Theoretical papers like Morris (1975) or Haberman (1977) suggest that the distribution of test-statistics for testing parametric models for the vector of cell probabilities in contingency tables depend also on the "disuniformity" of the expected frequencies in the table. In the context of testing independence in two-way contingency tables, this paper investigates the chi-square approximation to the conditional distribution of Pearson's X2 given the margins. Attention is focussed on on the effect of disuniformity as both the sample size and the number of cells of the table increase, for different levels of their ratio. This investigation is first carried out by analytically studying the ratio VR between the variance of the test-statistics X2 and that of the chi-square distribution of reference. The chi-square approximation is also studied by means of an extensive numerical work, guided by the results of the analytical study. The main outcome is that marginal disuniformity greatly affects the chi-square approximation, maily by affecting the X2 variance. Hence, the variance ratio VR may be used to assess the appropriateness of the traditional chi-square test; this criterion is in many circumstances less severe than the traditional ones.
The effect of marginal disuniformity on the chi-square approximation to the distribution of Pearson's X2 in sparse contingency tables.
CONTINI, Dalit;
1993-01-01
Abstract
Theoretical papers like Morris (1975) or Haberman (1977) suggest that the distribution of test-statistics for testing parametric models for the vector of cell probabilities in contingency tables depend also on the "disuniformity" of the expected frequencies in the table. In the context of testing independence in two-way contingency tables, this paper investigates the chi-square approximation to the conditional distribution of Pearson's X2 given the margins. Attention is focussed on on the effect of disuniformity as both the sample size and the number of cells of the table increase, for different levels of their ratio. This investigation is first carried out by analytically studying the ratio VR between the variance of the test-statistics X2 and that of the chi-square distribution of reference. The chi-square approximation is also studied by means of an extensive numerical work, guided by the results of the analytical study. The main outcome is that marginal disuniformity greatly affects the chi-square approximation, maily by affecting the X2 variance. Hence, the variance ratio VR may be used to assess the appropriateness of the traditional chi-square test; this criterion is in many circumstances less severe than the traditional ones.
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