Chapter 17 Causal Survival Analysis

文章目录

17.1 Hazards and risks
17.2 From hazards to risks
17.3 Why censoring matters
17.4 IP weighting of marginal structural models
17.5 The parametric g-formula
17.6 G-estimation of structural nested models
Fine Point
- Competing events
- The hazards of hazard ratios
- Models for survival analysis
Technical Point
- Approximating the hazard ratio via a logistic model
- Structural nested cumulative failure time (CFT) models and cumulative survival time (CST) models
- Artificial censoring

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这一章节主要介绍了如何在生存模型中进行因果分析.

17.1 Hazards and risks

Administrative censoring: 指开始追踪和停止追踪的时间有差异.

设生存时间为TTT, 并让kkk表示第kkk个时间单位.

生存概率: Pr[T>k]\mathrm{Pr}[T > k]Pr[T>k].

risk: 1−Pr[T>k]=Pr[T≤k]1 - \mathrm{Pr}[T > k] = \mathrm{Pr}[T \le k]1−Pr[T>k]=Pr[T≤k].

hazard: Pr[T=k∣T>k−1]\mathrm{Pr}[T=k|T>k-1]Pr[T=k∣T>k−1].

显然生存概率是单调递减的, 而risk是单调递增的, 但是hazard并不绝对.

17.2 From hazards to risks

用Dk=1D_k=1Dk=1表示个体在第k+1k+1k+1个时间单位状态为死亡(0则表示生存).
则
Pr[Dk=0]=∏m=1kPr[Dm=0∣Dm−1=0].\mathrm{Pr}[D_k = 0] = \prod_{m=1}^k \mathrm{Pr}[D_m=0|D_{m-1}=0]. Pr[Dk=0]=m=1∏kPr[Dm=0∣Dm−1=0].

注:
Pr[Dm=0∣Dm−1=0]=Pr[Dm=0]Pr[Dm−1=0]Pr[D1=0]=1.\mathrm{Pr}[D_m=0|D_{m-1}=0] = \frac{\mathrm{Pr}[D_m=0]}{\mathrm{Pr}[D_{m-1}=0]} \\ \mathrm{Pr}[D_1 = 0] = 1. Pr[Dm=0∣Dm−1=0]=Pr[Dm−1=0]Pr[Dm=0]Pr[D1=0]=1.

且容易证明:
Pr[Dk=0]=Pr[T>k]Pr[Dk=1]=Pr[T≤k]Pr[Dk=1∣Dk−1=0]=Pr[T=k∣T>k−1].\mathrm{Pr}[D_{k} = 0] = \mathrm{Pr}[T > k] \\ \mathrm{Pr}[D_{k} = 1] = \mathrm{Pr}[T \le k] \\ \mathrm{Pr}[D_{k} = 1|D_{k-1} = 0] = \mathrm{Pr}[T = k|T > k-1]. Pr[Dk=0]=Pr[T>k]Pr[Dk=1]=Pr[T≤k]Pr[Dk=1∣Dk−1=0]=Pr[T=k∣T>k−1].

这相当于, 只要我们建模出hazards, 就能够根据上面的公式推导出risks.

Pr[Dk=0∣A=0]\mathrm{Pr}[D_k=0|A=0] Pr[Dk=0∣A=0]
采用同样的方法.

17.3 Why censoring matters

如果考虑administrative censoring, 则我们实际上需要考虑
Pr[Dkcˉ=0ˉ=0∣A=a],cˉ=(c1,c2,⋯,ckend).\mathrm{Pr}[D_k^{\bar{c}=\bar{0}}=0|A=a], \bar{c} = (c_1, c_2, \cdots, c_{k_{end}}). Pr[Dkcˉ=0ˉ=0∣A=a],cˉ=(c1,c2,⋯,ckend).
其等价于
∏m=1kPr[Dm=0∣Dm−1=0,Cm=0,A=a].\prod_{m=1}^k \mathrm{Pr}[D_m=0|D_{m-1}=0, C_m=0, A=a]. m=1∏kPr[Dm=0∣Dm−1=0,Cm=0,A=a].
注意, 这是因为Cm=0C_m=0Cm=0表示
Cm′=0,∀m′≤m.C_{m'} = 0, \quad \forall m' \le m. Cm′=0,∀m′≤m.
所以在这种情况下, 我们需要对
Pr[Dm=0∣Dm−1=0,Cm=0,A=a].\mathrm{Pr}[D_m=0|D_{m-1}=0, C_m=0, A=a]. Pr[Dm=0∣Dm−1=0,Cm=0,A=a].

17.4 IP weighting of marginal structural models

和普通的IP weighting一样我们需要估计
Pr^[A=1∣L].\widehat{\mathrm{Pr}} [A=1|L]. Pr[A=1∣L].

17.5 The parametric g-formula

需要估计
Pr[Dm+1=0∣Dm=0,L=l,A=a].\mathrm{Pr}[D_{m+1}=0|D_m=0, L=l, A=a]. Pr[Dm+1=0∣Dm=0,L=l,A=a].

17.6 G-estimation of structural nested models

Fine Point

Competing events

censoring 的一种特殊情况.

The hazards of hazard ratios

Models for survival analysis

Technical Point

Approximating the hazard ratio via a logistic model

Structural nested cumulative failure time (CFT) models and cumulative survival time (CST) models

Artificial censoring