Conditional and marginal relative risk parameters for a class of recursive regression graph models

In linear regression modelling, the distortion of effects after marginalizing over variables of the conditioning set has been widely studied in several contexts. For Gaussian variables, the relationship between marginal and partial regression coefficients is well established and the issue is often addressed as a result of W. G. Cochran. Possible generalizations beyond the linear Gaussian case have been developed, nevertheless the case of discrete variables is still challenging, in particular in medical and social science settings. A multivariate regression framework is proposed for binary data with regression coefficients given by the logarithm of relative risks, and a multivariate Relative Risk formula is derived to define the relationship between marginal and conditional relative risks. The method is illustrated through the analysis of the morphine data in order to assess the effect of preoperative oral morphine administration on the postoperative pain relief.