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Overview
- 1. Preliminaries & Review
- 2. Marginal Structural vs Structural Nested Models
- 3. Practical MR Estimation of SNMMs
- 4. Simulation Study
- 5. Caveats and Conclusions
Effect Decomposition with Structural Nested Models A Practical - - PowerPoint PPT Presentation
Effect Decomposition with Structural Nested Models A Practical - - PowerPoint PPT Presentation
Effect Decomposition with Structural Nested Models A Practical Multiply Robust Approach Ashley I Naimi Stijn Vansteelandt 1 Overview 1. Preliminaries & Review 2. Marginal Structural vs Structural Nested Models 3. Practical MR Estimation
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Preliminaries
Path Analysis (Wright 1934) Mediation & Moderation (Judd, Baron, Kenny ~ 1980) Pure and Total Direct and Indirect Effects (Robins & Greenland 1992) Natural and Controlled Effects, npSEM, Mediation Formula (Pearl 2001, 2012) Four Way Decomposition (VanderWeele 2014)
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Preliminaries
Natural Indirect Effect
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Preliminaries
Natural Direct Effect
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Preliminaries
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Controlled Direct Effect
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Preliminaries
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Identifying Natural Effects
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Identifying Controlled Direct Effect
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MSM for Effect Decomposition
- Models for average of potential outcomes
- Estimated (usually) via inverse probability weighting
- Outcome and mediator model may be required
- Easily fit in standard regression
- IPWs obtained by modeling exposure & mediator density
10 VanderWeele 2009 Epidemiol. 20(1):18-26
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Inverse Probability Weighted MSMs
- Highly variable weights
- No effect modification by past time-varying covariates
- Correct specification of entire exposure / mediator density
- Current conventions limit to single robustness
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G Estimation of SNMMs
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- Highly variable weights not a problem
- Effect modification by past time-varying covariates
- Correct specification of exposure / mediator density moment
- Double and multiple robustness
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Objective
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Develop a modified direct and indirect effect g estimator that can be implemented using standard software routines
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Approach
14 Vansteelandt and Joffe 2015 Stat Sci. 29(4):07-31
Propensity scores for the mediator and exposure:
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Linear and Log-Linear SNMMs
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Estimation of a linear SNMM: Step 1
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Estimation of a linear SNMM: Step 2
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e Y = Y − ˆ ψ2M − ˆ ψ3XM
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Estimation of a linear SNMM: Step 3
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The estimated coefficient for X is a multiply-robust estimate of the controlled direct effect when M = 0. It is unbiased (consistent) if:
1. either the propensity score model for M is correct OR the model for Y (in step 1) up to but excluding the propensity score terms is correct and, 2. either the propensity score model for X is correct OR the model for the transformed outcome tilde Y (in step 3) up to but excluding the propensity score term is correct
Estimation of a linear SNMM
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Estimation of a linear SNMM
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Estimation of a log-linear SNMM: Step 1
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ˆ ψ(1)
2
and ˆ ψ(1)
3
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This model must also be weighted by Giving us “next step” estimates Iterate this until
Estimation of a log-linear SNMM: Step 2
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ˆ ψ(2)
2
and ˆ ψ(2)
3
exp(− ˆ ψ(1)
2 M − ˆ
ψ(1)
3 XM)
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Estimation of a log-linear SNMM: Step 3
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e Y = Y exp(− ˆ ψ2M − ˆ ψ3XM)
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Estimation of a log-linear SNMM: Step 4
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E[e Y | X, C1] = exp ψ1X + γ0 + γ1C1
- ˆ
ψ(1)
1
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This model must also be weighted by Giving us a “next step” estimate Iterate this until
Estimation of a log-linear SNMM: Step 5
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ˆ ψ(2)
1
| ˆ ψ(j)
1
− ˆ ψ(j−1)
1
| ≤ 1.0 × 10−6
exp(− ˆ ψ(1)
1 X)
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An iterating algorithm SAS macro
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All iid binary (Bernoulli) RVs Continuous mediator Strong exposure-outcome confounding Strong post-treatment mediator-outcome confounding
Simulation Study: Data Generation
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E(Y x0) = X
U
X
CXY
X
Z
P(Y | X = x,M = 0, Z, CXY , U) P(Z | X = x, U)P(CXY )P(U)
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Methods: Inverse probability weighting (MSM) Structural Transformation Method (ST) Modified G Estimation (mGE) Scenarios: All models correct Exposure/Mediator models mis-specified Outcome models mis-specified Results: Distribution of point estimates from 10,000 samples Mean and SD of point estimates Convergence of the iterative approach
Simulation Study: Analysis
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Simulation Study: Results
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Simulation Study: Results
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Simulation Study: Results
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Simulation Study: Results
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Convergence Results: All Models Correct
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Convergence Results: Exposure/Mediator Mis-Specified
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Convergence Results: Outcome Mis-specified
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- Simplified SNMMs in this talk
- Asymptotically equivalent to Robins’ g estimation
Caveats
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Use of g estimation has been relatively infrequent despite many advantageous properties Our modified multiply robust g estimator extends the range of approaches one can take for effect decomposition and direct effect estimation
Conclusion
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