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
加速失效模型，AFT 模型的特點是不管所謂的風險概念，而是對每個患者真正的生存時間進行模型化處理，其實個人更加喜歡這個模型，因為它很直觀地告訴你某類人的生存時間就是比另一類人短，而不是告訴你一個抽象的一組的風險低於或者高於另一組，因為很多人無法理解什麼是風險 (hazard)，正如很多人無法理解什麼是比值 (odds) 一樣。同樣還因為風險比例模型還要考慮是否對基線風險進行參數估計的取捨問題。
所以，風險比例模型：
\[ h_1(t) = \psi_{PH}h_0(t) \]
AFT 模型：
\[ T_1 = \psi_{AFT}T_0 \]
在 AFT 模型中，\(\psi_{AFT}\) 就是加速指數，它的直接涵義就是，治療組患者的死亡時間被“加速”或者“減緩”了。也就是它可以回答，治療組的患者是不是更快的痊癒？或者更快的死亡？這樣明了的問題。
通常情況下，這個加速指數用 \(e^{-\beta_{AFT}}\) 進行參數化分析：
\[ T_1 = T_0 e^{-\beta_{AFT}} \]
寫下此時的生存方程，和基線組生存方程的關係：
\[ \begin{aligned} S_1(t) & = \text{Pr}(T_1 > t) \\ & = \text{Pr}(e^{-\beta_{AFT}}T_0 > t) \\ & = \text{Pr}(T_0 > te^{\beta_{AFT}}) \\ & = S_0(e^{\beta_{AFT}}t) \end{aligned} \]
77.2.1 Weibull 模型也是一種 AFT 模型
一個 Weibull 模型在風險比例前提下的生存方程是：
\[ S(t|x) = \exp\{-\lambda e^{\beta_{PH}x} t^\kappa \} \]
一個 Weibull 模型在 AFT 模型下的生存方程是：
\[ S(t|x) = \exp \{ -\lambda e^{\kappa \beta_{AFT}x} t^\kappa \} \]
所以當你用 Weibull 模型時，其實可以自由在兩種類型 (PH or AFT) 之間自由切換：
\[ \exp(\beta^T_{PH}) = \exp(\kappa\beta^T_{AFT}) \]
所以，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。