Accelerated Failure Time (AFT) Model
As An Alternative to Cox Model
Nan Hu
Accelerated Failure Time (AFT) Model
• The effect of a fixed covariate Z is to act multiplicatively on
the failure time T or additively on Y = logT.
Y  logT     T Z  
• exp(β): regression parameter which can be interpreted as
the ratio of failure time per unit change in covariate.
• AFT model postulates a direct relationship between failure
time and covariates.
• “Accelerated failure time model are in many ways more
appealing because of their quite direct physical
interpretation” – Sir David Cox.
Accelerated Failure Models
(Some background & rationale)
• Often used in engineering for modeling reliability
(survival) of mechanical systems, but relatively
uncommon in medicine
• Posit uniform increase or decrease in the rate of
change in a system over time
• If the baseline hazard function is assumed to
follow a Weibul distribution, accelerated failure
and proportional hazards assumptions are
equivalent
Accelerated Failure Models
(Background & Rationale)
• In RCT setting,
– The coefficient of the treatment assignment indicator
variable represents the average causal effect of the
treatment on log survival over individuals
– The exponential of this coefficient represent the
geometric mean of individual causal effects
expressed as ratios
– By contrast, population hazard ratios do not have an
interpretation as an average of individual level causal
effects unless was assumes no frailty variation
Linear Rank Tests
• Let Yi = logTi (i = 1, 2, …,n) be an uncensored sample of log
failure times with corresponding covariates Z1, .., Zn, where Zi is
a vector of time-independent covariates for the ith subject. Y(1),
… Y(n) be the order statistic of Y, and Z(1),…Z(n) are the
corresponding covariates. A linear rank statistic is of the
follosing form:
n
v   Z ( i ) ci
i 1
n
where
c
i 1
•
i
0
Alternative Forms of AFTM
• 1. In terms of survival functions:
S (t | Z )  Pr(T  t | Z )  Pr(logT  log t | Z )
 Pr(   T Z    log t | Z )
 Pr{  (1 /  )[logt  (   T Z )] | Z }
 Pr(exp( )  [t / exp(   T Z )]1/  | Z )  S 0 {[t / exp(   T Z )]1/  }
• 2. In terms of quantile functions:
Q( p | Z )  Q0 ( p) exp( Z )
T
Alternative Forms of AFTM
• Two sample AFT models:
Alternative Forms of AFTM
• 3. In terms of hazard function
S (t | Z )  S 0 (t exp( T Z ))
d log S 0 (t exp( T Z ))
d log S (t | Z )


d
d
  (t | Z )  0 (t exp( T Z )) exp( T Z )
- cf. proportional hazards model
 (t | Z )  0 (t ) exp( T Z )
The only difference is the additional time scale change
on baseline hazard function.
Vaginal Cancer for Rats (Pike 1966)
• KM curve by treatment arms
0.00
0.25
0.50
0.75
1.00
Kaplan-Meier survival estimates
0
100
200
analysis time
trt = 0
300
trt = 1
400
AFT model with parametric baseline hazard(s)
data<- read.csv(“Pike1966.csv”, header=T)
library(eha)
mod1<- aftreg(Surv(log(Time-100),Surv)~Trt, data=data, shape=1)
mod2<- aftreg(Surv(log(Time-100),Surv)~Trt ,data=data)
library(survival)
mod3<- survreg(Surv(log(Time-100), Surv) ~ Trt, data=data, dist='weibull')
• The parametric baseline function for aftreg is given by:
S (t | Z )  Pr(T  t | Z )  S0{[t / exp(b   T Z )]a }
where a is the “shape” parameter and b is the “scale” parameter
The default baseline distribution is “weibull”, set “shape=1” *(a=1) for
the exponential. Other options include “loglogistic”, “lognormal” etc.
• The parametrization for survreg is: h(T )     T Z   . Hence,
T
1/ 
the baseline survival function will be S (t | Z )  S0{[t exp(   Z )] }
AFT model with parametric baseline hazard(s)
•
Output (model1)
aftreg(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data,
shape = 1)
Covariate
Trt
log(scale)
W.mean Coef Exp(Coef) se(Coef) Wald p
0.545
-0.020 0.980
0.330
0.952
1.658
5.251 0.245
0.000
Shape is fixed at 1
Events
37
Total time at risk
196.37
Max. log. likelihood
-98.755
LR test statistic
0
Degrees of freedom
1
Overall p-value
0.95212
•
Output (model2)
aftreg(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data)
Covariate
Trt
log(scale)
log(shape)
W.mean Coef
Exp(Coef) se(Coef) Wald p
0.545 -0.043 0.958
0.022
0.045
1.605
4.976
0.011
0.000
2.725 15.262
0.131
0.000
Events
37
Total time at risk
196.37
Max. log. likelihood
-20.488
LR test statistic
3.63
Degrees of freedom
1
Overall p-value
0.0567341
AFT model with parametric baseline hazard(s)
•
Comparing with Cox model:
Output (model3)
Call:
survreg(formula = Surv(log(Time - 100), Surv) ~ Trt,
data = data, dist = "weibull")
(Intercept)
Trt
Log(scale)
Value
1.5825
0.0433
-2.7253
Std. Error
z
0.0160 98.63
0.0216
2.01
0.1307 -20.85
p
0.00e+00
4.49e-02
1.60e-96
mod4<- coxph(Surv(log(Time-100), Surv)~ Trt, data=data)
•
Output (model4)
Call:
coxph(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data)
n= 41
Scale= 0.0655
Weibull distribution
Loglik(model)= -20.5 Loglik(intercept only)= -22.3
Chisq= 3.63 on 1 degrees of freedom, p= 0.057
Number of Newton-Raphson Iterations: 6
n= 41
Trt
---
coef exp(coef)
-0.6089 0.5440
se(coef)
z
0.3440 -1.77
Pr(>|z|)
0.0767
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
exp(coef) exp(-coef) lower .95 upper .95
Trt 0.544
1.838 0.2772 1.068
Rsquare= 0.072 (max possible= 0.994 )
Likelihood ratio test= 3.07 on 1 df, p=0.07993
Wald test
= 3.13 on 1 df, p=0.07673
Score (logrank) test = 3.22 on 1 df, p=0.07288
Least Square Regression for AFT (lss)
R Code:
library(lss)
data<- read.csv(“Pike1966.csv”, header=T)
mod5<- lss(cbind(log(Time-100),Surv) ~ Trt,data=data, gehanonly=FALSE,
maxiter=10,tolerance=0.001)
Output:
Gehan Estimator:
Estimate Std. Error Z value Pr(>|Z|)
[1,] 0.1914540 0.1147475 1.668481 0.09522027
Least-Squares Estimator:
Estimate Std. Error Z value Pr(>|Z|)
[1,] 0.1668167 0.1306339 1.276979 0.2016098
Discussion Topic
• Are conventional Cox proportional hazards
models over-used compared to other
regression methods in medical research?
• Other methods
– Additive hazards models
– Accelerated failure time models
– Proportional odds models
– Transformation models