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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

Author

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  • Manabu Kuroki

    (Yokohama National University)

  • Hisayoshi Nanmo

    (Yokohama National University)

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.

Suggested Citation

  • Manabu Kuroki & Hisayoshi Nanmo, 2020. "Variance formulas for estimated mean response and predicted response with external intervention based on the back-door criterion in linear structural equation models," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(4), pages 667-685, December.
    • Handle: RePEc:spr:alstar:v:104:y:2020:i:4:d:10.1007_s10182-020-00372-7
    DOI: 10.1007/s10182-020-00372-7
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    References listed on IDEAS

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    1. Manabu Kuroki & Masami Miyakawa, 2003. "Covariate selection for estimating the causal effect of control plans by using causal diagrams," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 209-222, February.
    2. Elena Stanghellini & Eduwin Pakpahan, 2015. "Identification of causal effects in linear models: beyond instrumental variables," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 489-509, September.
    3. Manabu Kuroki & Judea Pearl, 2014. "Measurement bias and effect restoration in causal inference," Biometrika, Biometrika Trust, vol. 101(2), pages 423-437.
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    1. Nanmo, Hisayoshi & Kuroki, Manabu, 2021. "Exact variance formula for the estimated mean outcome with external intervention based on the front-door criterion in Gaussian linear structural equation models," Journal of Multivariate Analysis, Elsevier, vol. 185(C).

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