# 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(死亡)模型 Accelerated failure time (AFT) model

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

$T_1 = \psi_{AFT}T_0$

### 77.2.1 詳細推導

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

\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}

### 77.2.2 再詳細推導

$S(t;\mathbf{x}) = S_0(te^{\beta_{AFT}^T\mathbf{x}})$

\begin{aligned} h(t;\mathbf{x}) & = h_0(te^{\beta_{AFT}^T\mathbf{x}})e^{\beta_{AFT}^T\mathbf{x}} \\ f(t;\mathbf{x}) & = f_0(te^{\beta_{AFT}^T\mathbf{x}})e^{\beta_{AFT}^T\mathbf{x}} \end{aligned}

### 77.2.3 風險比例模型(PH)和加速失效（死亡）模型(AFT)的比較

PH 對照組: $$h_0(t)$$ 對照組: $$S_0(t) = \exp\{-\int_0^th_0(u) du\}$$

AFT 對照組: $$h_0(t)$$ 對照組: $$S_0(t) = \exp\{-\int_0^th_0(u) du\}$$

### 77.2.4 Weibull 模型也是一種 AFT 模型

Weibull 模型下，對照組的生存方程 (survival function, $$S_0(t)$$) 被描述爲：

$S_0(t) = \exp\{-\lambda t^{\kappa}\}$

$S(t;\mathbf{x}) = \exp\{-\lambda(te^{\beta_{AFT}^T\mathbf{x}})^\kappa\}= \exp\{-\lambda e^{\kappa \beta_{AFT}^T\mathbf{x}} t^\kappa\}$

$S(t;\mathbf{x}) = \exp\{-\lambda e^{\beta^T_{PH}\mathbf{x}} t^\kappa \}$

$S(t;\mathbf{x}) = \exp \{ -\lambda e^{\kappa \beta^T_{AFT}\mathbf{x}} t^\kappa \}$

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

Weibull 模型和其特殊形態–指數模型，為唯二的兩個，可以在 PH 模型和 AFT 模型之間自由切換的模型。