exam1sol - g0 (“LN Exam I for Stat 4930 - Fall 2003...

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Unformatted text preview: g0 (“LN Exam I for Stat 4930 - Fall 2003 October 10, 2003 1. (15 points) Let S'U) be the Kaplan-Meier estimator of the survivor function and recall logmx) = log[$/(1 — Derive Vm:(logit(5'(t))) and use it to construct a 95% confidence interval for 3(t). Hint: Var(S(t)) = [S (10]2 2le “Had”, where k is the number of unique death times. ULSQ w‘kjlu’ 50‘”! among) m 3(9); 103.1(93= loje —-|oj(I-e‘) i i." | .. a + 1—0 GU13) 6(0‘0 fl ‘ E h SHl Z I \m CW5 Wm ‘ [amt-um] j.“ n-,I«;'»‘1l 5, .Nr l 2. (15 points) (a) [10 points) Show that h(t|X) = h0(t) exp(fi:z:) implies S(t|X) = SMQWPUW), (b) (5 points) Show that the Breslow estimator of the baseline cumulative hazard function in the Cox model, EUR) = 233:1 W simplifies to the N elson-Aalen estimator (3') if there are no covariates. a) Mm): tome?” @ KHHQAV “(Debit © Hum: e {5% Head LUQ \Lfiuw , um “HM, Bx 4mm - “emf: [33 C 3 6 g, L5 gum : EQUAL I [\o Cuuwr'hLC-J E Z E 1‘ Auhln/fi‘,‘ Ida. game-H) I at {(5) '” . L 0“ AM Me 5 mo- 2 r e w ~ 3. (20 points) Use the below data to answer the following questions. Note: 6 = 1 corresponds to failure, 6 = 0 corresponds to censoring. Subject Time 6 ‘l‘ M g 3; 5.1 l 2 l s " ' “’ 3 8 1 7) 3 O O 14 4 3 1 g l l 5 22 1 (o O O BK 5 3 0 s 1 l 2 a l u l (a) Compute the Kaplan-Meier and Nelson-Aalen estimator for S(t). Compare. (b) Estimate the median survival time using the two estimators and S (50). Compare. (c) Does patient 6 contribute to the estimator? Why or why not? ((1) Can you estimate the mean? Explain (Do NOT estimate it!). N- - -a‘jt‘j a) Mg“! H :5 N£[&A_A‘Lefl U8 “ A} Sflltl US Sill—‘6" ‘ll-e‘va— <25“ sews to» 7:33 EL— M— gcJCug general}: 633' gm .332 - (0(0 F a ‘/)__ M was m 5m W s {ma 5(E\ 3m9s(l_;_[): O “Scflr'qu .]: g 4. (50 points) The data is from the Mayo Clinic trial in primary biliary cirrhosis (PBC) of the liver conducted between 1974 and 1984. A total of 424 PBC patients, referred to Mayo Clinic during that ten—year interval, met eligibility criteria for the randomized placebo controlled trial of the drug D—penicillamine (drug). Age is measured in years. (3) Estimate the hazard ratio and give a 95% confidence interval for a 10 year change in age. > summary . coxph (pbc . age) Call: coxph(formula = Surv(time, status) " age, data = pbc) n= 312 coef expCcoaf) se(coef) z 13 age 0.04 1.04 0.0088 4.54 5.7e-06 ‘03 (not; {1r ,,\ 1 a,” (bub/3" p:‘oq inf a lo ([«anjg filo : ICE): “#0 féio: (:5. 34.01. 130007593 _. ( log?) L773) (b) The investigator wanted to fit the stratified log rank test to examine the impact of drug. stratifying on sex. He fit the below cox model. Give a test statistic and p—value that ’roughly’ corresponds to the stratified log rank test described (Be Carefull). What do you conclude? Justify your choice. Call: coxph(formu1a = Surv(time, status) " sex + drug, data = pbc) n= 312 coef exp(coef) se (coef) z p sex -0.4824 0.617 0.237 -2.039 0.041 drug -0.0503 0.951 0.179 -0.281 0.780 exp(coef) exp(-coef) lower .95 upper .95 sex 0.517 1.62 0.388 0.981 drug 0.951 1.05 0.669 1.351 quuare= 0.012 (max possible= 0.983 ) Likelihood ratio test: 3.85 on 2 df, p=0.146 Wald test = 4.27 on 2 df, p=0.119 Score (logrank) test = 4.35 on 2 df, p=0.114 Sign “wk LU? WU‘IJ (UN 23"“) +0 545b,.) .1’ Q's/“j; QJJ‘HLLJ £4” 39X. so 0L: (JUJJ dql 01' Glam] 3: “.3311 pate: .720 rm" fumbqu Io (fin; (no LL‘ Stoqll-jfu-L‘) L31“ coir 4/1“, Eel/Lam "HM? DMJPULJV at. in no Jo J m. .J Ara; we ((2) Test the hypothesis that Bl z 82 z 0. i.e., age and bilrubin are not needed in the model. State your conclusions. coxph(formula = Surv(time, status) " age + bili + drug, data = pbc) n= 312 coef expCcoef) seCcoef) z p age 0.0397 1.041 0.00913 4.355 1.3e—05 5111 0.1479 1.159 0.01308 11.307 0.0e+00 ( I) drug -0.0566 0.945 0.18724 -0.302 7.69-01 expCcoef) exp(-coef) lower .95 upper .95 age 1.041 0.961 1.022 1.06 bili 1.159 0.863 1.130 1.19 drug 0.945 1.058 0.655 1.36 quuare= 0.285 (max possible: 0.983 ) Likelihood ratio test= 105 on 3 df, Wald test = 150 on 3 df, Score (logrank) test = 214 on 3 df, "U'lfl‘d 000 ******************t** Call: coxph(formula = Surv(time, status) “ drug, data = pbc) n= 312 coef exp(coef) seCcoef) 2 p ( ) drug -0.0572 0.944 0.179 —0 319 0.75 a exp(coef) exp(—coef) lower .95 upper .95 drug 0.944 1.06 0.665 1.34 quuare= 0 (max possible= 0.983 ) Likelihood ratio test= 0.1 on 1 df, p=0.75 Wald test = 0.1 on 1 df, p=0.75 Score (logrank) test = 0.1 on 1 df, p=0.75 LRT~; -Q[L(n-L(&\]= |O§~.l:.loqfl “.01., 0.} 30”.) 7.. 7.11.3 U... (d) The investigator was interested in whether the effect of drug varied by sex. Test this hypothesis and state your conclusions. Estimate the hazard ratio for drug when sex=0 and when sex=1. Call: coxph(formula = Surv(time, status) ” sex + drug + sex * drug, data = pbc) n= 312 coef exp(coef) se(coef) z p sex -1.159 0.314 0.714 -1.622 0.10 drug -0.444 0.641 0.444 -1.000 0.32 sex:drug 0.475 1.607 0.486 0.977 0.33 exp(coef) exp(—coef) lower .95 upper .95 sex 0.314 3.185 0.0774 1.27 drug 0.641 1.559 0.2686 1.53 sex:drug 1.607 0.622 0.6204 4.16 quuare= 0.015 (max possible= 0.983 ) Likelihood ratio test= 4.83 on 3 df, p=0.185 Wald test = 5.63 on 3 df, p=0.131 Score (logrank) test = 5.82 on 3 df, p=0.121 uh" 65101 :0 'Z: .fij‘?) ~Up‘m5.33 (‘ Ll 00l_ QoflhjL" fiv'JAAL‘ LU r JP : Sex :0 q a “q HHS‘VO/D‘Ns”) 1 e. .- .(aul kttls'“:(0 JIU3:O) 60 Sex ~\ \‘(HS”:1. 3N3”) CHLIC'iHfluwA-is“ .0“ 77—“- 1 fl ——————— '—'—"——‘——~ '_ 6: 1 ('03 141.15“ I’ll/k3 O) ejllgc‘ (e) If the Breslow method had been used to handle ties (the Efron method was used in the problems above), describe how you would expect the estimated coefficients to look relative to those given above in part b). go LA LA) Exam I for Stat 6934 - Fall 2003 October 10, 2003 1. ( 15 points) Discuss the use of weights in the (weighted) log rank test, including when and why we should use particular sets of weights . Why does the (weighted) 10g rank test have low power when the hazards cross? Explain. Sue Mk 2. (15 points) Show that in the Cox model with t continuous covariate X2, i.e., h(t]X1, X2) 2 he ratio for X = 1 vs X we covariates, a binary covariate X1 and a (t) expwlzl + 6232), that [31 is the log hazard = 0 holding the continuous covariate X2 constant. +63%; molt—I x3} MEMB‘ 8 ..~__._.__________ 3 m ._ C l Mum), m tang“ [03 MU ): 6! MU ) ...
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This note was uploaded on 01/15/2012 for the course STA 6934 taught by Professor Young during the Fall '08 term at University of Florida.

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exam1sol - g0 (“LN Exam I for Stat 4930 - Fall 2003...

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