ABSTRACT
So far, we have only discussed multiple comparisons for experiments which consist of taking independent simple random samples under the treatments to be compared. However, in addition to measuring the response to the treatments, many real-life experiments incorporate the co-measurement of one or more quantitative variables that might impact on the response. For such experiments, the data analyst should first check whether the relative merits of the treatments depend on the values of these covariates, that is, whether the treatment effect and the covariates interact. Suppose they do not interact. Then one can meaningfully compare the treatments after adjusting for the covariates (which puts the treatments on an equal footing in the comparison). The experimenter might also attempt to increase the sensitivity of the comparison by blocking on one or more qualitative variables. Again, the data analyst should check whether the relative merits of the treatments depend on the blocks, that is, whether the treatment effect and the block effect interact. If they do not, then one can meaningfully compare the treatments after adjusting for block effects.
