Components of Pearson's statistic for at least partially ordered m-way contingency tables

For at least partially ordered three-way tables, it is well known how to arithmetically decompose Pearson's X²p 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 X²p 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 X²p into components. The asymptotic distributions of statistics related to the components are also addressed.