Exact variance formula for the estimated mean outcome with external intervention based on the front-door criterion in Gaussian linear structural equation models

Abstract

In this paper, we assume that cause–effect relationships among variables can be represented by a Gaussian linear structural equation model and a corresponding directed acyclic graph. For a set of intermediate variables that satisfies the front-door criterion, we provide the variance formula of the estimated mean outcome under an external intervention in which a treatment variable is set to a specified constant value. The variance formula proposed in this paper is exact, in contrast to those in most previous studies on estimating total effects. In addition, based on the variance formula, we formulate the mean squared error between a future sample and the estimated mean outcome with the external intervention.