Multiplicative Effect Modeling: The General Case
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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, it is hard to interpret the coefficients of a logistic regression model as these are log odds ratios. For example, it is hard to compare coefficients from different studies, even if treatment is randomized, since odds ratios are not collapsible. Coefficients from Poisson regressions are measures of multiplicative treatment effects and hence are collapsible. However, with a binary outcome the parameters in a Poisson regression are variation dependent, which can be undesirable for modeling, estimation and computation. Focusing on the special case where the treatment is also binary, Richardson et al. (2017) 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 categorical or continuous treatment on a binary outcome. A Monte Carlo simulation demonstrates the superior performance of our proposed methods. A data analysis further exemplifies our methods on real data.
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