Accuracy of Power-Divergence Statistics for Testing Independence and Homogeneity in Two-Way Contingency Tables

Abstract

The small-sample accuracy of seven members of the family of power-divergence statistics for testing independence or homogeneity in contingency tables was studied via simulation. The likelihood ratio statistic G ² and Pearson's X ² statistic are among these seven members, whose behavior was studied at nominal test sizes of.01 and.05 with marginal distributions that could be uniform or skewed and with a set of sample sizes that included sparseness conditions as measured through table density (i.e., the ratio of sample size to number of cells). The likelihood ratio statistic G ² rejected the null hypothesis too often even with large table density, whereas Pearson's X ² was sufficiently accurate and only presented a minor misbehavior when table density was less than two observations/cell. None of the other five statistics outperformed Pearson's X ². A nonasymptotic variant of X ² solved the minor inaccuracies of Pearson's X ² and turned out to be the most accurate statistic for testing independence or homogeneity, even with table densities of one observation/cell. These results clearly advise against the use of the likelihood ratio statistic G ².

Keywords:

• Contingency tables
• Homogeneity
• Independence
• Power-divergence statistics
• Small-sample accuracy

Mathematics Subject Classification:

• 62F03

Acknowledgments

This research was supported by grants SEJ2005-00485 (Ministerio de Educación y Ciencia), MTM2004-00341 (Ministerio de Educación y Ciencia and FEDER), MTM2007-60112 (Ministerio de Educación y Ciencia and FEDER), and IT-334-07 (Departamento de Educación del Gobierno Vasco - UPV/EHU Econometrics Research Group).