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Chapter_14

Aaron L9/17/25About 4 minCausal inferenceDeepSeek-R1-0528-2ndresponse_md
### Chapter 14: G-ESTIMATION OF STRUCTURAL NESTED MODELS **Link to Previous Chapters**: This chapter introduces g-estimation as a third method (alongside IP weighting and standardization from Chapters 12–13) to estimate the average causal effect of smoking cessation (\(A\)) on weight gain (\(Y\)) using the NHEFS dataset. G-methods (IP weighting, standardization, g-estimation) are designed for time-varying treatments but are applied here to a time-fixed treatment for pedagogical clarity.

14.1 The Causal Question Revisited

Technical Focus:

  • Data and Assumptions: Analysis restricted to 1,566 individuals with complete baseline (1971–75) and follow-up (1982) data on covariates (L): sex, age, race, education, smoking intensity/duration, physical activity, exercise, and alcohol use.
  • Causal Effect Definition: The average causal effect on the difference scale is (E[Y^{a=1,c=0}] - E[Y^{a=0,c=0}]), representing the difference in mean weight if all were treated (quit smoking) versus untreated, with no censoring ((c=0)).
  • Subgroup Effects: Interest extends to conditional effects within strata of (L) (e.g., age 45: (E[Y^{a=1,c=0}| \text{age}=45] - E[Y^{a=0,c=0}|\text{age}=45])). Marginal structural models (Chapter 12) or standardization (Chapter 13) can estimate subgroup effects.

Details:

  • Stratification Challenge: With many strata (e.g., {non-quitter, female, white, age 26, college dropout, etc.}), large datasets or model-based approaches (e.g., adding (L \times A) product terms to marginal structural models) are needed.

14.2 Exchangeability Revisited

Technical Focus:

  • Conditional Exchangeability: (Y^a \perp!!!\perp A | L) for (a = {0, 1}), implying equal outcome distributions under the same treatment within (L)-strata.
  • Equivalent Definition: (\text{Pr}[A=1|Y^{a=0}, L] = \text{Pr}[A=1|L]).
  • Logistic Model for G-Estimation:
    (\text{logit Pr}[A=1|Y^{a=0}, L] = \alpha_0 + \alpha_1 Y^{a=0} + \alpha_2 L),
    where (\alpha_2 L = \sum_{j=1}^p \alpha_{2j} L_j). Under exchangeability, (\alpha_1 = 0).

Details:

  • Purpose: This re-expression facilitates g-estimation by linking counterfactual outcomes to treatment assignment.

14.3 Structural Nested Mean Models

Technical Focus:

  • Model Definition:
    (E[Y^a - Y^{a=0} | A=a, L] = \beta_1 a + \beta_2 a L),
    where (\beta_1 + \beta_2 L) quantifies the conditional average causal effect within ((A, L)) strata.
  • Semiparametric Nature: Unlike parametric g-formula models, structural nested models omit intercept ((\beta_0)) and main effects of (L) ((\beta_3 L)), enhancing robustness.
  • Censoring Adjustment: For censoring (C), nonstabilized IP weights (W^C = 1 / \text{Pr}[C=0|L, A]) create a pseudo-population without censoring.

Key Formulas:

  • Additive causal effect: (E[Y^a - Y^{a=0}|L] = \beta_1 a + \beta_2 a L).
  • With censoring: Model becomes (E[Y^{a,c=0} - Y^{a=0,c=0}|A=a, L] = \beta_1 a + \beta_2 a L).

Details:

  • Comparison to Other Models:
    • Marginal structural models estimate population-average effects; structural nested models estimate within-(L) effects.
    • Semiparametric marginal structural models (leaving (E[Y^{a=0}|V]) unspecified) are equivalent to structural nested models when (V \subset L) (See Fine Point 14.1).

14.4 Rank Preservation

Technical Focus:

  • 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) for all (i).
  • Implausibility: Individual treatment effects (e.g., weight gain after smoking cessation) vary biologically, making rank preservation unrealistic.

Details:

  • Role in G-Estimation: Rank-preserving models simplify g-estimation intuition but are not required for validity. Structural nested mean models do not assume rank preservation.
  • Visualization:
    • Figure 14.1/14.2: Show rank preservation within strata (L=l) and (L=l') with constant shifts (\psi_1 + \psi_2 l).
    • Figure 14.3: Illustrates non-rank-preserving scenario with variable individual shifts.

14.5 G-Estimation

Technical Focus:

  • Rank-Preserving Model: (Y^{a=0} = Y - \psi_1 A).
  • Procedure:
    1. Compute candidate counterfactuals (H(\psi^\dagger) = Y - \psi^\dagger A).
    2. Fit logistic model: (\text{logit Pr}[A=1|H(\psi^\dagger), L] = \alpha_0 + \alpha_1 H(\psi^\dagger) + \alpha_2 L).
    3. Find (\psi^\dagger) such that (\alpha_1 = 0) (indicating (H(\psi^\dagger) \perp!!!\perp A | L)).
  • Estimation:
    • Point estimate: (\hat{\psi}_1 = 3.4) kg (for smoking cessation effect).
    • 95% CI: ([2.5, 4.5]) kg (by test inversion or bootstrapping).
  • Censoring Adjustment: Uncensored individuals ((C=0)) weighted by (W^C).

Key Insight:

  • Validity requires correct specification of the structural mean model, not rank preservation.
  • Sensitivity Analysis: For unmeasured confounding, test (\alpha_1 = \delta) (expected deviation from 0) instead of (\alpha_1 = 0) (See Fine Point 14.2).

14.6 Structural Nested Models with Two or More Parameters

Technical Focus:

  • Model with Effect Modification:
    (E[Y^a - Y^{a=0}|A=a, L] = \beta_1 a + \beta_2 a V) (e.g., (V = \text{smoking intensity})).
  • G-Estimation:
    • Define (H(\beta^\dagger) = Y - \beta_1^\dagger A - \beta_2^\dagger A V).
    • Fit logistic model: (\text{logit Pr}[A=1|H(\beta^\dagger), L] = \alpha_0 + \alpha_1 H(\beta^\dagger) + \alpha_2 H(\beta^\dagger)V + \alpha_3 L).
    • Solve for (\beta^\dagger) where (\alpha_1 = \alpha_2 = 0).
  • Closed-Form Solution: For linear models, estimators exist without parameter search.

Results:

  • Estimates: (\hat{\beta}_1 = 2.86), (\hat{\beta}_2 = 0.03).
  • Population effect: (\beta_1 + \frac{1}{n} \sum_i \sum_j \beta_{2j} L_{ij}).

Details:

  • Misspecification Risk: Omitting product terms (e.g., (\beta_2 a V)) biases estimates if effect modification exists.
  • Null Preservation: Structural nested models preserve the null hypothesis (no effect) correctly.

Technical Points Summary

  • 14.1 Multiplicative Models: For non-negative/binary outcomes, multiplicative structural nested models (e.g., (\log \frac{E[Ya|A=a,L]}{E[Y|A=a,L]} = \beta_1 a + \beta_2 a L)) avoid rare-outcome restrictions.
  • 14.2 G-Estimation Equations: Closed-form solutions exist for linear models. Doubly robust estimators combine models for (E[A|L]) and (E[Y^{a=0}|L]).

Figures

  • Figures 14.1–14.3: Illustrate rank preservation (constant shift within (L)-strata) vs. non-preservation (variable shifts). Descriptions align with bell-shaped distributions of (Y^{a=0}) and (Y^{a=1}).

Conclusion: G-estimation provides a flexible, semiparametric approach to estimate conditional causal effects, robust to model misspecification when combined with IP weighting for censoring. Its extension to sensitivity analyses enhances applicability in observational studies.