Design and Analysis of Clinical Trials MID-TERM EXAM
Q1. In a case-control study to examine the association between myocardial infarction (MI) and smoking, each of 262 Italian women with MI was matched with about 2 controls (MI-free women). Then their smoking history was identified. The data was summarized in the following table:
|
Ever Smoker
|
MI
|
Control
|
|
Yes
|
172
|
173
|
|
No
|
90
|
346
|
Answer the following questions:
(a) Can you estimate the probability that a random Italian woman is a ever smoker? State your reason. If you can, please provide your estimate.
(b) Can you estimate the probability of developing MI among smoking Italian women? State your reason. If you can, please provide your estimate.
(c) Can you estimate the odds ratio of developing MI between ever smoking Italian women and never smoking Italian women? If you can, provide your estimate and construct a 95% CI for the true odds ratio.
(d) Can you estimate or approximately estimate the relative risk of developing MI between ever smoking Italian women and never smoking Italian women? Under what assumption? Interpret the estimate.
Q2. Consider the following two stage decision rule: Three patients are given a new drug. If none respond, the study is terminated and the drug is declared a failure. If all 3 respond then the study is terminated and the drug declared a success. Otherwise, an additional 3 patients are treated. If the total number of responses among all eight patients is less than three then the drug is declared ineffective; otherwise, it is considered a success.
(a) Using this decision rule, compute the probability that the drug will be considered a success if the true probability of response is 0.5.
(b) For the same probability of response as in (a), compute the expected number of patients to be studied using the above two-stage decision rule.
Q3. We are to conduct a randomized clinical trial to compare treatment A to treatment B. In order to keep randomization codes secure, we decided to use varying block sizes 2, 4 and 6 with probabilities 0.2, 0.5 and 0.3. Then within each block half of patients will be assigned to treatment A and half to treatment B through permutation. Now suppose 24 patients are available for randomization. Using the following uniform numbers, assign these 24 patients to either treatment A or treatment B (you may not need all the numbers; but you are required to use the numbers sequentially):
- Uniform numbers for block sizes: 0.65, 0.06, 0.43, 0.93, 0.94, 0.15, 0.55, 0.77, 0.16, 0.08, 0.78, 0.41, 0.87, 0.85.
- Uniform numbers for blocks: 0.49, 0.87, 0.55, 0.01, 0.64, 0.52, 0.02, 0.38, 0.03, 0.49, 0.22, 0.09, 0.61, 0.51, 0.32, 0.84, 0.99, 0.73, 0.99, 0.30, 0.73, 0.38, 0.22, 0.36, 0.95, 0.75, 0.78, 0.41, 0.87, 0.85, 0.14, 0.03, 0.15, 0.56, 0.31, 0.84, 0.97, 0.52, 0.11, 0.91, 0.53, 0.22, 0.34, 0.44, 0.82, 0.92, 0.28.
Q4. Suppose we would like to compare a new treatment (treatment 1) to the standard treatment (treatment 2) in reducing weight where treatment assignment will be done by permuted block randomization so that the two-sample t-test will be appropriate to compare these two treatments. Assume the population variance of the weight distribution is 25. We wish to detect a 6 unit weight difference between the new treatment and the standard treatment with 90% power at the significance level α = 0.05 using a two-sided t-test.
(a) Find the necessary sample size for the study for the given design characteristics assuming equal allocation.
(b) If the size of the treatment effect turned out to be what you expected, what is the p-value?
(c) How large should the estimated treatment effect be to make the p-value significant (≤ 0.05)?
Q5. Suppose the dose-toxicity relationship is given by the following equation
π(x) = P[toxicity|x] = e-4+0.4x/1 + e-4+0.4x, 0 < x < ∞,
where x is a dose level and π(x) is the probability of having serious (but reversible) adverse events. You are going to use the traditional design to conduct a phase I trial to search for the maximum tolerated dose (MTD).
(a) Find the true MTD. That is, find a dose level x∗ such that P[toxicity|x∗] = 1/3.
(b) Starting with an initial dose level of x1 = 1, prepare a dose sequence of length 4 using the modified Fibonacci sequence.
(c) Assume that the dose level at which the trial stops is claimed to the (estimated) MTD (when the trial reaches the last dose level, that dose level is claimed to be the MTD). Define a random variable X = dose level claimed to be the MTD. Find the distribution of X.
(d) Find E(X) for the dose schedule. Is this expectation close the true MTD? What is the variance var(X) of X?