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