Abstract
This article compares 6 measures for the analysis of 2 ×2 tables: Pearson's X2, the correlation ρ, the standard normal z, the log-odds ratio θ, the log-linear interaction λ, and Goodman's weighted log-linear interaction λ. The comparison focuses on (a) the degree to which these measures maintain their nominal α level, (b) the nearness of the measures to their respective sampling distributions, (c) the degree to which these measures are influenced by discrepancies in marginals (i.e., marginal-dependency), and (d) the ability of the measures to depict the orientation of an association. Simulation results suggest that, for samples of size up to 48, (a) all measures tend to suggest conservative decisions and (b) the two measures z and X2 are closer than the other measures to α =. 05. For larger samples, the two λ measures are closer to. 05 whereas the z and X2 are biased toward nonconservative decisions. θ was consistently between these 2 groups (ρ was excluded from these analyses because it is a derivative of X2). The distributions of the measures X2, z, λ, and λ are near their respective sampling distributions. In contrast, the statistic θ, which is used to test the null hypothesis that the log-odds ratio is 0, is also symmetric but heavy-tailed. The log-odds ratio and the log-linear interaction are marginal-free; the other 4 measures are marginal-dependent. Only the always positive X2 is unable to depict the orientation of an association. It is shown that marginal-dependency becomes important when associations are strong in tables with discrepant marginals. For such tables, the correlations among the marginal-dependent measures and the marginal-free θ and λ can be very low, even negative. For tables with minor discrepancies between marginals, these correlations can be. 9 or higher. When the ability of depicting the orientation of an association is taken into account, the correlations between X2 and all other measures, including the other marginal-dependent ones, are about 0. Application and selection of measures are discussed.
