ABIVARIATE produces a bivariate analysis of variance with a graphical representation of the results, as described by Dear & Mead (1983, 1984). The procedure was developed from a Genstat 4 macro, further information about which is given by Poultney & Riley (1986), and is intended primarily for data from intercropping experiments. The variates to be analysed (specified by parameters Y1 and Y2) are measurements, usually yields, taken on the two crops. The final parameter, TITLE, defines a title for the graph.

The procedure will work for any of the designs that can be analysed by ANOVA, except that there must be no pseudo-factors. Option TREATMENTSTRUCTURE defines the treatment formulae for the analysis, and the block formula is defined by the BLOCKSTRUCTURE option. BLOCKSTRUCTURE can be omitted if there is a single error stratum (i.e. the analysis is of a completely randomized design). The FACTORIAL option controls the number of factors in each treatment term, as in the ANOVA directive.

First of all, ABIVARIATE calculates a univariate analysis of variance for each of the variates Y1 and Y2, with output controlled by the APRINT option. The settings are the same as those in the ANOVA directive; by default APRINT=aovtable.

Output from the bivariate analysis of variance, which follows, is controlled by the PRINT option. The setting error generates the error summary statistics from the bivariate analysis:

Error Sum of Products, Variances after Adjustment for Covariance, Correlation Coefficient between Y1 and Y2, Radius of Standard Errors, Radius of Confidence Regions, and Radius of Non-Significance Regions. The setting treatment produces the following statistics for each treatment term estimated within the specified error stratum: Treatment Sum of Products, Wilks' Lambda, Bivariate F-Statistic.

The stratum from which the means (and other information) are to be taken is defined by STRATUM option; if this is omitted, the lowest stratum is used. The significance level to use in the calculation of confidence regions is defined by the PROBABILITY option; by default this is 0.95.

The TERM option specifies a treatment term whose means are to be represented graphically. The means are plotted on axes transformed to allow for the variability in, and the correlation between, each crop variate. The plotting symbols can be defined as a factor or text using the option LABELS. Alternatively they will be taken to be the first n values of the series A to Z, a to z where n is the number of means to be plotted. The graph can be either line printer or high resolution, the default being high resolution. The external axes of a high-resolution graph can be suppressed by setting STYLE=none.

Problems arise in situations where the table of means to be plotted is incomplete; this can occur when a whole factor level is restricted out, or where the treatment structure is nested within a control. The length of the vector LABELS is calculated as the number of cells in the table, including missing values. If LABELS is declared, it must have length equal to the dimension of the table otherwise a fault will occur. Similarly, the calculation of the radius statistics is based on the assumption that the table of means is complete and has equal replication. These values, if printed, would be incorrect for a table with missing cells and so are suppressed. They can be calculated by hand as shown by Dear & Mead (1983).

Parameters: Y1, Y2, TITLE.

Variates Y1 and Y2 can be restricted, however this restriction must be identical for the two variates. Some problems may occur when whole levels of factors are restricted out leaving empty cells in the table of means to be plotted (see above).

Procedure AFALPHA generates the treatment, replicate, block and unit factors for an alpha design. The design can be printed by setting option PRINT=design, and the factors can be saved using the parameters TREATMENTS, REPLICATES, BLOCKS and UNITS. The generating array for the design must be specified as a k ´ r matrix using the GENERATOR parameter, and the number of levels of the treatment factor can be defined by the LEVELS parameter. If LEVELS is omitted, AFALPHA will see whether the TREATMENTS parameter has been set to a factor whose levels have already been defined; if not, AFALPHA will set LEVELS to the scalar value v = s ´ k. By default the design is unrandomized, but randomization can be requested by setting the SEED parameter.

Option: PRINT.

Parameters: GENERATOR, LEVELS, SEED, TREATMENTS, REPLICATES, BLOCKS, UNITS.

The INITIAL parameter specifies the initial blocks. If the design is to be generated from a single initial block, INITIAL should be set to a variate containing the levels corresponding to the treatments concerned; if there are several, the appropriate variates should be placed into a pointer. Similarly the INCREMENT parameter, which specifies the increment to be used, should be set to a scalar if the same increment is to be used for all the initial blocks, otherwise to a pointer of scalars. The levels of the treatment factor are specified by the LEVELS parameter and the SEED parameter allows the design to be randomized. As is customary in Genstat, if LEVELS is set to a scalar the levels are assumed to be represented by the integers 1 upwards, but LEVELS can be set to a variate to specify other numbers. LEVELS can be omitted if the TREATMENTS parameter is used to supply a factor to store the treatments, provided the levels of that factor have already been defined outside the procedure. The factors for blocks and units within blocks can be saved similarly by the BLOCKS and UNITS parameters respectively. The design can also be printed, by setting option PRINT=design.

Option: PRINT. Parameters: INITIAL, INCREMENT, LEVELS, SEED, TREATMENTS, BLOCKS.

AFORMS prints data forms which can be used to record data from an experimental design. Several measurements can be recorded, in separate columns across the page, and space is provided for a row of values for each unit of the design. The block factors to label the units can be supplied by setting the BLOCKSTRUCTURE option to the block formula of the design. If this is not set AFORMS will use the formula, if any, defined previously by the BLOCKSTRUCTURE directive.

The units can also be labelled with the treatments that have been used in the design, by setting the TREATMENTSTRUCTURE option to the appropriate treatment formula. However, to guard against bias, experimenters will often prefer not to know which treatments were applied to each unit when recording the results, so if this is omitted no treatment information is included.

The LABEL parameter supplies the column label to identify each column of measurements, and the FIELDWIDTH parameter can specify the width of the column. By default, a single line is provided for row of measurements but this can be increased using the NLINES option.

Options: BLOCKSTRUCTURE, TREATMENTSTRUCTURE, NLINES.

Parameters: LABEL, FIELDWIDTH.

AFORMS uses the standard Genstat directives for printing and manipulation.

AFORMS needs to use RESTRICT in order to organise the labelling of the forms, and so any existing restrictions will be cancelled.

When analysing experimental data in Genstat, it is usually unnecessary to specify the final stratum of the design. For example ANOVA, as explained on page 417 of the Genstat Reference Manual, will set up an internal factor called *Units* to define (along with the other block factors) the final stratum. However, it is then impossible, for example, to put the residuals into a table classified by the block factors, or to tabulate the treatment levels according to the block structure. Thus AFUNITS takes a set of block factors (specified in either a pointer or a model formula by the BLOCKSTRUCTURE option) and sets up the necessary extra factor, which is then returned by the UNITS parameter.

Option: BLOCKSTRUCTURE. Parameter: UNITS.

The FCLASSIFICATION and FORMULA directives are used, if necessary, to form a list of factors from the block formulae and then the factor values are set up using the standard Genstat facilities for calculations and manipulation.

The formation of an alpha design requires a generating array, as explained in the description of procedure AFALPHA, and the effectiveness of the design that is produced will be very dependent on the choice of array. Procedure AGALPHA selects an appropriate array from those presented by Patterson, Williams & Hunter (1978) and Williams (1975), and then calls AFALPHA to generate the design.

AGALPHA is easiest to use interactively. It then asks questions to determine the necessary information to select the generating array: for example, the number of treatments, the number of blocks per replicate and so on. The parameters allow you to anticipate questions, or to define all the necessary information if you want to use AGALPHA in batch.

The number of treatments can be defined using the LEVELS parameter. Similarly, the NREPLICATES and NBLOCKS parameters define the number of replicates and the number of blocks per replicate. If the number of blocks per replicate is greater than or equal to the number of units (or plots) per block, generators are available for either two, three or four replicates; otherwise there can only be two. The SEED parameter allows you to specify a seed to be used to randomize the design. In batch the default seed is zero, to suppress randomization. If you do not set SEED when running interactively AGALPHA will ask for a seed, and again a zero value suppresses any randomization. The remaining parameters, TREATMENTS, REPLICATES, BLOCKS and UNITS, allow you to specify identifiers for the treatment, replicate, block-within-replicate and unit-within-block factors. If these are not specified in a batch run, AGALPHA will use identifiers that are local within the procedure and thus lost at the end of the procedure. If you are running interactively, AGALPHA will ask you to provide identifiers, and these will remain available after AGALPHA has finished running.

AGALPHA has a PRINT option which can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGALPHA will ask whether or not you wish to print the design.

Option: PRINT.

The QUESTION directive is used to obtain the necessary details of the design. Procedure AFALPHA is then called to generate the design.

Incomplete block designs occur when the units in an experiment need to be divided into blocks that are not large enough to contain a unit for every treatment. In a balanced incomplete block design the contents of the blocks are arranged so that every pair of treatments occurs in an equal number of blocks. All comparisons between treatments are thus made with equal accuracy, so the design is balanced and, in particular, can be analysed by ANOVA.

AGBIB provides a range of balanced incomplete block designs, and is easiest to use interactively. It then asks questions to determine the necessary information to form the design. The options and parameters allow you to anticipate questions, or to define all the necessary information if you want to use AGBIB in batch.

First of all, AGBIB asks you to select the design. It lists those available, specifying the number of treatments, the number of blocks, the size of each block and the number of blocks containing each pair of treatments. Alternatively, if you set the LEVELS parameter to the required number of treatments, the NBLOCKS parameter to the number of blocks and the NUNITS parameter to the number of units per block, AGBIB will select the design (if available) automatically.

The SEED parameter allows you to specify a seed to be used to randomize the design. In batch the default seed is zero, to suppress randomization. If you do not set SEED when running interactively AGBIB will ask for a seed, and again a zero value suppresses any randomization.

Parameters TREATMENTS, BLOCKS and UNITS, allow you to specify identifiers for the treatment, the block and unit-within-block factors. If these are not specified in a batch run, AGBIB will use identifiers that are local within the procedure and thus lost at the end of the procedure. If you are running interactively, AGBIB will ask you to provide identifiers, and these will remain available after AGBIB has finished running.

The PRINT option controls printed output, with setting design to print a plan of the design, and catalogue to print a list of the available designs. By default, if you are running Genstat in batch, nothing is printed. If you do not set PRINT when running interactively, AGBIB will ask whether or not you wish to print the design, after it has been generated. Similarly the ANALYSE option governs whether or not AGBIB produces a skeleton analysis-of-variance table (containing just source of variation, degrees of freedom and efficiency factors). Again AGBIB assumes that this is not required if ANALYSE is unset in a batch run, and asks whether it is required if ANALYSE is unset in an interactive run.

Option: PRINT, ANALYSE.

Parameters: LEVELS, NBLOCKS, NUNITS, SEED, TREATMENTS, BLOCKS, UNITS.

AGBIB generates the designs from Hadamard matrices, as described by Hedayat & Wallis (1978). The QUESTION directive is used to obtain the necessary details of the design. The matrices are then recovered from a backing-store file and the standard Genstat manipulation directives are used to generate the design.

Box-Behnken designs are often used to study response surfaces. The design is usually formed to allow a quadratic response surface to be fitted. The factors are studied at three equally-spaced levels, below denoted by -1, 0 and 1. The construction uses a balanced incomplete block design to select successive sets of factors to be applied at all factorial combinations of -1 and +1, while other factors are held at 0. For example, with three factors A, B and C, the relevant balanced incomplete block design would have three blocks (A,B), (A,C) and (B,C). So the design would first have a section with A and B varying but C constant

then a section where B is held constant but A and C take all combinations of -1 and +1

and finally a section with A constant

In addition, there can be some "central points", where all the factors take the central value

The treatment factors are listed using the TREATMENTFACTOR parameter. If this is omitted in an interactive run, you will be asked how many factors you want and their names. The number of central points is specified by the NCENTRALPOINTS option; by default this is taken to be four. The LEVELS option can supply a variate to specify the outer treatment levels; the defaults are 1 and -1 (so the central point is at zero). The NCOMBINATIONS option defines the number of factors whose combinations of (outer) levels are to be varied at once. For the default of two, the relevant balanced incomplete block design is formed within AGBOXBEHNKEN. Other values can be supplied, but the corresponding balanced incomplete block design must be one of those obtainable from procedure AGBIB. You can find out the possibilities by putting

The SEED option allows you to specify a seed for randomization, with a default of zero indicating no randomization. The PRINT option can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGBOXBEHNKEN will ask whether or not you wish to print the design.

Options: PRINT, NCENTRALPOINTS, LEVELS, NCOMBINATIONS, SEED.

Parameters: TREATMENTFACTOR.

The QUESTION directive is used to obtain the necessary details of the design and this is then generated by the standard Genstat manipulation directives and procedure AGBIB.

The treatment factors are listed using the TREATMENTFACTOR parameter. If this is omitted in an interactive run, you will be asked how many factors you want and their names. The number of central points is specified by the NCENTRALPOINTS option; by default this is taken to be four. The LFACTORIAL option can supply a variate to specify the levels to be used in (a); the defaults are 1 and -1 (so the central point is at zero). Similarly, LSTAR specifies the levels for (b), which are taken, by default, to be the same as in (a). The star levels must, however, be equally spaced around the centre point. Option NSTARPOINTS defines how may replicates to have of each star point. The FRACTION supplies the denominator of a fractional design, if required for (a); the default of one indicates that a complete factorial design is to be used. The SEED option allows you to specify a seed for randomization, with a default of zero indicating no randomization. The PRINT option can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGCENTRALCOMPOSITE will ask whether or not you wish to print the design.

Options: PRINT, NCENTRALPOINTS, NSTARPOINTS, LFACTORIAL, LSTAR, FRACTION, SEED.

Parameters: TREATMENTFACTOR.

The QUESTION directive is used to obtain the necessary details of the design and this is then generated by the standard Genstat manipulation directives and procedure AGFRACTION.

Cyclic designs provide an effective way of assessing treatments using a block design where the blocks are each too small to hold all the treatments. In its simplest form, the cyclic method of generation starts with an initial block containing some subset of the treatments. This subset is represented by integers in the range 0...m-1 where m is the number of treatment levels. The second and subsequent blocks are then generated by successive addition modulo m of one to the numbers in the subset. Some designs have more than one initial block, and the increment need not be one. Further details of the method are given in the description of procedure AFCYCLIC.

The efficiency of the design depends very much on the choice of initial blocks. Procedure AGCYCLIC selects appropriate initial blocks from a repertoire obtained mainly from the program DSIGNX (Franklin & Mann 1986), and including designs from Davis & Hall (1969), Hall & Williams (1973) and John, Wolock & David (1972). It then calls AFCYCLIC to generate the design.

AGCYCLIC is easiest to use interactively. It then asks questions to determine the necessary information to form the design. In particular, it will tell you which block sizes are available for your chosen number of treatments. The options and parameters allow you to anticipate questions, or to define all the necessary information if you want to use AGCYCLIC in batch.

The first question, which can be anticipated by setting the METHOD option, determines the type of cyclic design. In addition to the standard cyclic designs, AGCYCLIC can also generate the cyclic change-over designs of Davis & Hall (1969) and the cyclic superimposed designs of Hall & Williams (1973). The change-over designs are used for trials in which subjects are given different treatments in different time periods; these thus have a crossed block structure subjects*periods. The extension in the cyclic superimposed design is that there are two treatment factors (each with the same number of levels); the design is intended to estimate their main effects but not their interaction.

The PRINT option controls whether AGCYCLIC prints a plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGCYCLIC will ask whether or not you wish to print the design, after it has been generated.

The number of treatments can be defined using the LEVELS parameter. Similarly, the NBLOCKS and NUNITS parameters define the number of blocks and the number of units per block (or the number of periods in a cyclic change-over design). The SEED parameter allows you to specify a seed to be used to randomize the design. In batch the default seed is zero, to suppress randomization. If you do not set SEED when running interactively AGCYCLIC will ask for a seed, and again a zero value suppresses any randomization.

Parameters, TREATMENTS, SUPERIMPOSED, BLOCKS and UNITS, allow you to specify identifiers for the treatment, the superimposed treatment (for a cyclic superimposed design), the block and unit-within-block factors. If these are not specified in a batch run, AGCYCLIC will use identifiers that are local within the procedure and thus lost at the end of the procedure. If you are running interactively, AGCYCLIC will ask you to provide identifiers, and these will remain available after AGCYCLIC has finished running.

Finally, the INITIAL parameter allows you to save the initial blocks, in a variate if there is only one, or in a pointer (to a list of variates) if there are several.

Option: PRINT METHOD.

The QUESTION directive is used to obtain the necessary details of the design. The initial blocks are then recovered from a backing-store file and procedure AFCYCLIC is called to generate the design.

AGDESIGN generates the factors and, if necessary, pseudo-factors required to define a generally balanced design. It also sets the block and treatment formulae (using the BLOCKSTRUCTURE and TREATMENTSTRUCTURE directives) to allow the design to be analysed by ANOVA. It can be accessed most conveniently through interactive Genstat design system, using the procedure DESIGN.

AGDESIGN relies upon a backing-store subfile that contains a repertoire of available designs, together with the information required to form them. FILENAME has a default file containing four subfiles.

If the default FILENAME is being used, the usual abbreviation rules are used to match SUBFILE with the names of the subfiles in the default file.

The backing-store file can be created by a procedure called FDESIGNFILE. This requires a data file, details of whose format can be obtained by setting option PRINT=filestructure when running FDESIGNFILE. You can thus provide additional files of designs which can be accessed by setting the FILENAME and SUBFILE options as appropriate.

AGDESIGN has two other options. The PRINT option can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGDESIGN will ask whether or not you wish to print the design. The other setting catalogue lists the designs in the subfile. Similarly the ANALYSE option governs whether or not AGDESIGN produces a skeleton analysis-of-variance table (containing just source of variation, degrees of freedom and efficiency factors). Again AGDESIGN assumes that this is not required if ANALYSE is unset in a batch run, and asks whether it is required if ANALYSE is unset in an interactive run.

The information required to select the design and give identifiers to its factors can be defined using the parameters of AGDESIGN. In an interactive run, AGDESIGN will ask questions to obtain any necessary information that is not supplied in this way; when running in batch, if any of the required information has not been specified, AGDESIGN will terminate with a warning message.

It is thus easiest to use AGDESIGN interactively. Then only the SUBFILE option need be set (assuming that you are happy to use the standard default design file), and the other information will be obtained by (clearly explained) questions. You need set the parameters only if you wish to anticipate some of the questions, or if you wish to use AGDESIGN in batch.

The DESIGN parameter can supply a variate whose first value selects the "type" of design: for example, in the LATTICE subfile, this would select between a 3´ 3 lattice, a 4´ 4 lattice, and so on. Some of these designs are available in several different "versions": for example, in lattice designs there are several ways of defining which treatment contrasts are to be confounded with blocks. If there is more than one version, the second and subsequent values of the DESIGN variate indicate which version, or versions, are required. These need not be distinct so, for example, you can replicate a basic design several times. If the variate has a single value, AGDESIGN will select the first version.

The TREATMENTFACTORS parameter can specify a pointer to supply identifiers for the treatment factors in the design. For example, if there are two factors you could define their identifiers to be A and B by forming the pointer Tf (say) with the statement

and then setting TREATMENTFACTORS=Tf. Alternatively, and more succinctly, you could put TREATMENTFACTORS=!p(A,B), where !p(A,B) is an unnamed pointer containing the required two identifiers. Similarly the BLOCKFACTORS parameter can specify a pointer to define the identifiers for the block factors in the basic design. If you have requested several versions, or several replicates, of the basic design AGDESIGN will also need a factor to represent the replicates. The identifier of this factor can be supplied using the REPLICATEFACTOR parameter. Partially balanced designs, such as lattices, will require pseudo-factors in the treatment formula to enable the design to be analysed by ANOVA. Identifiers can be supplied for these using the PSEUDOFACTORS parameter.

The UNITLABELS parameter can specify a variate to store a unique number to label each of the plots in the design. In the first replicate (or version) in the generated design, the variate contains the numbers one up to the number of plots per replicate. The second replicate (if any) contains these numbers plus the smallest power of ten greater than the number of plots per replicate, the third replicate contains the numbers plus twice this power of ten, and so on.

The SEED parameter allows you to specify a seed to randomize the design. In a batch run, this has a default of zero, to suppress randomization. If SEED is unset in an interactive run, you will be asked to provide a seed (and again a zero value will leave the design unrandomized).

Options: PRINT, ANALYSE, FILENAME, SUBFILE.

The QUESTION directive is used to obtain the details of the required design. The design is then generated using GENERATE and the other standard Genstat directives for calculation and manipulation.

AGFRACTION generates fractional factorial designs from stored keys & other information. It also sets the block and treatment formulae (using the BLOCKSTRUCTURE and TREATMENTSTRUCTURE directives) to allow the design to be analysed by ANOVA.

The procedure relies upon a backing-store file that contains a repertoire of available designs, together with the information required to form them. There is a standard file, used by default, but the FILENAME option allows you to specify another if you wish to form your own alternative file.

AGFRACTION has two other options. The PRINT option can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGFRACTION will ask whether or not you wish to print the design. Similarly the ANALYSE option governs whether or not AGFRACTION produces a skeleton analysis-of-variance table (containing just source of variation, degrees of freedom and efficiency factors). Again AGFRACTION assumes that this is not required if ANALYSE is unset in a batch run, and asks whether it is required if ANALYSE is unset in an interactive run. The FACTORIAL option sets a limit on the number of factors in the treatment terms in the analysis of variance; by default, this is two.

The information required to select the design and give identifiers to its factors can be defined using the parameters of AGFRACTION. In an interactive run, AGFRACTION will ask questions to obtain any necessary information that is not supplied in this way; when running in batch, if any of the required information has not been specified, AGFRACTION will terminate with a warning message.

It is thus easiest to use AGFRACTION interactively. Then all the information necessary to select and define the required design will be obtained by (clearly explained) questions. You need set the parameters only if you wish to anticipate some of the questions, or if you wish to use AGFRACTION in batch.

The number of levels of the treatment factors can be defined using the LEVELS parameter. The FRACTION parameter defines the denominator of the required fraction, and the NTREATMENTFACTOR parameter specifies how many treatment factors the design is to contain. Thus, for example,

For some of the designs it is possible also to allow a blocking factor (and you will be given details of what is feasible if you are running AGFRACTION interactively). The NUNITS parameter can then be used to define the number of units per block.

The SEED parameter allows you to specify a seed to be used to randomize the design. In batch the default seed is zero, to suppress randomization. If you do not set SEED when running interactively AGFRACTION will ask for a seed, and again a zero value suppresses any randomization.

The TREATMENTFACTORS parameter can specify a pointer to supply identifiers for the treatment factors in the design. For example, if there are two factors you could define their identifiers to be A and B by forming the pointer Tf (say) with the statement

and then setting TREATMENTFACTORS=Tf. Alternatively, and more succinctly, you could put TREATMENTFACTORS=!p(A,B), where !p(A,B) is an unnamed pointer containing the required two identifiers. The remaining parameters, BLOCKS and UNITS, allow you to specify identifiers for the block and unit-within-block factors. If the treatment, block or unit factors are not specified in a batch run, AGFRACTION will use identifiers that are local within the procedure and thus lost at the end of the procedure. If you are running interactively, AGFRACTION will ask you to provide identifiers, and these will remain available after AGFRACTION has finished running.

Options: PRINT, ANALYSE, FACTORIAL, FILENAME.

The QUESTION directive is used to obtain the details of the required design. The design is then generated using GENERATE and the other standard Genstat directives for calculation and manipulation.

TREATMENTFACTORS = factors or pointers

AGHIERARCHICAL forms orthogonal hierarchical designs: for example randomized blocks, split-plots, split-split-plots, and so on. The units of each stratum (or level of the hierarchy) are identified by a block factor: for example Replicates, Blocks, Plots, Subplots, Subjects &c.

AGHIERARCHICAL can be used either interactively or in batch. Interactively, there is no need to set any options or parameters - the procedure will ask questions to ascertain the necessary details of the design. The first question is to find out how many block factors (and thus strata) there are in the design. For a randomized block design there would be two, for example Blocks and Plots, defining strata for blocks and for plots within blocks. In a split-plot design there are three (for example Blocks, Wholeplots and Subplots) giving strata for blocks, whole plots within blocks and subplots within whole plots.

The questions involve each stratum in turn, and asking first for the name of the block factor to be used to identify the units of the stratum. Next it asks how many treatment factors are applied to the units of that stratum. In a randomized block design, there are no treatment factors applied to the blocks and one, or more, applied to the plots, whereas in a split-plot design treatments are applied to both the whole plots and the subplots. It then asks for the names of the treatment factors, and how many levels they are to have. Alternatively, if there are no treatments applied to the stratum, AGHIERARCHICAL asks how many levels the corresponding block factor should have - so, for example, it would how many blocks there should be in a randomized block or a split-plot design.

The parameters of AGHIERARCHICAL provide an alternative way of providing the details of the design. BLOCKFACTORS lists the block factors for the strata, TREATMENTFACTORS defines factors for the treatments applied to the units of the strata and LEVELS defines the levels of treatments and replication of block factors. For example

defines a randomized block design with three blocks, and a single treatment factor A (applied to the plots) with five levels. If there are several factors in a stratum, the identifiers should be placed into a pointer. For example,

for randomized block design with two treatment factors, A and B, both with two levels. Similarly, if the factors in a stratum have different numbers of levels, the LEVELS parameter may contain pointers.

defines A to have two levels and B to have three. The pointer can contain an extra element to indicate that there is to be replication (as well as treatments) in a stratum.

indicates that there are to be four replicates of the A and B combination on the plots of each block.

In an interactive run, AGHIERARCHICAL will ask about the treatment factors and the levels if these are not set. In a batch run all three parameters must be set.

The SEED option allows you to specify a seed to randomize the design. In a batch run, this has a default of zero, to suppress randomization. If SEED is unset in an interactive run, you will be asked to provide a seed (and again a zero value will leave the design unrandomized).

AGHIERARCHICAL has two other options. The PRINT option can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGHIERARCHICAL will ask whether or not you wish to print the design. Similarly the ANALYSE option governs whether or not AGHIERARCHICAL produces a skeleton analysis-of-variance table (containing just source of variation, degrees of freedom and efficiency factors). Again AGHIERARCHICAL assumes that this is not required if ANALYSE is unset in a batch run, and asks whether it is required if ANALYSE is unset in an interactive run.

Options: PRINT, ANALYSE, SEED.

Parameters: BLOCKFACTORS, TREATMENTFACTORS, LEVELS.

The QUESTION directive is used to obtain the details of the required design. The design is then generated using GENERATE and the other standard Genstat directives for calculation and manipulation.

AGMAINEFFECT generates designs for estimating main effects of factors with two levels, using a minimum number of experimental units; see Plackett & Burman (1946). The numbers of treatment factors for which designs are available can be printed by setting option PRINT=catalogue. They are, however, all expressible as 4n-1 for some integer n. The treatment factors are listed using the TREATMENTFACTOR parameter. If this is omitted in an interactive run, you will be asked how many factors you want and their names.

The basic design allows the main effects to be estimated, but has no residual degrees of freedom. This is fine if you merely want to screen the main effects to identify the largest. Otherwise you can generate a design for more factors than are needed, and then use the degrees of freedom of the unnecessary factors to provide the residual. Alternatively, if you set option FOLDED=yes, AGMAINEFFECT will include a "folded" replicate of the design: this is identical to the initial replicate except that the levels of the factors are swapped (level one instead of level two and vice versa). This particular arrangement has the advantage that no main effect is aliased with any first-order interaction.

The SEED option allows you to specify a seed for randomization, with a default of zero indicating no randomization. The PRINT option can be set to design to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set PRINT when running interactively, AGMAINEFFECT will ask whether or not you wish to print the design. Similarly the ANALYSE option governs whether or not AGMAINEFFECT produces a skeleton analysis-of-variance table (containing just source of variation, degrees of freedom and efficiency factors). Again AGMAINEFFECT assumes that this is not required if ANALYSE is unset in a batch run, and asks whether it is required if ANALYSE is unset in an interactive run.

Options: PRINT, ANALYSE, FOLDED, SEED.

Parameters: TREATMENTFACTOR.

The designs are based on the Hadamard matrices that are stored in the design system for forming balanced incomplete block designs (see procedure AGBIB). The QUESTION directive is used to obtain the necessary details of the design and this is then generated by the standard Genstat manipulation directives and procedure AGBIB.

Methods of constructing and randomizing neighbour-balanced designs for n treatments in either n blocks of n-1 plots or in n-1 blocks of n plots are described by Azais, Bailey and Monod (1993) together with generators for 3£ n£ 16 (other than for n=4 or 6 with n-1 blocks of size n, for which no designs are available). AGNEIGHBOUR uses these methods and generators, together with some further generators for blocks of n-1 plots formed using the method of Azais (1987).

AGNEIGHBOUR is easiest to use interactively. It then asks questions to determine the necessary information to form the design, and indicates the numbers of treatments for which designs are available. The options and parameters allow you to anticipate questions, or to define all the necessary information if you want to use AGNEIGHBOUR in batch.

The first question, which can be anticipated by setting the METHOD option, determines the type of design: n blocks of n-1 plots (METHOD=nblocks) or in n-1 blocks of n plots (METHOD=n_1blocks). The default in batch is n_1block. The PRINT option controls printed output, with setting design to print a plan of the design, and catalogue to print a list of the available designs. By default, if you are running Genstat in batch, nothing is printed. If you do not set PRINT when running interactively, AGNEIGHBOUR will ask whether or not you wish to print the design, after it has been generated.

The number of treatments can be defined using the LEVELS parameter. This can be set to zero to avoid constructing a design, as may be required if you merely wish to print the catalogue. The SEED parameter allows you to specify a seed to be used to randomize the design. If you do not set SEED when running interactively AGNEIGHBOUR will ask for a seed. In batch there is a default of 12345. Parameters TREATMENTS, BLOCKS and UNITS, allow you to specify identifiers to save the treatment, the block and unit-within-block factors. If these are not specified in a batch run, AGNEIGHBOUR will use identifiers that are local within the procedure and thus lost at the end of the procedure. If you are running interactively, AGNEIGHBOUR will ask you to provide identifiers and these will remain available after AGNEIGHBOUR has finished running. There are also parameters LEFTNEIGHBOUR and RIGHTNEIGHBOUR to allow you to save the treatments on the left and right neighbouring plots.

Some of the designs are such that each ordered pair of treatments occurs the same number of times as the left and right neighbours of some other treatment, the design is then said to be neighbour-balanced at distance 2. These designs have the further advantage that they are balanced if analysed with ANOVA with

Option: PRINT, METHOD.

The generation methods are described by Azais, Bailey & Monod (1993). The QUESTION directive is used to obtain the necessary details of the design and this is then generated by the standard Genstat manipulation directives.

Plots one- or two-way tables of means from ANOVA

AGRAPH plots tables of means. In its simplest form, if none of the options nor parameters are specified, AGRAPH plots a high-resolution graph for the first two-way table of means in the most recent ANOVA, or for the first one-way table if there were no two-way tables. Usually, each mean is represented by a point (using pens 1, 2, and so on for each level in turn of the second factor). However, with high-resolution plots, the METHOD option can be set to lines to draw lines between the points, or data to draw just the lines and then also plot the original data values. The GRAPHICS option controls whether a high-resolution or a line-printer graph is plotted; by default GRAPHICS=high.

The MEANS parameter allows the table of means to be specified explicitly. If this is not set, AGRAPH obtains the table automatically from ANOVA - from the save stucture specified by the SAVE option or from the most recent analysis if this too is unset. The BAR parameter can specify the size of an error bar (such as a standard error for differences between the means) to be plotted on the graph; if this is not specified, AGRAPH will calculate an average standard error of difference, if possible, for the body of the table from the SAVE structure (or from the most recent ANOVA).

The XFACTOR parameter indicates the factor against whose levels the means are plotted. With a two-way table, a separate line will be plotted for every level of a second factor. If the table is being obtained automatically, this second factor can be specified explicitly using the GROUPS parameter. If neither XFACTOR nor GROUPS are specified with a two-way table, AGRAPH will select the XFACTOR according to the following criteria (in decreasing order of importance): that the factor has no labels, that it has levels that are not the default integers 1 upwards, or that it has more levels than the other factor. Two-way tables can be obtained from ANOVA and plotted, even if the analysis contained only the main effects of the two factors; however, the lines will then all be parallel. The NEWXLEVELS parameter enables different levels to be supplied for XFACTOR if the existing levels are unsuitable. If XFACTOR has labels, these are used to label the x-axis unless option XFREPRESENTATION=levels.

Options: GRAPHICS, METHOD, XFREPRESENTATION, SAVE.

Parameters: XFACTOR, GROUPS, MEANS, BAR, NEWXLEVELS.

To set the various defaults (if necessary) AGRAPH uses the GET directive to obtain the ANOVA save structure, AKEEP to obtain the TREATMENT formula, FCLASSIFICATION to obtain the first two factors and GETATTRIBUTE to discover which have labels and so on.

The procedure AKAIKEHISTOGRAM has been designed as an alternative for the Genstat directive HISTOGRAM, for cases where the default settings are not optimal. Such cases may arise due to the following disadvantages of HISTOGRAM:

AKAIKEHISTOGRAM aims to avoid these disadvantages of HISTOGRAM. By default an "optimal" number of groups is determined using Akaike's Information Criterion.

Alternatively, own class limits can be specified using options LOWER and WIDTH instead of the option LIMITS of HISTOGRAM. In a FOR loop different values for the lower limit and/or for the interval width can be specified for different quantitative structures. Scalars with missing values can be used to specify default values for these options. Option LOWER is especially important if the observations have a "natural" lower limit, for example the value 0; then 0 is taken as the lower limit of the first group and the first group has the same interval width as the following groups.

The option TITLE and the parameters of HISTOGRAM have been transferred to AKAIKEHISTOGRAM. However, options NGROUPS and LABELS from HISTOGRAM have been omitted, because they are not in line with the style of AKAIKEHISTOGRAM.

Options: CHANNEL, TITLE, LOWER, WIDTH, SCALE.

Parameters: DATA, NOBSERVATIONS, GROUPS, SYMBOLS, DESCRIPTION.

The final limits of the starting histogram are obtained by a relatively strong rounding-off of the class limits (comparable with that in HISTOGRAM), where the width is always a multiple of the rounding-off interval.

The structures in DATA can be restricted, and in different ways; AKAIKEHISTOGRAM uses only those units that are not excluded by their respective restrictions.

AKEY generates the values of the block factors, if necessary, in systematic order and then generates the treatment factors from the block factors using a design key. It then allows you to print the design.

The design key method, described by Patterson (1976) and Patterson & Bailey (1978), provides a very flexible way of specifying the allocation of treatments in an experimental design. The method assumes that the units are identified by a set of what are termed "plot" factors. Generally these will be the same factors that are used in the block formula. Thus, in the procedure, they are specified by an option called BLOCKFACTORS which will take the factors from the formula already set by the BLOCKSTRUCTURE directive (outside the procedure) as its default. However, if any of these factors has a non-prime number of levels, it may need to be specified instead as the combination of two or more (pseudo) factors: for example, in a block design with blocks of size eight, the plots might need to be indexed by three factors with two levels (see Example 4). The method can also be used to set up pseudo-factors for use in the treatment formula, and then the "plot" factors may be the treatment factors themselves (Example 3). If these "plot" factors do not already have values, they will be generated in "standard order" using the GENERATE directive.

The factors whose values are to be generated are specified by the TREATMENTFACTORS parameter. Again this can be omitted, and AKEY will take the factors from the existing setting of the TREATMENTSTRUCTURE directive, in the order in which they occur there.

The generated values of the factors can be printed by setting option PRINT=design. The other options define how the values are generated. The KEY option specifies a matrix known as the design key, which indicates how the values of each treatment factor are to be calculated from the plot factors. The matrix has a row for each treatment factor and a column for each plot factor; below K_ij represents the element in row i and column j. (This is the transpose of the form used by Patterson 1976, but in Genstat it seems more convenient to specify the treatments by rows.) There is also an option called BASEVECTOR, which can specify a variate with an element B_i for each treatment factor to allow the levels of the factor to be shifted cyclically; by default this is a variate of zeros.

The calculation assumes that the values of the plot factors are represented by the integers zero upwards (and AKEY will perform this mapping automatically if necessary). The value q[i]_u in unit u of treatment factor i is then given by

where p[1]_u ... p[n]_u are the values of the plot factors in unit u, and t_i is the number of levels of treatment factor i. The calculated values are integers in the range 0, 1 ... t_i-1, but AKEY will again map these to the defined levels if necessary. However, all this takes place behind the scenes, within AKEY. The numbers of levels t_i must be prime numbers. They need not all be equal, but the key must be zero in any element where the row and column factors have different numbers of levels: that is, each treatment factor must be generated only from "plot" factors with the same number of levels as the treatment factor itself.

If any of the block or treatment factors has a non-prime number of levels, it must be specified as the combination of two or more (pseudo) factors: for example, in a block design with blocks of size eight, the plots would need to be specified by three factors with two levels (see Example 4). Thus the COLPRIMES option allows you to supply a variates listing the prime numbers for each column of the key, and the COLMAPPINGS option then to indicate the "plot" factor corresponding to each column. So, in Example 4, where we have

COLPRIME specifies that the prime for each column is 2, COLMAP specifies that the first column corresponds to the first "plot" factor (Block in the example) and that columns 2-4 correspond to the second "plot" factor (Plot in the example). The default for COLMAP is a variate containing the integers 1 up to the number of "plot" factors, so it can be omitted if no pseudo-factors are required. If COLPRIME is omitted, the primes for the columns are provided by the numbers of levels of the "plot" factors, as already explained. Options ROWPRIME and ROWMAP similarly allow you to specify pseudo-factors to generate the treatment factors.

and the skill when forming a design is in selecting the best set for each factor. The Genstat design system has a repertoire of keys, and these are used by procedures DESIGN and AGDESIGN to generate a range of designs, including factorials, fractional factorials, Latin squares and Lattices.

Parameter: TREATMENTFACTORS.

The FCLASSIFICATION and FORMULA directives are used, if necessary, to form lists of factors from the block or treatment formulae. The factor levels are then generated using the standard Genstat facilities for calculations and manipulation.

When a term is aliased in an analysis of variance, it is listed in the Information summary ( produced by ANOVA [PRINT=information] ) under the heading "Aliased model terms" (see the Genstat 5 Reference Manual, Section 9.7.1). However ANOVA does not indicate the terms with which it is aliased. This information can be obtained using procedure ALIAS.

The aliased term is specified by setting the TERM parameter to the list of factors that define it. The structure of the design can be specified either by options BLOCKSTRUCTURE and TREATMENTSTRUCTURE (together with option FACTORIAL, if necessary); alternatively you can save the design structure from the original analysis and supply this using the DESIGN option - this is the only way of specifying the design if there are weights or if the analysis is restricted. If an undeclared structure is specified for DESIGN (and BLOCKSTRUCTURE and TREATMENTSTRUCTURE are also specified), it will be set to the design structure for the analysis.

Options: TREATMENTSTRUCTURE, BLOCKSTRUCTURE, FACTORIAL, DESIGN. Parameter: TERM.

None of the options nor parameters are vectors. To indicate that the analysis is of a restricted set of units you must use the DESIGN option to specify the design structure from the original analysis.

NEWUNITS = factors, variates or scalars

AMERGE provides a convenient way of adding extra units into an experimental design. In the simplest case, this can be used to add control treatments to an already generated factorial design. More complicated uses may join together two completely different designs, from example a randomized block design to a balanced incomplete block design. These are both illustrated in the example.

The factors of the design which is to be augmented are specified using the FACTOR parameter, and the units that are to be added to each one are specified by the NEWUNITS parameter. The same number of units must be added to every FACTOR, and their levels (and labels) will be extended, if necessary, according to those defined on the units that are added. New units of a factor that are to receive different levels should be specified in a factor or a variate. Alternatively, if every new unit is to receive the same level of the FACTOR, NEWUNIT can be set to a scalar.

The SORT option can be set to yes to request that the FACTOR values are sorted after the new units have been added. Otherwise, they are simply placed at the end of the existing values.

Option: SORT.

Parameters: FACTOR, NEWUNITS.

AMERGE uses the standard Genstat manipulation facilities.

Suppose that we have a set of experimental units, or plots, within which observations are made in several locations at a sequence of times. Data from some of the locations may be missing at various times. The observed data values are specified in separate variates, one for each time point, using the DATA parameter. The factor identifying the experimental unit on which each observation was made is specified using the GROUPS option.

ANTMVESTIMATE assumes that the data have an ante-dependence (AD(r)) covariance structure whose order can be specified using the ORDER option; if this is not set, ANTMVESTIMATE takes the maximum possible order, number of times minus one. Using this assumption, ANTMVESTIMATE estimates the missing values and calculates the mean profiles for each unit. These can be saved, in tables indexed by the GROUPS factor, using the MEANPROFILES parameter, or printed by setting the PRINT option to meanprofiles. Also, the NEWDATA parameter allows new variates to be saved with the missing values replaced by their estimates.

Options: PRINT, GROUPS, ORDER. Parameters: DATA, NEWDATA, MEANPROFILE.

A repeated measures experiment is one in which the same set of units, or subjects, is observed at a sequence of times to investigate treatment effects over a period of time. The set of variates observed at the successive times is said to have an ante-dependence structure of order r if each ith variate (i>r), given the preceding r, is independent of all further preceding variates (Gabriel 1961, 1962). Procedure ANTORDER calculates statistics to assist in the selection of an appropriate order of ante-dependence structure for sets of repeated measures data, using the method of Kenward (1987). Once the order of ante-dependence structure has been established, the individual variates can be analysed individually by analysis of covariance, adjusting for the r previous variates, to assess the times at which treatment effects occurred. Also, procedure ANTTEST can be used to perform overall tests of treatment effects.

The data variates are supplied, as a list, by the DATA parameter. The variates may contain missing values, provided these are "dropouts", that is, once a unit becomes missing it should remain missing at all subsequent times.

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, ANTORDER 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 ANTORDER 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). Option MAXORDER specifies the maximum order of ante-dependence structure to be tested; by default, this is taken as the maximum possible order (the smaller of the number of times minus one or the number of residual degrees at each time; see Kenward 1987). The FACTORIAL option can be used to set a limit on the number of factors in the terms generated from the treatment formula.

Options: TREATMENTSTRUCTURE, BLOCKSTRUCTURE, MAXORDER, FACTORIAL.

Parameter: DATA.

Any restriction on the DATA variates will be applied to all of them.

A repeated measures experiment is one in which the same set of units, or subjects, is observed at a sequence of times to investigate treatment effects over a period of time. The set of variates observed at the successive times is said to have an ante-dependence structure of order r if each ith variate (i>r), given the preceding r, is independent of all further preceding variates (Gabriel 1961, 1962). Procedure ANTTEST calculates overall tests of treatment terms based on a specified order of ante-dependence structure (see Kenward, 1987).

The data variates are supplied, as a list, by the DATA parameter. The variates may contain missing values, provided these are "dropouts", that is, once a unit becomes missing it should remain missing at all subsequent times.

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, ANTTEST 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 ANTTEST 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). Option ORDER specifies the order of ante-dependence structure to be assumed for the tests; by default, this is taken as the maximum possible order (the smaller of the number of times minus one or the number of residual degrees at each time). A suitable order can be established using the ANTORDER procedure. The FACTORIAL option can be used to set a limit on the number of factors in the terms generated from the treatment formula.

Options: TREATMENTSTRUCTURE, BLOCKSTRUCTURE, ORDER, FACTORIAL.

Parameter: DATA.

Any restriction on the DATA variates will be applied to all of them.

This procedure provides one-way analysis of variance, avoiding the technicalities of the Genstat analysis-of-variance directives, but with extra options for residual plots and to check homogeneity of variances. Also, by setting option EXPLAIN=yes, the procedure can be requested to print an explanation of the statements required for the analysis, helping users to learn how to run the analysis directly in Genstat, and go beyond the facilities in the procedure; see the Genstat 5 Release 3 Reference Manual, Chapter 9, or Genstat 5: an Introduction, Chapter 7. This illustrates the sort of procedures that might be provided to assist inexperienced users.

The data values to be analysed are specified by the parameter of the procedure, in a variate. The groups to be compared are indicated using the GROUPS option. The easiest way of doing this is to give either a variate or a text, each of whose distinct values represents one of the groups; a more efficient method is to use a factor.

Printed output from the analysis of variance is requested by listing the required components in the PRINT option. The relevant possibilities are: aovtable to print the analysis-of-variance table, means to print the table of means, effects to print the effects (means minus grand mean), %cv to print the coefficient of variation, and missingvalues to print estimates for missing values (if any).

Homogeneity of the variances within the different groups can be tested by setting option HOMOGENEITY=yes, and settings of the PLOT option allow various residual plots to be requested: fittedvalues for a plot of residuals against fitted values, normal for a Normal plot, halfnormal for a half Normal plot, and histogram for a histogram of residuals.

Residuals and fitted values cannot be saved using the procedure, but the directive AKEEP can be used to get those from the analysis of the last data variate to be analysed by the procedure. For example

Options: PRINT, GROUPS, HOMOGENEITY, PLOT, EXPLAIN. Parameter: Y.

If the setting of the GROUPS option is a variate or a text, the GROUPS directive is used to form a factor, with one level for each distinct value. The factor is then used to treatment model with a single factor, for example

(Genstat 5 Release 3 Reference Manual, page 466). The data variate is then analysed using ANOVA (page 469), for example

Homogeneity of variance can be tested using procedure VHOMOGENEITY with option GROUPS set to the factor and parameter DATA to the y-variate. Plots of residuals are provided by procedure APLOT, for example

If the Y variate is restricted, only the units not excluded by the restriction will be analysed.

Plots residuals from an ANOVA analysis

Procedure APLOT provides four types of plots of residuals from an ANOVA analysis. These are selected using the METHOD parameter with settings: fittedvalues for residuals versus fitted values, normal for a Normal plot, halfnormal for a half-Normal plot, and

histogram for a histogram of residuals.

The residuals and fitted values are accessed automatically from the structure specified by the SAVE option. If the SAVE option is not set, they are taken from the SAVE structure of the last y-variate to have been analysed by ANOVA.

Option: SAVE. Parameter: METHOD.

Residuals and fitted values are accessed, using AKEEP, from the latest ANOVA, or from that specified by the SAVE option. For a Normal plot, the Normal quantiles are calculated as follows:

If the y-variate in the ANOVA was restricted, only the units not excluded by the restriction will be included in the graphs.

APPEND provides a convenient way of putting the values from several vectors, all into a single vector. The new vector will contain all the values of the first vector, then all those from the second vector, and so on. The vectors can thus contain different numbers of values but they must all be of the same type: all variates, all factors or all texts.

The vectors whose values are to be appended together are specified by the OLDVECTOR parameter, and the NEWVECTOR option supplies the vector to store the appended values. If NEWVECTOR is omitted, the values are placed into the first OLDVECTOR.

For factors, the FREPRESENTATION option indicates how the levels are to be matched amongst the old vectors. If this is set to labels and the levels of the old factors are compatible (that is if each label corresponds to the same level in all the old factors), then the level definitions are transferred to the new factor; if not, the levels are defined to be the default values 1, 2... and a warning is printed. Similarly, with the default setting FREPRESENTATION=levels, the labels are retained if they are compatible, but no warning is printed if they are not. For FREPRESENTATION=ordinals, the levels of all the factors are taken as the ordinal values 1, 2... (and no labels are defined). Finally, the renumbered setting assumes that the old factors all have independent sets of levels, and renumbers these from one upwards for the first factor, from number of levels of the first factor plus one upwards for the second factor, and so on; the new factor will thus have a different level for every level of the original factors.

The GROUPS option allows a factor to be formed indicating the OLDVECTOR to which each unit of the appended vector originally belonged. The levels are labelled by the identifier of the corresponding OLDVECTOR. This factor could be used in the CONDITION option of the SUBSET procedure subsequently to recover the original vectors.

Options: NEWVECTOR, FREPRESENTATION. Parameter: OLDVECTOR.

APPEND defines the lengths and all other relevant attributes of the NEWVECTOR and then uses EQUATE to transfer the values.

APRODUCT forms an experimental design by taking the product of two other designs. The METHOD option controls whether the product is formed by nesting the second design within the first, or by crossing the two designs together. For example, suppose that the first design has a single factor Units in the block structure and a single treatment factor A, while the second design is a Latin square with block structure Rows*Colums and treatment factor B. If we nest the second design within the first, we would obtain a design with block structure Units/(Rows*Columns) in which each unit of the first design has been subdivided into a row by column array of subplots to contain a Latin square of the sort defined by the second design. Nesting is thus useful when you want to subdivide the units of a design and apply further treatments (in this case those defined by the factor B) to the resulting subplots. Similarly, if we cross the two designs, the new design will have a block structure of Units*(Rows*Columns), or Units*Rows*Columns, in which we have duplicated the second design for every level of Units. Crossing is useful if you need to introduce a new blocking structure into an existing design. For example, the Units factor might represent different time periods or different locations in which the latin square design was to be used, and the factor A the different systematic conditions that might apply on each occasion.

With both nesting and crossing, the new design will contain a unit for every combination of the block factors in the two original designs, and so every combination of the treatment factors in the first design will occur with every combination of the treatment factors in the second design. The treatment structure is thus defined for the new design by crossing the treatment structures of the two original designs, to estimate all the original treatment terms and their interactions. So, in the example above, the treatment structure is defined to be A*B.

APRODUCT redefines the values of the factors as required for the new design, and executes BLOCKSTRUCTURE and TREATMENTSTRUCTURE directives with the new block and treatment formulae. The new formulae can then be accessed, outside the procedure, using the GET directive or procedure ASTATUS. The PRINT option can be set to design to print the new design, and the ANALYSE option can be set to yes to produce a skeleton analysis of variance from ANOVA. Options BF1, TF1, BF2, and TF2 define the block structure and treatment structure of the first and then the second design.

Options: PRINT, ANALYSE, METHOD, BF1, TF1, BF2, TF2. Parameters: none.

APRODUCT uses the standard Genstat manipulation directives such as FCLASSIFICATION, CALCULATE and DUPLICATE.

ARANDOMIZE provides a convenient way of randomizing the treatment allocations in an experimental design. It has several advantages over the RANDOMIZE directive (which is used inside the procedure).

First of all, the BLOCKSTRUCTURE option, which (as in RANDOMIZE) specifies the block model formula to indicate how the randomization is to take place, will use any setting that has already been defined by the BLOCKSTRUCTURE directive as its default. Moreover, the formula need not index all the units of the design, as would be required by RANDOMIZE; if necessary ARANDOMIZE will set up an extra factor _units_ simular to the factor *units* used by ANOVA.

ARANDOMIZE allows the original (unrandomized) values to be retained. There are two parameters: OLDVECTOR to specify the factors or variates to be randomized, and NEWVECTOR to allow new structures to be supplied to store the randomized values. If no NEWVECTOR is specified, the randomized values replace the original values of the corresponding OLDVECTOR. By default, NEWVECTOR is assumed to contain the list of factors in the model formula (if any) specified by the previous TREATMENTSTRUCTURE directive.

The levels of the treatment factors can be randomized by setting option LPERMUTE=yes; ARANDOMIZE then randomly permutes the numbering of the levels of each treatment factor on the units of the design. There is also a PRINT option which can be set to design to print the design. The other two options, EXCLUDE and SEED, are as in RANDOMIZE. EXCLUDE lists block factors whose levels are not to be permuted during the randomization; for example the period factor might need to be excluded in the randomization of a trial to study carry over effects. SEED defines the seed used to generate the random numbers used for the randomization; the default of 0 ensures that a seed will be chosen at random if SEED is not set.

Options: PRINT, BLOCKSTRUCTURE, EXCLUDE, SEED, LPERMUTE.

Parameters: OLDVECTOR, NEWVECTOR.

The GET directive is used to access any existing settings defined by the BLOCKSTRUCTURE or TREATMENTSTRUCTURE directives. AFUNITS, if necessary, forms the extra _units_ factor, and DUPLICATE generates new copies of the original vectors, if these are to be kept, before RANDOMIZE is used to produce the randomized values. Finally, if required, PDESIGN is used to print the design.

RESTRICT can be used, as usual, to restrict the set of units to be randomized.

The design might thus seem analogous to a split-plot design, with subjects corresponding to whole plots, and the occasions of observation to the sub-plots. There are, however, some important differences between the two situations. With repeated measurements, there is likely to be a greater correlation between observations that are made at adjacent time points than between those that are more greatly spaced. Furthermore, the Times factor cannot, by its very nature, be allocated at random to the occasions within subjects. In the customary split-plot situation we can usually assume that there is an equal correlation between the sub-plots of each whole plot and, even if this were not so, the sub-plot treatment should have been allocated at random to the sub-plots within each whole plot. The formal conditions for the validity of the split-plot analysis will be discussed in more detail below, together with advice on how to proceed if they do not hold.

It is worth pointing out first, though, that this problem affects only the Subjects.Times stratum. The Subjects stratum contains an analysis of variance of the measurements totalled over the subjects, and this part of the analysis will be valid whatever the within-subject correlation structure. A further point is that, when measurements are taken on only two occasions, the analysis in the Subjects.Times stratum will also be valid; there can then be only one within-subject correlation, and the analysis in the Subjects.Times stratum is of the difference between the observations at time 2 and time 1 on each subject.

Another potential problem arising from the systematic nature of the Times factor is that effects arising from the "length of treatment time" will be confounded with any effects arising from the duration of the experiment, such as age of subject (which may be important with short-lived material such as aphids), season of year, time of day, and so on. This does not affect the validity of the analysis, and some of the confusion may be capable of being unravelled by running the experiment during more than one period. Nevertheless, care needs to be taken in drawing conclusions about time-effects.

The Subjects.Times information, describing the way in which the treatment effects change differentially with time, is often the aspect of most interest in the study. The formal requirement for the validity of the analysis in the sub-plot stratum of a split-plot design is that all the normalised contrasts in that stratum have an equal variance. The only practical arrangement of covariances between times that satisfies this condition would have a single variance down the diagonal and a single covariance off-diagonal. This pattern is known as a uniform covariance structure or, equivalently, the matrix is said to show compound symmetry; Box (1950) describes how this can be tested. In the usual split-plot analysis, the Subjects.Times sum of squares is assumed to be distributed as s² ´ c²_r where s² is a constant and c²_r has a chi-square distribution on r degrees of freedom. Similarly, under the assumption that there is no Treatments.Times interaction, the Treatments.Times sum of squares is assumed to be distributed as s² ´ c²_t where c²_t has a chi-square distribution on t degrees of freedom. If the variance-covariance structure does not exhibit compound symmetry, it is possible to show that the distributions can still be approximated by chi-squared distributions, but the degrees of freedom are instead epsilon ´ r and epsilon ´ t. The correction factor epsilon lies between one, which would give the ordinary split-plot analysis, and 1/(number of times minus one), which would leave just one degree of freedom within each subject (remember that when there are only two observation on each subject, and thus just one within-subject degree of freedom, the analysis is valid); epsilon can be estimated by maximum likelihood, as described by Greenhouse & Geisser (1959).

The printing of information about the covariances is controlled by the strings listed for the PRINT option: vcovariance variance-covariance matrix, correlation correlation matrix, epsilon Greenhouse-Geisser epsilon, test test for compound symmetry.

The output from the analysis of variance is controlled by the APRINT option, with settings identical to those in the PRINT option of the ANOVA directive.

The treatments applied to the subjects can be specified (as a model formula) using the TREATMENTSTRUCTURE option, the block structure (if any) on the subjects can be specified by the BLOCKSTRUCTURE option, and the COVARIATE option can be used to list any covariates. If any of these options is unset, the default is taken from any existing setting defined by the directives TREATMENTSTRUCTURE, BLOCKSTRUCTURE or COVARIATE, respectively. The FACTORIAL option can be used to set a limit on the number of factors in the terms generated from the TREATMENTSTRUCTURE option.

The observed data for the procedure should be specified in a set of variates, each one containing the measurements made on the subjects at one of the occasions on which they were observed, and input using the DATA parameter.

Parameter: DATA.

Provides information about the settings of ANOVA models and variates

ASTATUS allows information to be printed and saved about the model settings and other information involved in an ANOVA analysis.

By default ASTATUS prints the current settings defined by the directives TREATMENTSTRUCTURE, BLOCKSTRUCTURE and COVARIATE. This is governed by the default setting, model, of the PRINT option. The other setting y prints the name of the y-variate from the most recent ANOVA. Alternatively, if the INSAVE parameter is set to the save structure from an ANOVA analysis, the y-variate and model settings will be those used to form the save structure.

If the INSAVE parameter is not set, the Y parameter can be used to save the identifier of the y-variate most recently analysed by ANOVA, in a pointer of length one. The TREATMENTSTRUCTURE parameter saves the current setting defined by the TREATMENTSTRUCTURE directive (in a formula structure), and the BLOCKSTRUCTURE parameter similarly saves the current setting defined by the BLOCKSTRUCTURE directive. The COVARIATE parameter saves the current setting defined by the COVARIATE directive (in a pointer). Alternatively, if INSAVE is set to an ANOVA save structure, the parameters Y, TREATMENTSTRUCTURE, BLOCKSTRUCTURE and COVARIATE save the settings used to form INSAVE.

The SAVE parameter saves the save structure from the most recent ANOVA (regardless of the setting of INSAVE).

Options: PRINT.

Parameters: Y, TREATMENTSTRUCTURE, BLOCKSTRUCTURE, COVARIATE, SAVE, INSAVE.

ASTATUS uses the GET directive to obtain the current settings of BLOCKSTRUCTURE, TREATMENTSTUCTURE and COVARIATE, and the save structure from the most recent ANOVA. It uses specialist knowledge of the ANOVA save structure to obtain information from an ANOVA save structure input using the INSAVE option.

The analysis-of-variance algorithm in the Genstat ANOVA directive involves a series of sweep operations performed on a working variate which initially contains the data values and finally contains the residuals. Sweeps may have two parts. The first involves the estimation of the effects of a particular term. For a term that is orthogonal to the terms that preceed it in the model, the effects are estimated simply by the tables of means for that term, calculated from the working variate; for non-orthogonal terms, the effects are the means divided by an efficiency factor. In the second part, the working variate is modified. Usually this involves subtracting the estimated effects. Alternatively there is a special sweep, known as a pivot, which is used to initiate the analysis within a stratum. In this, the value in each unit of the working variate is replaced by the corresponding effect of the term. Further details can be found in the Genstat 5 Release 3 Reference Manual, page 523, or in the paper by Payne & Wilkinson (1977). Procedure ASWEEP is provided as a research tool for studying the algorithm and its properties.

The values initially in the working variate are specified by the Y parameter. The procedure can sweep for a single term or, if several terms have the same efficiency factor, these can all be swept together. The TERM option specifies the term (or terms) and the efficiency factor is defined by the EFFICIENCY option. The EFFECTS parameter allows the estimated effects of the term(s) to be stored if option EMETHOD=calculated, or to be supplied if EMETHOD=given. The values in the working variate after the sweep can be saved using the RESIDUALS parameter, and the RMETHOD option indicates whether these are to be formed by an ordinary sweep (RMETHOD=subtract) or by a pivot (RMETHOD=replace). The SS parameter saves the sum of squares due to the term(s), and the RSS parameter saves the sum of squares of the working variate after the sweep.

Options: TERM, EFFICIENCY, EMETHOD, RMETHOD.

Parameters: Y, EFFECTS, RESIDUAL.

If the working variate (specified by the Y parameter) is restricted, the sweep will use only the units not excluded by the restriction.

Produces further output for an unbalanced design (after AUNBALANCED)

This procedure can be used, following the use of procedure AUNBALANCED, to produce further output for the analysis of variance of an unbalanced design.

The output to be printed is controlled by the PRINT option, with settings: aovtable to print the analysis-of-variance table, effects to print the effects (as estimated by Genstat regression), residuals to print residuals and fitted values, means to print tables of predicted means with standard errors, and %cv to print the coefficient of variation. The default is to print the analysis-of-variance table and tables of means.

The model is fitted sequentially, first any covariates and then the treatments. Thus, any sums of squares produced for covariates are ignoring treatments (not after eliminating treatments, as with the ANOVA directive). Tables of means are produced with the PREDICT directive using the default weighting (Genstat 5 Release 3 Reference Manual, page 395).

The PFACTORIAL option limits the number of factors in terms for which predicted means are printed. Probabilities can be printed for variance ratios by setting option FPROBABILITY=yes. Finally, there is a PLOT option which allows various residual plots to be requested: fittedvalues for a plot of residuals against fitted values, normal for a Normal plot, and halfnormal for a half Normal plot.

The SAVE parameter can be set to the save structure from the analysis for which further output is required. If SAVE is not set, output will be produced for the most recent analysis from AUNBALANCED; however, none of the Genstat regression directives (MODEL, TERMS, FIT, ADD, DROP and so on) must then have been used in the interim.

Options: PRINT, PFACTORIAL, FPROBABILITY, PLOT. Parameter: SAVE.

The output is produced mainly using the directives RKEEP and PREDICT.

If the y-variate originally analysed by AUNBALANCED was restricted, only the units not excluded by the restriction will have been analysed.

This procedure carries out analysis of variance using the regression directives in Genstat. It is particularly useful for designs that are unbalanced and which thus cannot be analysed by the ANOVA directive.

The method of use is similar to that for ANOVA. The treatment terms to be fitted must be specified, before calling the procedure, by the TREATMENTSTRUCTURE directive. Similarly, any covariates must be indicated by the COVARIATE directive. However, the procedure does not cater for blocking, so any settings of the BLOCKSTRUCTURE directive are ignored.

The parameters of the procedure are identical to those of ANOVA. The variates to be analysed are specified by the Y parameter. Residuals and fitted values can be saved using the RESIDUALS and FITTEDVALUES parameters respectively. Finally, the SAVE parameter allows details of the analysis to be saved so that further output can be obtained using the AUDISPLAY procedure. (Note that this is a regression save structure, not an ANOVA structure, so it cannot be used with the directives ADISPLAY or AKEEP).

Printed output is controlled by the PRINT option, with settings: aovtable to print the analysis-of-variance table, effects to print the effects (as estimated by Genstat regression), residuals to print residuals and fitted values, means to print tables of predicted means with standard errors, and %cv to print the coefficient of variation. The default is to print the analysis-of-variance table and tables of means.

The model is fitted sequentially, first any covariates and then the treatments. Thus, any sums of squares produced for covariates are ignoring treatments (not after eliminating treatments, as with the ANOVA directive). Tables of means are produced with the PREDICT directive using the default weighting (Genstat 5 Release 3 Reference Manual, page 395).

The FACTORIAL option sets a limit on the number of factors that a higher-order term, such as an interaction, can contain; any terms with more factors are deleted from the analysis. Similarly, the PFACTORIAL option limits the number of factors in terms for which predicted means are printed. Probabilities can be printed for variance ratios by setting option FPROBABILITY=yes. Finally, there is a PLOT option which allows various residual plots to be requested: fittedvalues for a plot of residuals against fitted values, normal for a Normal plot, and halfnormal for a half Normal plot.

Options: PRINT, FACTORIAL, PFACTORIAL, FPROBABILITY, PLOT.

Parameters: Y, RESIDUALS, FITTEDVALUES, SAVE.

The y-variate is specified using the MODEL directive, along with any variates to save residuals and fitted values. The current settings of the TREATMENTSTRUCTURE and COVARIATE directives are recovered using the SET directive, and used to define the terms in the analysis (using the TERMS directive). The model is then fitted (using FIT), AUDISPLAY is called to print the output and any plots of residuals.

If the Y variate is restricted, only the units not excluded by the restriction will be analysed.

The treatment effects to be studied should be specified using the TREATMENTSTRUCTURE directive before using A2PLOT. They are grouped according to the error strata as specified by a previous BLOCKSTRUCTURE statement. Normal or half-Normal plots, according to the METHOD option, are made in either lineprinter or high-resolution quality (option GRAPHICS). By default plots are made for each error stratum. Alternatively, option STRATUM can be set to a formula defining the strata from which the Yates effects are to be plotted. The EXCLUDE option specifies the number of largest effects to be exclude from half-Normal plots (the option does not work with Normal plots). The titles of the plots can be provided using option TITLE. Setting GRAPHICS=* suppresses the plots. Options FACTORIAL, PRINT and CHANNEL, are as in ADISPLAY. Note, however, that effects are printed as Yates effects, and that CHANNEL also controls the line-printer graphics.

When the number of effects in the plot is in the range 7 to 127, robust estimators are calculated for the standard error of the effects. The robustness of the estimators against contamination with active effects is specified through option ROBUSTNESS. A vertical line in the plot indicates the least significant Yates effect (LSE). The type I error is controlled by option ALPHA. Effects larger than the LSE are labelled in the plot.

The data variates are specified using the Y parameter. The EFFECTS parameter can save a pointer holding a variate of effects, sorted from small to large, for each error stratum. Effects are either the usual Yates effects (METHOD=normal) or their absolute values (METHOD=halfnormal). Parameter SIGNIFICANT can save a formula with the joint significant effects of all the strata. Parameter SE holds scalars with the standard errors of the effects in the respective strata.

Parameters: Y, EFFECTS, SE, SIGNIFICANT.

A2PLOT accesses the current BLOCKSTRUCTURE and TREATMENTSTRUCTURE settings using the GET directive. If the STRATUM option is unset, separate plots for each of the strata are to be produced. A2PLOT checks, therefore, whether all strata are set explicitly. If this is not the case it augments the current BLOCKSTRUCTURE with a bottom stratum using procedure AFUNITS. If no BLOCKSTRUCTURE is set, it generates an explicit Units stratum and sets the BLOCKSTRUCTURE and STRATUM options to this stratum.

Yates effects for each stratum are saved using AKEEP. They are ordered and plotted against either Normal or half-Normal quantiles. Normal quantiles are calculated asq_i = NED( (i-0.375) / (n+0.25) ) i=1...n

For ROBUSTNESS=low, ASE based standard errors are calculated with the initial standard error calculated from the median of all effects, a cut-off of 2.5 times this value, and a final standard error from the sum of squares of the remaining effects. For ROBUSTNESS=medium, PSE based standard errors are calculated with the same cut-off as for ASE and a final standard error is calculated from the median of the remaining effects. For ROBUSTNESS=high, PSE based standard errors are calculated using the 45th percentile instead of the median for the initial estimate, and 1.25 instead of 2.5 as a multiplication factor to establish the cut-off. The final estimate uses the median of the retained effects.

Significant Yates effects are labelled in the half-Normal plots using the factor names from the TREATMENT statement.

The authors thank Peter Lane for suggesting and sketching procedure A2PL_EXPAND.

AFUNITS (which may be called by A2PLOT if the STRATUM option is unset and no explicit bottom error stratum is specified in the current BLOCKSTRUCTURE setting) requires that none of the blocking factors be restricted.