Calculates efficiency factors for experimental designs (R.W. Payne).
Options
FACTORIAL = scalar |
Limit on the number of factors in each treatment term generated from TERMS; default 3 |
|---|---|
METHOD = string token |
Whether to eliminate or ignore earlier model terms from the TERMS formula (eliminate, ignore); default elim |
FORCED = formula |
Terms to be eliminated before fitting TERMS; default * i.e. none |
Parameters
TERMS = formula |
Model terms |
|---|---|
DF = pointer or scalar |
Saves the degrees of freedom of the terms |
EFFICIENCY = pointer or variate |
Saves the efficiency factors of the terms |
DFALIASED = pointer or scalar |
Saves the number of aliased degrees of freedom of the terms |
Description
The efficiency factors of a model term represent the proportion of the information about various contrasts amongst its effects that remains available for estimating the contrasts, after fitting the earlier terms in the analysis. If the term is balanced, the efficiency factors will all be equal. If not, their range gives an indication of the degree of imbalance.
The model terms of interest are specified by the TERMS parameter. You can also use the FORCED option to specify a set of model terms that must be eliminated before those in TERMS are fitted. By default, the efficiency factors are calculated under the assumption that the model terms in TERMS are to be fitted sequentially. So, each term is estimated eliminating the earlier terms in TERMS. Alternatively, you can set option METHOD=ignore to calculate the efficiency factors for the terms eliminating only their marginal terms and the terms in the FORCED formula. (Marginal terms are terms whose factors are a subset of those in the term: e.g. the main effects A and B are marginal terms of the interaction A.B.)
The EFFICIENCY parameter saves the efficiency factors. If the TERMS parameter specifies a single term, EFFICIENCY must be undeclared or set to a variate. If TERMS specifies several terms, you must supply a pointer which will then be set up to contain as many variates as there are terms. Similarly the DF parameter can save the numbers of degrees of freedom of each term, and the DFALIASED parameter can save the numbers of degrees of freedom of each term that are aliased either with terms in the FORCED formula or with terms that come before it in the TERMS formula.
Options: FACTORIAL, METHOD, FORCED.
Parameters: TERMS, DF, EFFICIENCY, DFALIASED.
Method
The efficiency factors are the eigenvalues of the matrix TST, where T is the projection matrix for the model term, and S is the projection matrix into the space orthogonal to the previous terms. The corresponding contrasts are the eigenvectors of the matrix. See Payne & Tobias (1992), Section 4.
AEFFICIENCY uses the FPROJECTIONMATRIX procedure to form projection matrices for the model terms. Marginal terms are eliminated using Equation (2.7) of Payne & Tobias (1992), amd the efficiency factors are calculated by an eigenvalue decomposition as in Equation (4.9).
Action with RESTRICT
AEFFICIENCY takes account of any restrictions on the y-variate.
Reference
Payne, R.W. & Tobias, R.D. (1992). General balance, combination of information and the analysis of covariance. Scandinavian Journal of Statistics, 19, 3-23.
See also
Directive: ANOVA.
Procedure: ASWEEP.
Commands for: Analysis of variance.
Example
CAPTION 'AEFFICIENCY example',\
'Data in the Guide to Genstat, Part 2, Example 4.7.1';\
STYLE=meta,plain
FACTOR [NVALUES=32; LEVELS=8] Blocks
& [LEVELS=4] Plots
& [LEVELS=2; LABELS=!T(_,n)] N
& [LABELS=!T(_,k)] K
& [LABELS=!T(_,d)] D
GENERATE Blocks,Plots
READ [PRINT=errors] N,K,D; FREPRESENTATION=labels
_ _ _ n k _ n _ d _ k d n _ _ _ k _ _ _ d n k d
n _ _ _ k _ n _ d _ k d _ _ _ _ _ d n k _ n k d
n _ _ _ _ d n k _ _ k d _ _ _ _ k _ n _ d n k d
_ k _ _ _ d n k _ n _ d _ _ _ n _ _ _ k d n k d :
VARIATE Yield
READ [SETNVALUES=yes] Yield
101 291 373 398 106 265 312 450
89 272 338 407 106 324 306 449
128 323 334 423 87 279 324 471
302 324 272 361 131 103 445 437 :
BLOCKSTRUCTURE Blocks/Plots
TREATMENTSTRUCTURE N * K * D
ANOVA [PRINT=aov,info; FPROB=yes] Yield
AEFFICIENCY [FORCED=Blocks] N*K*D; EFFICIENCY=ef
PRINT ef[]; FIELD=8