2013 STAT3955 Past Paper 1

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The University of Hong Kong *

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3811

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Nov 24, 2024

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3811 SURVIVAL ANALYSIS May 21, 2013 Time: 9:30 a.m. - 12:30 p.m. Only approved calculators as announced by the Examinations Secretary can be used in this examination. It is candidates' responsibility to ensure that their calculator op- erates satisfactorily, and candidates must record the name and type of the calculator used on the front page of the examination script. Answer ALL questions. Marks are shown in square brackets. 1. The following table contains data from a study on the survival times T 1 (in days) of twenty male rats exposed to high level of radiation for 2 hours. Right censored observations are denoted by "+". 6 6 11 11 Days to death s s+ s+ 9 9+ 13 14 15 16 16+ 9+ Is+ (a) Estimate the survival function S(t) using the Kaplan-Meier estimator. Compute the standard error of the estimate at each of the uncensored failure time points. [6 marks] (b) Estimate the median and the 75th-percentile of the survival time based on the Kaplan-Meier estimates. [1 mark] (c) Provide an estimate and construct a 95% confidence interval for S ( t) at . t = 13 days. [2 marks] (d) Provide an estimate and construct a 95% confidence interval for S(t) at t = 18 days. Is this a reliable estimate? Why or why not? [2 marks] (e) Estimate the mean survival time fl defined forT:::; T = 16 based on the relationship f1 = J; S(u)du. [3 marks] (f) Compute the Nelson-Aalen (NA) estimates for the cumulative hazard function, and hence the survival function. [4 marks] (g) Calculate the estimated variance of S(t) based on the NA estimator and determine a 95% confidence interval for the survival probability S(t) at t = 13 days. How does this confidence interval compare to the one in (c)? [3 marks] (h) Based on the uniform kernel function given by K(u) =~for -1:::; u:::; 1 and a bandwidth of h = 3, estimate .\(13) with a pointwise 95% confi- dence interval where .\(t) is the hazard function. [4 marks]

S&AS: STAT3811 Survival Analysis 2 (i) The survival times T 2 (in days) of an independent group of 20 female rats exposed to the same level of radiation for 2 hours are recorded in the following table. Days to death 4 4 4 6 5+ 5+ 5+ 8 3+ 9 11 11+ 13+ 16 16+ 18 18+ 18+ 18+ 18+ Without further information and additional assumption, carry out an appropriate test to determine whether the survival distributions of the male and female rats are the same at the 0.05 level of significance. State the null and alternative hypotheses, the name of the test, the test statistic, the rejection region and your conclusion clearly. [6 marks] (j) Assume that Tj follows an exponential distribution with parameter Aj, density function for Aj, t > 0 and median Mj for j = 1 (male rats), 2 (female rats). i. Let 1 be the hazard ratio of group 2 to group 1 (female to male). Show that 1 = MI/M 2 • [1 mark] ii. Using the above data sets, estimate log(!) with an asymptotic 95% confidence interval. Can you conclude at the 0.05 level of signifi- cance that the distributions of T 1 and T 2 are the same based on this confidence interval? Explain briefly. [4 marks] (k) Let Z = 0 if the rat is male and Z = 1 otherwise. An investigator assumes that the survival time T follows a semiparametric proportional hazards model with hazard function :\(t I z) = Ao(t) exp(,Bz). i. Write down the first 4 terms (i.e., the terms for t = 4, 6, 8 and 9) of the partial likelihood in the simplest form for ,B using Breslow's adjustment for ties based on the data from the above two tables. [2 marks] ii. The maximum partial likelihood estimate for ,B is found to be S = -0.3410 with observed information i(S) = 4.7027. A. Provide an estimate and construct a 95% confidence interval for 1, the hazard ratio of female to male. Are your estimates here similar to those found in part [(j). ii.] above? Explain briefly. [3 marks] B. Carry out a Wald test to test Ho : 1 = 1/3 against H1 : 1 > 1/3. Compute the test statistic, and state the rejection region and your conclusion at the 0.05 level of significance. [4 marks] [Total: 45 marks]

S&AS: STAT3811 Survival Analysis 3 2. An experiment similar to that in Question 1 is conducted to study the survival times (in days) of the male and female rats exposed to high level of radiation for 2 hours. Let Iij and Cij be the failure time and the right censored time of the j-th rat in the i-th group [j = 1, · · ·, ni; i = 1 (Male), 2 (Female)] and that T and Care independent. The observed failure time is Xij = min(Tij, Cij) and the censoring indicator is 5ij = I { Xij < Cij}. Assume that the failure times, T, from the two gender groups are exponentially distributed with failure rates >. 1 = exp(,8 1 ) and >. 2 = exp(,8 1 + ,8 2 ), and survival functions Si(t) = exp{ -A;t} (i = 1, 2), respectively. Suppose the observed quantities from independent random samples of n 1 male and n 2 female rats are the total number of observed failures di = L;';, 1 5ij and the total observed event time 'T/i = L;]~ 1 Xij in sample i (i = 1, 2). (a) Construct the likelihood function and express the log-likelihood function £(,81, ,82; ry, d) in terms of ,Bi, 'T/i, and di for (i = 1, 2). [4 marks] (b) Let the score vector be U(,8 1 , ,8 2 ) = [U 1 (,8 1 , ,8 2 ), U2(,81, ,82W where Ui(,8 1 , ,8 2 ) = 8 ~,£(,81,,82;77,d). Show that U1(,81, ,82) d1 + d2- exp(,81J'T/1- exp(,81 + ,82)772; U2(,81, ,82) = d2- exp(,81 + ,82)712· [2 marks] (c) Find the maximum likelihood estimators of ,Bi in terms of 'T/i and di for i = 1, 2. [4 marks] (d) Find the observed information matrix i(,6), where i(,B) = 8 /' 81 /(,81, ,82; ry, d). Show that its inverse is [6 marks] (e) Find the maximum likelihood estimator of ,8 1 if it is known that ,8 2 = 0. [2 marks] (f) Show that the observed information matrix knowing that ,8 2 = 0 is given by [3 marks] (g) Suppose we are interested in testing the hypothesis Ho: ,82 = 0 vs Show that the test statistic based on the score test is given by 2 (d2771- d1772J2 Xsc = (d1 + d2) 7!17!2 ·

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S&AS: STAT3811 Survival Analysis 4 State the asymptotic distribution of the test statistic and the rejection region using the 5% level of significance. [8 marks] (h) Suggest a Wald statistic to test H 0 : /3 2 = 0 against H1 : /3 2 > 0. State the asymptotic distribution of the test statistic and the rejection region using the 5% level of significance. [4 marks] [Total: 33 marks] 3. With reference to Questions 1 and 2, a small scale experiment is carried out to study the effects of the exposure time to high level of radiation. Five male rats were randomized to each of the 4 levels of exposure duration, namely 1-hour (X 1 = 0, X 2 = 0, X 3 = 0), 2-hour (X 1 = 1, X2 = 0, Xs = 0), 3- hour (X 1 = 0, X 2 = 1, X 3 = 0), and 4-hour (X1 = 0, X2 = O, Xs = 1), respectively. A semiparametric Cox proportional hazards regression model with hazard function ofT, the failure time, is specified as where ,\ 0 ( t) is the unspecified, arbitrary baseline hazard function. Some sum- mary statistics like the maximum partial likelihood estimates for the f3's, their associated standard errors and the estimated variance-covariance matrix of b = (g1,$2,g3)' are given below. Model Fit Statistics Criterion Without Covariates With Covariates -2 LOG L 66.422 53.155 AIC 66.422 59.155 SBC 66.422 61.279 Analysis of Maximum Likelihood Estimates Parameter Standard Hazard Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio xl 1 0.8332 0.7874 1.1196 0.2900 2.301 x2 1 2.7449 0.9966 7.5863 0.0059 15.563 Xs 1 3.4585 1.1876 8.4808 0.0036 31.768

S&AS: STAT3811 Survival Analysis 5 Estimated Variance-Covariance Matrix Variable Xl X2 Xs X! 0.6200 0.4152 0.4191 X2 0.4152 0.9931 0.8319 Xs 0.4191 0.8319 1.4104 (a) Carry out an appropriate test for the null hypothesis that the 4 levels of exposure duration have no effect on survival at the 0.05 level of signifi- cance. State the null and alternative hypotheses, the name of the test, the test statistic, the rejection region and your conclusion clearly. [4 marks] (b) Carry out an appropriate test for the hypothesis that the rats exposed to high dose of radiation for 3 hours are at a higher risk of failure than those exposed for 1 hour, at the 0.05 level of significance. State the null and alternative hypotheses, the name of the test, the test statistic, the rejection region and your conclusion clearly. [3 marks] (c) Carry out an appropriate test for Ho : f32 = f3s vs H 1 : Ho is not true at the 0.05 level of significance. State the name of the test, the test statistic, the rejection region and your conclusion clearly. [5 marks] (d) Carry out an appropriate test for Ho : f31 = 0 and f32 = f3s vs H 1 : Ho is not true at the 0.05 level of significance. State the name of the test, the test statistic, the rejection region and your conclusion clearly. Discuss the interpretation of this test result in at least 50 words. [10 marks] [Total: 22 marks] ********** END OF PAPER **********

S&AS: STAT3811 Survival Analysis 6 A LIST OF FORMULAE 1. A.(t) = fffi = -t, logS(t), A(t) = J~ A.(u)du, S(t) = exp{-A(t)}, f(t) - A.(t)S(t) = A.(t) exp{ -A(t)}. 2. MLE theorem - Let (jn be the maximum likelihood estimator for the param- eter of interest e. Under some mild regularity condition, we have where iy, (e) is the Fisher information matrix about e based on Y 1 . 3. Newton Raphson iterative procedure for MLE- {j(k+l) = {j(k) _ [t'({j(k);y)rl .f.'({j(k);y). 4. Delta method or Slutsky's theorem - Let Tn be an estimator for the unknown parameter e, and fo(Tn- e) ~ N (0, u 2 (B)). For any continuous function g, we have 5. Kaplan-Meier Estimator A rrj ( A ) rrj (n•- d•) S(tu)) = i=l 1- A; = i=l 'n; ' . Var(5- 1 -) = d;(n~;-d;) ·, Var [s(tl] - S(t) 2 "' <4 - L..ilt(<)9 n, (n, <4) · 6. Nelson-Aalen Estimator A(t) = L ~' Var [A(tJ] ilt(i) St ~ 7. Log-rank Test (2-sample case) ~ x( 1 ) under Ho 8. Mantel-Haenzel Test

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S&AS: STAT3811 Survival Analysis 9. Ramlau-Hansen Estimator A (1- a) x 100% confidence interval for .>.(t) is given by - [ Zl-a/2 x Jvar[~(t)]] .>.(t) exp ± - . .>.(t) 10. Partial Likelihood In cases with no ties, the baseline survivor function can be estimated by where. i So(t(i)) = IT ci; j=l . [1- exp(x(i),B) ] exp(-x(,).6) 'L:zER; exp(xi,B) The baseline cumulative hazard function can be estimated by - - " ( d· ) Ao(t) = -logSo(t) = L..., ' - H(<)<t 'L:zER; exp(xi,B) even when there are ties with d; > 1. 7 In the cases with ties, let s; be the sum of vectors x 1 over all individuals who die at time t(i). (a) Breslow's Partial Likelihood L p, ((3) = IT exp ( sj,B) d, . i=l {'L:jER; exp(xj,B)} (b) Efron's Partial Likelihood

S&AS: STAT3811 Survival Analysis 8 (c) Discrete Method k II exp( s;,e) Lp 3 (,8) = 2::: exp(s*',8) i=l qEQi q with Qi being the set of all possible di-tuple of individuals who could have been one of the di failures at time t(i). 11. Global Tests - Testing for H 0 : ,8 = ,So. (a) Wald's Test X~= ($-,So)' i($)($- ,8o). (b) Likelihood ratio test (LRT) (c) Score Test 12. Local Tests - Testing for Ho : ,81 = ,81o (a) Wald's Test X~= (/h- ,8w)'i 11 ($)- 1 (f1- ,8w). (b) Likelihood ratio test (LRT) XZR = -2{ R[(,B~o' $;o)'- C(f)]}. (c) Score Test (d) Test on Linear Combinations of the Regression parameters Testing for Ho : C,8 = C,8 0 , the Wald's statistic is given by ********** END OF LIST **********

2013 STAT3955 Past Paper 1

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