It is tempting to assume that confounding bias is eliminated by choosing controls that are identical to the cases on the matched confounder(s). We used causal diagrams to explain why such matching not only fails to remove confounding bias, but also adds colliding bias, and why both types of bias are removed by conditioning on the matched confounder(s). As in some publications, we trace the logic of matching to a possible tradeoff between effort and variance, not between effort and bias. Lastly, we explain why the analysis of a matched case-control study - regardless of the method of matching - is not conceptually different from that of an unmatched study.
Keywords: case-control study; causal diagrams; colliding bias; confounding bias; directed acyclic graphs; matching; variance.