Biometrika Advance Access originally published online on February 4, 2008
Biometrika 2008 95(1):17-33; doi:10.1093/biomet/asm092
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Articles |
Distortion of effects caused by indirect confounding
Department of Mathematical Statistics, Chalmers/Göteborgs Universitet, Gothenburg, Sweden wermuth{at}math.chalmers.se
Nuffield College, Oxford OX1 1NF, U.K. david.cox{at}nuffield.ox.ac.uk
Received for publication 1 April 2006.
Revision received 1 June 2007.
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Abstract |
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Undetected confounding may severely distort the effect of an explanatory variable on a response variable, as defined by a stepwise data-generating process. The best known type of distortion, which we call direct confounding, arises from an unobserved explanatory variable common to a response and its main explanatory variable of interest. It is relevant mainly for observational studies, since it is avoided by successful randomization. By contrast, indirect confounding, which we identify in this paper, is an issue also for intervention studies. For general stepwise-generating processes, we provide matrix and graphical criteria to decide which types of distortion may be present, when they are absent and how they are avoided. We then turn to linear systems without other types of distortion, but with indirect confounding. For such systems, the magnitude of distortion in a least-squares regression coefficient is derived and shown to be estimable, so that it becomes possible to recover the effect of the generating process from the distorted coefficient.
Key Words: Graphical Markov model • Identification • Independence graph • Linear least-squares regression • Parameter equivalence • Recursive regression graph • Structural equation model • Triangular system