A study by Rikkers et al. (1978) involved a prospective randomized surgical trial that compared cirrhotic patients who had bled from use of either a nonselective shunt (a standard operation) or a selective shunt (a new operation). The response is a measure of the maximal rate of urea synthesis (MRUS), a measure of liver function; low values of MRUS indicate poor liver function. The study sample consisted of 8 selective shunt patients and 13 nonselective shunt patients. MRUS was measured both preoperatively and early postoperatively on each of the 21 patients. The purpose of the study was to compare the change in liver function in each of the two groups. The data are described below:
Pre- and postoperative maximal rate of urea synthesis level (MRUS) by group

a. Decide whether each of the factors Group (Selective Shunt versus Nonselective Shunt) and Time (PreOp versus PostOp) should be considered as a fixed or as a random factor. Explain.
b. State a subject-specific scalar model for analyzing the above data, and specify the assumptions typically made about the random effects (including the error term) in the model.
c. State the subject-specific matrix model that corresponds to the model you provided in your answer to part (b).
d. State the null hypothesis that there is no significant effect of the factor Group in terms of a statement about parameters in your model given in part (b). What other null hypothesis should be tested prior to testing whether there is a significant effect of Group?
e. Based on a comparison of sample means, does there appear to be a meaningful difference in the change in liver function between the two groups? Explain.
f. Below are given computer results for a repeated measures analysis of these data. Based on these results, is there a significant effect of the factor Group? Explain by describing the F statistic, its degrees of freedom, and its P-value appropriate for these data.
g. Use the output to test whether there is a significant random effect for Subjects.