AGNATURALBLOCK procedure

Forms 1- and 2-dimensional designs with blocks of natural size (P.D. Johnstone & D.B. Baird).

Options

Parameters

Description

This procedure uses random generation for obtaining block and row-column designs and selects the best of those designs using a criterion similar in effect to the M,S-optimality criterion of Shah (1960). Unlike some other design generators, this generator has no restrictions on the layout of plots in the array (which can be incomplete), nor on the replicates per treatment (which may be unequal).

The number of treatments can be defined by the LEVELS parameter, as either a scalar or a variate, as in the FACTOR directive. Alternatively, you can set the TREATMENTS option to a factor whose levels have already been defined. You can also use the TREATMENTS option to request unequally replicated treatments, by specifying a factor whose values have been defined with the desired replications. For example:

FACTOR [LEVELS=3] Trt; VALUES=!(6(1),3(2,3),4(4))

The DESIGNTYPE option specifies whether to form a row-column or a block design. With a row-column design (DESIGNTYPE=rowcolumn), the layout of the plots can be defined in two factors supplied by the ROWS and COLUMNS parameters. If the ROWS or COLUMNS options are set to factors whose values have not been defined, their levels will be set up to define a regular grid of plots. Alternatively, if ROWS and COLUMNS have no levels defined, the numbers of rows and columns can be specified by the NROWS and NCOLUMNS parameters. The NUNITS parameter defines the number of plots in the design. If this is not specified, the number of plots for the treatments is defined by the number of values in ROWS or COLUMNS, if available, or alternatively as NROWS × NCOLUMNS. Thus, you need to specify NUNITS only if you are not using the full grid of NROWS × NCOLUMNS values, or if you wish to limit the units used in ROWS and COLUMNS.

For a one-way block design (DESIGNTYPE=block) with unequal block sizes, the BLOCKS parameter must be set to a factor giving the block number for each plot. The block design can allocated to a two dimensional layout by specifying the position of each unit in two factors supplied by the ROWS and COLUMNS parameters. Alternatively, you can specifying the size of the grid with the NROWS and NCOLUMNS parameters, in which case the LAYOUT option is used to allocate units to the grid. Again, if the ROWS or COLUMNS option is set to a factors whose values have not been defined, the factor values will be set up to define a regular grid of plots. With a block design, if NUNITS is unset, the number of plots is taken from the number of values of the BLOCKS factor. The FIRSTPLOT and FILLMETHOD options control how the plots are allocated to the row and column locations when these are not defined by ROWS and COLUMNS factors. The FIRSTPLOT option defines the starting location as follows:

The FILLMETHOD option defines the order in which the plots are then filled:

The NSIMULATIONS option defines the number of random designs to be searched. The SEED option specifies the starting seed for the randomization process; the default of zero continues an existing sequence of random numbers if any have already been used in this Genstat job, or initializes the seed automatically. The resulting optimal treatment allocation can be saved, either in a factor specified by the TREATMENTS parameter, or by setting the PLAN parameter to a matrix to save the two-dimensional layout of the treatments. If the value of the ROWS and COLUMNS factors have not already been defined, these will be defined by the procedure.

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

Options: PRINT, DESIGN, NSIMULATIONS, SEED, FIRSTPLOT, FILLMETHOD.
Parameters: LEVELS, NROWS, NCOLUMNS, NUNITS, TREATMENTS, ROWS, COLUMNS, BLOCKS, PLAN.

Method

For a row-column design the treatments are allocated in random order down the columns, while for a block design they are allocated one replicate at a time. A number of designs (specified by the NSIMULATIONS option) are generated. The design is chosen that minimizes both the range and the average of the standard errors of differences for all pairwise comparisons between elements of the generalized least squares estimates of treatment effects.

Action with RESTRICT

If any of the factors is restricted, only the part of the design not excluded by the restriction will be generated.

References

Johnstone, P.D. (2003). Random generation and selection of one- and two-dimensional designs for experiments on blocks of natural size. Journal of Agricultural, Biological and Environmental Statistics, 8, 67-74.

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

See also

Commands for: Design of experiments.

Example

CAPTION 'AGNATURALBLOCK example',!t(\
        'Generate a design for 8 treatments in an incomplete',\
        '16 row by 11 column array.'); STYLE=meta,plain
FACTOR  [LEVELS=16; VALUES=1,3,4...9,11,12...14,16,\ 
        1,2...5,7,8...16,1,2...11,13,14...15,1,2...15,\
        1,2...15,1,2...11,1,2...7,9,10...14,1,2...7,9,\
        10...14,1,2...3,5,6,8,9...13,1,3,4...13,1,2...8,\ 
        10,11...13] Row1
FACTOR  [LEVELS=11; VALUES=13(1),15(2),14(3),15(4,5),11(6),\ 
        13(7,8),11(9),12(10,11)] Col1
AGNATURALBLOCK [PRINT=search; DESIGNTYPE=rowcolumn; SEED=234567]\ 
        LEVELS=8; TREATMENT=Trt1; ROWS=Row1; COLUMNS=Col1
PRINT   Row1,Col1,Trt1; FIELDWIDTH=5; DECIMALS=0

CAPTION !t('Generate a design for 5 treatments with blocks of',\
        'size 3, 6 and 4, and allocate the treatments to a 3 x 5',\
        'layout, starting in the top left corner, running across',\
        'rows in a serpentine manner (i.e. at the end of the first',\
        'row, drop directly down to the plot in the row below and',\
        'work back to the start of the row, etc.')
FACTOR  [LEVELS=4; VALUES=3(1),6(2),2(3),4(4)] Block
AGNATURALBLOCK [PRINT=design; DESIGNTYPE=block; SEED=7856;\
        FIRSTPLOT=lowright; FILLMETHOD=rowserpentine] LEVELS=5;\ 
        TREATMENT=Trt2; BLOCK=Block; NROWS=3; NCOLUMNS=5
PRINT   Block,Trt2; FIELDWIDTH=5; DECIMALS=0

CAPTION 'Generate a design for 13 treatments in a 8 row by 6 column array.'
AGNATURALBLOCK [NSIMULATIONS=500] LEVELS=13; NROWS=8; NCOLUMNS=6;\ 
        TREATMENT=Trt3