1/5/2009
Monday morning

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Suppose you are interested in differences in weight loss as a function of the several diets. For each person, you have 3 weight-loss values, being the difference between measurements at #1 and #3 above, between #5 and #7, and between #9 and #11. For conceptual simplicity, lets take those as differences between the averaged values. (What _kind_ of averages may also be up for grabs; the median of each set of 42 values might be best, as being relatively uninfluenced by unexpected, and perhaps unaccountable, extreme values for an individual.) These values are in a Latin square structure that can be displayed thus:
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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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1. Pretest weight (42 readings, averaged) 2. 10 weeks of Diet A (or Diet B, or Diet C, in the other two groups)_ 3. Post-diet weight #1 (42 readings, averaged) 4. 10 weeks of uncontrolled non-diet 5. New non-diet baseline weight #2 (42 readings, averaged) 6. 10 weeks of Diet B (or Diet C, or Diet A, respectively) 7. Post-diet weight #2 (42 readings, averaged) 8. 10 weeks of uncontrolled non-diet 9. New non-diet baseline weight #3 (42 readings, averaged) 10. 10 weeks of Diet C (or Diet A, or Diet B, respectively) 11. Post-diet weight #3 (42 readings, averaged)
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