where k is the average number of times that treatment pairs occur together in a block.


An excellent approximation to the average variance for treatment pair differences is then

 VAR = MINVAR + (MAXVAR MINVAR)/t

 where t is the number of treatments.

 For the example r = 3. The number of pairwise comparisons in each block is 15, so that over the set of
 blocks there are 6x15 = 90 pairwise comparisons within blocks and there are 66 treatment pairs. Thus k =
 90/66. Hence

 MINVAR = 02 x2/3 = 0.66702

 MAXVAR = 02 x(2x66/90) = 1.467o2

 VAR = 02 0.667 + o2 (1.467 0.667)/12

 = 0.73302

giving an approximate standard error of 0.856 a.

10. Loss of Plots/Sites

The loss of individual plot data causes, if anything, rather less problem for incomplete block designs than
for complete block designs. For complete block designs the pattern of block-treatment structure is no
longer complete when plots are missing and the data should be analysed as an incomplete block design.
The alternative of estimating missing values is only approximately correct, is a throwback to the pre-
computer days, and should not be necessary. For incomplete block designs the loss of plot data produces a
different incomplete block design but no change in principle. We simply analyse the data we do have.

In either situation, if several, or all, plots of a particular treatment are lost the information about that
treatment is badly affected, but the two design types suffer equally. The loss will be reduced when factorial
treatment structure is used (complete or incomplete) because of the hidden replication benefits.

When different treatment subsets are used at different locations the total loss of some sites should not cause
problems provided the different treatment subsets have been chosen so that each location provides
information on all or most of the factor main effects and two-factor interactions.

11. Analysis and Computers

The analysis of experimental data is, rightly, increasingly handled by the use of computer packages. These
vary from those which can analyse a very restricted set of tightly specified designs to general statistical
packages which can handle almost any design structure.

Where computer facilities are available the analysis of incomplete block designs can be managed using a
general package, the most powerful being REML (the form of analysis information being illustrated in the
example attached to this document), GENSTAT and, rather less informatively, by SAS. In the absence of
any of these packages any block-treatment design structure can be analysed by a multiple regression
package, as illustrated in my report on my 1989 consultancy, by defining a regression variable for each
block and each treatment except block I and treatment 1. The regression coefficients then estimate the
difference of each block, or treatment, from the first block, or treatment.