MATH : Design and Analysis of Clinical Trials

Transcription

1 MATH : Design and Analysis of Clinical Trials MID-TERM EXAM Spring, 2012 (Time allowed: TWO AND HALF HOURS) INSTRUCTIONS TO STUDENTS: 1. This test contains FIVE questions and comprises SIX printed pages. 2. Answer ALL questions for a total of 100 marks. 3. This is a open-book, open-note test; you can use any materials you have. 4. Write your name on the front of your answer booklet and on any additional sheets you write on. 1

2 1. In a case-control study conducted in France sometime ago to investigate the association between esophageal cancer and alcohol assumption, the following data was collected from 200 cases and 775 controls (so the total sample size is 975) Alcohol assumption Case Control Heavy Light where Heavy is defined as Average daily alcohol assumption 80 g, Light is defined as Average daily alcohol assumption < 80 g. An investigator produced following estimates. Indicate which is correct and which is wrong. State your reasons. (a) The proportion of heavy drinkers in French was estimated to be 205/975=0.21. (b) The proportion of French people having an esophageal cancer was estimated to be 200/975 = (c) The probability of having an esophageal cancer among heavy drinkers was estimated to be 96/205=0.47. (d) The probability of having an esophageal cancer among light drinkers was estimated to be 104/775 = (e) The relative risk of having an esophageal cancer between heavy drinkers and light drinkers was estimated to be 0.47/0.13=3.62. So heavy drinkers are 3.62 times as likely to have an esophageal cancer as light drinkers. (f) The odds-ratio of having an esophageal cancer between heavy drinkers and light drinkers was estimated to be 96*666/(104*109) = If we assume that an esophageal cancer is a rare disease, then the relative risk of having an esophageal cancer between heavy drinkers and light drinkers would be very closely estimated by This is different from (e). Then which assumption is wrong? 2

3 2. 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 5 patients are treated. If the total number of responses among all eight patients is less than or equal to four 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. 3

4 3. 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, 0.41, 0.55, 0.77, 0.16, 0.08, 0.78, 0.41, 0.87, 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,

5 4. 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)? 5

6 5. Suppose the dose-toxicity relationship is given by the following equation π(x) = P [toxicity x] = e 4+0.4x, 0 < x <, 1 + e 4+0.4x 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 x 1 = 2, prepare a dose sequence of length 5 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? 6