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. 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 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

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

Alternative Forms of AFTM Two sample AFT models:

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.

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

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 .

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 0.545 -0.020 0.980 0.330 0.952 log(scale) 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 W.mean Coef Exp(Coef) se(Coef) Wald p Trt 0.545 -0.043 0.958 0.022 0.045 log(scale) 1.605 4.976 0.011 0.000 log(shape) 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: 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 -0.6089 0.5440 0.3440 -1.77 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 Output (model3) Call: survreg(formula = Surv(log(Time - 100), Surv) ~ Trt, data = data, dist = "weibull") Value Std. Error z p (Intercept) 1.5825 0.0160 98.63 0.00e+00 Trt 0.0433 0.0216 2.01 4.49e-02 Log(scale) -2.7253 0.1307 -20.85 1.60e-96 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

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