The data set for this problem derives from the posture measurement study described in the main body of this chapter. Here we consider the data on shoulder flexion (SF) for 19 subjects that were each observed by 2 different raters on each of the 3 days (Monday, Wednesday, and Friday) and at 2 time periods (AM and PM) during each day. Note that our previous version of this example considered only one rater, but now there are two raters who have independently taken the posture measurements. Thus, we now consider 12 observations to have been made on each subject, \ observation corresponding to one of the 12 combinations of 3 days, 2 times, and 2 raters. In this problem, we assume that the investigator is interested only in assessing whether the SF measurements vary significantly by day of measurement. For this purpose, we have provided in the table below the average of the 4 SF scores taken on each subject (at 2 times and by 2 raters) for each of the 3 days.

a. Assuming that the only important effect is that of Day (i.e., the factors Time and Rater are assumed not to have important effects), state the subject-specific scalar form of a random intercept model for analyzing the above data. In stating this model, make sure to describe the assumptions made on the random effects (including the error term) in the model.
b. State the null hypothesis that there is no significant effect of the factor Day in terms of a statement about parameters in yout model given in part (a).
c. Based on a comparison of averages at the bottom of the table, does there appear to be a meaningful effect of the factor Day? Explain.
d. Describe how the data in the above table need to be reorganized in order for the MIXED procedure to carry out the analysis. (Use the format given in Table 25.2 of Chapter 25.)
e. Based on the computer output provided below, is there a significant effect of the factor Day? Explain by specifying the F statistic, its degrees of freedom, and its P-value appropriate for these data.
f. Based on the computer output, compute the estimate of the (exchangeable) correlation assumed by the model. (The output provides estimates of the variance of the subject-specific random intercept effect and the variance of the error term; you need to combine this information to compute the correlation coefficient estimate.)
g. Use the output to test whether there is a significant random effect for Subjects. If such a test was nonsignificant, why might you be concerned regarding the model that has been fit, and how might you redo the analysis?

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Average SF score by day of week Subject Monday Wednesday Friday 10.50 4.00 25.75 20.00 4.75 17.25 24.00 45.50 10.00 28.25 21.75 23.00 10.50 22.00 7.25 19.50 23.25 28.50 7.78 2.50 28.00 22.50 8.25 22.00 9.25 50.00 10.00 31.25 22.25 29.00 7.00 19.50 5.00 9.00 26.50 35.25 6.00 22.75 20.00 650 18.00 1250 41.75 550 41.25 15.50 26.00 7.00 2250 6 10 12 13 14 15 16 17 18 19 9.00 1750 875 250 3.00 1.75 Mean of Averages: 18.36 18.25 15.78