On causal parameters in recursive systems for binary random variables

The relationship between parameters of marginal and conditional logistic regression models is presented when no conditional independence assumptions can be made. We show that the marginal parameters decompose into the sum of terms that vanish whenever the parameters of the conditional model vanish, in parallel with the Cochran’s formula for the linear case. The derivations can be applied to decompose the marginal effectofabinary/continuoustreatmentonabinaryresponseintoadirecteffectandanindirecteffectthroughabinarymediatorandcanbeextended to a recursive system of binary random variables, leading to results similar to the well-known path analysis. The interest lies on several research questions. Given a data-generating process, a researcher may wish to quantify how much of the total effect of a covariate on a response is due to intermediate variables and can be removed after conditioning on their values. From a different, though related, point of view, one may wish to quantify the distortion on some regression coefficients of interest due to the omission of relevant unmeasured covariates, and use this information to build reasonable bounds or to conduct sensitivity analysis. We further show how these derivations can used in the context of causal inference, and propose a decomposition of the total effect into terms that are due to direct and indirect effects.