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# AGBOXBEHNKEN procedure

Generates Box-Behnken designs (R.W. Payne).

### Options

`PRINT` = string token Controls printed output (`design`); if unset in an interactive run `AGBOXBEHNKEN` will ask whether the design is to be printed, in a batch run the default is not to print anything Defines the number of central points to include; default 4 Defines the outer levels to be used; default `!(-1,1)` Number of factors to vary in combination at once; default 2 Seed to be used to randomize each design; a negative value implies no randomization Saves a command to recreate the design (useful if the design information has been specified in response to questions from `AGBOXBEHNKEN`)

### Parameter

`TREATMENTFACTOR` = factors Treatment factors

### Description

Box-Behnken designs are often used to study response surfaces. The design is usually formed to allow a quadratic response surface to be fitted. The factors are studied at three equally-spaced levels, below denoted by -1, 0 and 1. The construction uses a balanced incomplete block design to select successive sets of factors to be applied at all factorial combinations of -1 and +1, while other factors are held at 0. For example, with three factors `A`, `B` and `C`, the relevant balanced incomplete block design would have three blocks (`A`,`B`), (`A`,`C`) and (`B`,`C`). So the design would first have a section with `A` and `B` varying but `C` constant

` A B C`

`-1 -1 0`

`-1 +1 0`

`+1 -1 0`

`+1 +1 0`

then a section where `B` is held constant but `A` and `C` take all combinations of -1 and +1

` A B C`

`-1 0 -1`

`-1 0 +1`

`+1 0 -1`

`+1 0 +1`

and finally a section with `A` constant

` A B C`

` 0 -1 -1`

` 0 -1 +1`

` 0 +1 -1`

` 0 +1 +1`

In addition, there can be some “central points”, where all the factors take the central value

` A B C`

` 0 0 0`

` 0 0 0`

` 0 0 0`

` 0 0 0`

The treatment factors are listed using the `TREATMENTFACTOR` parameter. If this is omitted in an interactive run, you will be asked how many factors you want and their names. The number of central points is specified by the `NCENTRALPOINTS` option; by default this is taken to be four. The `LEVELS` option can supply a variate to specify the outer treatment levels; the defaults are 1 and -1 (so the central point is at zero). The `NCOMBINATIONS` option defines the number of factors whose combinations of (outer) levels are to be varied at once. For the default of two, the relevant balanced incomplete block design is formed within `AGBOXBEHNKEN`. Other values can be supplied, but the corresponding balanced incomplete block design must be one of those obtainable from procedure `AGBIB`. You can find out the possibilities by putting

`AGBIB [PRINT=catalogue]`

The `SEED` parameter allows you to specify a seed to be used to randomize the design. In batch the default seed is -1, to suppress randomization. If you do not set `SEED` when running interactively `AGBOXBEHNKEN` will ask for a seed, and again a negative value suppresses any randomization. The `PRINT` option can be set to `design` to print the plan of the design. By default, if you are running Genstat in batch, the plan is not printed. If you do not set `PRINT` when running interactively, `AGBOXBEHNKEN` will ask whether or not you wish to print the design.

The `STATEMENT` option allows you to save a Genstat text structure containing a command to recreate the design. This is particularly useful if `AGBOXBEHNKEN` is being used interactively, and the information to define the design has been provided in response to questions from the procedure.

Options: `PRINT`, `NCENTRALPOINTS`, `LEVELS`, `NCOMBINATIONS`, `SEED`, `STATEMENT`.

Parameter: `TREATMENTFACTOR`.

### Method

The `QUESTION` procedure is used to obtain the necessary details of the design and this is then generated by the standard Genstat manipulation directives and procedure `AGBIB`.

Directive: `AFRESPONSESURFACE`.

Procedures: `AFNONLINEAR`, `AGCENTRALCOMPOSITE`, `AGMAINEFFECT`, `RQUADRATIC`.

Commands for: Design of experiments, Regression analysis.

### Example

```CAPTION      'AGBOXBEHNKEN example'; STYLE=meta
AGBOXBEHNKEN [PRINT=design] A,B,C,D
```
Updated on June 20, 2019