1. Extending Cox Regression Accelerated Failure Time Models Tom Greene & Nan Hu

2. Quick Review from Last Time

3. Relationship of Population and Individual Hazard Ratios Multiplicative frailty model for individual hazard: α(t|W) = W × α(t)if assigned to control α r (t|W) = r × W × α(t)if assigned to treatment W ~ Gamma with mean 1 and variance δ. Hence the individual HR is r. The population HR is A(t) is the cumulative hazard associated with α(t) Attenuates towards 1 as t increases

4. Relative Mortality for Norwegian Men 1901-1905 compared to 1991 Horizontal line indicates relative risk of 1 From “Statistics Norway” as reproduced in Aalen O, Survival and Event History Analysis

5. Practical Implications of Frailty and Changing Risk Sets Over Time Variation in population hazards ratio over time depends both on variation in individual hazards ratio and frailty effects It is often found that HRs attenuate towards 1 or reverse over time, or among “survivors” who reach an advance stage of a chronic disease (e.g., those with end stage renal disease) Frailty effects are less of an issue when the fraction of pts with events is low

6. Practical Implications Prevailing practice is to avoid covariate adjustment for survival outcomes in RCTs. But adjustment for strong prognostic factors can: – Reduce conservative bias in estimated treatment effect – Increase power – Reduce differential survival bias if the analysis censors competing risks

7. Practical Implications Two approaches to estimating effects on individual hazards: – Analysis of repeat event data – Joint analysis of longitudinal and time-to-event outcomes

8. Evaluation of Proportional Hazards Parametric models for change in hazard ratios over time Non-parametric smooths of Schoenfeld residuals Non-parametric models for multiplicative hazards

9. Parametric models for change in hazard ratios over time Example: Cox Regression of Effects of Dose Group (Ktv_grp) and baseline serum albumin (Balb) in the HEMO Study RCT with 871 deaths in 1871 patients; planned follow-up 1.5 to 7 years.

10. Parametric models for change in hazard ratios over time 1) Test for linear interactions of predictors with follow-up time proc phreg data=demsum01 ; model fu_yr * ev_d(0) = ktv_grp balb ktv_grpt balbt; ktv_grpt = ktv_grp*fu_yr; balbt = balb*fu_yr; Parameter Standard Parameter DF Estimate Error Chi-Square Pr > ChiSq KTV_GRP 1 -0.06933 0.12112 0.3277 0.5670 BALB 1 -1.50469 0.17279 75.8320 <.0001 Ktv_grpt 1 0.00686 0.04406 0.0243 0.8762 Balbt 1 0.16874 0.06378 7.0001 0.0082 HR for baseline albumin (in g/dL) is 0.22 at time 0, but attenuates by a factor of exp(0.1687) = 1.184 per year

11. Parametric models for change in hazard ratios over time 2) Test for interactions of predictors with time period (> 1 yr vs. < 1yr) proc phreg data=demsum01 ; model fu_yr * ev_d(0) = Ktv_grp1 Balb1 Ktv_grp2 Balb2; if fu_yr > 1 then period =1; if. < fu_yr <= 1 then period = 0; Ktv_grp1 = ktv_grp*(1-period); Balb1 = balb*(1-period); Ktv_grp2 = ktv_grp*period; Balb2 = balb*period; PropHazKtv: test Ktv_grp1=Ktv_grp2; PropHazBalb: test Balb1 = Balb2;

12. Parametric models for change in hazard ratios over time 2) Test for interactions of predictors with time period (> 1 yr vs. < 1yr) Parameter Standard Parameter DF Estimate Error Chi-Square Pr > ChiSq Ktv_grp1 1 -0.07569 0.13413 0.3185 0.5725 Balb1 1 -1.58168 0.18855 70.3661 <.0001 Ktv_grp2 1 -0.04680 0.07861 0.3545 0.5516 Balb2 1 -0.95959 0.11553 68.9860 <.0001 Wald Label Chi-Square DF Pr > ChiSq PropHazKtv 0.0345 1 0.8526 PropHazBalb 7.9139 1 0.0049

13. Plots of Schoenfeld Residuals R Code: require(survival) HEMOCox<-coxph(Surv(fu_yr,EV_D) ~ KTV_GRP+BALB,data=hemodat) HEMOPropchk<-cox.zph(HEMOCox) HEMOPropchk plot(HEMOPropchk,var="KTV_GRP") plot(HEMOPropchk,var="BALB") plot(HEMOPropchk,var=“KTV_GRP",resid=FALSE) plot(HEMOPropchk,var="BALB",resid=FALSE) rho chisq p KTV_GRP 0.0049 0.0209 0.88510 BALB 0.0968 8.2343 0.00411 GLOBAL NA 8.2570 0.01611

14. Schoenfeld Residual Plots with Cubic Spline Smooths: R Output For KTV_GRPFor Baseline Albumin

15. Schoenfeld Residual Plots with Cubic Spline Smooths: R Output (Omitting the residuals) For KTV_GRPFor Baseline Albumin

16. Multiplicative Hazards with Time Varying Coefficients Standard Cox Proportional Hazards Model λ(t|Z) = λ 0 (t) × exp(β Z) Cox Proportional Hazards Model with Time-Dependent Covariates λ(t|Z) = λ 0 (t) × exp(β Z(t)) Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients λ(t|Z) = λ 0 (t) × exp(β(t) Z) Multiplicative Hazards Model with Time-Dependent Covariates and Time-Varying Coefficients λ(t|Z) = λ 0 (t) × exp(β(t) Z(t))

17. Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients Model λ(t|Z) = λ 0 (t) × exp(β(t) Z) Useful if proportional hazards assumption in doubt, and you don’t want to assume a particular parametric model for change in HR over time Estimands are cumulative Cox regression coefficients Can use timereg package in R (if you are careful to center predictor variables)

18. Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients > cBALB<-BALB – mean(BALB) > cKTV_GRP <- KTV_GRP – mean(KTV_GRP) > require(timereg) > fit<-timecox(Surv(fu_yr,EV_D)~KTV_GRP+cBALB,max.time=5) > summary(fit) > par(mfrow(1,2) > plot(fit,c(2,3)) R Code:

19. Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients summary(fit) Multiplicative Hazard Model Test for nonparametric terms Test for non-significant effects Supremum-test of significance p-value H_0: B(t)=0 cKTV_GRP 1.38 0.931 cBALB 11.30 0.000 Test for time invariant effects Kolmogorov-Smirnov test p-value H_0:constant effect cKTV_GRP 0.163 0.986 cBALB 0.833 0.038 Cramer von Mises test p-value H_0:constant effect cKTV_GRP 0.015 0.993 cBALB 1.040 0.026

20. Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients