On marginal and conditional parameters in logistic regression models

TL;DR

This paper derives an exact formula linking marginal and conditional logistic regression parameters with binary mediators, enabling sensitivity analysis and bounds estimation without assuming conditional independence.

Contribution

It provides a novel exact formula connecting marginal and conditional parameters in logistic models with mediators, extending path analysis to binary variables.

Findings

Exact formula linking marginal and conditional parameters

Enables sensitivity analysis and bounds estimation

Extends path analysis to binary variables

Abstract

A fundamental research question is how much a variation in a covariate influences a binary response variable in a logistic regression model, both directly or through mediators. We derive the exact formula linking the parameters of marginal and conditional regression models with binary mediators when no conditional independence assumptions can be made. The formula has the appealing property of being the sum of terms that vanish whenever parameters of the conditional models vanish, thereby recovering well-known results as particular cases. It also permits to quantify the distortion induced by omission of some relevant covariates, opening the way to sensitivity analysis. Also in this case, as the parameters of the conditional models are multiplied by terms that are always positive or bounded, the formula may be used to construct reasonable bounds on the parameters of interest. We assume that, conditionally on a set of covariates, the data-generating process can be represented by a Directed Acyclic Graph. We also show how the results here presented lead to the extension of path analysis to a system of binary random variables.