Analyses of variance to show SS for individual effects can also be constructed though the t values provide
equivalent information.

There are some correlations between the different effect estimates and these could be checked by
requesting the correlation matrix for the estimates. For the model fitted here, the correlations are not large
and can be ignored.

8. (Non-complete) Block Designs

We consider here the situation where one or more replicates of a set of treatments are to be divided into
blocks and where the block size is less than the number of treatments. First, suppose that the experimental
treatments are simply an unstructured set or, if there is some structure the important comparisons are
between particular combinations rather than main effects and interactions. Then the division of each
replicate into two or more blocks should be such that the divisions in different replicates are as different as
possible and those treatments whose comparison is more important should tend to occur together in a
block. The sense of "as different as possible" is that the treatments occurring together in a block in one
replicate should be distributed evenly between the various blocks in each other replicate.

For structured treatment sets we first identify the treatment contrasts which are important. In a factorial
structure these will almost always be the main effects, and probably also two-factor interactions. In other
treatment structures the treatment contrasts will correspond to the questions which prompted the choice of
the particular treatments. For these important contrasts we must arrange that each block provides maximal
information. Thus, for a main effect, al-a2, each block should include equal numbers of al and a2
observations. For an interaction effect, (al-a2)(bl-b2), the four combinations, albl, alb2, a2bl, a2b2
should all occur equally frequently in each block. For a contrast between a control group of treatments and
an innovative set of treatments, each block should contain the same proportion of control:innovative.

In some cases the obvious block size does not allow a complete replicate of the set of experimental
treatments to be contained in a set of blocks. In these cases we try to arrange that each pair of treatments
occurs together in a block as nearly equally frequently as possibly. The requirement of equal occurrence for
main effects still applies.

Examples of the construction of designs are included in part B.

9. Precision in Incomplete Block Designs

In incomplete block designs we trade the hope of a reduced value of sigma, the random variance (estimated
by the error mean square) against some loss of information because we cannot compare each treatment
with every other treatment in each block. One exception to this balancing act is confounded designs where
the effects that can be estimated in each block suffer no loss of information to offset the gain from a
smaller value of sigma.

Although we cannot estimate in advance the gain achieved through a reduction in sigma, we can assess the
loss of information from having to compare treatments occurring in different blocks indirectly. If treatment
A occurs in block 1 and treatment B in block 2 and if treatments C,D and E occur in both then A and B
may be compared by comparing each with the average of (C,D,E). The use of the intermediary treatments
reduces the precision of the A-B difference by 33%. In incomplete block designs each treatment occurs
several times and the web of comparisons through intennediaries becomes very complex.

To assess the loss of information from the use of a proposed design we can pre-test the design using a
statistical analysis package. Any package capable of handling the analysis of a general block-treatment
design would provide the information, but the simplest method available at CIMMYT is to use the