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In Applied Survival Analysis: Regression Modeling of Time-to-Event Data, 2nd edition, Ch8, page262. Hosmer, Lemeshow and May obtained the time ratio of Weibull regression model, according to the following procedure:

The survival function of the AFT Weibull regression model is $$S(t,x,\beta,\sigma)=\exp\left\{-t^\lambda\exp[(-1/\sigma)(\beta_0+\beta_1x)]\right\}.$$ Setting $S(t,x,\beta,\sigma)=0.5$ and solving for $t$, then we get the median survival time $$t_{50}(x,\beta,\sigma)=[-\ln(0.5)]^{\sigma}e^{\beta_0+\beta_1x}.$$ Suppose that $x$ is a dummy variable, then the time ratio at the median survival time is $$\text{TR}(x=0,x=1)=\frac{t_{50}(x=1,\beta,\sigma)}{t_{50}(x=0,\beta,\sigma)}=\frac{[-\ln(0.5)]^{\sigma}e^{\beta_0+\beta_1}}{[-\ln(0.5)]^{\sigma}e^{\beta_0}}=e^{\beta_1}.$$ I am trying to solve the median survival time ratio of the AFT generalized gamma parametric regression model (the case of $\kappa>0$), which is not in the book. I have known that the survival function of the AFT generalized gamma parametric regression is $$\left\{\begin{matrix} 1-I(\gamma,u) \quad \text{if}\ \kappa >0 \\ 1-\Phi(z) \quad \text{if}\ \kappa =0\\ I(\gamma,u) \quad \text{if}\ \kappa <0 \end{matrix}\right.$$ where $\gamma=\left|\kappa \right|^2$, $u=\gamma\exp(\left|\kappa \right|z)$, $z=\text{sign}(\kappa)\left\{\log(t_j)-\mu_j\right\}/\sigma$ and $I(a,x)$ is the incomplete gamma function, which is $$\dfrac{1}{\Gamma(a)}\int^x_0e^{-t}t^{a-1}dt$$ with shape parameter $a$. Furthermore, if $x<0$ then $I(a,x)=0$.

This question takes me a lot of time, if anyone who is familiar with Survival Analysis can help me. I am grateful.

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