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Chapter_21

Aaron L9/17/25About 4 minCausal inferenceDeepSeek-R1-0528response_md
### Chapter 21: G-METHODS FOR TIME-VARYING TREATMENTS **Context**: This chapter addresses limitations of traditional methods for time-varying treatments with treatment-confounder feedback (e.g., bias under null effects). It introduces three g-methods—g-formula, IP weighting, and g-estimation—that correctly estimate causal effects under sequential exchangeability, positivity, and consistency.

21.1 The g-formula for time-varying treatments

Technical Points:

  • Core Theory: The g-formula standardizes mean outcomes to the confounder distribution in the study population. For time-varying treatments, it generalizes to:

    ∑lˉE[Y∣Aˉ=aˉ,Lˉ=lˉ]∏k=0Kf(lk∣aˉk−1,lˉk−1) \sum_{\bar{l}} E[Y | \bar{A} = \bar{a}, \bar{L} = \bar{l}] \prod_{k=0}^K f(l_k | \bar{a}_{k-1}, \bar{l}_{k-1})

    where aˉ\bar{a} is the treatment strategy, lˉ\bar{l} is confounder history, and f(lk∣aˉk−1,lˉk−1)f(l_k | \bar{a}_{k-1}, \bar{l}_{k-1}) is the confounder density at time kk given prior history.
  • Key Formula: Under sequential exchangeability, this equals E[Yaˉ]E[Y^{\bar{a}}], the counterfactual mean under strategy aˉ\bar{a}.
  • Positivity Requirement: The g-formula requires f(lk∣aˉk−1,lˉk−1)>0f(l_k | \bar{a}_{k-1}, \bar{l}_{k-1}) > 0 for all aˉ,lˉ\bar{a}, \bar{l} (Technical Point 19.2).
  • Application: In Table 21.1 (sequentially randomized experiment), the g-formula correctly estimates E[Ya0=1,a1=1]−E[Ya0=0,a1=0]=0E[Y^{a_0=1,a_1=1}] - E[Y^{a_0=0,a_1=0}] = 0 (null effect).

Fine Points:

  • Confounder Dependency: Omitting necessary confounders (e.g., L1L_1 in Figure 20.8) invalidates the g-formula’s causal interpretation.
  • Component Interpretation: Individual components (e.g., E[Y∣Aˉ,Lˉ]E[Y|\bar{A},\bar{L}]) may lack causal meaning, but their combination identifies E[Yaˉ]E[Y^{\bar{a}}] under sequential exchangeability.
  • Simulation View: The g-formula simulates counterfactual outcomes by assigning treatment aˉ\bar{a} and preserving observed covariate distributions (Figures 21.1–21.2).

21.2 IP weighting for time-varying treatments

Technical Points:

  • Weights Definition:
    • Nonstabilized: WAˉ=∏k=0K1f(Ak∣Aˉk−1,Lˉk)W^{\bar{A}} = \prod_{k=0}^K \frac{1}{f(A_k | \bar{A}_{k-1}, \bar{L}_k)}
    • Stabilized: SWAˉ=∏k=0Kf(Ak∣Aˉk−1)f(Ak∣Aˉk−1,Lˉk)SW^{\bar{A}} = \prod_{k=0}^K \frac{f(A_k | \bar{A}_{k-1})}{f(A_k | \bar{A}_{k-1}, \bar{L}_k)}
  • Estimation: E[Yaˉ]E[Y^{\bar{a}}] is the mean of YY in the pseudo-population weighted by WAˉW^{\bar{A}} or SWAˉSW^{\bar{A}}, among those with Aˉ=aˉ\bar{A} = \bar{a}.
  • Marginal Structural Models (MSMs): For high-dimensional treatments, specify E[Yaˉ]=β0+β1cum(aˉ)E[Y^{\bar{a}}] = \beta_0 + \beta_1 \text{cum}(\bar{a}). Parameters are estimated via weighted least squares.

Fine Points:

  • Equivalence to g-formula: Nonparametric IP weighting and g-formula yield identical results, even without causal validity.
  • Model Robustness: MSMs avoid the g-null paradox (Technical Point 21.3) and allow effect modification by baseline variables (e.g., E[Yaˉ∣V]=β0+β1cum(aˉ)+β2V+β3cum(aˉ)VE[Y^{\bar{a}} | V] = \beta_0 + \beta_1 \text{cum}(\bar{a}) + \beta_2 V + \beta_3 \text{cum}(\bar{a})V).
  • Software: Implemented in R (gfoRmula) and SAS (GFORMULA).

21.3 A doubly robust estimator for time-varying treatments

Technical Points:

  • Doubly Robust Estimator: Combines treatment and outcome models. Consistent if either model is correct.
  • Steps for "Always Treat" (aˉ=1\bar{a} = 1):
    1. Fit treatment model πk(Lˉk;α)\pi_k(\bar{L}_k; \alpha) for Pr⁡(Ak=1∣Aˉk−1=1,Lˉk)\Pr(A_k = 1 | \bar{A}_{k-1} = 1, \bar{L}_k).
    2. Compute time-varying IP weights W^1m\widehat{W}^{1_m}.
    3. Fit sequential outcome models bm(Lˉm;βm)b_m(\bar{L}_m; \beta_m) with W^1m\widehat{W}^{1_m} as covariate, starting from t=Kt = K to t=0t = 0.
    4. Estimate E[Yaˉ=1]E[Y^{\bar{a}=1}] as the sample mean of b0(Lˉ0;β^)b_0(\bar{L}_0; \hat{\beta}).
  • Robustness: K+2K+2-robust—unbiased if treatment models for times 00 to mm and outcome models for m+1m+1 to KK are correct for any mm.

Fine Points:

  • Extensions: Targeted Minimum Loss-Based Estimators (TMLE) handle dynamic strategies (Technical Point 21.6).
  • Practical Use: Requires correct specification of either treatment or outcome models, reducing bias in observational studies.

21.4 G-estimation for time-varying treatments

Technical Points:

  • Structural Nested Mean Model (SNMM): Models the effect of a "blip" of treatment at time kk:

    E[Yak−1,ak,0k+1−Yak−1,0k∣aˉk−1,lˉk]=akγk(aˉk−1,lˉk;β) E[Y^{a_{k-1}, a_k, 0_{k+1}} - Y^{a_{k-1}, 0_k} | \bar{a}_{k-1}, \bar{l}_k] = a_k \gamma_k(\bar{a}_{k-1}, \bar{l}_k; \beta)

  • G-estimation:
    1. Compute Hk(ψ†)=Y−∑j=kKAjγj(Aˉj−1,Lj;ψ†)H_k(\psi^\dagger) = Y - \sum_{j=k}^K A_j \gamma_j(\bar{A}_{j-1}, L_j; \psi^\dagger).
    2. Solve estimating equations for ψ†\psi^\dagger such that Hk(ψ†)H_k(\psi^\dagger) is independent of AkA_k given past history.
  • Closed-Form Solution: For linear SNMMs, β^\hat{\beta} has an analytic expression (Technical Point 21.8).

Fine Points:

  • Rank Preservation: Optional assumption; g-estimation remains valid for SNMMs without it.
  • Efficiency vs. Robustness: SNMMs increase efficiency if no effect modification by past covariates, but may introduce bias if misspecified (vs. MSMs).

21.5 Censoring is a time-varying treatment

Technical Points:

  • Censoring Weights: Extend IP weighting to handle censoring CkC_k:

    WCˉ=∏k=1K+11Pr⁡(Ck=0∣Cˉk−1=0,Aˉk−1,Lˉk−1) W^{\bar{C}} = \prod_{k=1}^{K+1} \frac{1}{\Pr(C_k = 0 | \bar{C}_{k-1} = 0, \bar{A}_{k-1}, \bar{L}_{k-1})}

  • G-formula Adjustment: For censored data, condition on Cˉ=0\bar{C} = 0:

    ∑lˉE[Y∣Cˉ=0,Aˉ=aˉ,Lˉ=lˉ]∏k=0Kf(lk∣ck=0,aˉk−1,lˉk−1) \sum_{\bar{l}} E[Y | \bar{C} = 0, \bar{A} = \bar{a}, \bar{L} = \bar{l}] \prod_{k=0}^K f(l_k | c_k = 0, \bar{a}_{k-1}, \bar{l}_{k-1})

Fine Points:

  • Survival Analysis: For failure-time outcomes, use pooled logistic models with IP weights or g-formula (Technical Point 21.10).
  • Selection Bias: Censoring can induce bias if not adjusted (e.g., when CC is a collider).

21.6 The big g-formula

Technical Points:

  • Definition: The big g-formula includes all variables (observed LL and unobserved UU) and identifies E[Yg]E[Y^g] under any causal DAG.
  • Reduction to Observed Data: If d-separation holds, it simplifies to formulas using only observed data (e.g., front-door criterion).

Fine Points:

  • Front-Door Formula Proof: Under Figure 7.14, E[Ya]=∑mPr⁡(m∣a)∑a′E[Y∣m,a′]Pr⁡(a′)E[Y^a] = \sum_m \Pr(m | a) \sum_{a'} E[Y | m, a'] \Pr(a') (Technical Point 21.11).
  • SWIG Property: Single-world counterfactual graphs (SWIGs) enable d-separation proofs without cross-world assumptions (Technical Point 21.12).

Key Takeaways:

  • g-methods (g-formula, IP weighting, g-estimation) resolve bias from treatment-confounder feedback.
  • Doubly/Multiply Robust Estimators enhance reliability in high-dimensional settings.
  • Censoring requires explicit adjustment as a time-varying treatment.
  • Software (e.g., gfoRmula, GFORMULA) implements these methods for applied research.