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
(σ2 × 2) / (r × meanv)
| where σ2 | 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