Path Analysis for Binary Random Variables

TL;DR

This paper develops a method for decomposing effects in binary variable systems, extending to multiple mediators and connecting to counterfactual definitions, with application to a real-world educational dataset.

Contribution

It introduces a novel decomposition of effects in binary systems with multiple mediators, linking traditional and counterfactual approaches.

Findings

Effect decomposition formulas for binary variables

Extension to multiple mediators and marginalized effects

Application to educational data with statistical estimates

Abstract

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.