  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.
