The Type II Error of Measures for the Analysis of 2 � 2 Tables

Abstract

The results of a simulation study on the performance of measures for the analysis of 2 � 2 tables are reported. The simulations included 11 measures for 2 � 2 tables: (a) Pearson's α; (b) the standard normal z; (c) the log-odds ratio; (d) the log-linear interaction; (e) Goodman's (1991) weighted log-linear interaction; (f) Vogel's z; (g) the binomial test; (h) Lehmacher's (1981) asymptotic hypergeometric test; (i) Perli, Hommel, and Lehmacher's (1985) asymptotic test; (j) Lindner's (1984) exact hypergeometric test; and (k) Lautsch and von Weber's (2003) adaptation of Dunkl and von Eye's (1990) test. The factors varied in the simulations were (a) type of sampling distribution, (b) sample size, (c) strength of association in the 2 � 2 table, (d) symmetry of the distribution in the 2 � 2 table, and (e) the nominal α. Results suggest that the distribution of these 11 tests is very near the normal under all conditions. Of the 5 factors, only the type of the sampling distribution has no strong effects on the Type II error curves of the 11 measures. Finally, it was found that the 11 measures respond differently to the 5 factors such that the rank order of performance varies with the simulated conditions. In the discussion, issues concerning the binomial effect size and application implications of the simulation results are addressed.