Extending Cox Regression
Accelerated 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 control
αr(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
2 (t )
1  A(t )
r
1(t )
1  rA(t )
Attenuates
towards 1 as t
increases
A(t) is the cumulative hazard associated with α(t)
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
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
DF
Parameter
Estimate
KTV_GRP
BALB
Ktv_grpt
Balbt
1
1
1
1
-0.06933
-1.50469
0.00686
0.16874
Standard
Error
Chi-Square
Pr > ChiSq
0.12112
0.17279
0.04406
0.06378
0.3277
75.8320
0.0243
7.0001
0.5670
<.0001
0.8762
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
DF
Parameter
Estimate
Ktv_grp1
Balb1
Ktv_grp2
Balb2
1
1
1
1
-0.07569
-1.58168
-0.04680
-0.95959
Label
PropHazKtv
PropHazBalb
Standard
Error
Chi-Square
Pr > ChiSq
0.13413
0.18855
0.07861
0.11553
0.3185
70.3661
0.3545
68.9860
0.5725
<.0001
0.5516
<.0001
Wald
Chi-Square
DF
Pr > ChiSq
0.0345
7.9139
1
1
0.8526
0.0049
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
rho
KTV_GRP 0.0049
BALB
0.0968
GLOBAL
NA
chisq
p
0.0209 0.88510
8.2343 0.00411
8.2570 0.01611
plot(HEMOPropchk,var="KTV_GRP")
plot(HEMOPropchk,var="BALB")
plot(HEMOPropchk,var=“KTV_GRP",resid=FALSE)
plot(HEMOPropchk,var="BALB",resid=FALSE)
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
t
B j (t )    j (u )du
0
• Can use timereg package in R (if you are careful to
center predictor variables)
Multiplicative Hazards Model with Fixed
Covariates and Time-Varying Coefficients
R Code:
>
>
>
>
>
>
>
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))
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
Multiplicative Hazards Model with Fixed
Covariates and Time-Varying Coefficients