Testing for homogeneity. I. The binomial and multinomial distributions.

SUMMARY If we have k binomial samples of different sizes, we may sometimes be interested in the question of homogeneity, i.e. we may want to know whether the k samples all came from binomial distributions with the same parameter p. A similar question of homogeneity may arise if we have k samples from multinomial or Poisson distributions. This paper, which is the first of a series of two, treats the binomial and multinomial situations; the second paper will treat the Poisson case. Homogeneity tests already exist for the problems just mentioned, but these existing tests apparently were not constructed with any optimal power properties explicitly in mind. These papers approach the problem of homogeneity testing by attempting to construct tests having maximal power against certain reasonable alternative hypotheses. Some new tests result from this approach; these tests will be described and numerical illustrations will be presented. The traditional tests (i.e. the usual x2 tests) will also be discussed. For the binomial problem, all tests which are considered are applicable to the situation (frequently arising in genetics) in which some or all of the k sample sizes are small numbers (even as small as 2 or 3). Section 1 of this paper is concerned with testing for homogeneity for the binomial case; a number of biological applications are presented. In ? 3, we show briefly how the new test which is introduced in ? 1 may be generalized to the multinomial case. All of the more technical details have been relegated to the Mathematical Appendices, which form the last part of the paper.

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