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RESEARCH PRODUCT

The size of Simes’ global test for discrete test statistics

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Abstract To increase the power of the Bonferroni–Holm procedure several modified Bonferroni procedures have been proposed (for example, Hochberg, 1988. Biometrika 75, 800–802; Hommel, 1988. Biometrika 75, 383–386), which are based on Simes’ global test (Simes, 1986. Biometrika 73, 751–754). By several simulation studies which, in particular, considered multinormal test statistics, it has been suggested that the Simes test is a level α test. However, an exact proof exists for only few situations one of them assuming independence of test statistics. We studied the behaviour of Simes’ test for discrete test statistics. Due to discreteness one can expect more conservative decisions whereas dependence structure can become more complicated and therefore cause liberalism. By means of analytical investigations we considered multivariate versions of the sign test, the exact Fisher test and the Poisson test. According to results obtained for multinormal test statistics we never observed exceedance of the nominal level α as long as two-sided tests were performed. For one-sided tests we did not find an exceedance when correlations were positive. If correlations are negative, one obtains liberal results; these are of negligible order for realistic choices of α , say α ⩽0.10, but become dramatic for larger α values.