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Chapter_12

Aaron L9/17/25About 5 minCausal inferencegpt-4.1response_md

Chapter 12: IP Weighting and Marginal Structural Models

12.1 The Causal Question

Main Point 主旨

  • Goal: Estimate the average causal effect of smoking cessation (AA) on weight gain (YY).
  • 目标：估算戒烟（AA）对体重增加（YY）的平均因果效应。

Definitions 定义

  • Counterfactual means:
    • E[Ya=1]E[Y^{a=1}] — mean weight gain if everyone quit smoking
    • E[Ya=0]E[Y^{a=0}] — mean weight gain if no one quit
  • Additive average causal effect:
    • E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}]
  • Association is not causation:
    • The observed difference E[Y∣A=1]−E[Y∣A=0]E[Y|A=1] - E[Y|A=0] is generally not equal to E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}] due to confounding.

重点

  • 必须调整混杂变量（如年龄、性别、种族、教育、吸烟强度等）以获得真正的因果效应。
  • Observed difference in weight gain between quitters and non-quitters likely underestimates causal effect due to confounding.

12.2 Estimating IP Weights via Modeling

Main Technical Points 技术要点

  • IP Weighting creates a pseudo-population where A⊥LA \perp L (treatment independent of covariates).
  • Properties:
    1. AA and LL independent
    2. Eps[Y∣A=a]=∑lE[Y∣A=a,L=l]P(L=l)E^\text{ps}[Y|A=a] = \sum_l E[Y|A=a, L=l]P(L=l) (standardized mean)
    3. If Ya⊥A∣LY^a \perp A|L, then: Association is causation in pseudo-population.

Formula for Individual Weights

  • For each individual:

    WA=1f(A∣L)W^A = \frac{1}{f(A|L)}

    where f(A∣L)f(A|L) is the probability of receiving their observed treatment given their covariates (i.e., the propensity score).

  • For a binary treatment:

    WA={1P(A=1∣L)A=11P(A=0∣L)A=0W^A = \begin{cases} \frac{1}{P(A=1|L)} & A=1\\[1em] \frac{1}{P(A=0|L)} & A=0\\ \end{cases}

  • Propensity scores P(A=1∣L)P(A=1|L) are modeled, e.g., by logistic regression including all relevant confounders.

Weighted Estimation

  • Fit the model:

    E[Y∣A]=θ0+θ1AE[Y|A] = \theta_0 + \theta_1 A

    Using observed YY, AA, weighted by WAW^A.
  • θ1\theta_1 estimates the average causal effect: θ1≈E[Ya=1]−E[Ya=0]\theta_1 \approx E[Y^{a=1}] - E[Y^{a=0}]

Horvitz-Thompson and Hajek estimators

  • Unbiased estimator under positivity and conditional exchangeability.
  • Horvitz-Thompson Estimator:

    E^[I(A=a)YP(A∣L)]\hat{E}\left[\frac{I(A=a)Y}{P(A|L)}\right]

  • Hajek Estimator (generally preferred, especially for bounded outcomes):

    E^[I(A=a)YP(A∣L)]E^[I(A=a)P(A∣L)]\frac{\hat{E}\left[\frac{I(A=a)Y}{P(A|L)}\right]}{\hat{E}\left[\frac{I(A=a)}{P(A|L)}\right]}

12.3 Stabilized IP Weights

Main Technical Points 要点

  • Non-stabilized weights: WA=1P(A∣L)W^A = \frac{1}{P(A|L)}.
  • Stabilized weights: SWA=P(A)P(A∣L)SW^A = \frac{P(A)}{P(A|L)}.
    • P(A)P(A) = marginal probability of treatment in the study population.
  • Stabilized weights have mean 1, reduce variance of the weights, thus typically give narrower confidence intervals.
  • Both types of weights give the same point estimate for saturated models (like binary AA with E[Y∣A]=θ0+θ1AE[Y|A] = \theta_0+\theta_1A), but stabilized preferred for efficiency in non-saturated models.

Positivity Violations

  • 结构性违背：某些LL值下AA概率为0（如某些人群永远不会被处理），则不能对全体人群推断，需限制于有正概率的亚群体。
  • 随机性违背：样本有限，不代表总体真实概率为0，可以通过建模平滑“空格”。

12.4 Marginal Structural Models (MSMs)

Definition 定义

  • Marginal structural mean model for binary AA:

    E[Ya]=β0+β1aE[Y^a] = \beta_0 + \beta_1 a

    • β1\beta_1 = E[Ya=1]−E[Ya=0]E[Y^{a=1}] - E[Y^{a=0}] is the average causal effect.

Non-saturated MSM for continuous/polytomous treatment:

  • E.g., for dose-response (continuous AA):

    E[Ya]=β0+β1a+β2a2E[Y^a] = \beta_0 + \beta_1 a + \beta_2 a^2

  • Use IP weights (SWA=f(A)f(A∣L)SW^A = \frac{f(A)}{f(A|L)}), where f(A)f(A) is the marginal density of AA and f(A∣L)f(A|L) is the conditional density modeled as, e.g., Gaussian for continuous AA.
  • For simulated pseudo-population, fit E[Y∣A]=θ0+θ1A+θ2A2E[Y|A]=\theta_0+\theta_1A + \theta_2A^2 with IP weights.

Marginal Structural Logistic Model for Binary YY:

  • logit P(Da=1)=α0+α1a\text{logit} \, P(D^a=1) = \alpha_0 + \alpha_1 a

    So exp⁡(α1)\exp(\alpha_1) gives the causal odds ratio.

12.5 Effect Modification and MSMs

Main Concepts 主要概念

  • MSMs usually do not include covariates unless to model effect modification.
  • To assess effect modification (e.g., by sex VV):

    E[Ya∣V]=β0+β1a+β2Va+β3VE[Y^a|V] = \beta_0 + \beta_1 a + \beta_2 V a + \beta_3 V

    If β2≠0\beta_2 \neq 0, effect modification exists.
  • Fitting via weighted least squares: E[Y∣A,V]=θ0+θ1A+θ2VA+θ3VE[Y|A,V]=\theta_0+\theta_1 A+\theta_2 VA+\theta_3 V with IP weights.
  • Stabilized weights for subgroups: SWA(V)=P(A∣V)P(A∣L)SW^A(V)=\frac{P(A|V)}{P(A|L)}.
  • 若模型中条件于全部混杂变量LL，权重SWA(L)=1SW^A(L) = 1，此时IP weighting等价于对LL直接回归调整。

12.6 Censoring and Missing Data 缺失与删失

Main Points 要点

  • Only analyzing individuals with observed outcomes can introduce selection bias if censoring (CC) depends on AA or LL.
  • Define C=0C=0 if observed, C=1C=1 if censored.
  • Interest: E[Ya=1,c=0]−E[Ya=0,c=0]E[Y^{a=1, c=0}] - E[Y^{a=0, c=0}] (if no one censored in either treatment).

Joint IP Weighting for Censoring and Treatment

  • Total weight: WA,C=WA×WCW^{A,C} = W^A \times W^C, with

    WC=1P(C=0∣L,A)W^C = \frac{1}{P(C=0|L,A)}

  • Or, stabilized:

    SWA,C=SWA×SWC,SWC=P(C=0∣A)P(C=0∣L,A)SW^{A,C} = SW^A \times SW^C, \quad SW^C = \frac{P(C=0|A)}{P(C=0|L,A)}

  • Weights correct for both confounding and selection bias (provided Ya,c=0⊥(A,C)∣LY^{a,c=0} \perp (A,C)|L holds).

Fine Points & Technical Insights 细节和技术要点

Horvitz-Thompson & Hajek Estimators

  • Horvitz-Thompson:

    E^[I(A=a)Yf(A∣L)]\hat{E}\left[\frac{I(A=a)Y}{f(A|L)}\right]

  • Hajek Estimator:

    E^[I(A=a)Yf(A∣L)]E^[I(A=a)f(A∣L)]\frac{\hat{E}\left[\frac{I(A=a)Y}{f(A|L)}\right]}{\hat{E}\left[\frac{I(A=a)}{f(A|L)}\right]}

    • Hajek estimator is preferred for bounded outcomes and is less sensitive to model misspecification.

Stabilized Weights — Generalization

  • General form: g(A)/f(A∣L)g(A)/f(A|L), g(A)g(A) unrelated to LL.
  • In practice, often g(A)=f(A)g(A)=f(A).

Positivity Check

  • Always check mean of stabilized weights is ~1.
  • Deviations indicate possible model misspecification or positivity violations.
  • Strict positivity violations preclude valid causal inference in certain strata.

Continuous Treatments

  • Estimation of conditional densities f(A∣L)f(A|L) much harder.
  • Assumed normality for f(A∣L)f(A|L); results can be very sensitive.
  • 需警惕模型假设的影响，特别是在连续处理上。

总结 Summary Table (中英双语)

ConceptFormula/Definition中英文解释
IP Weight (binary)$W^A = 1/P(AL)$
Stabilized IP Weight$SW^A = P(A)/P(AL)$
MSM (binary)E[Ya]=β0+β1aE[Y^a] = \beta_0 + \beta_1 a边际结构模型
MSM (continuous)E[Ya]=β0+β1a+β2a2E[Y^a] = \beta_0 + \beta_1 a + \beta_2 a^2连续处理的边际结构模型
Marginal Structural Logistic Model\logitP(Da=1)=α0+α1a\logit P(D^a=1)=\alpha_0 + \alpha_1 a边际结构logistic模型
Hajek estimator$\dfrac{\hat\left[\frac{I(A=a)Y}{f(AL)}\right]}{\hat\left[\frac{I(A=a)}{f(A
Joint Weight (Treatment &\& Censoring)WA,C=WA×WCW^{A,C} = W^A \times W^C， $W^C = 1/P(C=0L,A)$
Stabilized Censoring Weight$SW^C = P(C=0A)/P(C=0
Positivity condition$P(A=aL=l)>0 \ \forall a,l$
Effect modification in MSM$E[Y^aV] = \beta_0 + \beta_1 a + \beta_2 V a + \beta_3 V$

结论 Key Takeaways

  • IP weighting (including stabilized weights) allows estimation of causal effects from observational data under specific assumptions (exchangeability, positivity, consistency).
  • Marginal structural models provide a framework for causal inference, especially when traditional regression would be biased by time-varying confounding or missing data.
  • Choice of model (and assumptions, especially normality for continuous AA) critically affects estimates.
  • Always check positivity, stabilizing, and model specification sensitivity.
  • Selection bias (including from censoring/missing outcome) can be corrected using joint IP weighting.

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