Multiplicative SNMMs in Causal Inference

• Multiplicative SNMMs are models that estimate causal effects by applying a multiplicative parameterization to handle time-dependent confounding in longitudinal settings.
• They use G-estimation techniques to identify causal effects even when conventional methods fail due to intermediate confounding.
• The approach incorporates doubly robust procedures, making it valuable for non-probability surveys and dynamic treatment regimes.

Multiplicative Structural Nested Mean Models (SNMMs) are a class of structural nested models used to estimate causal effects in longitudinal settings, particularly when exposures and confounders vary over time and conventional adjustment methods fail due to time-dependent confounding by intermediate variables. The "multiplicative" formulation models the effect of an intervention as acting multiplicatively on the mean or conditional mean of the outcome, rather than additively. These models formalize and identify causal effects in contexts such as longitudinal intervention studies, time-varying treatment regimens, and complex survey designs with selection or missingness mechanisms related both to covariates and outcomes.

1. Formal Definition and Model Structure

Multiplicative SNMMs define the potential outcomes for each individual at each time point as functions of their actual and counterfactual treatment histories, using a multiplicative tilting parameterization to characterize the effect of hypothetical interventions. Let $Y_t$ denote the outcome at time $t$, $A_t$ the treatment or exposure at time $t$, and $L_t$ the history of observed covariates. The canonical multiplicative SNMM specifies that

$$\mathbb{E}[Y_t^{\bar a_t} \mid \bar L_t, \bar A_{t-1}] = \mathbb{E}[Y_t^{\bar a_{t-1}, 1} \mid \bar L_t, \bar A_{t-1}] \cdot \exp\{ \gamma_t(A_t, L_t; \psi) \},$$

where $Y_t^{\bar a_t}$ is the potential outcome under treatment history $\bar a_t$, and $\gamma_t$ is a structural function, often parametrized as a log-relative effect $\gamma_t(a_t, l_t; \psi) = \psi a_t$, analogous to a conditional log-rate ratio or log odds ratio in generalized linear models. This formulation encodes how the effect of treatment acts multiplicatively on mean potential outcomes, enabling identification and estimation even in the presence of time-dependent confounding that may itself be affected by past treatment.

2. Identification Assumptions and Causal Inference

Correct identification using multiplicative SNMMs requires a set of assumptions extending standard ignorability and positivity conditions to longitudinal and complex data structures:

• Sequential Exchangeability (No Unmeasured Confounding): At each time $t$0, conditional on observed covariate and treatment history, the future counterfactual outcomes are independent of current treatment assignment:

$t$1

• Positivity: There is a nonzero probability of observing each history of treatment given past data.
• Correct Model Specification: Either the structural model for $t$2 (i.e., the multiplicative causal effect) or the models for the treatment and outcome processes must be correctly specified, as in the "double robustness" property of SNMM estimation (Chen et al., 2018).

When these assumptions hold, multiplicative SNMMs enable estimation of causal effects even in highly nonrandomized settings, using methods such as G-estimation and doubly robust augmentations that account for time-dependent selection or missingness.

3. Estimation Strategies: G-Estimation and Doubly Robust Procedures

Estimation of multiplicative SNMMs commonly employs G-estimation, which exploits the structural model equations to solve for $t$3 by constructing estimating equations insensitive to nuisance model misspecification. In finite population or non-randomized survey settings, the approach is closely related to augmented inverse probability weighting (AIPW):

• Propensity Score/G-Estimation Approach: Define models for the conditional probability of selection or treatment at each time (e.g., logistic regression for $t$4) and for the conditional mean outcome ($t$5).
• Doubly Robust Estimation: Construct an estimator that combines propensity weighting and outcome regression, such as

$t$6

where $t$7 is a regression estimate (possibly incorporating a multiplicative parameterization), and $t$8 is the estimated selection/treatment mechanism (Chen et al., 2018).

The estimator is consistent and asymptotically normal if either the outcome regression model or the treatment mechanism model is correct, and is optimal (achieves the semiparametric efficiency bound) if both are correct.

4. Connections to Modern Causal Inference and Survey Methodology

Multiplicative SNMMs generalize classical marginal structural models (MSMs) by focusing on conditional mean models with multiplicative effects and are particularly aligned with contemporary approaches in complex surveys and non-probability samples. As detailed in Chen, Li & Wu (Chen et al., 2018), doubly robust methods for finite population inference with non-probability samples are operationally equivalent to SNMM-based G-estimation, both being robust to misspecification of one, but not both, of the selection (propensity) or mean structure (outcome regression) models.

The procedure addresses general settings where outcome data are observed in only a nonrandom sample (opt-in, missing outcome, or survey selection), and corresponding covariate data are available from a probability (reference) sample. The DR estimator integrates both outcome modeling and inverse probability of selection, yielding unbiased estimates provided at least one nuisance model is correct, a defining feature of the SNMM class (Chen et al., 2018).

5. Extensions, Implementation, and Practical Recommendations

Recent developments apply multiplicative SNMMs and their estimation paradigms to settings far beyond traditional randomized trials, including:

• Non-probability Survey Integration: Using auxiliary covariate information from probability samples with non-probability outcome samples to recover finite population means, leveraging the double robustness property (Chen et al., 2018).
• Longitudinal or Panel Data: Estimating dynamic causal effects where treatments and intermediate confounders are time-varying, often using stabilized weights or incorporating random effects.
• Complex Variance Estimation: Employing sandwich estimators or empirical variance estimates that retain consistency even when some model components are misspecified, per the doubly robust variance framework.

For empirical implementation, it is recommended to:

• Collect rich covariate data predictive of both selection and outcomes.
• Fit flexible parametric or semiparametric models for propensity and outcome processes (e.g., via logistic regression, generalized linear models, or machine learning).
• Prefer the Hájek-type estimator for improved stability and to compute conservative (plug-in) variance estimates, supplementing with more robust sandwich or reweighted approaches when model specification is in doubt (Chen et al., 2018).

6. Empirical Evidence and Simulation Results

Simulation studies indicate that multiplicative SNMMs, when estimated using doubly robust procedures, outperform estimators relying solely on IPW or regression adjustment.