Chapter 14 G-estimation of Structural Nested Models

目录

• 14.1 The causal question revisited
• 14.2 Exchangeability revisited
• 14.3 Structural nested mean models
• 14.4 Rank preservation
• 14.5 G-estimation
• 14.6 Structural nested models with two or more parameters
• Fine Point
  • Relation between marginal structural models and structural nested models
  • Sensitivity analysis for unmeasured confounding
• Technical Point
  • Multiplicative structural nested mean models
  • G-estimation of structural nested mean models

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

前面已经介绍过了Standardization 和 IP weighting, 这里在介绍另外一种方法: G-estimation.

14.1 The causal question revisited

14.2 Exchangeability revisited

\(Y^a \amalg A | L\), 即

\[\mathrm{Pr}[Y^a|A,L] = \mathrm{Pr}[Y^a|L]. \]

但是我们也可以这么理解:

\[\mathrm{Pr}[A|Y^a,L] = \mathrm{Pr}[A|L]. \]

于是我们可以假设, 此概率为一个logistic模型:

\[\mathrm{logit} \mathrm{Pr} [A=1|Y^{a=0}, L] = \alpha_0 + \alpha_1Y^{a=0} + \alpha_2 L. \]

既然我们已经假设了条件可交换性, 那么正常来讲, \(\alpha_1 = 0\).
g-estimation方法就是利用了这一点, 通过构造\(Y^{a=0}\)和我们所要求的causal effect之间的联系来估计参数.

14.3 Structural nested mean models

之前介绍过marginal model

\[\mathbb{E}[Y^a] = \beta_0 + \beta_1 a. \]

此时我们需要估计两个参数, 但是倘若我们关注的只是causal effect, 那么我们完全可以

\[\mathbb{E}[Y^{a} - Y^{a=0}] = \beta_1a, \]

此时就只需要估计一个参数, 这是能够防止bias的.

在扩展到strata中:

\[\mathbb{E}[Y^a - Y^{a=0}|A=a, L] = \beta_1a + \beta_2 a L. \]

注: \(\beta_2 L = \sum_{j} \beta_{2j} L_j\).

14.4 Rank preservation

指的是, intervention \(A\)对于每个个体所带来的变化都是固定, 即

\[Y_i^a - Y_i^{a=0} = \psi_1a. \]

相应的还有 conditional rank preservation

\[Y_i^a -Y_i^{a=0} = \psi_1a + \psi_2a L_i. \]

所以, rank preservation的意思就是, 经过intervention的干预, 个体间的排名不发生变化(因为效果是相同的).

14.5 G-estimation

这里我们关注:

\[\mathbb{E}[Y^a - Y^{a=0}|A=a, L] = \beta_1a, \]

则根据条件可交换性可知, \(\beta_1\)就是我们所关注的causal effect.
当然这个情况是简化的, 直接舍去的\(\beta_2 \alpha L\), 但是思想是类似的.

倘若

\[Y^{a=0} = Y^a - \psi_1a, \]

即满足rank preservation.
注意到, 等式右边实际上就是

\[Y^{a=0} = Y - \psi_1 A. \]

回到最开始的logistic模型, 我们得到:

\[\mathrm{logit} \mathrm{Pr} [A=1|Y^{a=0}, L] = \alpha_0 + \alpha_1(Y-\psi_1 A)+ \alpha_2 L. \]

对于不同的\(\psi_1\), 关于上式我们有不同的估计, 但是该估计可能\(\alpha_1\not=0\)或者不在0附件, 这说明我们假设的\(\psi_1\)可能并不合理.
换言之, 我们认为\(\psi_1\)的合适的估计应该使得\(\alpha_1\)的估计在0附近, 且越接近我们认为估计的越好.
这就是g-estimation.
特殊的情况也是可以的:

\[\mathrm{logit} \mathrm{Pr} [A=1|Y^{a=0}, L] = \alpha_0 + \alpha_1(Y-\psi_1 A - \psi_2 L)+ \alpha_2 L. \]

此时要估计多个参数了.
另外其实我没能很好get到数学的内核.

14.6 Structural nested models with two or more parameters

Fine Point

Relation between marginal structural models and structural nested models

Sensitivity analysis for unmeasured confounding

Technical Point

Multiplicative structural nested mean models

G-estimation of structural nested mean models

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