[Paper Review] An Alternative Perspective on the Robust Poisson Method for Estimating Risk or Prevalence Ratios

Denis Talbot, Miceline Mésidor+3 more|arXiv (Cornell University)|Dec 1, 2021

Advanced Causal Inference Techniques4 citations

TL;DR

This paper presents a semiparametric reinterpretation of the robust Poisson method, showing it does not require assuming a Poisson distribution for binary outcomes. Instead, it relies solely on a log-linear relationship between exposure variables and outcome risk/prevalence, offering a theoretically sound alternative to traditional Poisson modeling while maintaining interpretability as risk or prevalence ratios and avoiding convergence issues seen in log-binomial models.

ABSTRACT

The robust Poisson method is becoming increasingly popular when estimating the association of exposures with a binary outcome. Unlike the logistic regression model, the robust Poisson method yields results that can be interpreted as risk or prevalence ratios. In addition, it does not suffer from frequent non-convergence problems like the most common implementations of maximum likelihood estimators of the log-binomial model. However, using a Poisson distribution to model a binary outcome may seem counterintuitive. Methodological papers have often presented this as a good approximation to the more natural binomial distribution. In this paper, we provide an alternative perspective to the robust Poisson method based on the semiparametric theory. This perspective highlights that the robust Poisson method does not require assuming a Poisson distribution for the outcome. In fact, the method only assumes a log-linear relationship between the risk/prevalence of the outcome and the explanatory variables. This assumption and consequences of its violation are discussed. Suggestions to reduce the risk of violating the modeling assumption are also provided. Additionally, we discuss and contrast the robust Poisson method with other approaches for estimating exposure risk or prevalence ratios.

Motivation & Objective

• To provide a new theoretical foundation for the robust Poisson method that moves beyond the assumption of a Poisson-distributed outcome.
• To clarify that the method's validity rests not on the distributional assumption of the outcome, but on the correct specification of the log-linear relationship between exposures and risk/prevalence.
• To address concerns about the method's validity when applied to binary outcomes by grounding it in semiparametric theory.
• To compare the robust Poisson method with alternative approaches for estimating risk or prevalence ratios, including log-binomial and other regression techniques.
• To offer practical guidance on reducing the risk of model misspecification when using the robust Poisson method.

Proposed method

• The authors adopt a semiparametric framework to reframe the robust Poisson method, focusing on the estimating equation approach rather than full parametric likelihood.
• They show that the robust Poisson estimator is consistent under a weaker assumption: the correct specification of the conditional mean function via a log-linear link, not the distributional assumption of the outcome.
• The method uses a working Poisson likelihood with robust variance estimation (sandwich estimator) to correct for potential misspecification of the variance structure.
• The key identifying assumption is that the expected outcome given covariates follows a log-linear model, i.e., log(E[Y|X]) = X^Tβ.
• The approach does not require the outcome to be Poisson-distributed; it only requires the mean structure to be correctly modeled.
• The paper discusses how violations of the log-linear assumption affect inference and provides recommendations to assess or reduce such risks.

Experimental results

Research questions

• RQ1 Does the robust Poisson method truly require the outcome to follow a Poisson distribution, or is this assumption unnecessary?
• RQ2 Can the robust Poisson method be justified under a semiparametric framework that does not rely on distributional assumptions for the outcome?
• RQ3 How does the robust Poisson method compare to the log-binomial model in terms of model assumptions, convergence, and interpretability?
• RQ4 What are the consequences of violating the log-linear mean structure assumption in the robust Poisson method, and how can they be mitigated?
• RQ5 What are the practical implications of using the robust Poisson method when estimating risk or prevalence ratios in epidemiological studies?

Key findings

• The robust Poisson method does not require the outcome to follow a Poisson distribution; its validity stems from the correct specification of the log-linear mean structure.
• The method remains consistent and interpretable as a risk or prevalence ratio estimator even when the outcome is binary, provided the log-linear model is correctly specified.
• The robust Poisson method avoids the frequent non-convergence issues common in maximum-likelihood log-binomial models.
• The paper demonstrates through simulation that the robust Poisson method maintains good performance in terms of bias and coverage under correct model specification.
• Violations of the log-linear assumption lead to biased estimates, and the paper recommends diagnostic checks such as residual analysis and model comparison to detect such issues.
• The authors conclude that the robust Poisson method is a reliable and practical alternative to log-binomial models for estimating risk or prevalence ratios in binary outcome settings.