Simple tests of homogeneity of controls in matched studies

Consider matched case-control studies with a pair of distinguishable controls, say neighbourhood and hospital, wherein one of the controls may be missing. Liang and Stewart derive an efficient test for the possible difference in controls for the complete data case. They suggest inefficient tests for the situation where some triplets are incomplete. Levin, and Risch and Tibshirani, derive efficient tests for the incomplete triplet case by the methods of maximum likelihood estimator (Wald) tests and likelihood ratio tests, respectively. All these tests require the use of compute algorithms. Using score theory, we derive a simple efficient test for complete triplets which only requires solving a quadratic equation. Furthermore, we show that the usual normal approximation to McNemar's test is equivalent to the score test and is thus fully efficient as well. For incomplete triplets, score theory leads to a fully efficient test which only requires the solution of a cubic equation. We propose a nearly efficient test which is even more simply computed. The example of Liang and Stewart illustrates application of the results.