Chapter 21 G-Methods for Time-Varying Treatments
目录
- 21.1 The g-formula for time-varying treatments
- 21.2 IP weighting for time-varying treatments
- 21.3 A doubly robust estimator for time-varying treatments
- 21.4 G-estimation for time-varying treatments
- 21.5 Censoring is a time-varying treatment
- Fine Point
- Technical Point
- The g-formula density for static strategies
- The g-null paradox
- A doubly estimator of \(\mathbb{E}[Y^{\bar{a}}]\) for time-varying treatments
- Relation between marginal structural models and structural nested models (Part II)
- A closed form estimator for linear structural nested mean models
- Estimation of \(\mathbb{E}[Y^g]\) after g-estimation of a structural nested mean model
这一章介绍了如何估计time-varying 下的causal effect.
21.1 The g-formula for time-varying treatments
求静态的\(\mathbb{E}[Y^{\bar{a}}]\),
\[\sum_l \mathbb{E}[Y|\bar{A}=\bar{a}, \bar{L}=\bar{l}]\prod_{k=0}^K f(l_k|\bar{a}_{k-1}, \bar{l}_{k-1}).
\]
至于动态的\(Y^g\),总感觉书上给的公式缺了一块.
21.2 IP weighting for time-varying treatments
同样是静态的:
\[W^{\bar{A}} = \prod_{k=0}^K \frac{1}{f(A_k|\bar{A}_{k-1}, \bar{L}_k)},\\
SW^{\bar{A}} = \prod_{k=0}^K \frac{f(A_k|\bar{A}_{k-1})}{f(A_k|\bar{A}_{k-1}, \bar{L}_k)}.\\
\]
21.3 A doubly robust estimator for time-varying treatments
一种doubly robust的估计方法.
21.4 G-estimation for time-varying treatments
\[H_k(\psi^{\dagger}) = Y - \sum_{j=k}^K A_j \gamma_j(\bar{A}_{j-1}, \bar{L}_{j}, \psi^{\dagger}).
\]
通过下式来估计:
\[\mathrm{logit}\:\mathrm{Pr} [A_k=1|H_k(\psi^{\dagger}), \bar{L}_k, \bar{A}_{k-1}] = \alpha_0 + \alpha_1 H_k(\psi^{\dagger}) + \alpha_2 W_k.
\]
21.5 Censoring is a time-varying treatment
当censoring也是一个time-varying变量的时候.
\[\sum_{\bar{l}} \mathbb{E}[Y|\bar{A}=a, \bar{C}=\bar{0}, \bar{L}=\bar{l}] \prod_{k=0}^K f(l_k|\bar{a}_{k-1}, c_{k-1}=0, \bar{l}_{k-1}).
\]
\[W^{\bar{C}} = \prod_{k=1}^{K+1} \frac{1}{\mathrm{Pr}(C_k=0|\bar{A}_{k-1}, C_{k-1}=0,\bar{L}_k)}, \\
SW^{\bar{C}} = \prod_{k=1}^{K+1} \frac{\mathrm{Pr}(C_k=0|\bar{A}_{k-1}, C_{k-1}=0)}{\mathrm{Pr}(C_k=0|\bar{A}_{k-1}, C_{k-1}=0,\bar{L}_k)}, \\
\]
