Log-mean linear regression models for binary responses with an application to multimorbidity

In regression models for categorical data a linear model is typically related to the response variables via a transformation of probabilities called the link function. We introduce an approach based on two link functions for binary data named the log-mean and the log-mean linear methods. The choice of the link function plays a key role in the interpretation of the model, and our approach is especially appealing in terms of interpretation of the effects of covariates on the association of responses. Similarly to Poisson regression, the log-mean and log-mean linear regression coefficients of single outcomes are log-relative-risks, and we show that the relative risk interpretation is maintained also in the regressions of the association of responses. Furthermore, certain collections of zero log-mean linear regression coefficients imply that the relative risks for joint responses factorize with respect to the corresponding relative risks for marginal responses. This work is motivated by the analysis of a data set obtained from a case–control study aimed at investigating the effect of human immunodeficiency virus infection on multimorbidity, i.e. simultaneous presence of two or more non-infectious comorbidities in one patient.