Approximating the distribution of maximally selected McNemar's statistics

It is common in epidemiologic analyses to summarize continuous outcomes' as falling above or below a threshold. With paired data and with a threshold chosen without reference to the outcomes, McNemar's test of marginal homoge neity may be applied to the resulting dichotomous pairs when testing for eq uality of the marginal distributions of the underlying continuous outcomes. If the threshold is chosen to maximize the test statistic, however, referr ing the resulting test statistic to the nominal chi(2) distribution is inco rrect; instead, the p-value must be adjusted for the multiple comparisons. Here the distribution of a maximally selected McNemar's statistic is derive d, and it is shown that an approximation due to Durbin (1985, Journal of Ap plied Probability 22, 99-122) may be used to estimate approximate p-values. The methodology is illustrated by an application to measurements of insuli n-like growth factor-I (IGF-T) in matched prostate cancer cases and control s from the Physicians' Health Study. The results of simulation experiments that assess the accuracy of the approximation in moderate sample sizes are reported.