Case-control matching on confounders revisited

Matching by a confounder in a case-control study nearly always produces a control-selection bias that mixes with the confounding to produce a net bias. Previous theoretical work has assumed that control for a single confounder, the matching factor, is sufficient to remove all the confounding and that the confounder-exposure, confounder-outcome and exposure-outcome associations are monotonic. Under these conditions: (a) The net bias is toward the null if the exposure affects the outcome and nil if it does not. (b) If the confounding is away from the null, the selection bias is toward the null. (c) If the confounding is toward the null, the selection bias can be in any direction or even nil. If more than one confounder needs to be controlled to remove all the confounding, the net bias from matching by one of them can be away from the null, whether the exposure affects the outcome or not. An influential heuristic, that matching controls to cases by a variable associated with exposure always brings the marginal exposure distributions of the case and control groups closer together, turns out to be faulty. The implications of matching by confounders in case-control studies are less straightforward than previously thought. Suggestions are offered for advancing the methodologic literature on this topic.