Relating parameters in conditional, marginalized, and marginal logistic models when the mediator is binary

Stanghellini and Doretti (2019) studied the exact formulae relating parameters in conditional and marginalized logistic models when the mediator is binary. Those formulae generally do not hold for the reduced model as the reduced model is generally not the same as the marginalized model. For a conditional model that allows for treatmentmediator interaction, I present 1) alternative exact formulae relating parameters in the conditional model to those in the marginalized logistic model. They are equivalent to but simpler and easier to interpret than those given in Stanghellini and Doretti (2019); 2) a decomposition of the total treatment effect into the natural direct effect and the natural indirect effect without assuming the outcome is rare; 3) exact formulae relating parameters in the conditional model to those in the reduced logistic model by using likelihood equations; 4) a bound on the size of the natural direct effect regardless of whether the treatment is numeric or discrete; and 5) a numerical assessment of the bias of the approximate formulae reported in Valeri and VanderWeele (2013). The relative bias can be greater than 15% even when the prevalence is less than 10%.

Keywords

mediation, likelihood equation, indirect effect, direct effect

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Received 26 December 2019

Accepted 30 April 2020

Published 22 December 2020