Variance formulas for estimated mean response and predicted response with external intervention based on the back-door c
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Variance formulas for estimated mean response and predicted response with external intervention based on the back‑door criterion in linear structural equation models Manabu Kuroki1 · Hisayoshi Nanmo1 Received: 13 February 2019 / Accepted: 18 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Causal effect · Identification · Path diagram · Structural causal model
1 Introduction In various fields of practical science, it is very important to evaluate the causal effect of a treatment variable on a response variable based on observed data (e.g., Bollen 1989; Duncan 1975; Pearl 2009). Statistical causal inference began with path analysis (Wright 1923, 1934) and advanced to structural causal models (Pearl 2009) to
* Manabu Kuroki kuroki‑manabu‑[email protected] Hisayoshi Nanmo nanmo‑hisayoshi‑[email protected] 1
Department of Mathematical Science, Yokohama National University, 79‑1 Tokiwadai, Hodogaya‑ku, Yokohama 240‑8501, Japan
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evaluate the causal effect using qualitative causal information along with observed data. In the framework of statistical causal inference based on linear structural equation models (linear SEMs), the total effect is an important measure for evaluating the causal effect. The total effect can be interpreted as the change of the mean of the response variable (called a mean response) when the treatment variable is changed by one unit through the external intervention. To evaluate the total effect, statistical researchers in the field of linear SEMs have provided various identification conditions (e.g., Bowden and Turkington 1984; Brito 2004; Cai and Kuroki 2008; Chan and Kuroki 2010; Chen 2017; Garcia et al. 2010; Kuroki and Miyakawa 2004; Kuroki and Pearl 2014; Pearl 2009; Stanghellini and Pakpahan 2015; Tian 2004, 2007a, b). Herein, “identifiable” indicates that the total effect can be uniquely determined based on the variance–covariance parameters of observed variables. Kuroki (2000) investigated how intermediate variables influence the estimation accuracy of the total effect based on the front-door criterion (Pearl 2009). In addition, Kuroki and Miyakawa (2003) dis
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