# williams: Construction of Williams Designs In crossdes: Construction of Crossover Designs

## Description

The function constructs williams designs. Williams designs are row-column designs. They are used if each of the treatments in the study is given to each of the subjects. If the number of treatments to be tested is even, the design is a latin square, otherwise it consists of two latin squares.

## Usage

 `1` ```williams(trt) ```

## Arguments

 `trt` An integer > 1, giving the number of treatments in the design.

## Details

The resulting design is a (generalized) latin square that is also balanced for first order carryover effects. Carryover balance is achieved with very few subjects. In the experimental design the treatments are numbered 1,...,trt. The entry (i,j) of the design corresponds to the treatment the i-th subject gets in the j-th period.

## Value

A matrix representing the experimental design.

Oliver Sailer

## References

Wakeling, I.N. and MacFie, H.J.H. (1995): Designing consumer trials balanced for first and higher orders of carry-over effect when only a subset of k samples from t may be tested. Food Quality and Preference 6, 299-308.

Williams, E. J. (1949): Experimental designs balanced for the estimation of residual effects of treatments. Australian Journal of Scientific Research, Ser. A 2, 149-168.

`get.plan`

## Examples

 ```1 2``` ```williams(3) williams(10) ```

### Example output

```Loading required package: AlgDesign
[,1] [,2] [,3]
[1,]    1    2    3
[2,]    2    3    1
[3,]    3    1    2
[4,]    3    2    1
[5,]    1    3    2
[6,]    2    1    3
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    1    2   10    3    9    4    8    5    7     6
[2,]    2    3    1    4   10    5    9    6    8     7
[3,]    3    4    2    5    1    6   10    7    9     8
[4,]    4    5    3    6    2    7    1    8   10     9
[5,]    5    6    4    7    3    8    2    9    1    10
[6,]    6    7    5    8    4    9    3   10    2     1
[7,]    7    8    6    9    5   10    4    1    3     2
[8,]    8    9    7   10    6    1    5    2    4     3
[9,]    9   10    8    1    7    2    6    3    5     4
[10,]   10    1    9    2    8    3    7    4    6     5
```

crossdes documentation built on May 30, 2017, 4:14 a.m.