Testing for homogeneity. I. The binomial and multinomial distributions.
Testing for homogeneity. I. The binomial and multinomial distributions.
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测试同质性。
DOI：
10.1093/biomet/53.1-2.167
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发表时间：
1966
期刊：
影响因子：
2.7
通讯作者：
Maurice Whittinghill
中科院分区：
文献类型：
--
作者：
Richard F. Potthoff;Maurice Whittinghill
摘要如果我们有不同大小的k个二项式样本，我们有时可能会对齐性问题感兴趣，即我们可能想知道k个样本是否都来自具有相同参数p的二项分布。如果我们有k个样本来自多项分布或泊松分布，可能会出现类似的齐性问题。这是两篇系列文章中的第一篇，讨论的是二项式和多项式的情况；第二篇文章将讨论泊松情况。对于刚才提到的问题，同质性测试已经存在，但这些现有的测试显然没有明确考虑到任何最佳的功率特性。这些论文通过尝试构造对某些合理的替代假设具有最大能力的测试来探讨同质性测试的问题。这种方法产生了一些新的测试；这些测试将被描述，并将提供数值说明。还将讨论传统测试(即通常的x2测试)。对于二项式问题，所考虑的所有检验都适用于k个样本量中的一些或全部是小数字(甚至小到2或3)的情况(经常出现在遗传学中)。本文的第一部分涉及二项式情形的齐性检验；提出了一些生物学应用。在？3中，我们简要地展示了如何将在？1中引入的新检验推广到多项式的情形。所有更多的技术细节都归入了数学附录，这是论文的最后一部分。
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.