Exercise 1 Answer the following with regard to multisite trials.

1. What is meant by heterogeneity of the treatment effect in a multisite trial?

2. Under what conditions could a multisite trial require a higher total number of participants as a trial with independent subjects to achieve the same power? Under what conditions could it require a lower total number?

Exercise 2 Investigators are planning a multicenter trial with a continuous outcome. They will randomize patients to intervention and control conditions 1:1 within center. They want \(80\%\) power to detect a standardized effect size of \(d=0.3\). They assume that \(\rho_0 = 0\) and \(\rho_1 = 0.05\).

1. Provide an interpretation of the parameters \(\rho_0\) and \(\rho_1\).

2. What is the total sample size that would be needed for this study if all of the observations were independent?

3. The number of patients per site is uncertain and may range from 10 to 20. Compute the design effect associated with the multisite design when the number of patients per site equal to 10, 15 or 20.

4. Using your answers to the previous question, compute the number of centers needed for the trial when the number of patients per site is equal to 10, 15 or 20.

5. The efficiency of a multisite trial is reduced when there is variation in the number of participants per site. Suppose that we expect the number of participants per site to vary uniformly from 10 to 100. What would be the impact on the sample size requirement?

Exercise 3 What is the variance ratio in a multisite trial? How do the variance ratio, number of sites, and number of subjects per site affect power for a test of heterogeneity of the treatment effect?

Exercise 4 Investigators are planning a multicenter trial with a binary outcome. They will randomize patients to intervention and control conditions 1:1 within center. The expected probabilities of the outcome are 0.5 in the control condition and 0.36 in the treatment condition. There are a total of 10 sites available to enroll patients in the study.

1. What is the expected odds ratio? What are plausible values for \(\sigma_{u1}\) in this study?

2. How many patients per site are needed to provide at least 80% power, assuming one-sided \(\alpha\) of 0.025? Include a sensitivity analysis using a plausible range of values for \(\sigma_{u1}\).