Simulating complex survival data

Transcription

1 Simulating complex survival data Stata Nordic and Baltic Users Group Meeting 11 th November 2011 Michael J. Crowther 1 and Paul C. Lambert 1,2 1 Centre for Biostatistics and Genetic Epidemiology Department of Health Sciences University of Leicester, UK. 2 Department of Medical Epidemiology and Biostatistics Karolinska Institutet Stockholm, Sweden. michael.crowther@le.ac.uk Michael J. Crowther Stata Users Group Meeting 1 / 20

2 Outline Outline Standard parametric distributions 2-component mixture distributions Cause-specific competing risks Time-dependent effects Extensions Michael J. Crowther Stata Users Group Meeting 2 / 20

3 Background Survival times are often generated using the exponential or Weibull distributions h 0 (t) = λ h 0 (t) = λγt γ 1 Michael J. Crowther Stata Users Group Meeting 3 / 20

4 Background Survival times are often generated using the exponential or Weibull distributions h 0 (t) = λ h 0 (t) = λγt γ 1 Are these distributions biologically realistic? Are they complex enough to fully assess statistical models? Michael J. Crowther Stata Users Group Meeting 3 / 20

5 Parametric distributions Simulating under parametric distributions We can use the method of Bender et al. (2005): h(t) = h 0 (t) exp(x β) Michael J. Crowther Stata Users Group Meeting 4 / 20

6 Parametric distributions Simulating under parametric distributions We can use the method of Bender et al. (2005): h(t) = h 0 (t) exp(x β) H(t) = H 0 (t) exp(x β), S(t) = exp[ H 0 (t) exp(x β)] Michael J. Crowther Stata Users Group Meeting 4 / 20

7 Parametric distributions Simulating under parametric distributions We can use the method of Bender et al. (2005): h(t) = h 0 (t) exp(x β) H(t) = H 0 (t) exp(x β), S(t) = exp[ H 0 (t) exp(x β)] F (t) = 1 exp[ H 0 (t) exp(x β)] Michael J. Crowther Stata Users Group Meeting 4 / 20

8 Parametric distributions Simulating under parametric distributions We can use the method of Bender et al. (2005): h(t) = h 0 (t) exp(x β) H(t) = H 0 (t) exp(x β), S(t) = exp[ H 0 (t) exp(x β)] F (t) = 1 exp[ H 0 (t) exp(x β)] U = exp[ H 0 (t) exp(x β)] U(0, 1) Michael J. Crowther Stata Users Group Meeting 4 / 20

9 Parametric distributions Simulating under parametric distributions We can use the method of Bender et al. (2005): h(t) = h 0 (t) exp(x β) H(t) = H 0 (t) exp(x β), S(t) = exp[ H 0 (t) exp(x β)] F (t) = 1 exp[ H 0 (t) exp(x β)] U = exp[ H 0 (t) exp(x β)] U(0, 1) T = H 1 0 [ log(u) exp( X β)] Michael J. Crowther Stata Users Group Meeting 4 / 20

10 Example Simulate Weibull distributed survival times. set obs gen age = rnormal(50,5). gen trt = rbinomial(1,0.5). survsim stime, dist(weibull) lambda(0.1) gamma(1.5) cov(age 0.2 trt -0.5). gen event = stime<5. replace stime = 5 if event == 0. stset stime, failure(event=1) (output omitted ). streg age trt, dist(w) nohr nolog noheader failure _d: event == 1 analysis time _t: stime _t Coef. Std. Err. z P> z [95% Conf. Interval] age trt _cons /ln_p p /p Michael J. Crowther Stata Users Group Meeting 5 / 20

11 Increasing complexity Increasing complexity We wish to increase the complexity of the baseline hazard function beyond standard and sometimes biologically implausible shapes Michael J. Crowther Stata Users Group Meeting 6 / 20

12 Increasing complexity Increasing complexity We wish to increase the complexity of the baseline hazard function beyond standard and sometimes biologically implausible shapes In many cancer trial datasets a turning point in the hazard function is observed Michael J. Crowther Stata Users Group Meeting 6 / 20

13 Increasing complexity Increasing complexity We wish to increase the complexity of the baseline hazard function beyond standard and sometimes biologically implausible shapes In many cancer trial datasets a turning point in the hazard function is observed We propose to use 2-component mixture distributions (see McLachlan and McGiffin (1994)). For example the Weibull-Weibull mixture distribution: S 0 (t) = p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 ) Michael J. Crowther Stata Users Group Meeting 6 / 20

14 Increasing complexity Simulating survival times S 0 (t) = p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 ) We can induce proportional hazards by: S(t X ) = {p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 exp(x β) )} Michael J. Crowther Stata Users Group Meeting 7 / 20

15 Increasing complexity Simulating survival times S 0 (t) = p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 ) We can induce proportional hazards by: S(t X ) = {p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 exp(x β) )} We therefore have: {p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 )} exp(x β) U(0, 1) Michael J. Crowther Stata Users Group Meeting 7 / 20

16 Increasing complexity Simulating survival times S 0 (t) = p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 ) We can induce proportional hazards by: S(t X ) = {p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 exp(x β) )} We therefore have: {p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 )} exp(x β) U(0, 1) This cannot be directly solved for t... Michael J. Crowther Stata Users Group Meeting 7 / 20

17 Increasing complexity Newton-Raphson A simple solution is to use Newton-Raphson iterations where t n+1 = t n f (t n) f (t n ) f (t n ) = {p exp( λ 1 t γ 1 ) + (1 p) exp( λ 2 t γ 2 )} exp(x β) u and u U(0, 1) Michael J. Crowther Stata Users Group Meeting 8 / 20

18 Increasing complexity λ1 = 1, γ1 = 1.5, λ2 = 1,γ2 = 0.5, p = 0.5 λ1 = 0.1, γ1 = 3, λ2 = 0.1,γ2 = 1.6, p = 0.8 Hazard function Hazard function Follow up time Follow up time λ1 = 1.4, γ1 = 1.3, λ2 = 0.1,γ2 = 0.5, p = 0.9 λ1 = 1.5, γ1 = 0.2, λ2 = 0.5,γ2 = 0.1, p = 0.1 Hazard function Hazard function Follow up time Follow up time Figure: Example baseline mixture-weibull hazard functions. Michael J. Crowther Stata Users Group Meeting 9 / 20

19 Example Fit stmix model using Crowther and Lambert (2011). webuse brcancer, clear (German breast cancer data). stset rectime, failure(censrec==1) scale(365.25). stmix hormon, dist(ww) nolog Mixture Weibull-Weibull proportional hazards regression Log likelihood = Number of obs = 686 Haz. Ratio Std. Err. z P> z [95% Conf. Interval] xb hormon logit_p_mix _cons ln_lambda1 _cons ln_gamma1 _cons ln_lambda2 _cons ln_gamma2 _cons Michael J. Crowther Stata Users Group Meeting 10 / 20

20 Example Weibull model Mixture Weibull model Survival probability 0.6 Survival probability Follow-up (years) Follow-up (years) KM, hormone=0 KM, hormone=1 Predicted survival, hormone==0 Predicted survival, hormone==1 Figure: Fitted survival function. Michael J. Crowther Stata Users Group Meeting 11 / 20

21 Example Simulation study. survsim stime, mixture lambdas(`l1 `l2 ) gammas(`g1 `g2 ) pmix(`pmix ) /// > cov(trt `loghr ). simulate s2w = r(s2w) s2ww = r(s2ww) trt_w = r(trt_w) setrt_w = r(setrt_w) /// > trt_ww = r(trt_ww) setrt_ww = r(setrt_ww), reps(500): /// > simstudy2, pmix(`pmix ) l1(`l1 ) l2(`l2 ) g1(`g1 ) g2(`g2 ) loghr(`loghr ). gen s2 = `pmix * exp(-`l1 * 2^(`g1 )) + (1-`pmix )*exp(-`l2 * 2^(`g2 )). /* Bias */. gen bias_trt_w = trt_w - (`loghr ). gen bias_trt_ww = trt_ww - (`loghr ). su bias* Variable Obs Mean Std. Dev. Min Max bias_trt_w bias_trt_ww su bias_s2* Variable Obs Mean Std. Dev. Min Max bias_s2w bias_s2ww Michael J. Crowther Stata Users Group Meeting 12 / 20

22 Competing risks Simulating competing risks We can use the method of Beyersmann et al. (2009): Specify K cause-specific hazard functions, h k (t) Michael J. Crowther Stata Users Group Meeting 13 / 20

23 Competing risks Simulating competing risks We can use the method of Beyersmann et al. (2009): Specify K cause-specific hazard functions, h k (t) Simulate survival times with all-cause hazard: h all (t) = K h k (t) k=1 Michael J. Crowther Stata Users Group Meeting 13 / 20

24 Competing risks Simulating competing risks We can use the method of Beyersmann et al. (2009): Specify K cause-specific hazard functions, h k (t) Simulate survival times with all-cause hazard: h all (t) = K h k (t) k=1 For a simulated time t we run a multinomial experiment, with probability for each cause j: Prob(Cause = j t) = h j (t) K k=1 h k(t) Michael J. Crowther Stata Users Group Meeting 13 / 20

25 Example Example using Coviello (2008). set obs obs was 0, now gen trt = rbinomial(1,0.5). survsim stime event, dist(weibull) ncr(2) lambdas( ) gammas( ) > cov(trt ). replace event = 0 if stime>15 (78 real changes made). stset stime, failure(event==1) (output omitted ). streg trt, dist(w) nohr nolog noheader failure _d: event == 1 analysis time _t: stime _t Coef. Std. Err. z P> z [95% Conf. Interval] trt _cons /ln_p p /p stcompet ci1 = ci, compet1(2) by(trt) Michael J. Crowther Stata Users Group Meeting 14 / 20

26 Example 1.0 Cause Cause Cumulative Incidence Cumulative Incidence Follow up time Follow up time Treatment = 0 Treatment = 1 Figure: Cumulative incidence. Michael J. Crowther Stata Users Group Meeting 15 / 20

27 Time-dependent effects We want to incorporate time-dependent effects, such as a diminishing treatment effect. Under standard parametric models this can be achieved simply: h(t) = λγt γ 1 exp(βx i + φx i log(t)) = λγt γ 1+φX i exp(βx i ) Michael J. Crowther Stata Users Group Meeting 16 / 20

28 Example Example using Lambert and Royston (2009). set obs obs was 0, now gen trt = rbinomial(1,0.5). survsim stime, dist(weibull) lambdas(0.1) gammas(1.5) /// > cov(trt -0.5) tde(trt 0.15). gen died = stime <= 5. replace stime = 5 if died == 0 (3869 real changes made). stset stime, f(died = 1) (output omitted ). stpm2 trt, scale(h) df(3) tvc(trt) dftvc(1). predict hr, hrnumer(trt 1) ci. stpm2 trt, scale(h) df(3). predict hr2, hrnumer(trt 1) ci Michael J. Crowther Stata Users Group Meeting 17 / 20

29 Example 0.8 Time dependent hazard ratio Hazard Ratio Time 95% Confidence Interval Time dependent HR Constant HR Figure: Time-dependent hazard ratio. Michael J. Crowther Stata Users Group Meeting 18 / 20

30 Cure proportions Frailty distributions Time-dependent effects in mixture distributions Joint longitudinal and survival data Michael J. Crowther Stata Users Group Meeting 19 / 20

31 References I R. Bender, T. Augustin, and M. Blettner. Generating survival times to simulate Cox proportional hazards models. Stat Med, 24(11): , Jan Beyersmann, Aurélien Latouche, Anika Buchholz, and Martin Schumacher. Simulating competing risks data in survival analysis. Stat Med, 28(6): , Mar doi: /sim URL Enzo Coviello. Stcompet: Stata module to generate cumulative incidence in presence of competing events. Statistical Software Components, Boston College Department of Economics, URL Michael J. Crowther and Paul C. Lambert. Stmix: Stata module to fit two-component parametric mixture survival models. Statistical Software Components, Boston College Department of Economics, October URL P. C Lambert and P. Royston. Further development of flexible parametric models for survival analysis. The Stata Journal, 9: , G. J. McLachlan and D. C. McGiffin. On the role of finite mixture models in survival analysis. Stat Methods Med Res, 3(3): , Michael J. Crowther Stata Users Group Meeting 20 / 20