Why There Is No Statistical Test for Confounding, Why Many Think There Is, and Why They Are Almost Right

Confounding is a simple concept. If we undertake to estimate the e ect of one variable (X) on another (Y ) by examining the statistical association between the two, we ought to ensure that the association is not produced by factors other than the e ect under study. The presence of spurious association, due for example to the in uence of extraneous variables, is called confounding as it tends to confound our reading and to bias our estimate of the e ect studied. Conceptually, therefore, we can say that X and Y are confounded when there is a third variable Z that in uences both X and Y ; such a variable is then called a \confounder" of X and Y . As simple as this concept is, it has resisted formal treatment for several decades, and for a good reason: The very notions of \e ect" and \in uence", relative to which \spurious association" is to be de ned, has resisted mathematical formulation. The empirical de nition of e ect as an association that would prevail in a controlled randomized experiment, cannot easily be expressed in the standard language of probability theory, because that theory deals with static conditions, and does not permit us to predict, even from a full speci cation of a population density function, what relationships would prevail if conditions were to change, say from observational to controlled studies. Such predictions require extra information, in the form of causal or counterfactual assumptions [Greenland and Robins 1986; Wickramaratne and Holford 1987], which is not discernible from density functions. These di culties notwithstanding, epidemiologists, biostatisticians, social scientists and economists have made numerous attempts to express confounding in statistical terms, partly because statistical de nitions, free of theoretical terms of \e ect" or \in uence," can be expressed in conventional mathematical form, and partly because such de nitions may lead to