Chi-Squared-Type Tests for Ordered Alternatives in Contingency Tables

Abstract

It is well known that the classical Pearson's chi-squared test for contingency tables does not have good power against ordered alternatives. Such alternatives arise naturally when there is an ordering in the rows or columns of the table. This article considers a general class of tests called cumulative chi-squared—type (CCS-type) tests for ordered alternatives. Different members of this class correspond to the choice of different weighting schemes. The properties of the CCS-type statistics are examined, and their relationships to Cramér-von Mises goodness-of-fit tests, Neyman—Barton smooth tests, and asymptotically optimal score tests are discussed. The properties of T_E , Taguchi's (1966, 1974) “cumulative-sum” statistic obtained by assigning a weight to each term that is inversely proportional to its (conditional) expectation under the null hypothesis, and a simpler statistic, T_c , which assigns constant weights, are investigated in detail.

The main tool for studying the properties of the CCS-type tests is to decompose them into orthogonal components and express them as a weighted sum of these components. Pearson's statistic can be represented as the sum of the components of each CCS-type test. One can get a good insight into the properties of a CCS-type test by interpreting the behavior of the components to which it assigns most of its weight. For example, in a 2 × K table and for a special case, the first component of Taguchi's statistic T_E is in fact Wilcoxon's rank test applied to grouped data. Under a multinomial model it tests for location differences, and under a binomial model it tests for the linear component in logistic regression. Since T_E puts most of its weight to the first component, its behavior is similar to that of the two-sided Wilcoxon test, thus explaining the results in Takeuchi and Hirotsu (1982). The higher-order components also have interpretations. The second component corresponds to Mood's (1954) rank test applied to grouped data. Under a multinomial model it tests for scale differences, and under a binomial model it tests for the quadratic component in logistic regression. The components of the constant weight statistic T_c can also be interpreted similarly. In the binomial case, they are optimal for testing for cosinusoidal components in logistic regression. In the multinomial case, they have good power for testing for differences in location, scale, and so on. The behavior of T_c is very similar to its first component, to which it assigns most of its weight.

Key Words:

• Location—scale models
• Logistic regression
• Orthogonal decomposition
• Rank tests
• Smooth tests