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# AGNONORTHOGONALDESIGN procedure

Generates non-orthogonal split-plot and other hierarchical designs (B. M. Parker).

### Options

`PRINT` = string token Controls printed output (`design`, `debug`); default `*` i.e. nothing Specifies the algorithm to use (`jonesgoos`, `trincagilmour`); default `trin` Optimality criterion (`a`, `d`); default `a` Defines the model to be estimated Number of random starts for the `jg` algorithm; default 10 Number of exchanges to try from each start; default 10000 Minimum value for levels; default -1 Maximum value for levels; default 1 Seed for the random numbers used by the algorithms; default 0

### Parameters

`BLOCKFACTOR` = factors Specifies the identifier for the block factor used to index the units of the whole-plots, the sub-plots and, if required, the sub-sub-plots Specifies the identifier of the treatment factor or factors applied to the whole, sub-plots and sub-sub-plots Numbers of levels for the block factors Numbers of levels for the treatment factors Variances for the strata

### Description

Many industrial and agricultural experiments involve some factors whose levels are harder to set than others. For example, in an agriculture experiment, some factors may only be applied to whole-plots, whereas some may be applied to sub-plots. This agricultural background means these are referred to as “split-plot” experiments. Where a further sub-sub-plot factor is investigated, these are known as split-split-plot experiments. In industrial or laboratory applications, the designs are known as “multi-strata” designs. The factors that are hardest to set should be in stratum 1, the next hardest in stratum 2, and so on. The results of these experiments are typically analysed using a mixed-model analysis. As randomization is restricted, care must be taken to find designs which are efficient, in the sense that the unknown parameters in a model are estimated well. (Note, though, that this restriction on randomization often precludes other desirable qualities, such as orthogonality.)

The algorithm to use is specified by the `METHOD` option. The default setting, `trincagilmour`, uses the algorithm of Trinca & Gilmour (2014). This can be used for designs with any number of strata (although in practice, designs with more than three strata are rare). It works on each stratum in turn to find a design which is optimal for that stratum, together with any terms that appear in that stratum and higher strata. The next lowest stratum is then considered, until all strata are exhausted. The `NTRIES` option specifies how many random exchanges to attempt; default 10,000.

The `CRITERION` option specifies the optimality criterion to use in the Trinca & Gilmour algorithm to assess the quality of designs: either As (default) or Ds. The aim is to estimate a function of the parameters of the model with maximum efficiency. Block effects are considered nuisance parameters, and in the case of As optimality in a second order model, quadratic effects are weighted as 0.25 and other effects are weighted as 1.

An important unknown factor in designing multi-strata experiments correctly is the ratio of (usually unknown) variances at each stratum level. A locally optimal design could, in theory, be found for each value of this variance ratio. Whilst the Trinca & Gilmour algorithm does not find an absolutely locally optimal design, in practice locally optimal designs can perform poorly if the variance ratio is mis-specified. The algorithm finds designs that are robust against mis-specification of the variance ratio, and which should perform well when the ratio is unknown (which is the usual situation in practice). As well as being robust, the algorithm is quick to run, even for large designs.

Alternatively, setting `METHOD=jonesgoos`, selects the algorithm of Jones & Goos (2007). This implements an exchange algorithm to calculate (locally) D-optimal designs for split-plot designs. It is a candidate-set free algorithm, which helps to make it relatively fast to run. An important parameter that must be specified, is the variance ratio between the whole-plots and the sub-plots. In general, if a D-optimal split-plot design is required, the variance ratio should be known. The Jones and Goos design should then be better than a Trinca and Gilmour design. The `NSTARTS` option specifies the number of random starts to use with the Jones & Goos algorithm; default 10

The `BLOCKFACTOR` parameter lists the block factors: first the factor to index the units of the whole plots, then a factor to index the sub-plots, and so on, as required. The `BLEVELS` parameter must be set to specify the numbers of levels of the block factors, and the `VARIANCES` parameter can be set to supply the stratum variances. The number of plots (or runs) in the experiment is specified by the `NUNITS` option.

The `TREATMENTFACTORS` parameter defines factors for the treatments applied to the units of the strata, and the `LEVELS` defines their numbers of levels. If several factors are to be applied to a particular stratum, the factors and their levels should each be put into a pointer.

The `MINIMUM` and `MAXIMUM` options specify minimum and maximum possible values, respectively, for the treatment factors; default -1 and 1.

The `MODELMATRIX` option defines a polynomial model that will be fitted to the results of the experiment. This is a matrix, with a row for each model term, and a column for each treatment factor. The entries in the rows specify the powers of the factors involved in the corresponding polynomial term.

The `SEED` option specifies the seed for the random numbers used by the algorithms. The default of 0 continues an existing sequence or, if none, obtains a seed automatically from the system clock.

The `PRINT` option can be set to `design`, to print the design. There is also a setting `debug` to provide debugging information for the algorithm.

Options: `PRINT`, `METHOD`, `CRITERION`, `MODELMATRIX`, `NSTARTS`, `NTRIES`, `MINIMUM`, `MAXIMUM`, `SEED`.

Parameters: `BLOCKFACTOR`, `TREATMENTFACTORS`, `BLEVELS`, `LEVELS`, `VARIANCES`.

Jones, B. & Goos, P. (2007). A candidate set free algorithm for generating D optimal split plot designs. Applied Statistics, 56, 347-364.

Trinca, L.A. & Gilmour, S.G. (2014). Improved Split-Plot and Multi-Stratum Designs. Technometrics, DOI:10.1080/00401706.2014.915235.

Procedure: `AGHIERARCHICAL`.

Commands for: Design of experiments, Analysis of variance.

### Example

```CAPTION 'AGNONORTHOGONALDESIGN example',\
'Non-resolvable split-plot design for 5 factors.';\
STYLE=meta,plain
" all linear 2-way interactions (linear main effects included automatically) "
MATRIX  [ROWS=10; COLUMNS=5; VALUES=\
1,1,0,0,0, 1,0,1,0,0, 1,0,0,1,0, 1,0,0,0,1,\
0,1,1,0,0, 0,1,0,1,0, 0,1,0,0,1,\
0,0,1,1,0, 0,0,1,0,1, 0,0,0,1,1] modelmat
AGNONORTHOGONAL [PRINT=design; MODEL=modelmat; NTRIES=1000; SEED=129896]\
BLOCKFACTOR=wplot,subplot; TREATMENTFACTORS=a,!p(b,c,d,e);\
BLEVELS=21,2; LEVELS=3,!p(3,3,3,3)
```