Find an expression for the variance of jr and $2»when 20 subjects are assigned to each treatment.

A psychologist is planning a three-group study in which he wants to test the following two comparisons: Group 1 versus Group 3 (1^1) and Group 2 versus Group 3 (V^)- Sixty subjects are available to participate in the study. His initial thought was to assign 20 subjects at random to each condition. However, after further thought, he has decided to assign twice as many subjects to the third group as to the first two groups because the third group is involved in both comparisons. (Notice that subjects are still randomly assigned to conditions.) Is this a good idea? To explore the answer to this question, we must consider the variances of the two contrasts. Why the variances? Both sample size-allocation schemes produce unbiased estimates of the population value ^ of the contrast in question. However, the two schemes differ in the imprecision—that is, the variance—of the estimate. It can be shown that (assuming homogeneity of variance) the population variance of a contrast is given by a. Find an expression for the variance of jr and $2»when 20 subjects are assigned to each treatment. b. Find an expression for the variance of $1 and $2 when 15 subjects are assigned to Group 1, 15 to Group 2, and 30 to Group 3. c. Which method of allocating subjects to groups is better for testing jr and 1/^2 if homogeneity holds? d. Will any allocation scheme yield a smaller variance than the two schemes already proposed? Consider the possibility of assigning 18 subjects to Group 1,18 subjects to Group 2, and 24 subjects to Group 3. Find the variance of fr and $2 a°d compare your answer to the answers you obtained in parts a and b. e. All other things being equal, variance is minimized by assigning an equal number of subjects to each group. How does this help explain why the 18, 18, 24 scheme results in a lower variance than does the 15, 15, 30 scheme?


 

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