Using the Exterior Match to Compare Two Entwined Matched Control Groups
When comparing outcomes, such as survival, in two groups- say a focal group and a comparison group-a common question is whether an adjustment for certain baseline differences that separate these two groups actually matters for the difference in outcomes. Did the adjustment matter? If it did matter, to what quantitative extent did it matter? This question is quite distinct from whether the baseline variables predict the outcome: baseline variables may predict the outcome, yet explain no part of the difference in outcomes in two groups. The question is also distinct from whether a difference between the groups remains after adjustment: an adjustment may matter quite a bit, yet fail to explain a substantial part of the difference in outcomes, and, indeed, adjustment may increase the difference. Whether an adjustment for (<bold>x</bold> <sub>1</sub>, <bold>x</bold> <sub>2</sub>) matters over and above an adjustment for <bold>x</bold> <sub>1</sub> alone can be addressed by comparing outcomes in two control groups formed from the comparison group, one matched to the focal group for <bold>x</bold> <sub>1</sub> alone, the other matched to focal group for (<bold>x</bold> <sub>1</sub>, <bold>x</bold> <sub>2</sub>). How do outcomes differ in these two matched control groups? If two control groups are each pair-matched to the same focal group, then the result is a set of matched triples, so controls in the two groups are implicitly matched to each other by virtue of being matched to the same person in the focal group. When the comparison group is vastly larger than the focal group and their distributions exhibit extensive overlap on (<bold>x</bold> <sub>1</sub>, <bold>x</bold> <sub>2</sub>), it may be possible to construct nonintersecting matched control groups, but quite often the comparison group is large enough to yield closely matched groups one at a time, but is not large enough to produce several nonintersecting matched control groups. How can one compare two matched control groups that are entwined, with some of the same controls in both groups? Two entwined control groups have a nonempty intersection: some of the same controls appear in both groups as duplicates. These duplicates may appear in the same matched triple, but more commonly they appear in different matched triples. This structure yields a new nonintersecting match that we call the exterior match. Properties of the exterior match are discussed. Our on-going study of black-versus-white disparities in survival following breast cancer in Medicare motivated this work and is used to illustrate.
