  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.
