Structural properties of UMPU-tests for 2x2 tables and some applications

The paper is concerned with structural properties of the acceptance regions of uniformly most Powerful unbiased tests (UMPU-tests) for one- and two-sided hypotheses for 2 x 2 tables as, for instance. the comparison of two proportions or testing for association. These tests can be considered as randomized versions of Fisher's exact tests, A series of monotonicity and unimodality properties will be proved. These properties are equivalent to a symmetry and convexity condition often required for powerful unconditional tests, Knowledge of such properties allow's a fast and in some sense recursive calculation of the critical values of the UMPU-tests which is important if a repeated calculation of all critical values for different sample sizes or different levels is required. This is, for example, the case if the unconditional power has to be controlled over a certain subset of the alternative, or, if one is interested in powerful unconditional non-randomized tests generated by a UMPU-test. Our results also imply some useful properties of the two-dimensional unconditional power function. On the other hand. we found some less nice properties of the UMPU-tests, too. (C) 2002 Elsevier Science B.V. All rights reserved.