  The decomposition of the overall effect of a treatment into direct and
indirect effects is here investigated with reference to a recursive system of
binary random variables. We show how, for the single mediator context, the
marginal effect measured on the log odds scale can be written as the sum of the
indirect and direct effects plus a residual term that vanishes under some
specific conditions. We then extend our definitions to situations involving
multiple mediators and address research questions concerning the decomposition
of the total effect when some mediators on the pathway from the treatment to
the outcome are marginalized over. Connections to the counterfactual
definitions of the effects are also made. Data coming from an encouragement
design on students' attitude to visit museums in Florence, Italy, are
reanalyzed. The estimates of the defined quantities are reported together with
their standard errors to compute p-values and form confidence intervals.
