By D. J. Hand, C. C. Taylor (auth.)
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Additional resources for Multivariate Analysis of Variance and Repeated Measures: A practical approach for behavioural scientists
Contrast 3(i) described the (treatment-control) difference. Contrast 20(i) describes the way this difference itself differs between females and males. It is thus comparing how the (treatment-control) effect changes as we change levels of the sex factor - that is, it describes an interaction between the two factors. Since there are also possibly differences between the three treatment categories, these differences also might differ according to sex. That is, there might be treatment category-by-sex interactions.
6 we described the process of testing the value of a single pattern of differences. This pattern may have been part of a complete comparison, so that the contrast elements were applied directly to the sample means, or the pattern may have involved more-complicated weighting procedures to derive the estimate. In either case, exactly the same procedures and tests apply in a multi-factor design as in the single-factor case. 7, also applies to the multi-factor case. In particular, we also have the choice of whether or not to pool error variation and within-group variance for the denominator of the F-test.
We have already seen some of the advantages of having equal numbers of subjects. Obviously simple averages over artificially manipulated numbers of subjects will have little meaning and such a situation will preclude the use of the simpler analysis. Thus, while both questions have meaning and may be relevant, it is our experience that the second kind, where effects are explored controlling for other factors, is the more common in the behavioural sciences. Note that we can put this notion of 'controlling for the sex effect' the other way round and ask: is sex alone adequate to explain the observed treatment effect?