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Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 21Page 22Page 23Page 24Page 25Page 26Page 27Page 28Page 29Page 30Page 31Page 32Page 33Page 34Page 35Page 36Page 37Page 389.3Examine kaplan-meier plots:Crossing survival curves: evidence against proportional hazards even without looking at log-logor other such plotslog-log plot, natural time scale; curves not parallel for 250 days or so, then slopes similar orslightly reversed until late (after 600 days), then move even more (although not much data there)a. simple proportional hazards model: Analysis of Maximum Likelihood Estimates Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits grp 1 -0.10749 0.22336 0.2316 0.6304 0.898 0.580 1.391little evidence of association; test fails to reject; but, as seen above, proportional hazardsseems invalid, so test using time varying covariate (linear)b. model with time-varying covariate/interaction Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits grp 1 -0.92288 0.34976 6.9623 0.0083 0.397 0.200 0.789 grp_tim 1 0.00164 0.0005632 8.4911 0.0036 1.002 1.001 1.003test of proportional hazards rejects (p = 0.0036)overall test for covariate:proc phreg;model t*delta(0) = grp grp_tim / rl;grp_tim = t*grp;t1: test grp,grp_tim; Linear Hypotheses Testing Results Wald Label Chi-Square DF Pr > ChiSq t1 8.9417 2 0.0114test for effect of arm given above, rejects; treatment affects survival in complex fashionc. time-varying dichotomous covariate with different cutpoint:code (courtesy of Ben Leiby)%macro cut(lower,upper,interval); %do i=&lower %to &upper %by &interval; proc phreg data = one; model t*delta(0) = z1 z2; if grp = 1 then do; z1 = (t le &i); z2 = (t gt &i); end; else do; z1 = 0; z2 = 0; end; ods output fitstatistics = c&i(keep = withcovariates); ods listing select none; run; %end; data cuts; set %do i=&lower %to &upper %by &interval; c&i(obs=1) %end;; retain cutpoint; if _n_=1 then cutpoint = &lower; else cutpoint = cutpoint+&interval; loglikelihood = -.5*withcovariates; symbol i = none v=dot h=.25; proc gplot data=cuts; plot loglikelihood*cutpoint; run; %mend; %cut(10,3000,10); %cut(100,400,5); %cut(200,300,1);plot likelihood against cutpointso maximized in range 254-260fit model:proc phreg data = one;model t*delta(0) = grp_t_254_0 grp_t_254_1 / rl;if t <= 254 then do; grp_t_254_0 = grp; grp_t_254_1 = 0; end;else do; grp_t_254_0 = 0; grp_t_254_1 = grp; end; */ Analysis of Maximum Likelihood Estimates Parameter Standard Hazard 95% Hazard Ratio Variable DF Estimate Error Chi-Square Pr > ChiSq Ratio Confidence Limits grp_t_254_0 1 -1.42289 0.43271 10.8131 0.0010 0.241 0.103 0.563 grp_t_254_1 1 0.63878 0.30481 4.3918 0.0361 1.894 1.042 3.442the hazard of failing in group 1 is substantially lower (0.24 times) that in group 0 during thefirst 254 days. Among subjects in that group surviving beyond that time, the hazard issubstantially higher than in group 0 beyond that time.Some Stata code:. gen id = _n /* create an id variable, necessary for stsplit */. stset t delta, id(id) id: id failure event: delta != 0 & delta < .obs. time interval: (t[_n-1], t] exit on or before: failure------------------------------------------------------------------------------ 90 total obs. 0 exclusions------------------------------------------------------------------------------ 90 obs. remaining, representing 90 subjects 82 failures in single failure-per-subject data 63713 total analysis time at risk, at risk from t = 0 earliest observed entry t = 0 last observed exit t = 2988. stsplit tint, at(254)(60 observations (episodes) created). list +--------------------------------------------------------+ | t grp delta id _st _d _t _t0 tint | |--------------------------------------------------------| 1. | 1 0 1 1 1 1 1 0 0 | 2. | 63 0 1 2 1 1 63 0 0 | 3. | 105 0 1 3 1 1 105 0 0 | 4. | 129 0 1 4 1 1 129 0 0 | 5. | 182 0 1 5 1 1 182 0 0 | |--------------------------------------------------------| 6. | 216 0 1 6 1 1 216 0 0 | 7. | 250 0 1 7 1 1 250 0 0 | 8. | 254 0 . 8 1 0 254 0 0 | 9. | 262 0 1 8 1 1 262 254 254 | 10. | 254 0 . 9 1 0 254 0 0 | |--------------------------------------------------------| 11. | 301 0 1 9 1 1 301 254 254 |_t0 is stata’s internal variable for left end of the interval, _t is the right end; each subjectwith follow-up at least 254 split into 2 rows. gen intt = 0. replace intt = 1 if tint == 254(60 real changes made). gen grpint1 = grp*intt /* generate 2 time-varying indicators/interactions */. gen grpint0 = grp*(1-intt). stcox grpint0 grpint1 /* run Cox model */9.6. stcox txNo. of subjects = 150 Number of obs = 150No. of failures = 40Time at risk = 13418 LR chi2(1) = 7.86Log likelihood = -181.8874 Prob > chi2 = 0.0051------------------------------------------------------------------------------ _t | _d | Haz. Ratio Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- tx | 2.453399 .7787254 2.83 0.005 1.317016 4.570306------------------------------------------------------------------------------the drug is associated with an


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Penn BSTA 653 - BSTA 653 LECTURE NOTES

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