Which approach do you think would generally be preferable—that of part a or part b? Why?
I. B. Normal, a graduate student at Skew U., conducted a study with four groups. The first three groups are treatment groups, and the fourth group is a control group. The following data are obtained: a. Normal’s adviser says that the first question Normal should address is whether the mean of the treatment subjects differs from the mean of the control subjects. The adviser tells her to perform a t test comparing the 18 treatment subjects to the 6 control subjects. In other words, the adviser recommends that the three treatment groups be combined into one group, ignoring (for this analysis) the distinction among the three treatment groups. What did Normal find? (HINT: It will be helpful for parts c and d that follow if you analyze these data as a one-way ANOVA, using the principles discussed in Chapter 3.) b. Normal was rather disappointed with the result she obtained in part a. Being the obsessive type, she decided also to test a contrast whose coefficients were 1,1,1, and —3. What did she find? c. Why are the results to parts a and b different? After all, they both compare treatment subjects to control subjects. To see why the results differ, we look at the numerator and the denominator of the F statistic individually. How does the value of the sum of squares for the contrast in part b compare to the value of the sum of squares between groups in part a? d. How does the value of the within-group sum of squares in part b compare to the value of the within-group sum of squares in part a? Notice that the within-group sum of squares in part b is based on four groups, whereas the within-group sum of squares in part a is based on only two groups. As a consequence, the full model in part b has four parameters to be estimated, whereas the full model in part a has only two parameters. e. Verify that the following expressions provide the correct sums of squares (within rounding error) for the full models in parts a and b. For part a: f. The between-group sum of squares for differences among the three treatment groups equals 16 for these data. How does this relate to the difference in the two approaches? Why? g. Which approach do you think would generally be preferable—that of part a or part b? Why?
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