Counts how often each pair of treatments occurs in the same block (W. van den Berg).

### Options

`PRINT` = string tokens |
Controls printed output (`concurrences` , `efficiency` ); default `conc` , `effi` |
---|---|

`DIAGONAL` = string token |
What to store on the diagonal of the concurrence matrix (`missingvalues` , `replication` ); default `repl` |

### Parameters

`TREATMENTS` = factors |
Supplies the treatment factor |
---|---|

`REPLICATES` = factors |
Supplies the replicates factor |

`BLOCKS` = factors |
Supplies the block factor |

`CONCURRENCES` = symmetric matrices |
Saves the concurrence matrix, recording the number of times each pair of treatments occurs together in a block |

`EFFICIENCY` = scalars |
Save the efficiency of the design |

### Description

`FOCCURRENCES`

forms a symmetric “concurrence” matrix recording the number of times that each pair of treatments occurs together in the same block. If the treatments all have the same replication, it can also calculate the efficiency of the design, namely the average efficiency factor of the treatment contrasts after eliminating blocks.

The treatment and block factors are supplied by the `TREATMENTS`

and `BLOCKS`

factors, respectively, and the concurrence matrix and the efficiency can be saved by the `CONCURRENCES`

and `EFFICIENCY`

parameters, respectively. For resolvable designs the replicate factor can be supplied using the `REPLICATES`

parameter.

Printed output is controlled by the `PRINT`

option, with settings:

`concurrences` |
to print the concurrence matrix, and |
---|---|

`efficiency` |
to print the efficiency. |

By default, both are printed.

The diagonal of the concurrence matrix usually contains the replication of each treatment i.e. its concurrence with itself. Alternatively, if you are interested only in the concurrences of pairs of different treatments, you can put missing values in the diagonal by setting option `DIAGONAL=missingvalues`

.

Options: `PRINT`

, `DIAGONAL`

.

Parameters: `TREATMENTS`

, `REPLICATES`

, `BLOCKS`

, `CONCURRENCES`

, `EFFICIENCY`

.

### Method

First the treatments-by-blocks incidence matrix `N`

is formed. This contains one in row *i* and column *j* if treatment *i* occurs in block *j*, otherwise it contains zero. The symmetric matrix of concurrences can then be calculated as

`N *+ T(N)`

See John & Williams (1995).

The efficiency is calculated by analysing a y-variate of Normally-distributed random numbers, using `REML`

, with fixed model

`BLOCKS + TREATMENTS`

The efficiency of the design is then calculated as

(σ^{2} × 2) / (*r* × *meanv*)

where σ^{2} |
is the residual variance, |
---|---|

r |
is the replication of the treatments, and |

meanv |
is the mean of the squares of the standard errors of differences of the treatment effects. |

For resolvable designs the block factor Blocks is used and constructed using

`FACPRODUCT !P(REPLICATES, BLOCKS); Blocks.`

### Action with `RESTRICT`

`FOCCURRENCES`

takes account of restrictions on `BLOCKS`

or `TREATMENTS`

.

### Reference

John, J.A. & Williams, E.R. (1995). *Cyclic and Computer Generated Designs, 2nd edition*. Chapman & Hall, London.

### See also

Procedures: `AFCYCLIC`

, `AGBIB`

, `AGCYCLIC`

.

Commands for: Design of experiments.

### Example

CAPTION 'FOCCURRENCES example'; STYLE=meta AGALPHA [PRINT=design] 24; NREPLICATES=3; NBLOCKS=6; TREATMENTS=Treat;\ REPLICATES=Rep; BLOCKS=Block; UNITS=Plot; SEED=-1 FOCCURRENCES Treat; REPLICATES=Rep; BLOCK=Block