1. Accelerated Failure Time (AFT) Model As An Alternative to Cox Model
Nan Hu

2. 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.
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.

3. 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

4. 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

5. Linear Rank Tests where
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:
where

6. Alternative Forms of AFTM
1. In terms of survival functions:
2. In terms of quantile functions:

7. Alternative Forms of AFTM
Two sample AFT models:

8. Alternative Forms of AFTM
3. In terms of hazard function
cf. proportional hazards model
The only difference is the additional time scale change on baseline hazard function.

9. Vaginal Cancer for Rats (Pike 1966)
KM curve by treatment arms

10. 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:
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: Hence, the baseline survival function will be .

11. AFT model with parametric baseline hazard(s)
Output (model1)
aftreg(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data, shape = 1)
Covariate W.mean Coef Exp(Coef) se(Coef) Wald p
Trt
log(scale)
Shape is fixed at 1
Events
Total time at risk
Max. log. likelihood
LR test statistic
Degrees of freedom
Overall p-value
Output (model2)
aftreg(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data)
Covariate W.mean Coef Exp(Coef) se(Coef) Wald p
Trt
log(scale)
log(shape)
Events
Total time at risk
Max. log. likelihood
LR test statistic
Degrees of freedom
Overall p-value

12. AFT model with parametric baseline hazard(s)
Comparing with Cox model:
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
coef exp(coef) se(coef) z Pr(>|z|)
Trt
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
exp(coef) exp(-coef) lower .95 upper .95
Trt
Rsquare= (max possible= )
Likelihood ratio test= on 1 df, p=
Wald test = on 1 df, p=
Score (logrank) test = on 1 df, p=
Output (model3)
Call:
survreg(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data, dist = "weibull")
Value Std. Error z p
(Intercept) e+00
Trt e-02
Log(scale) e-96
Scale=
Weibull distribution
Loglik(model)= Loglik(intercept only)= -22.3
Chisq= 3.63 on 1 degrees of freedom, p= 0.057
Number of Newton-Raphson Iterations: 6
n= 41

13. 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,]
Least-Squares Estimator:
Estimate Std. Error Z value Pr(>|Z|)
[1,]

14. 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