Chapter 13 Standardization and The Parametric G-formula

目录

• 13.1 Standardization as an alternative to IP weighting
• 13.2 Estimating the mean outcome via modeling
• 13.3 Standardizing the mean outcome to the confounder distribution
• 13.4 IP weighting or standardization
• 13.5 How seriously do we take our estimates?
• Fine Point
  • Structural Positivity
  • A doubly robust estimator
• Technical Point
  • Bootstrapping
  • The bias of doubly robust estimators

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

这一节来讲怎么结合standardization.

13.1 Standardization as an alternative to IP weighting

\[\sum_l \mathbb{E}[Y|A=a, C=0, L=l] \times \mathrm{Pr}[L=l]. \]

13.2 Estimating the mean outcome via modeling

当数据的维度比较大的时候, 直接通过非参数方法估计

\[\mathbb{E} [Y|A=a, C=0, L=l] \]

是很困难的(甚至会出现positivity不满足的情况).
因此我们可以用参数模型来建模, 并用最小二乘估计参数.

13.3 Standardizing the mean outcome to the confounder distribution

实际上, 在实际估计的时候, 我们不需要估计\(\mathrm{Pr}[L=l]\), 只需通过

\[\frac{1}{n} \sum_{i=1}^n \hat{\mathbb{E}} [Y|A=a, C=0, L=l], \]

即可.
这个等价于一种奇妙的思路, 详情回见原文.

13.4 IP weighting or standardization

当然模型变换的时候, 二者的估计值可能会有所不同, 但并没有孰优孰劣的分别.
实际上, 有一些 doubly robust estimator 可以兼用二者, 并获得一个更加鲁棒的估计器.

13.5 How seriously do we take our estimates?

Fine Point

Structural Positivity

A doubly robust estimator

Technical Point

Bootstrapping

如何估计标准差.

The bias of doubly robust estimators

\[\hat{\mathbb{E}} [Y^{a=1}]_{DR} = \frac{1}{n} \sum_{i=1}^n[ \hat{b}(L_i) + \frac{A_iY_i}{\hat{\pi}(L_i)} ( Y_i - \hat{b}(L_i)). ] \]

\[b(L) = \mathbb{E} [Y|A=1, L], \\ \pi(L) = \mathrm{Pr}[A=1|L]. \]