A maximally selected test of symmetry about zero

The problem of testing symmetry about zero has a long and rich history in the statistical literature. We introduce a new test that sequentially discards observations whose absolute value is below increasing thresholds defined by the data. McNemar's statistic is obtained at each threshold and the largest is used as the test statistic. We obtain the exact distribution of this maximally selected McNemar and provide tables of critical values and a program for computing p-values. Power is compared with the t-test, the Wilcoxon Signed Rank Test and the Sign Test. The new test, MM, is slightly less powerful than the t-test and Wilcoxon Signed Rank Test for symmetric normal distributions with nonzero medians and substantially more powerful than all three tests for asymmetric mixtures of normal random variables with or without zero medians. The motivation for this test derives from the need to appraise the safety profile of new medications. If pre and post safety measures are obtained, then under the null hypothesis, the variables are exchangeable and the distribution of their difference is symmetric about a zero median. Large pre-post differences are the major concern of a safety assessment. The discarded small observations are not particularly relevant to safety and can reduce power to detect important asymmetry. The new test was utilized on data from an on-road driving study performed to determine if a hypnotic, a drug used to promote sleep, has next day residual effects.