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