In this problem, we consider the analysis of the combined information from both raters on the shoulder flexion (SF) scores in the posture measurement study. Thus, the questions below concern the data on 19 subjects, where there are 12 observations on each subject, with each observation corresponding to one of the 12 combinations of 3 days, 2 times, and 2 raters. The design is a balanced repeated measures design. The data are given below:
a. Using the information provided in the data layout, supply the values of the sample means in the following tables:

b. Based on the results obtained for the tables in part (a), does there appear to be a meaningful main effect of either Day, Time, and/or Rater? Explain.
c. Based on the results obtained for the tables in part (a), does there appear to be a meaningful interaction effect between Day and Time, between Day and Rater, and/or between Time and Rater? Explain.
Shoulder flexion scores by day, time, and rater

d. Based on the results obtained for the tables in part (a), does there appear to be a meaningful three-way interaction effect among Day, Time, and Rater? Explain.
e. We will now consider the analysis of the SF data in which we have treated the Day, Time, and Rater factors all as fixed.
i. How can the investigators justify a decision to treat the Rater factor as fixed?
One set of analyses that was carried out for these data is described by the following computer code and edited output using SAS's MIXED procedure.
ii. State the subject-specific scalar model that is being fit by each of the two program statements ((T) and (2)). In what way do the program statements differ?
iii. Is the correlation structure assumed by the model stated in part (e.ii) an exchangeable correlation structure? Explain.
iv. Assuming (perhaps incorrectly) that the F test for each effect is "orthogonal," what would you conclude about which effects are significant and which effects are not significant?
v. Based on the information provided by the "NOTE" and by the "Covariate Parameter Estimates" in the output, should you be concerned about the appropriateness of these analyses? Explain.
Another set of analyses that was carried out for these data is described by the following computer code and edited output using SAS's MIXED procedure.
vi. State the subject-specific scalar model that is being fit by each of the two program statements ((3) and (?)). In what way do these program statements differ?
vii. Is the correlation structure assumed by the model stated in part (e,vi) an exchangeable correlation structure? Explain.
viii. Assuming (perhaps incorrectly) that the F test for each effect is "orthogonal," what would you conclude about which effects are significant and which effects are not significant?
ix. Based on the information provided by the "Covariate Parameter Estimate," in the output, should you be concerned about the appropriateness of these analyses?
f. We will now consider the analysis of the SF data in which we have treated the Day and Time factors as fixed and the Rater factor as random,
i. How can the investigators justify a decision to treat the Rater factor as random? Consider the following computer program code:
ii. State the subject-specific scalar model that is being fit by the above program statements ((5)).
iii. Is the correlation structure assumed by the model stated in part (f.ii) an exchangeable correlation structure? Explain.
The code described in (5) can be shown from the output to provide an inappropriate analysis. Another code and its corresponding output are given as follows:
iv,. State the subject-specific scalar model chat is being fit by the above program code(6)
v. Is the correlation structure assumed by the model stated in part (f.iv) an exchangeable correlation structure? Explain,
vi. Assuming that the above analysis is appropriate, what can you conclude about whether there are significant effects of Day, Time, Day X Time, and/or Rater? Explain.
vii. What other approaches might you take to carry out the analysis when considering Day and Time as fixed effects and Rater as a random effect?

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