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
  • Treatment and covariate history
  • Representations of the g-formula
  • G-estimation with a saturated structural nested model
• Technical Point
  • The g-formula density for static strategies
  • The g-null paradox
  • A doubly estimator of 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 after g-estimation of a structural nested mean model

Hern\(\'{a}\)n M. and Robins J. Causal Inference: What If.

这一章介绍了如何估计time-varying 下的causal effect.

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.

同样是静态的:

\[W^{\bar{A}} = \prod_{k=0}^K \frac{1}{f(A_k|\bar{A}_{k-1}, \bar{L}_k)},\\
SW^{\bar{A}} = \prod_{k=0}^K \frac{f(A_k|\bar{A}_{k-1})}{f(A_k|\bar{A}_{k-1}, \bar{L}_k)}.\\
\]

一种doubly robust的估计方法.

\[H_k(\psi^{\dagger}) = Y - \sum_{j=k}^K A_j \gamma_j(\bar{A}_{j-1}, \bar{L}_{j}, \psi^{\dagger}).
\]

通过下式来估计:

\[\mathrm{logit}\:\mathrm{Pr} [A_k=1|H_k(\psi^{\dagger}), \bar{L}_k, \bar{A}_{k-1}] = \alpha_0 + \alpha_1 H_k(\psi^{\dagger}) + \alpha_2 W_k.
\]

当censoring也是一个time-varying变量的时候.

\[\sum_{\bar{l}} \mathbb{E}[Y|\bar{A}=a, \bar{C}=\bar{0}, \bar{L}=\bar{l}] \prod_{k=0}^K f(l_k|\bar{a}_{k-1}, c_{k-1}=0, \bar{l}_{k-1}).
\]

\[W^{\bar{C}} = \prod_{k=1}^{K+1} \frac{1}{\mathrm{Pr}(C_k=0|\bar{A}_{k-1}, C_{k-1}=0,\bar{L}_k)}, \\
SW^{\bar{C}} = \prod_{k=1}^{K+1} \frac{\mathrm{Pr}(C_k=0|\bar{A}_{k-1}, C_{k-1}=0)}{\mathrm{Pr}(C_k=0|\bar{A}_{k-1}, C_{k-1}=0,\bar{L}_k)}, \\
\]

Treatment and covariate history

Representations of the g-formula

G-estimation with a saturated structural nested model

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