Repeat Exercise 17.4 with the “Type × Sex” interactions omitted from all models. Compare the sums of squares, the ordinary means, and the least squares means with those obtained with interaction effects in the models.
Exercise 17.4
Exercise 9.13 used data on survival time of patients with different types of cancer (Cameron and Pauling, 1978). The data are crossclassified with unequal numbers if both sex of patient and cancer type are considered. Use the logarithm of the ratio of days survival of the treated patient to the mean days survival of his or her control group as the dependent variable. (In the following analyses, include interaction effects between sex of patient and type of cancer in your models, but ignore differences in age.)
(a) Do an unweighted analysis of cell means to investigate the effects of sex, cancer type, and their interaction. Compute the within-cell variance and the harmonic mean of the numbers of observations, and summarize the results in an analysis of variance table. Note that Type I and Type III sums of squares are equal and that the ordinary means are equal to the least squares means. Can you explain why?
(b) Do a weighted analysis of cell means, weighted by nij, using PROC GLM or a similar program. Do any of the sums of squares agree with those obtained from the unweighted analysis of cell means? Do the ordinary means or the least squares means agree with those from the unweighted analysis?
(c) Use the general linear models approach (PROC GLM or similar program) to analyze the data. Compare this analysis with the weighted analysis of cell means. Compare the least squares means with those from the unweighted and weighted analysis of cell means.
Exercise 9.13
Reconstruct the table developed in Exercise 17.2 assuming there are three blocks but that treatment (2,3) is missing in Block 3. Identify the contrasts on cell treatment means and on marginal treatment means that are free of block effects. Would the analysis of cell means be appropriate for these data? Show why or why not.
Exercise 17.2
Table 17.1 gives the expectations of the cell means for a 2 × 3 factorial in a completely random experimental design. Construct a similar A × B table but for a randomized complete block design with balanced data. Assume the block effects are fixed effects. Include A × B interactions but do not include interactions with blocks. Demonstrate that the expectation of any contrast on treatment means, cell means, or marginal means does not involve block effects.
Table 17.1