Forms design keys for multi-stratum experimental designs, allowing for confounded and aliased treatments.
Options
BASICFACTORS = factors |
Factors indexing the units of the design |
|---|---|
ADDEDFACTORS = factors |
Factors to be allocated to the units of the design |
KEY = matrix |
Stores the design key (ADDEDFACTORS × BASICFACTORS) |
INKEY = matrix |
Can be used to input existing allocations for some of the added factors |
HIERARCHIES = matrix |
Can be used to specify that some of the factors must be constant within each combination of levels of other factors; the matrix has a row for each added factor and columns first for the basic factors and then for the added factors, ones in the entries where the row factor must be constant within the combinations of the column factors, zero elsewhere |
SEED = scalar |
Can provide a seed to generate a random permutation of the sets of basic effects that may be allocated to each added factor, thus producing design randomly selected from all those that might be possible; default * i.e. no permutation |
ROWPRIMES = variate |
Prime numbers for the rows of the KEY matrix |
COLPRIMES = variate |
Prime numbers for the columns of the KEY matrix |
ROWMAPPINGS = variate |
Mappings from the rows of the KEY to the TREATMENTFACTORS |
COLMAPPINGS = variate |
Mappings from the columns of the KEY to the BLOCKFACTORS |
SAVE = identifier |
Structure to save all the information about the formation of the design; this can then be input later to give a different design (if possible) with the same properties |
Parameters
REQUIRED = formula structures |
Formulae each defining a list of terms that are to be estimated in the analysis |
|---|---|
NONNEGLIGIBLE = formula structures |
Formulae each specifying terms that cannot be ignored in the context of the corresponding REQUIRED formula |
Description
Design keys can be used in the GENERATE directive to generate values of treatment factors from block factors. They also provide the basis of the representation used to store the repertoire of designs obtainable from procedure AGDESIGN (see Payne and Franklin 1994). This covers a range of standard situations, but cannot allow for every eventuality. FKEY allows you to form keys for other circumstances and, if these are likely to occur frequently, you can extend or replace the standard repertoire using procedure FDESIGNFILE.
The assumption in FKEY is that the units of the design are indexed by a set of factors known as the basic factors. The key allows the values of another set of factors, known here as the added factors, to be calculated from the basic factors. These factors are listed using the BASICFACTORS and ADDEDFACTORS options. They must all have been declared previously as factors, and their numbers of levels must have been defined. Usually the basic factors are the factors that will be used to define the block formula of the design (for example, blocks, plots, rows, columns, subplots and so on) and the added factors are the treatment factors, but in partial replicates, for example, the basic factors may be the treatment factors and the added factors the block factors.
If the basic and added factors all have prime numbers of levels the key is saved, by the KEY option, as a matrix with a row for each added factor and a column for each basic factor. However, if the levels are not all prime, FKEY will break up factors that do not have prime numbers of levels into “pseudo-factors”. Thus, a factor with six levels will be represented by the combinations of levels of two pseudo-factors, one with two levels and one with three levels. When pseudo-factors are required for the added factors, the ROWPRIMES option can be used to save a variate storing the (prime) number of levels corresponding to each row of the key, and the ROWMAPPINGS option can save a variate with an element for each row containing the number of the corresponding added factor. So, if we had two added factors, one with five and one with six levels, the ROWPRIMES variate might contain the values 5, 2 and 3, and the ROWMAPPINGS variate the values 1, 2 and 2. The second added factor (with six levels) would then be represented by two pseudo-factors, corresponding to the second and third rows of the key. The COLPRIMES and COLMAPPINGS options can similarly save details of the pseudo-factors required for basic factors with non-prime numbers of levels. The variates saved by ROWPRIMES, COLPRIMES, ROWMAPPINGS and COLMAPINGS can be used in the AKEY procedure, together with the key, to form the added factors automatically without the need to worry about the pseudo-factoring.
The main properties of the design are derived from the REQUIRED and NONNEGLIGIBLE parameters. Suppose we have a block design containing three blocks of nine plots. The experiment is to have three treatment factors, A, B and C, and these will be the added factors. The design has a block structure of plots nested within blocks
Blocks/Plots
In the analysis we wish to be able to estimate all main effects and interactions of the factors A, B and C, except the three-factor interaction A.B.C; these terms are specified by the formula structure supplied using the REQUIRED parameter. The NONNEGLIGIBLE parameter specifies model terms that cannot be ignored in the analysis: that is, the model terms with which these required terms cannot be confounded. Here we have the main effect Blocks and all main effects and interactions of the factors A, B and C. To form the design key K, we thus need to put
FACTOR [NVALUES=27; LEVELS=3] Block,A,B,C
& [LEVELS=9] Plot
FKEY [BASIC=Block,Plot; ADDED=A,B,C; KEY=K;\
COLPRIMES=Bplev; COLMAPPINGS=Bmap]\
REQUIRED=!f(A*B*C-A.B.C); NONNEGLIGIBLE=!f(Block+A*B*C)
If the design has more than two strata suitable for the estimation of treatment effects, the REQUIRED and NONNEGLIGIBLE parameters can specify lists of formulae, in parallel, one pair of formulae for each stratum. Each REQUIRED formula specifies the terms that must be estimated in one of the strata (or in a stratum below it), and the corresponding NONNEGLIGIBLE formula specifies the terms that cannot be ignored there. Suppose we put
FACTOR [NVALUES=81; LEVELS=3] Block,Wplot,A,B,C,D,E
& [LEVELS=9] Subplot
FKEY [BASIC=Block,Wplot,Subplot; ADDED=A,B,C,D,E; KEY=K;\
COLPRIMES=Bplev; COLMAPPINGS=Bmap]\
REQUIRED=!f((A+B+C)*(A+B+C)),!f((A+B+C+D+E)*(A+B+C+D+E));\
NONNEGLIGIBLE=!f(Block+Block.Wplot),!f(Block)
Here we have a block formula
Block / Wplot / Subplot1
which produces three strata
Block + Block.Wplot + Block.Wplot.Subplot
The first formula in the REQUITRED list !f((A+B+C)*(A+B+C)), in parallel with the formula !f(Block+Block.Wplot) in the NONNEGLIGIBLE list, indicates that we do not want the main effects or two-factor interaction of factors A, B and C to be confounded with each other nor with Block or Block.Wplot; this ensures that they will be estimated in the Block.Wplot.Subplot stratum. The second pair of formulae, !f((A+B+C+D+E) * (A+B+C+D+E)) and !f(Block), indicate that we want to estimate the main effects and two-factor interactions of all the five treatment factors A, B, C, D and E in the Block.Wplot stratum or below; in effect this means that we are willing to have D and E and any of their interactions estimated in the Block.Wplot stratum.
The algorithm that FKEY uses to construct the key is based on the method developed by Franklin & Bailey (1977), Franklin (1985) and Kobilinsky (1995). Essentially this considers the possible orthogonal sets of contrasts amongst the main effects and interactions of the basic factors, and tries in turn to find a feasible set against which to confound each added factor. Often there are several feasible ways in which this can be done. To avoid FKEY selecting the same key every time, you can set the SEED option to an integer that will be used to generate a random permutation of the order in which the sets of basic contrasts are considered, thus producing design randomly selected from all those that might be possible; by default no permutation takes place. Alternatively, you can use the SAVE option to save all the information about the formation of the design; this can then be input later to provide the next possible key (if available) with the requested properties.
In a multi-stratum design, you may wish to insist that some factors are applied to complete units of one of the strata. This can be done using the HIERARCHIES option, which allows you to indicate that some of the added factors must be constant within each combination of levels of other factors. These constraints are specified, if required, by supplying a matrix with a row for each added factor and columns first for the basic factors and then for the added factors. The matrix contains ones in the entries where the row factor must be constant within the combinations of the column factors, and zeros elsewhere.
FKEY can also be used to extend an existing design, by allocating further factors to the units. The existing key should then be input using the INKEY option, with zeros in the rows for the new added factors.
FKEY can form keys for small designs fairly quickly, but for complicated arrangements you may find that it takes some time to check the various possibilities.
Options: BASICFACTORS, ADDEDFACTORS, KEY, INKEY, HIERARCHIES, SEED, ROWPRIMES, COLPRIMES, ROWMAPPINGS, COLMAPPINGS, SAVE.
Parameters: REQUIRED, NONNEGLIGIBLE.
References
Franklin, M.F. (1985). Selecting defining contrasts and and confounded effects in pn-m factorial experiments. Technometrics, 27, 165-172.
Franklin, M.F. & Bailey, R.A. (1977). Selection of defining contrasts and confounded effects in two-level experiments. Applied Statistics, 26, 321-326.
Kobilinsky, A. (1995). PLANOR: Programme de Génération Automatique de Plans d’Expériences Réguliers. INRA, Versailles.
Payne, R.W. & Franklin, M.F. (1994). Data structures and algorithms for an open system to design and analyse generally balanced designs. In: COMPSTAT 94 Proceedings in Computational Statistics (ed. R. Dutter & W. Grossmann), pp. 429-434. Physica-Verlag, Hiedelberg.
See also
Directives: AFMINABERRATION, GENERATE, FPSEUDOFACTORS.
Procedures: AKEY, ARANDOMIZE, ASAMPLESIZE, FACPRODUCT, FBASICCONTRASTS.
Commands for: Design of experiments, Analysis of variance.
Example
" Examples 2:4.13.5a-c "
" Augmented design based on a 4x4 Latin square,
as in Lin & Poushinsky (1983, Biometrics)."
AGLATIN [PRINT=*; ANALYSE=no] NROWS=4; NSQUARES=1; SEED=584578;\
TREATMENTFACTORS=!p(Genotype); ROWS=Row; COLUMNS=Column
AFAUGMENTED [PRINT=design; BLOCKSTRUCTURE=Row*Column;\
LEVTEST=!(5...132); LEVCONTROL=5; GENOTYPES=Genotype;\
NSUBPLOTS=9; SUBCONTROL=5; TESTVSCONTROL=TvsC; CONTROLS=Control
PRINT TvsC,Genotype,Control,Row,Column
" Augmented design based on a balanced-incomplete-block design to
show how to form a design with more than one control per whole-plot."
FACTOR [LEVELS=3; VALUES=1,1,2,2,3,3] Blocks
FACTOR [LEVELS=3; VALUES=1,3,2,3,1,2] Genotypes
AFAUGMENTED [PRINT=design; BLOCKSTRUCTURE=Blocks; LEVTEST=!(101...118);\
GENOTYPES=Genotypes; NSUBPLOTS=8; SUBCONTROL=!(3,6)
" Augmented design with a null basic design, to show how
to form a design with systematic repeating controls."
" design with systematic repeating controls "
FACTOR [LEVELS=32; VALUES=2,6...30] plots
FACTOR [LEVELS=2; VALUES=(1,2)4] genotypes
AFAUGMENTED [SUBPLOTS=plots; LEVTEST=!(3...26);\
GENOTYPES=genotypes; CONTROLS=controls
PRINT plots,genotypes,controls