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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 log-mean (LM) and log-mean linear (LML), respectively. The choice of the link function plays a key role for 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 LM and LML 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 LML 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 dataset obtained from a case-control study aimed to investigate the effect of HIV-infection on multimorbidity, that is simultaneous presence of two or more noninfectious commorbidities in one patient.