Statistical Methods in Medical Research 4ed

256 Experimental design
interactions between any two of the factors concerned are rather artificial
concepts.
Random effects
If, in the previous example, A and C were fixed-effect factors and B was a
random-effect factor, the presence of an interaction between B and C would
not preclude an interest in the main effect of CÐregarded not as an average over
the particular levels of B chosen in the experiment, but as an average over the
whole population of potential B levels. Under certain conditions (discussed
below) the null hypothesis for the main effect of C is tested by comparing the
MSq for C against the MSq for the interaction BC. If C has more than two levels,
it may be more informative to concentrate on a particular contrast between the
levels of C (say, a comparison of level 1 with level 4), and obtain the interaction
of this contrast with the factor B.
If one of the factors in a multifactor design is a blocking system, it will
usually be natural to regard this as a random-effect factor. Suppose the other
factors are controlled treatments (say, A, B and C). Then each of the main effects
and interactions of A, B and C may be compared with the appropriate interaction
with blocks. Frequently the various interactions involving blocks differ by
no more than might be expected by random variation, and the SSq may be
pooled to provide extra DF.
The situations referred to in the previous paragraphs are examples in which a
mixed model is appropriateÐsome of the factors having fixed effects and some
having random effects. If there is just one random factor (as with blocks in the
example in the last paragraph), any main effect or interaction of the other factors
may be tested against the appropriate interaction with the random factor; for
example, if D is the random factor, A could be tested against AD, AB against
ABD. The justification for this follows by interpreting the interaction terms
involving D in the model like (9.7) as independent observations on random
variables with zero mean. The concept of a random interaction is reasonable;
if, for example, D is a blocking system, any linear contrast representing part of a
main effect or interaction of the other factors can be regarded as varying
randomly from block to block. What is more arguable, though, is the assumption
that all the components in (9.7) for a particular interaction, say, AD, have
the same distribution and are independent of each other. Hence the suggestion,
made above, that attention should preferably be focused on particular linear
contrasts. Any such contrast, L, could be measured separately in each block and
its mean value tested by a t test.
When there are more than two random factors, further problems arise
because there may be no exact tests for some of the main effects and interactions.
For further discussion, see Snedecor and Cochran (1989, §16.14).