2011 STAT3955 Past Paper 1

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT3811 SURVIVAL ANALYSIS May 13, 2011 Time: 2:30 p.m. - 5:30 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 (in days) of twenty male rats exposed to high level of radiation for 2 hours. Right censored observations are denoted by "+". 2 2 5+ 16 16 17+ Days to death 6 6 s+ u+ 12 12+ 15+ 17+ 19 21 22 24 25+ 30+ (a) Estimate the survival function S(t) using the Kaplan-Meier estimator. Compute the standard error of the estimate at each of the uncensored time points. [6 marks] (b) Estimate the median and the 60th-percentile of the survival time based on the Kaplan-Meier estimates. [1 mark] (c) Construct a 95% confidence interval for S(t) at t = 20 days. [2 marks] (d) Construct a 95% confidence interval for S(t) at t = 30 days. Is this a reliable estimate? Why or why not? [3 marks] (e) Calculate the mean survival time f.J, defined for T :::; T = 24 based on the relationship f.J, = J; S(u)du. [3 marks] (f) Construct 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 N A estimator and determine a 95% confidence interval for the survival probability S(t) at t = 20 days. How does this confidence interval compare to the one in (c)? [3 marks]

S&AS: STAT3811 Survival Analysis 2 (h) Based on the biweight kernel function given by K( u) = i~ (1 - u 2 ) 2 for -1 :::; u :S 1 and a bandwidth of h = 4, estimate >.(20) with a pointwise 95% confidence interval. [3 marks] (i) The survival times (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 1 1 1 2 3+ 3+ 4+ 6 lQ+ 12 16 18+ 20+ 24 25+ 30 30+ 30+ 30+ 30+ 1. By just looking at the two sets of data, do you think the survival function of the two groups are different intuitively? Explain briefly. [2 marks] 11. Carry out an appropriate test to determine whether the gender of the rats will have an effect on the risk of dying. Use a 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] m. Is your conclusion in (ii) in line with your observation in (i)? Why or why not? [2 marks] [Total: 35 marks] 2. A clinical trial has been conducted to study how the survival time (in months) of the heart transplant patients is related to age. For simplicity, the patients were simply classified into 3 age groups, and mathematically represented by two dummy variables X = (X 1 , X 2 ). The three groups are respectively the old age group (age 60 or above) with X = (0, 0), the middle age group (age between 30 to 59) with X = (1, 0) and the young age group (age under 30) with X= (0, 1). Let y be the observed death time or censoring time, 6 be the censoring indicator with 6 = 0 representing a censored observation and 6 = 1 representing an uncensored observation. The following data are obtained from a random sample of n = 7 patients in a pilot study. 12.0 1 0 0 13.0 0 1 0 15.0 1 0 1 17.0 0 0 0 19.0 1 1 0 21.0 0 0 1 23.0 0 0 1

S&AS: STAT3811 Survival Analysis (a) Find the probability distribution of o. (b) Find the distribution of Y. 4 [2 marks) [3 marks) (c) Show that o and Y are statistically independent. (2 marks] (d) Let W 2 = T 1 + T2. Show that the density function of W 2 has the form fw(w) = A 2 wexp(-Aw) for w, 0 < A < oo which is a Gamma(k, A) distribution with density function f ( . k ..\) = >.kwk-1 exp( ->.w) w w, , r(k) for w, A, k > 0 and k = 2. Also state the distribution of W m = L; 1 1j for some m 2: 1. [5 marks] (e) Show that the maximum likelihood estimator of ). is , = 2:~ 1 oi 1\ "n . L,.,i=1 Yi [2 marks) (f) Find E(~) and show that ~ is asymptotically unbiased as n--+ oo. · [5 marks] (g) Find the exact variance of the estimator ~ (not the asymptotic variance based on the Fisher's information). [4 marks] (h) Explain how to construct a 95% confidence interval for A based on a large sample and hence a 95% confidence interval for the median ofT. (4 marks) (i) Without doing any calculation, state E(B) and Var(B) based on the re- sults in (f) and (g) where B is the maximum likelihood estimator of B. (1 mark) (j) Derive an estimator of (..\+B). From a random sample of n = 10, we have 2:~ 1 Yi = 60 and 2:~ 1 oi = 6. Construct a 95% confidence interval for A+ B. [5 marks) [Total: 33 marks] 4. Show that >.(t) = exp( -et), 0 < (} < oo, t > 0 is NOT a valid hazard function of a proper statistical distribution. (Total: 5 marks] ********** END OF PAPER ********** \

S&AS: STAT3811 Survival Analysis 5 A LIST OF FORMULAE 1. >.(t) = fffi = -!log S(t), A(t) = J; >.(u)du, S(t) = exp{ -A(t)}, f(t) - >.(t)S(t) = >.(t) exp{ -A(t)}. 2. MLE theorem- Let 8n be the maximum likelihood estimator for the param- eter of interest 8. Under some mild regularity condition, we have Vn(Bn- 8) ~ N(O, i¥ 1 1 (8)) where iy 1 ( 8) is the Fisher information matrix about 8 based on Y 1 . 3. Newton Raphson iterative procedure for MLE- {j(k+I) = {j(k) _ [.e" ( {j(k); Y) J -I .e' ( {j(k); Y). 4. Delta method or Slutsky's theorem - Let Tn be an estimator for the unknown parameter 8, and .Jri(Tn- 8) ~ N (0, l7 2 (8)). For any continuous function g, we have Jn(g(Tn)- g(8)) ~ N (0, [g'(8)] 2 0" 2 (8)). 5. Kaplan-Meier Estimator 6. Nelson-Aalen Estimator 7. Log-rank Test (2-sample case) _ I: (di) ilt . <t n; . (•)- "'"'XJ 1 ) under H 0 with e · = nli dj and v · = nljno~ dj (nj-dj) for some known weight function w. lJ nj lJ nj (nj-1)

S&AS: STAT3811 Survival Analysis 7 (c) Discrete Method Lp. (!3) - [lk exp(s~/3) 3 - I: exp(s*' /3) 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 Ho : /3 = f3o. (a) Wald's Test X~ = (/3- f3o)' i(/3)(/3- f3o). (b) Likelihood ratio test (LRT) x~R = -2{l(f3o)- £(/3)}. (c) Score Test 12. Local Tests - Testing for Ho : (3 1 = /3 10 (a) Wald's Test X~= (/31- f310)'i 11 ({3)- 1 (/31- fJ10). (b) Likelihood ratio test (LRT) x~R = -2{ l[(f3~o, /3~o)'- £(/3)]}. (c) Score Test (d) Test on Linear Combinations of the Regression parameters Testing for H 0 : C/3 = C/3 0 , the Wald's statistic is given by ********** END OF PAPER **********