Multiplicative Effect Modeling: The General Case

TL;DR

This paper develops a flexible modeling framework for multiplicative effects of treatments on binary outcomes, extending existing models to continuous or categorical treatments with better parameter independence.

Contribution

It introduces a general approach to model multiplicative effects for various treatment types, building on and extending previous models like the Richardson2017 binomial regression.

Findings

Monte Carlo simulations show improved performance of the proposed methods.

The methods allow direct modeling of relative risks with parameter independence.

Application to real data demonstrates practical utility.

Abstract

Generalized linear models, such as logistic regression, are widely used to model the association between a treatment and a binary outcome as a function of baseline covariates. However, the coefficients of a logistic regression model correspond to log odds ratios, while subject-matter scientists are often interested in relative risks. Although odds ratios are sometimes used to approximate relative risks, this approximation is appropriate only when the outcome of interest is rare for all levels of the covariates. Poisson regressions do measure multiplicative treatment effects including relative risks, but with a binary outcome not all combinations of parameters lead to fitted means that are between zero and one. Enforcing this constraint makes the parameters variation dependent, which is undesirable for modeling, estimation and computation. Focusing on the special case where the treatment is also binary, Richardson2017 propose a novel binomial regression model, that allows direct modeling of the relative risk. The model uses a log odds-product nuisance model leading to variation independent parameter spaces. Building on this we present general approaches to modeling the multiplicative effect of a continuous or categorical treatment on a binary outcome. Monte Carlo simulations demonstrate the desirable performance of our proposed methods. A data analysis further exemplifies our methods.