2006 STAT3955 Past Paper 1

I I THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3811 Survival Analysis May 19, 2006 Time: 9:30 a.m. - 12:30 p.m. Candidates taking examinations that permit the use of calculators may use any cal- culator which fulfils the following criteria: (a) it should be self-contained, silent, battery-operated and pocket-sized and (b) it should have numeral-display facilities only and should be used only for the purposes of calculation. It is the candidate's responsibility to ensure that the calculator operates satisfactorily and the candidate must record the name and type of the calculator on the front page of the examination scripts. Lists of permitted/prohibited calculators will not be made available to candidates for reference, and the onus will be on the candidate to ensure that the calculator used will not be in violation of the criteria listed above. Answer ALL SIX questions. Marks are shown in square brackets. 1. The following data consist of the time to relapse of 21 leukemia patients (in months) 6 17+ 6 19+ 6 20+ 7 23 13 35+ 16 (i) Construct the Kaplan-Meier Product-Limit (PL) estimator and plot the estimated survival curve. (ii) Calculate the estimated variance of the PL estimator and determine the 95% confidence interval for the survival probability S(t) at t = 16 months. (iii) Calculate the mean survival time to 35 months. (iv) Find the median remission survival time. [Total: 20 marks] 2. The following data consist of the survival times (days to death) of twenty patients with inoperative lung cancer 22 139 27 144 68 193 gg+ 193+ 10s+ 203 121+ 210 134 231+ 136 248

S&AS: STAT3811 Survival Analysis 2 (a) Estimate the cumulative hazard rate using the Nelson-Aalen estimator. (b) Estimate the hazard rate at t = 60 days, using the uniform kernel with a bandwidth of 30 days. (c) Estimate the hazard rate at t = 210 days, using the uniform kernel with a bandwidth of 30 days. Hint: The uniform kernel is given by K(x) = 1/2 for -1 ::; x ::; 1, and equal to zero otherwise. [Total: 15 marks] 3. The following data consists of the survival time in weeks of 7 mice with a given tumor: 1, 7, 8, 11 +, 14, 19+, 21 +. Assume the significance level a = 0.05. Carry out an appropriate test that the survival time follows an exponential distribution with ,.\ = 0.08. The null and alternative hypotheses are H 0 : S(t) = So(t) vs Hi : S(t) =I= So(t) where So(t) = e-o.ost. [Total: 10 marks] 4. 10 patients with acute leukemia are randomized to receive either Treatment A or Treatment B. The remission durations in weeks are Treatment group 8, 3, 15+, 27+, 1 Control group 3, 2, 1, 5+, 6 (a) Suppose the significance level a = 0.05. Use the Gehan test to test the null hypothesis S 2 ( t) (two treatments equally effective) against H 1 : S 1 (t) =/:. S 2 (t) (two treatments not equally effective) (b) Assume that the two distributions are exponential with parameters ,.\ 1 and ..\ 2 , respectively. Under the same significance level as (a), use the likelihood ratio test to test the null hypothesis Ho : ..\1 - ..\2 (two treatments equally effective) against H 1 : ..\ 1 =/:. ..\ 2 (two treatments not equally effective) [Total: 20 marks]

S&AS: STAT3811 Survival Analysis 4 (ii) Use the score test to test the null hypothesis /3 = 0 at the 0.05 significance · level. (iii) Use (i) and the Newton-Raphson algorithm with an initial guess at /3 = 0 to compute a two step iterative estimate for /3. (iv) Use the likelihood ratio test to test for the null hypothesis Ho : /3 = 0 at the 0. 05 significance level. (v) Use the Wald test to test for the null hypothesis Ho : f3 = 0 at the 0.05 significance level. [Total: 25 marks] ********** END OF PAPER********** . . • I