Statistica Sinica 13(2003), 555-570

SIZE AND POWER CONSIDERATIONS FOR TESTING

LOGLINEAR MODELS USING $\phi$-DIVERGENCE TEST

STATISTICS

Noel Cressie$^1$, Leandro Pardo$^2$ and Maria del Carmen Pardo$^2$

$^1$The Ohio State University and $^2$Complutense University of Madrid

Abstract: In this article, we assume that categorical data are distributed according to a multinomial distribution whose probabilities follow a loglinear model. The inference problem we consider is that of hypothesis testing in a loglinear-model setting. The null hypothesis is a composite hypothesis nested within the alternative. Test statistics are chosen from the general class of $\phi$-divergence statistics. This article collects together the operating characteristics of the hypothesis test based on both asymptotic (using large-sample theory) and finite-sample (using a designed simulation study) results. Members of the class of power divergence statistics are compared, and it is found that the Cressie-Read statistic offers an attractive alternative to the Pearson-based and the likelihood ratio-based test statistics, in terms of both exact and asymptotic size and power.

Key words and phrases: Chi-squared distribution, contiguous alternatives, multinomial distribution, nested hypotheses, noncentral chi-squared distribution, power-divergence statistic.