Extending Cox RegressionAccelerated Failure Time Models

Tom Greene & Nan Hu

Quick Review from Last Time

Relationship of Population and Individual Hazard Ratios

• Multiplicative frailty model for individual hazard:α(t|W) = W × α(t) if assigned to controlαr(t|W) = r × W × α(t) if assigned to treatmentW ~ 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)

)(1

)(1

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1

2

tAr

tAr

t

t

Attenuates towards 1 as t increases

Relative Mortality for Norwegian Men1901-1905 compared to 1991

Horizontal line indicates relative risk of 1

From “Statistics Norway” as reproduced in Aalen O, Survival and Event History Analysis

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

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

Practical Implications • Two approaches to estimating effects on

individual hazards:– Analysis of repeat event data– Joint analysis of longitudinal and time-to-event

outcomes

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

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.

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

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;

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

Plots of Schoenfeld ResidualsR 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 pKTV_GRP 0.0049 0.0209 0.88510BALB 0.0968 8.2343 0.00411GLOBAL NA 8.2570 0.01611

Schoenfeld Residual Plots with Cubic Spline Smooths: R Output

For KTV_GRP For Baseline Albumin

Schoenfeld Residual Plots with Cubic Spline Smooths: R Output(Omitting the residuals)

For KTV_GRP For Baseline Albumin

Multiplicative Hazards with Time Varying Coefficients

• Standard Cox Proportional Hazards Model λ(t|Z) = λ0 (t) × exp(β Z)

• Cox Proportional Hazards Model with Time-Dependent Covariates λ(t|Z) = λ0 (t) × exp(β Z(t))

• Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients λ(t|Z) = λ0 (t) × exp(β(t) Z)

• Multiplicative Hazards Model with Time-Dependent Covariates and Time-Varying Coefficients λ(t|Z) = λ0 (t) × exp(β(t) Z(t))

Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients• Model

λ(t|Z) = λ0 (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)

t

jj duutB0

)()(

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:

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)=0cKTV_GRP 1.38 0.931cBALB 11.30 0.000

Test for time invariant effects Kolmogorov-Smirnov test p-value H_0:constant effectcKTV_GRP 0.163 0.986cBALB 0.833 0.038 Cramer von Mises test p-value H_0:constant effectcKTV_GRP 0.015 0.993cBALB 1.040 0.026

Multiplicative Hazards Model with Fixed Covariates and Time-Varying Coefficients