Performs screening tests for designs with orthogonal block structure (R.W. Payne).
Options
PRINT = string tokens |
Which tests to print (conditional, marginal, efficiency); default cond, marg |
|---|---|
FACTORIAL = scalar |
Limit on the number of factors in each treatment term; default 3 |
EXCLUDEHIGHER = string token |
Whether to exclude higher-order interactions in the initial model for the conditional test of each term (yes, no); default no |
FORCED = formula |
Terms that must be included (together with any covariates) in the initial models for every term; default * i.e. none |
Parameter
Y = variates |
Variates to be analysed |
|---|
Description
ASCREEN can be used to assess the treatment terms in an analysis of variance when the design is unbalanced but its error terms that are all orthogonal to one another. This includes any design with a hierarchical block structure, for example
Blocks / Plots
or
Replicates / Wholeplots / Subplots
ASCREEN thus provides a way of testing treatment terms in designs that cannot be analysed by ANOVA. Once ASCREEN has been used to decided which terms need to be included in the treatment model, the treatment effects and means can be estimated using REML.
Before using ASCREEN, the block and treatment models for the design must be defined by the BLOCKSTRUCTURE and TREATMENTSTRUCTURE directives, in exactly the same way as for an analysis by ANOVA. As in ANOVA, the FACTORIAL option sets a limit on number of factors in each treatment term (default 3). You can also define covariates using the COVARIATE directive. The y-variate is specified by the Y parameter of ASCREEN.
ASCREEN forms marginal and conditional tests for the treatment terms similar to those produced by the RSCREEN procedure. These are produced for the analysis of each stratum of the design (i.e. for the variation associated with each error term).
In a marginal test, each term is assessed by adding it to the simplest possible model. So, with a treatment model of
A + B + C + D + A.B + A.C + A.D + B.C + C.D + A.B.C + A.B.D + A.C.D + B.C.D + A.B.C.D
the main effect of A is added it to the null model, while the interaction term A.B is added to a model containing only the main effects of A and B.
In a conditional test, each term is added to the most complex possible model. So the main effect A is added to an initial model excluding any term that has A as one of its margins. A is a margin of any term that contains A as one of its factors. So the terms to exclude for A are A.B, A.C, A.D, A.B.C, A.B.D, A.C.D and A.B.C.D. Similarly the interaction A.B is added to a model excluding any term that has A.B as a margin; i.e. any term that contains A and B amongst its factors. So A.B.C, A.B.D and A.B.C.D are excluded with A.B. The other terms to be included in the initial model depend on the setting of the EXCLUDEHIGHER option. With the default setting of no, all other terms are included in the initial model. So, the initial model for A would be
B + C + D + B.C + C.D + B.C.D
Alternatively, if EXCLUDEHIGHER=yes, the initial model contains only terms with no more factors than the term being tested. So, the initial model for A would be
B + C + D
The FORCED option lets you specify a model formula with terms that must be included in the initial model for the conditional and marginal tests of every treatment term. The forced model automatically includes any covariates.
The PRINT option controls printed output. The settings marginal and conditional control which tests are produced if there is more than one stratum (or error term); by default both types of test are printed. However, if there is only one error term, ASCREEN uses procedure RSCREEN, which always prints both. There is also a setting, efficiency, which prints the minimum, maximum and harmonic mean efficiency factor of the terms in each of the strata if there is more than one. These efficiency factors show the amount of information available to construct the marginal test for each of the terms in the strata where it can be estimated. The harmonic mean is presented, rather than an ordinary average, as this corresponds to the average variance of differences amongst the effects of the term (remember that the variance is proportional to the reciprocal of the efficiency factor).
Options: PRINT, FACTORIAL, EXCLUDEHIGHER, FORCED.
Parameter: Y.
Method
ASCREEN uses RSCREEN if there is only one error term. Otherwise, it first uses ANOVA to check that the design has orthogonal block structure. Then, if so, it calculates the relevant sums of squares by regression with matrices of weights calculated using FPROJECTIONMATRIX. The weight matrix for each stratum is its projection matrix; for further details see Payne & Tobias (1992).
Action with RESTRICT
ASCREEN 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
Directives: ANOVA, REML.
Procedure: RSCREEN, VSCREEN.
Commands for: Analysis of variance, REML analysis of linear mixed models.
Example
CAPTION 'ASCREEN 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; FPROB=yes] Yield
CAPTION !t('The treatments are orthogonal,',\
'so the marginal and conditional tests are identical and',\
'the results match those in the analysis of variance table.')
ASCREEN Yield
CAPTION !t('Analysing only blocks 1, 3, 5 and 7,',\
'the treatments become non-orthogonal.')
RESTRICT Yield; Blocks.IN.!(1,3,5,7)
ASCREEN Yield