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Biometrika 2006 93(3):601-611; doi:10.1093/biomet/93.3.601
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© 2006 Biometrika Trust

On recovering a population covariance matrix in the presence of selection bias

Manabu Kuroki1 and Zhihong Cai2

1 Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama-cho, Toyonaka, Osaka, 560-8531, Japan. mkuroki{at}sigmath.es.osaka-u.ac.jp, 2 Department of Biostatistics, Graduate School of Public Health, Kyoto University, Yoshida-Konoe-cho, Sakyo-ku, Kyoto, 606-8501, Japan. cai{at}pbh.med.kyoto-u.ac.jp

This paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models. We propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable. When the answer is affirmative, we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population. The results of this paper provide guidance for reliable causal inference, based on the recovered covariance matrix obtained from the statistical information with selection bias.

Key Words: Directed acyclic graph; Path diagram; Single factor model; Tetrad difference.

Received June 2004. Revised February 2006.


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