Chapter_14

### Chapter 14: G-Estimation of Structural Nested Models

14.1 The Causal Question Revisited

• Technical Points:
  • Reiterates the causal effect of smoking cessation ((A)) on weight gain ((Y)) as (E[Y^{a=1,c=0}] - E[Y^{a=0,c=0}]), with censoring indicator (C).
  • Conditional exchangeability assumed given covariates (L): sex, age, race, education, smoking intensity/duration, physical activity, weight.
• Fine Points:
  • Subgroup analysis (e.g., age, sex) can use marginal structural models with product terms or standardization restricted to subsets.
  • With large datasets, strata can be defined by unique combinations of (L) to estimate conditional effects (E[Y^{a=1,c=0}|L] - E[Y^{a=0,c=0}|L]).

14.2 Exchangeability Revisited

• Technical Points:
  • Conditional exchangeability formalized as (Y^a \perp!!!\perp A \mid L) for (a = 0, 1).
  • Equivalent definition: (\text{Pr}[A=1 \mid Y^{a=0}, L] = \text{Pr}[A=1 \mid L]).
  • Logistic model proposed:

    $$\text{logit } \text{Pr}[A=1 \mid Y^{a=0}, L] = \alpha_0 + \alpha_1 Y^{a=0} + \alpha_2 L$$

    where (\alpha_1 = 0) under exchangeability.
• Fine Points:
  • Inclusion of counterfactual (Y^{a=0}) as a covariate distinguishes this model from IP weighting denominator models.

14.3 Structural Nested Mean Models

• Technical Points:
  • Structural nested mean model (SNMM):

    $$E[Y^a - Y^{a=0} \mid A=a, L] = \beta_1 a + \beta_2 a L$$

    where (\beta_1, \beta_2) quantify the conditional average causal effect.
  • Semiparametric: No intercept (\beta_0) or main effect (\beta_3 L), enhancing robustness.
  • Censoring adjustment: Use IP weights (W^C = 1 / \text{Pr}[C=0 \mid L, A]) to handle selection bias.
• Fine Points (Fine Point 14.1):
  • Semiparametric marginal structural models (e.g., (E[Y^a - Y^{a=0} \mid V] = \beta_1 a + \beta_2 a V)) avoid bias from misspecifying (E[Y^{a=0} \mid V]).
  • When (V \subset L), use IP weights (SW^A(V) = f(A \mid V) / f(A \mid L)) for g-estimation.
• Technical Points (Technical Point 14.1):
  • Multiplicative SNMM for positive outcomes:

    $$\log \left( \frac{E[Y^a \mid A=a, L]}{E[Y^{a=0} \mid A=a, L]} \right) = \beta_1 a + \beta_2 a L$$

  • Avoid structural nested logistic models for non-rare binary outcomes due to non-collapsibility.

14.4 Rank Preservation

• Technical Points:
  • Additive rank preservation: Assumes constant individual causal effect within (L)-strata:

    $$Y_i^a - Y_i^{a=0} = \psi_1 a + \psi_2 a L_i \quad \forall i$$

  • Implausible in practice (e.g., individual susceptibility to treatment varies).
• Fine Points:
  • Figures 14.1–14.3 illustrate rank preservation (constant shift within strata) vs. realistic effect heterogeneity.
  • Marginal structural models and SNMMs do not require rank preservation.

14.5 G-Estimation

• Technical Points:
  • G-estimation procedure:
    1. Define (H(\psi^\dagger) = Y - \psi^\dagger A).
    2. Solve for (\psi^\dagger) such that (\alpha_1 = 0) in logistic model:

       $$\text{logit Pr}[A=1 \mid H(\psi^\dagger), L] = \alpha_0 + \alpha_1 H(\psi^\dagger) + \alpha_2 L$$

    3. For censoring, restrict to uncensored ((C=0)) and weight by (W^C).
  • Estimate: (\hat{\psi}_1 = 3.4) kg (95% CI: 2.5, 4.5) for smoking cessation.
  • Confidence intervals via test inversion or bootstrapping.
• Fine Points (Fine Point 14.2):
  • Sensitivity analysis for unmeasured confounding: Assume known (\alpha_1 \neq 0) (e.g., (\alpha_1 = 0.1)) and re-estimate.

14.6 Structural Nested Models with Two or More Parameters

• Technical Points:
  • Model with effect modification:

    $$E[Y^a - Y^{a=0} \mid A=a, L] = \beta_1 a + \beta_2 a V$$

    where (V \subseteq L) (e.g., smoking intensity).
  • G-estimation: Fit logistic model with (H(\beta^\dagger) = Y - \beta_1^\dagger A - \beta_2^\dagger A V) and solve (\alpha_1 = \alpha_2 = 0).
  • Closed-form estimators exist for linear SNMMs (Technical Point 14.2).
• Technical Points (Technical Point 14.2):
  • Doubly robust g-estimation: Replace (H(\beta^\dagger)) with (H(\beta^\dagger) - E[H(\beta^\dagger) \mid L]) to gain robustness against misspecification of (E[A \mid L]) or (\text{Pr}[C=1 \mid A, L]).
  • Estimating equation:

    $$\sum_{i=1}^n I[C_i=0] W_i^C H_i(\beta^\dagger) (A_i - E[A \mid L_i]) q(L_i) = 0$$

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Key Formulas and Concepts

Concept	Formula/Definition
Causal Effect	(E[Y^{a=1,c=0} \mid L] - E[Y^{a=0,c=0} \mid L])
Exchangeability	(Y^a \perp!!!\perp A \mid L)
SNMM	(E[Y^a - Y^{a=0} \mid A=a, L] = \beta_1 a + \beta_2 a L)
G-Estimation	Solve (\alpha_1=0) in (\text{logit Pr}[A=1 \mid H(\psi^\dagger), L] = \alpha_0 + \alpha_1 H(\psi^\dagger) + \alpha_2 L)
Doubly Robust G-Estimation	Use (H(\beta^\dagger) - E[H(\beta^\dagger) \mid L]) in estimating equation

Summary of Visual Aids

• Figures 14.1–14.3: Illustrate rank preservation (constant treatment effect within strata) vs. heterogeneous effects. Descriptions indicate:
  • Fig 14.1: Constant shift (\psi_1 + \psi_2 l) in stratum (L=l).
  • Fig 14.2: Different shift (\psi_1 + \psi_2 l') in stratum (L=l').
  • Fig 14.3: Non-rank-preserving scenario (variable shifts within stratum).

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Note: All conclusions strictly derived from the provided text. No subjective interpretations added.