Essentials

SevenFACTORIAL ANOVATWO-WAY ANOVAInteraction ContrastsSuppose you are testing a completely new drug for depression. You are happy thata t test shows the drug group to be significantly lower in depression than theplacebo group at the end of the study. To explore your results further you decideto rerun your t test just for the men in the study, and then again just for thewomen. Although you have equal numbers of men and women in both conditions,you find that the t test for men just misses significance at the .05 level,whereas the t test for women is easily significant. Does this mean that the femalepopulation is more responsive to the drug than the male population? As wepointed out in the previous chapter, you cannot conclude this. In fact, the true effectsizes for men and women could easily be identical with accidental differencesin sampling explaining the difference in results. However, even if the t tests forboth genders were significant (or both not significant), the effect sizes for menand women could be quite different, and it would be useful to know if this werethe case. There should be a way to test this, and there is. One way is to test the differencein the male and female results by means of a contrast, as described inChapter 5.First, let us look at some possible means for the drug/placebo experimentbroken down by gender (see Table 7.1). The relevant contrast is to take the femaledifference of 18 points (i.e., 40 – 22) and subtract from it the male difference of8 points (i.e., 36 – 28); the difference of these two differences is 10 points, so Lequals 10 for this contrast. More formally, the contrast can be written as followsL 1 X male/placebo– 1 X male/drug– (1 X female/placebo– 1 X female/drug). Noticethat all four of the coefficients are 1 or –1, so Σc i2 4. If there are 20 participantsper group, SS contrast nL 2 / Σc i2 20 100/4 500, which also equalsMS contrast. The error term for this contrast is MS w, which in this case would be anaverage of the four group variances.145