This paper considers a situation in which cause–effect relationships among variables can be described by a linear structural equation model (linear SEM) and the corresponding directed acyclic graph (DAG). By considering a set of covariates that satisfies the back-door criterion, we formulate (1) the variances of the estimated mean response and (2) the mean squared error (MSE) of the predicted response, with external intervention in which a treatment variable is set to be a certain constant value. The variance and MSE formulas proposed in this paper are exact, unlike those in most previous studies regarding the problem of estimating total effects. In addition, we compare the performance of the simple regression model with that of the predicted response with the external intervention. Furthermore, we apply the present results to statistical quality control.

本文考虑了一种情况，其中变量之间的因果关系可以通过线性结构方程模型（linear SEM）和相应的有向无环图（DAG）来描述。通过考虑一组满足后门准则的协变量，我们采用外部干预（其中包括治疗变量）来制定（1）估计平均反应的方差和（2）预测反应的均方误差（MSE）。设置为某个常数。本文提出的方差和MSE公式是精确的，这与以往大多数关于总效应估算问题的研究不同。此外，我们将简单回归模型的性能与外部干预的预期响应进行了比较。此外，