Chapter_21

### Chapter 21: G-METHODS FOR TIME-VARYING TREATMENTS **Technical Focus**: Solutions to bias in traditional methods for time-varying treatments with treatment-confounder feedback, using g-methods (g-formula, IP weighting, g-estimation, doubly-robust estimators).

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21.1 The g-formula for time-varying treatments

Context: Extends the g-formula (Chapter 13) to time-varying treatments under sequential exchangeability, positivity, and consistency.
Technical Points:

• Core Theory: For time-varying treatment $A = (A_0, A_1)$, the g-formula for $E[Y^{a_0,a_1}]$ is:

  $$\sum_{l_1} E[Y \mid A_0=a_0, A_1=a_1, L_1=l_1] \cdot f(l_1 \mid a_0)$$

  where $f(l_1 \mid a_0) = \Pr(L_1=l_1 \mid A_0=a_0)$.
• Key Formula: Standardizes mean outcome to confounder distribution ($L_1$). Requires positivity: $f(l_1 \mid a_0) \neq 0$ for observed $(a_0, a_1, l_1)$.
• Application: In Table 21.1 (sequentially randomized experiment), estimates $E[Y^{a_0=1,a_1=1}] - E[Y^{a_0=0,a_1=0}] = 0$ (null effect), correcting traditional methods' bias.
• Generalization: For $K$ time points:

  $$\sum_{\bar{l}} E[Y \mid \bar{A}=\bar{a}, \bar{L}=\bar{l}] \prod_{k=0}^K f(l_k \mid \bar{a}_{k-1}, \bar{l}_{k-1})$$

Details Points:

• Simulation View: Under sequential exchangeability, g-formula simulates counterfactual outcomes under strategy $\bar{a}$.
• Critical Notes:
  • Omitting confounders (e.g., $L_1$) invalidates causal interpretation.
  • Components (e.g., $E[Y \mid \cdot]$) may lack causal meaning individually.
  • Fine Point 21.1: "History" includes confounders needed for exchangeability, not strictly temporal order.

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21.2 IP weighting for time-varying treatments

Technical Points:

• Weights:
  • Nonstabilized: $W^{\bar{A}} = \prod_{k=0}^K \frac{1}{f(A_k \mid \bar{A}_{k-1}, \bar{L}_k)}$
  • Stabilized: $SW^{\bar{A}} = \prod_{k=0}^K \frac{f(A_k \mid \bar{A}_{k-1})}{f(A_k \mid \bar{A}_{k-1}, \bar{L}_k)}$
• Estimation: $E[Y^{\bar{a}}]$ is $E_{\text{ps}}[Y \mid \bar{A}=\bar{a}]$ in pseudo-population created by weights.
• Marginal Structural Models (MSM): For high-dimensional strategies, specify $E[Y^{\bar{a}}] = \beta_0 + \beta_1 \text{cum}(\bar{a})$. Fit via weighted least squares.

Details Points:

• Application: In Table 21.1, IP weighting yields correct null effect (60 vs. 60).
• Robustness: MSMs avoid g-null paradox (unlike parametric g-formula).
• Software: gfoRmula R package, GFORMULA SAS macro.
• Technical Point 21.2: For dynamic strategies, use $E[Y I(\bar{A}_K = A_K^g) W^{\bar{A}}]$.

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21.3 Doubly robust estimator for time-varying treatments

Technical Points:

• Approach: Combines g-formula and IP weighting. Consistent if either treatment or outcome model is correct.
• **Steps for "always treat" ($\bar{a}=1$):
  1. Fit treatment model $\pi_k(\bar{L}_k; \alpha)$ for $\Pr(A_k=1 \mid \bar{A}_{k-1}=1, \bar{L}_k)$.
  2. Compute time-varying IP weights $\widehat{W}^{1_m}$.
  3. Fit sequential outcome models $b_m(\bar{L}_m; \beta_m)$ with $\widehat{W}^{1_m}$ as covariate.
  4. Estimate $E[Y^{\bar{a}=1}]$ as sample mean of $b_0(\bar{L}_0; \hat{\beta})$.
• Robustness: $K+2$-robust (correct if treatment model for times $0:m$ and outcome model for $m+1:K$ hold).

Details Points:

• Extensions: Targeted Minimum Loss-Based Estimator (TMLE) for static/dynamic strategies (Technical Point 21.6).
• Challenges: Computational complexity; emerging software for machine learning integration.

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21.4 G-estimation for time-varying treatments

Technical Points:

• Structural Nested Mean Model (SNMM): Models blip effect of treatment at $k$:

  $$E[Y^{a_{k-1},a_k,0_{k+1}} - Y^{a_{k-1},0_k} \mid \cdot] = a_k \gamma_k(a_{k-1}, l_k, \beta)$$

• G-estimation:
  1. Compute $H_k(\psi^\dagger) = Y - \sum_{j=k}^K A_j \gamma_j(A_{j-1}, L_j, \psi^\dagger)$.
  2. Solve estimating equation for $\psi^\dagger$ such that $A_k \perp H_k(\psi^\dagger) \mid L_k, A_{k-1}$.
• Closed Form: For linear SNMM, use $\widehat{\beta} = \left( \sum X_{i,k} Q_{i,k} S_{i,k}^T \right)^{-1} \left( \sum Y_i X_{i,k} Q_{i,k} \right)$ (Technical Point 21.8).

Details Points:

• Saturated Example: In Table 21.1, g-estimation yields $\beta = 0$ (null effect).
• Optimal Strategies: SNMMs can estimate $g^{\text{opt}} = \arg\max E[Y^g]$ (Technical Point 21.13).

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21.5 Censoring as a time-varying treatment

Technical Points:

• Handling Censoring: Treat censoring indicators $C_k$ as time-varying treatment.
• g-formula with Censoring:

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

• IP Weighting: Use weights $W^{\bar{C}} = \prod_{k=1}^{K+1} \frac{1}{\Pr(C_k=0 \mid \bar{C}_{k-1}=0, \bar{A}_{k-1}, \bar{L}_{k-1})}$.

Details Points:

• Survival Analysis: Extend to failure-time outcomes via hazards models (Technical Point 21.10).
• Stabilized Weights: Reduce variance by conditioning on past treatment.

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21.6 The big g-formula

Technical Points:

• Concept: g-formula including all variables (measured $L$, unmeasured $U$) under causal DAGs.
• Identification: If DAG is causal, big g-formula identifies $E[Y^g]$.
• Front-Door Formula Example: Reduces to $\sum_m \Pr(M=m \mid A=a) \sum_{a'} E[Y \mid M=m, A=a'] \Pr(A=a')$.

Details Points:

• Non-Observational Formulas: e.g., front-door criterion valid under d-separation (Technical Point 21.11).
• Limitation: Requires correct causal DAG specification (unverifiable in practice).

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Summary Concluded