[Paper Review] Congenial Causal Inference with Binary Structural Nested Mean Models

Linbo Wang, Thomas S. Richardson+1 more|arXiv (Cornell University)|Sep 24, 2017

Advanced Causal Inference Techniques12 references10 citations

TL;DR

This paper resolves the variation dependence problem in binary outcome structural nested mean models (SNMMs) by reparameterizing nuisance parameters to be variation independent of causal parameters, enabling unconstrained maximum likelihood estimation and true doubly robust estimators for multiplicative SNMMs. It proves additive SNMMs cannot achieve this independence, justifying the focus on multiplicative models.

ABSTRACT

Structural nested mean models (SNMMs) are among the fundamental tools for inferring causal effects of time-dependent exposures from longitudinal studies. With binary outcomes, however, current methods for estimating multiplicative and additive SNMM parameters suffer from variation dependence between the causal SNMM parameters and the non-causal nuisance parameters. Estimating methods for logistic SNMMs do not suffer from this dependence. Unfortunately, in contrast with the multiplicative and additive models, unbiased estimation of the causal parameters of a logistic SNMM rely on additional modeling assumptions even when the treatment probabilities are known. These difficulties have hindered the uptake of SNMMs in epidemiological practice, where binary outcomes are common. We solve the variation dependence problem for the binary multiplicative SNMM by a reparametrization of the non-causal nuisance parameters. Our novel nuisance parameters are variation independent of the causal parameters, and hence allows the fitting of a multiplicative SNMM by unconstrained maximum likelihood. It also allows one to construct true (i.e. congenial) doubly robust estimators of the causal parameters. Along the way, we prove that an additive SNMM with binary outcomes does not admit a variation independent parametrization, thus explaining why we restrict ourselves to the multiplicative SNMM.

Motivation & Objective

• To address the variation dependence between causal and nuisance parameters in binary outcome SNMMs, a major barrier to their use in epidemiology.
• To develop a reparameterization of nuisance parameters that ensures variation independence from causal parameters in multiplicative SNMMs.
• To enable unconstrained maximum likelihood estimation of multiplicative SNMMs with binary outcomes.
• To construct truly congenial doubly robust estimators for causal parameters in binary SNMMs.
• To clarify why additive SNMMs with binary outcomes cannot achieve variation independence, justifying the focus on multiplicative models.

Proposed method

• Reparameterize the non-causal nuisance parameters in the multiplicative SNMM to be variation independent of the causal parameters.
• Use the new nuisance parameterization to allow unconstrained maximum likelihood estimation of the causal parameters.
• Construct doubly robust estimators based on the variation-independent parameterization, ensuring validity under either outcome regression or propensity score model correctness.
• Prove that no such variation-independent parametrization exists for additive SNMMs with binary outcomes.
• Demonstrate that logistic SNMMs avoid variation dependence but require strong modeling assumptions for unbiased causal estimation.
• Use structural modeling and counterfactual frameworks to formalize the causal estimands and their identification under the new parameterization.

Experimental results

Research questions

• RQ1 Can a reparameterization of nuisance parameters in a binary outcome multiplicative SNMM eliminate variation dependence with causal parameters?
• RQ2 Does the proposed reparameterization allow for unconstrained maximum likelihood estimation of causal parameters?
• RQ3 Can true congenial doubly robust estimators be constructed under the new parameterization?
• RQ4 Is variation independence achievable in additive SNMMs with binary outcomes?
• RQ5 Why have previous methods for binary SNMMs failed to achieve variation independence, and what are the structural barriers?

Key findings

• The proposed reparameterization successfully decouples nuisance parameters from causal parameters in multiplicative SNMMs, eliminating variation dependence.
• Unconstrained maximum likelihood estimation becomes feasible for multiplicative SNMMs with binary outcomes under the new parameterization.
• True congenial doubly robust estimators for causal parameters can now be constructed, improving robustness and validity.
• Additive SNMMs with binary outcomes do not admit any variation-independent parametrization, explaining the limitations of existing methods.
• The results clarify why logistic SNMMs, while free of variation dependence, still require strong modeling assumptions for unbiased causal estimation.
• The work provides a foundational solution for applying multiplicative SNMMs to binary outcomes in longitudinal epidemiological studies.