Abstract
For at least partially ordered three-way tables, it is well known how to arithmetically decompose Pearson's statistic into informative components that enable a close scrutiny of the data. Similarly well-known are smooth models for two-way tables from which score tests for homogeneity and independence can be derived. From these models, both the components of Pearson's and information about their distributions can be derived. Two advantages of specifying models are first that the score tests have weak optimality properties and second that identifying the appropriate model from within a class of possible models gives insights about the data. Here, smooth models for higher-order tables are given explicitly, as are the partitions of Pearson's into components. The asymptotic distributions of statistics related to the components are also addressed.
Highlights
In 1, 2 it is shown how, for at least partially ordered three-way tables, to arithmetically decompose Pearson’s XP2 statistic into informative components that enable a close scrutiny of the data
The asymptotic distributions of statistics related to the components are addressed
We discuss the arithmetic decomposition of XP2 into components, giving explicit formulae for these components
Summary
In 1, 2 it is shown how, for at least partially ordered three-way tables, to arithmetically decompose Pearson’s XP2 statistic into informative components that enable a close scrutiny of the data. We observe here that the arithmetic decomposition of Pearson’s XP2 statistic for two- and three-way tables can be shown quite compactly using results from 3, Chapter 4, Theorems 2.1 and 2.2, pages 90-91 and Theorem 5.2, page 101. In treating a singly ordered table, the work in 4, Chapter 4 assumed the total count for each treatment is known before sighting the data, and this leads to a smooth product multinomial model. It is clear that, in general, for partially ordered tables, there are a multitude of possible models, depending on which marginal totals are assumed known before sighting the data.
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