statistical package REML. The method is illustrated in the attached output for a design comparing twelve
treatments in six blocks of six plots per block which is attached to the end of this document (3A).

The information required by REML is the block and treatment identification for each plot and a set of data
values. For the illustration the plot allocation to blocks is (in plot order)

 S 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3
 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6

 and the treatment allocation (same order) is

 1 2 7 8 9 11 3 4 5 6 10 12 1 3 6 8 10 11
 2 4 5 7 9 12 1 4 6 7 9 10 2 3 5 8 11 12

 For data we use any set of simple numbers (all different)

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

 The output summarises the standard errors for comparing pairs of treatments by giving the average,
 maximum and minimum standard errors. Since we have used nonsense data these will be nonsense
 standard errors. However, the relative values will be correct because relative precision depends only on the
 design and not on the data. Further REML prints the value of a2, the random variance. If we divide each
 standard error by the square root of sigma squared we obtain the standard errors as multiples of sigma. We
 can therefore assess exactly the loss of information which we expect to more than outweigh by the
 reduction in the value of sigma

 In the attached example the standard errors for the nonsense data are

 AVERAGE 0.385 MAXIMUM 0.404 MINIMUM 0.361

 The value of a2 is 0.196 so that a is 0.443 and the standard errors are a multiplied by

 AVERAGE 0.869 MAXIMUM 0.912 MINIMUM 0.815

 Note first that the range of standard errors is about 5% either side of the average so that a single standard
 could be used in summarising the results of the experiment. Second the minimum possible standard error
 for comparing two treatments with three observations each is

 a4(2/3) = 0.816o

which is the same (apart from rounding error) as the minimum achieved in our example design.

In general incomplete block designs are much more efficient than might initially be expected. A rough
guide to precision in incomplete block designs is derived by considering the minimum possible variance
between two treatment means

 MINVAR = 02 x 2/r

where r is the replication per treatment;
and the maximum possible variance which is

 MAXVAR = 02 x 2/k