1/5/2009
Monday morning

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The Group main effect represents any systematic differences due to whether the sequence of diets was (A,B,C) or (B,C,A) or (C,A,B). The Diet main effect represents the effects presumably of interest, that is, whether the three diets differed in amount of weight loss observed. The Sequence effect shows whether being the 1st, 2nd, or 3rd diet in the temporal sequence makes any difference. The remaining 2 d.f. for assorted interactions is unlikely to be interesting, but if significant might indicate something that might be worth pursuing. (The problem with interactions in a Latin square is that GD interactions are confounded with the S main effect, GS with D, and DS with G.) (The mean square for Respondents within Group is the error term, or denominator mean square, against which the mean square for Group is tested; the mean square for Diet X Respondents within Group is the error mean square for Diet, Sequence, and the remaining interaction mean square.)
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So far weve considered weight loss as the simple difference between average pre-diet weight and average post-diet weight. But you migh wish also to consider relative weight loss: that difference as a proportion of pre-diet weight. Same structure and analysis, possibly different results. Or use pre-diet weight as a covariate for the simple weight loss as a response variable. There may also be interesting comparisons to be made between the several pre-diet average weights; one can imagine that these might differ depending on which diet preceded them, or on their position in the temporal sequence.
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[I might have argued for a further 10 weeks of non-diet, with another weighing at the end of it; but that would depend on whether your interest be only in the weight loss attributable to a particular diet, or to the persistence of that loss after one has stopped dieting.]
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