Consider again the study described in the main body of this chapter concerning the effect of an air pollution episode on pulmonary function measurements (FEV1) taken on each of K= 40 school children over five weeks. The subject-specific scalar and matrix versions of the model defined by expressions (25.7) and (25.8) use week 5 as the reference level. In this exercise, you are to consider how the model and analysis would be modified if week 1 is used as the reference level.
a. State the subject-specific scalar form of the model (i.e., analogous to expression (25.7)) in which week 1 is used as the reference level. (Denote the coefficients in this model by βʹg, g = 0, 1, 2, 3, 4.)
b. State the subject-specific matrix form of the model (i.e., analogous to expression
(25.8)) in which week 1 is used as the reference level, making sure to specify the components and corresponding dimensions of the matrices involved.
c. Using the subject-specific scalar model defined in part (a), express each of the estimated regression coefficients (βʹg, g = 0, 1, 2, 3, 4) in terms of sample mean FEV1 levels
FEVL, j = 1, 2, 3, 4, 5
d. Express the null hypothesis
H0: μ1 = μ2 + μ3 + μ4 + μ5/4 = 0
in terms of the regression coefficients βʹg(g = 0, 1, 2, 3, 4) for the model specified in part (a).
e. Will the computed test statistic resulting from testing H0using the model in part (a) be identical to the computed test statistic resulting from testing H0using model (25.2)—or, equivalently, model (25.7)—in which the reference level is week 5? Explain briefly.
f. What are the predicted FEV1 mean values for weeks 1 through 5, respectively?
g. What is the estimate of the contrast that compares week 1 with the average of the other four weeks? Is this estimated contrast statistically significantly different from zero?
h. The output shown above does not provide the results for testing whether or not there is a significant difference among the estimated mean FEV1 levels for all five weeks.
i. Specify a "contrast" statement (SAS code) that would perform such a test. Will the results obtained from the contrast statement specified in part (h.i) be the same as the results previously obtained in Table 25.7, for which the referent group is week 5? Explain briefly.
J. What correlation structure has been assumed to generate the above output? What is the estimated
ii. Correlation between any two responses on the same subject?
iii. How can the correlation in part (i.ii) be computed using the output information provided for "Covariance Parameter Estimates"? Does the output shown above provide test results that allow for the possibility that the assumed correlation structure may be incorrectly specified? Explain briefly.
v. Based on the output summarized in Table 25.7, would you expect the numerical test results obtained for different working correlation structures to be identical when the referent group is week 1? Explain briefly.