# 第 77 章 生存分析的其他手段

## 77.1 分層Cox生存分析 stratified Cox proportional hazards model

Under the Cox proportional hazards model, the effect of each explanatory variable on the hazard is assumed to be such that the ratio of hazards is constant accross the time scale (the proportional hazards assumption). In applications with several explanatory variabls, the effect of some of these variables may not be proportional. When the aim of the analysis is not focussed on these particular variables, for example if they are just being used as adjustment variables and are not the main exposures of interest, then the proportionality assumption can be relaxed just for those variables by fitting a stratified Cox proportional hazards model.

In the stratified Cox proportional hazards model, instead of assuming that the proportional hazards model holds overall, we assume that the proportional hazards assumption holds within groups (or strata) of individuals.

$h(t|x,s) = h_{0s} (t)e^{\beta^T x}$

Each stratum, s, has a separate baseline hazard $$h_{0s}(t)$$. However, the other explanatory variables x are assumed to act in the same way on the baseline hazard in each stratum, i.e. the $$\beta$$ are the same accross strata.

## 77.2 加速失效模型 Accelerated failure time (AFT) model

$h_1(t) = \psi_{PH}h_0(t)$

AFT 模型：

$T_1 = \psi_{AFT}T_0$

$T_1 = T_0 e^{-\beta_{AFT}}$

\begin{aligned} S_1(t) & = \text{Pr}(T_1 > t) \\ & = \text{Pr}(e^{-\beta_{AFT}}T_0 > t) \\ & = \text{Pr}(T_0 > te^{\beta_{AFT}}) \\ & = S_0(e^{\beta_{AFT}}t) \end{aligned}

### 77.2.1 Weibull 模型也是一種 AFT 模型

$S(t|x) = \exp\{-\lambda e^{\beta_{PH}x} t^\kappa \}$

$S(t|x) = \exp \{ -\lambda e^{\kappa \beta_{AFT}x} t^\kappa \}$

$\exp(\beta^T_{PH}) = \exp(\kappa\beta^T_{AFT})$