and steplengths to be specified for the optimization. The order of parameters is: LD50(s), slope(s), mortality parameters (if any), and immunity parameters (if any). Parameter estimates, fitted values, residuals, and so on, can be saved after running the procedure, by using the RKEEP directive in the usual way.

Options: PRINT, TRANSFORMATION, MORTALITY, IMMUNITY, GROUPS, SEPARATE, LD.

Parameters: Y, DOSE, NBINOMIAL, INITIAL, STEPLENGTHS.

Initial values are obtained, if necessary, using the Genstat facilities for generalized linear models, ignoring any mortality or immunity. Expressions specifying the model are defined in sets of nested IF-blocks, taking account of the settings for example of TRANSFORMATION and GROUPS. The fitting is carried out by the FITNONLINEAR directive, and any extra LD values are estimated using RFUNCTION.

Probit analysis can also be performed using the Genstat facilities for generalized linear models, but these do not cover Wadley's problem, nor allow for natural mortality and immunity. The results obtained from this procedure may differ slightly from those that are obtained (where possible) from the use of GLMs (by the FIT directive) or using the WADLEY procedure, due to the use here of maximum likelihood for the fitting, rather than iterative weighted linear regression.

The Y variate, the DOSE variate, or the GROUPS factor can be restricted to indicate that the model is to be fitted only to a subset of the units.

This procedure takes as input two variates containing the coordinates of a spatial point pattern (specified by the X and Y parameters) and returns the coordinates of either a bounding or a surrounding box, according to the setting of the METHOD option. The default, METHOD=bounding, provides a bounding box, defined as the smallest rectangle such that all the events in the spatial point pattern lie inside the box or on its boundary. The coordinates of the bounding box are the coordinates of its four corners, in the order lower left, lower right, upper right, and upper left. The surrounding box (METHOD=surrounding) is a rectangle which contains all the points. It is obtained by extending the vertical and horizontal edges of the bounding box by specified fractions of the range of values in X and Y, respectively. The parameters XFRACTION and YFRACTION can be used to specify the proportional extension required in each direction; the default value of both parameters is 0.1. The coordinates of the surrounding box are the coordinates of its four corners in the order lower left, lower right, upper right, upper left. The coordinates of the bounding or surrounding box can be saved using the parameters XBOX and YBOX.

Printed output is controlled by the PRINT option. The default setting of summary prints the coordinates of the bounding or surrounding box under the headings XBOX and YBOX.

Option: PRINT.

Parameters: Y, X, YBOX, XBOX, YFRACTION, XFRACTION.

A procedure PTCHECKXY is called to check that X and Y have identical restrictions. The minimum, maximum and range of the horizontal (X) and vertical (Y) coordinates are then calculated. For a bounding box, the coordinates are calculated as (min(X), min(Y)), (max(X), min(Y)), (max(X), max(Y)), (min(X), max(Y)). For a surrounding box the coordinates are

(min(X) - XFRACTION ´ range(X), min(Y) - YFRACTION ´ range(Y)),

(max(X) + XFRACTION ´ range(X), min(Y) - YFRACTION ´ range(Y)),

(max(X) + XFRACTION ´ range(X), max(Y) + YFRACTION ´ range(Y)),

(min(X) - XFRACTION ´ range(X), max(Y) + YFRACTION ´ range(Y)).

If X and Y are restricted, only the subset of values specified by the restriction will be included in the calculations.

A polygonal region of two-dimensional space is represented in Genstat by the coordinates of a sequence of points which define the boundary of the polygon with the last point implicitly connected to the first point. If the first and last pairs of coordinates are the same then the polygon is said to be closed, otherwise it is open. Sometimes it is necessary to work with a closed polygon, for example, when drawing a polygon onto a graphics device as a series of line segments. This procedure takes as input a set of coordinates which define a polygon. The parameters OLDXPOLYGON and OLDYPOLYGON specify variates containing the coordinates. The output of the procedure is a closed polygon, which is identical to the input polygon if it is already closed and otherwise consists of the input polygon with the first pair of coordinates repeated at the end. The coordinates of the closed polygon may be saved using the parameters NEWXPOLYGON and NEWYPOLYGON.

Printed output is controlled by the PRINT option. The default setting of summary prints the horizontal and vertical coordinates of the closed polygon under the headings NEWXPOLYGON and NEWYPOLYGON.

Option: PRINT.

Parameters: OLDYPOLYGON, OLDXPOLYGON, NEWYPOLYGON, NEWXPOLYGON.

A procedure PTCHECKXY is called to check that OLDXPOLYGON and OLDYPOLYGON have identical restrictions. It then checks whether the first and last pairs of coordinates are the same. If they are, the DUPLICATE directive is used to copy OLDXPOLYGON to NEWXPOLYGON and OLDYPOLYGON to NEWYPOLYGON. If they are different, NEWXPOLYGON and NEWYPOLYGON are declared as variates with one more value than their old counterparts, and the EQUATE directive is used to copy the values from OLDXPOLYGON to NEWXPOLYGON and OLDYPOLYGON to NEWYPOLYGON (so that the first element of each old variate is repeated at the end of the corresponding new one).

If OLDXPOLYGON and OLDYPOLYGON are restricted, only the subset of values specified by the restriction will be included in the calculations.

A comprehensive account of methods for analysing point processes is given by Cox & Lewis (1966). PTDESCRIBE implements many of the test and summary statistics they give and should be used in conjunction with the text for a full discussion of the motivation and context of their use. All equations referred to below are from Cox & Lewis (1966).

The DATA variate may contain either the times at which events occur, the intervals between events, or a sequence of 0's and 1's, with 1's indicating the times of events on an integer time scale. The option REPRESENTATION specifies which of these is used. If REPRESENTATION=time and the process is measured from some time other than zero, the initial time should be given in the parameter START. Otherwise the START time is assumed to be zero. The first interval is taken to lie between the START time and the first event. If the process is observed beyond the last event, the total duration of the process should be given in the parameter LENGTH. Checks are carried out on START, LENGTH and the length of each interval, and the procedure terminates if these are inconsistent. If REPRESENTATION=time, the DATA variate may be restricted, facilitating the analysis of truncated or thinned point processes.

If SAVE is set, time and interval are saved, together with summary interval or second order statistics specified by SELECTION as detailed below. SAVE sets up a pointer, with each element labeled by the name of the relevant statistics saved. For example, if SAVE=clstats, then the intervals between the events will be saved in clstats['interval'].

The option SELECTION can be used to obtain any combination of eight available analyses, with the PRINT and GRAPHICS options controlling the output. The default setting is SELECTION=interval, while SELECTION=all gives all eight analyses. In what follows, the number of events is denoted by N and the variate carrying the times of events by time. The rate of a point process is calculated as the reciprocal of the average interval length.

interval - plots data and summarises the interval distribution

trend - tests for trend in the process

poisson - tests whether the point process is Poisson

icorrelation - autocorrelations for the interval sequence.

ispectrum - periodogram for the interval process

cspectrum - periodogram for the count process

cintensity - intensity function for the counting process

vtcurve variance-time curve V(t) and index of dispersion I(t)

Options: PRINT, SELECTION, REPRESENTATION, GRAPHICS.

Parameters: DATA, START, LENGTH, CITAU, VTTAU, SAVE.

The procedure tests of whether a point process is a Poisson process and calculates summary statistics in the time and frequency domains for a point process following Cox & Lewis (1966). Most statistics are obtained using CALCULATE, with FOURIER being used for ispectrum and CORRELATE for the pre-adjusted autocorrelations.

DATA may be restricted only if REPRESENTATION=time, in which case only the units not excluded by the restriction are involved in the analysis.

PTREMOVE uses the DREAD directive to delete points from a spatial point pattern. The coordinates of the existing points must be supplied using the parameters OLDX and OLDY. These points will be plotted on the current graphics device using DPTMAP with a pen setting of SYMBOLS=1. The WINDOW option may be used to specify the graphics window to use for the plot.

The operation of DREAD may vary slightly from one system to another. The Users' Note supplied with Genstat explains how to read points and terminate input on specific devices. The usual method for reading points is to click the left mouse button at the required position. The usual way to terminate input is to click the right mouse button. The points read using DREAD will be echoed using a pen setting of SYMBOLS=2. The coordinates of the new spatial point pattern containing the original points minus any points which have been deleted may be saved using the parameters NEWX and NEWY.

Printed output is controlled using the PRINT option. The settings available are monitoring (which prints the coordinates of the points to be deleted) and summary (which prints the coordinates of the new pattern consisting of the original points minus any that have been deleted under the headings NEWX and NEWY). The default setting is for both monitoring and summary.

Options: PRINT, WINDOW.

Parameters: OLDY, OLDX, NEWY, NEWX.

A procedure PTCHECKXY is called to check that OLDX and OLDY have identical restrictions. DPTMAP is then used to draw a map of the original point pattern. The DREAD directive is used to read the coordinates of points to be deleted. Finally, the coordinates for the deleted points are removed from the original points using the SUBSET procedure and the coordinates of the undeleted points are stored in new variates.

If OLDX and OLDY are restricted, only the subset of values specified by the restriction will be included in the calculations.

Thus the quantile for proportion 0.5 is the median; for 0.0 it is the minimum and for 1.0 the maximum of the sample. By default, QUANTILE produces the five quantiles called the "five-number summary" of a sample, corresponding to the proportions 0.0, 0.25, 0.5, 0.75, 1.0. The option PROPORTION can be set to a scalar or variate to request other single quantiles or sets of quantiles. By default, QUANTILE prints the statistics, but this can be suppressed by setting option PRINT=*. The quantiles can be stored in a variate using the parameter QUANTILE.

Options: PRINT, PROPORTION. Parameters: DATA, QUANTILE.

If the DATA variate is restricted, the quantiles are formed only using the units that are not restricted out. The PROPORTION and QUANTILE variates must not be restricted.