Small-Sample Comparisons of Exact Levels for Chi-Squared Goodness-of-Fit Statistics

Abstract The small-sample properties of three goodness-of-fit statistics for the analysis of categorical data are examined with respect to the adequacy of the asymptotic chi-squared approximation. The approximate tests based on the likelihood ratio and Freeman-Tukey statistics yield exact levels that are typically in excess of the nominal levels for moderate expected values. In contrast, the Pearson statistic attains exact levels that are quite close to the nominal values. The reason for the large number of rejections for the likelihood ratio and Freeman-Tukey statistics is related to their handling of small observed counts.