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  2. AFRCRESOLVABLE procedure


Form doubly resolvable row-column designs, with output (D.B. Baird).


PRINT = string tokens Controls printed output (design, plotnumbers, factors, efficiency; default desi, effi
DESIGNPLOT = string token What factors to display in the design plot (treatment, plotandtreatment); default * i.e. no plot
FIRSTPLOT = string token Defines the starting location for allocating plots to the row-by-column grid (lowleft, lowright, upleft, upright); default uple
PLOTORDER = string token Defines the order in which the blocks are filled (colserpentine, colbycol, rowserpentine, rowbyrow); default rowb
TIME = scalar Time in seconds to spend searching for an optimal design; default 60
SEED = scalar Seed for the randomization; default 0
MAXITERATIONS = scalar The number of random designs to search for an optimal design; default 10000
SPREADSHEET = string tokens What to save in a spreadsheet (data, plan); default *


NROWS = scalars Number of rows in the layout of each design
NCOLUMNS = scalars Number of columns in the layout of each design
LEVELS = scalar, variate or text Defines the number of levels or labels of the TREATMENT factor for each design
TREATMENTS = factors Saves the treatment allocation in each design
ROWREPLICATES = factors Saves the row replicates in each design
COLREPLICATES = factors Saves the column replicates in each design
ROWS = factors Saves the row locations of the plots in each design
COLUMNS = factors Saves the column locations of the plots in each design
PLOTNUMBERS = factors Saves the plot numbers
TITLE = texts The title for the design plot; default an automatic description of the design
OUTFILE = texts Gives a file name (with extension .gsh, .gwb, or .xlsx) to save the factors in each design
EXIT = scalars Saves the exit code from the design search program (0 for success, greater than 0 for failure)


AFRCRESOLVABLE creates approximately optimal row-column designs. They are formed into replicates in both the row and column directions so that they are doubly resolvable, i.e. resolvable in both row and column directions. The layout of plots must be a complete rectangular array, and the treatments must be equally replicated. This requires that the number of rows multiplied by the number of columns in the array must be equal to the number of treatments multiplied by the number of replicates. The row replicates consist of units in adjacent rows, and the column replicates consist of units in adjacent columns. This design can be thought of as a generalization of a Latin square, with each treatment occurring once in each row and column replicate.

An example design with four replicates of five genotypes in a five-row by four-column array is shown below. As the number of treatments is the same as the number of rows, the column replicates are the same as the columns, so each treatment occurs once in each column. The row replicates are shaded in different colours and consist of five plots from adjacent rows. This is the optimal design, as the treatments in the five rows form a balanced incomplete block design (within the rows, each treatment occurs three times with every other treatment).

The number of rows and columns in the row-column array must be specified by the NROWS and NCOLUMNS parameters respectively. The number of treatments is specified by the LEVELS parameter, either as a scalar (defining the number explicitly), or as a variate giving the levels for the treatments, or by a text defining a name for each treatment. The maximum number of rows, columns or treatments is 4000, with at most 8000 plots in the array, and the maximum number of replicates is 20. There must be at least two rows, columns, treatments and replicates in the design. The number of columns must be less than or equal to the number of treatments. The number of rows can be greater than the number of treatments. It must then be a multiple of the number of treatments, and multiple replicates are stacked in the columns.

The PLOTNUMBER parameter saves the plot numbers, in a factor. The FIRSTPLOT option specifies where the numbering should start, as follows:

lowleft left-hand plot at the bottom of the design, i.e. in the final row and first column;
lowright right-hand plot at the bottom of the design, i.e. in the final row and final column;
upleft left-hand plot at the top of the design, i.e. in the first row and first column (default);
upright right-hand plot at the top of the design, i.e. in the first row and final column.

The PLOTORDER option defines the order in which the row or column replicates are formed and the order the plots are numbered from the first plot:

colserpentine column-by-column in a serpentine way, e.g. top-to-bottom, and then bottom to top;
colbycol column-by-column taking the same direction for every column;
rowserpentine in a serpentine way, e.g. left-to-right, and then right-to-left;
rowbyrow row-by-row taking the same direction for every row (default).

The TIME option (default 60 seconds) specifies the maximum time in seconds to spend searching for an optimal design. For large designs, TIME should be increased. For example, 1000 seconds is recommended for more than 100 treatments, and 4000 seconds for more than 200 treatments.

The SEED parameter specifies the starting seed for the randomization process; the default of zero initializes the seed automatically.

The MAXITERATIONS option (default 10000) sets the maximum number of random starting designs to use in the search. The search stops when either the TIME or the MAXITERATIONS limit is reached.

The factors for the resulting design can be saved by the TREATMENTS, ROWREPLICATES, COLREPLICATES, ROWS and COLUMNS parameters.

The EXIT parameter can save a scalar which is set to 0 if the design search has found a valid design, 1 if the design limits have been exceeded, 2 if a design is not possible, 3 if no design has been found, and 9 if not enough memory could be allocated for the design search.

Printed output is controlled by the PRINT option, with settings:

design to print treatments in the design in a row-column layout;
plotnumbers to print the plot numbers in a row-column layout;
factors to print the design factors for the best design; and
efficiency to print the combined design efficiency (using row and column replicates, rows and columns as fixed terms) for the design.

The design can be displayed in a spreadsheet using the SPREADSHEET option: the data setting creates a sheet containing the design factors (plot numbers, rows, columns, row replicates, column replicates and treatments); plan setting will display the plot numbers and treatments in a rectangular array in a sheet. The spreadsheet can be saved to a file if the OUTFILE parameter specifies a Genstat or Excel spreadsheet filename (.gsh, .gwb or .xlsx).

The design layout can be plotted in graph using the DESIGNPLOT option: the treatment setting just displays the treatments in each cell in the array; plotandtreatment displays both the plot numbers and treatments. The row and column replicates are demarked in the graph by blue and red lines respectively. The title for this graph can be set with the TITLE parameter. If this not set, a descriptive title will be created from the design parameters.




The procedure calls the directive AGRCRESOLVABLE to form the design which is chosen to optimise the row-column efficiency using the MS criterion (Shah 1960). The procedure FPLOTNUMBER is used to form the plot numbers and the procedure AEFFICIENCY to calculate the design’s efficiency. The ADSPREADSHEET procedure is used to save or display the design in a spreadsheet.


Shah, K.R. (1960). Optimality criteria for incomplete block designs. Annals of Mathematical Statistics, 22, 235-247.

See also

Commands for: Design of experiments.



"Simple design of 5 treatments in 5 rows by 4 columns (gives a Youden square)"
         LEVELS=5; TREAT=Treat; ROWS=Row; COLUMNS=Column
"15 Genotypes x 4 replicates in 10 x 6 array"
TEXT     [VALUES='AB14.33','AB15.1','AB15.32','AB18.1','AB19.2','AB19.27', \
         'AB31.11','AB32.4'] Gtype
AFRCRESOLVABLE [PLOTORDER=rowserpentine; SEED=12457; SPREADSHEET=data,plan] \
         NROWS=10; NCOLUMN=6; LEVELS=Gtype; TREATMENTS=Genotype; \
         COLUMNS=Column; PLOTNUMBER=Plot
"16 cultivars in a 8 x 8 array (pairs of rows & columns make replicates)"
TEXT     [VALUES='Apollo','Bumper','Chronicle','Diamond','Earner','Fleet',\
         'Orator','Phoenix','Queen'] Names
AFRCRESOLVABLE [PRINT=design,plotnumbers,efficiency,factors; \
         FIRSTPLOT=lowleft; PLOTORDER=colserpentine; SEED=75421; \
         MAXITERATIONS=20000; DESIGN=plotandtreatment] \
         NROWS=8; NCOLUMNS=8; LEVELS=Names; \
         ROWS=Row; COLUMNS=Column; PLOTNUMBER=Plot

"Create dummy data"
                        NEWLEVELS(ColRep;CUMULATE(GRNORMAL(4;0;1))) + \
                        NEWLEVELS(Row;CUMULATE(GRNORMAL(8;0;1))) + \
                        NEWLEVELS(Column;GRNORMAL(8;0;0.5)) + GRNORMAL(64;9;1)

"Analyse dummy data by ANOVA removing both rows and column replicates effects
as in this design these are both orthogonal to treatments as they are 
comprised of full rows and columns - however most designs are not orthogoanal"
BLOCKS RowRep + ColRep
TREAT Cultivar
ANOVA [FPROB=yes] Yield

"Analyse data by automatic REML"
VAROWCOLUMN [PBEST=model,means,components; FIXED=Cultivar; RANDOM=ColRep; \
    VCONSTRAINTS=positive; RSTRATEGY=optimal] Yield
Updated on October 30, 2020

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