Accelerated Life Model

Definition

Consider the random variable $T_0$ represents survival time with $\mathbf{x}=0$ Hazard Function with $f_0(t)$, ${\lambda }_0(t)$, and $S_0(t)$ And assume that the survival time of individual with covariate $\mathbf{x}$ is defined as $T_{\mathbf{x}}=T_0\exp ({\mathbf{x}}^{⊺}\mathbf{\beta })$ If ${\mathbf{x}}^{⊺}\mathbf{\beta }<0$ then the covariate $\mathbf{x}$ accelerates the time to failure. The model based on this assumption is called accelerated failure time (AFT) model.

Under AFT model $S(t:\mathbf{x})=S_0(t\exp (-{\mathbf{x}}^{⊺}\mathbf{\beta }))$ $\lambda (t:\mathbf{x})={\lambda }_0(t\exp (-{\mathbf{x}}^{⊺}\mathbf{\beta }))\exp (-{\mathbf{x}}^{⊺}\mathbf{\beta })$

Let $Y=\ln T_{\mathbf{x}}$, then $E(Y)=E(\ln T_0)+{\mathbf{x}}^{⊺}\mathbf{\beta }\equiv \alpha +{\mathbf{x}}^{⊺}\mathbf{\beta }$ $Y=\alpha +{\mathbf{x}}^{⊺}\mathbf{\beta }+ϵ\equiv E(\ln T_0)+{\mathbf{x}}^{⊺}\mathbf{\beta }+\underset{ϵ}{\underbrace{\ln T_0-E(\ln T_0)}}$ where $ϵ=\ln T_0-E(\ln T_0)$.

Assume that $ϵ=\sigma W$, where $W$ is a Random Variable represents error term. Then, the AFT model becomes $Y\equiv \ln T_{\mathbf{x}}=\alpha +{\mathbf{x}}^{⊺}\mathbf{\beta }+\sigma W$

Relationship between $Y=\ln T_{\mathbf{x}}$ and $W$. $F_T(t)=P_W(\frac{\ln t-\mu }{\sigma })$ $f_T(t)=\frac{1}{\sigma t}{f}_W(\frac{\ln t-\mu }{\sigma })$ where $\mu =\alpha +{\mathbf{x}}^{⊺}\mathbf{\beta }$

If $W\sim N(0,1)$ then $\ln T\sim N(\mu ,{\sigma }^2)$, and if $W\sim \mathrm{Gumbel}(0,1)$, then $T\sim \mathrm{Exp}(\lambda )$ where $\lambda =\exp [-(\alpha +{\mathbf{x}}^{⊺}\mathbf{\beta })]$.