Estimation of the average correlation coefficient for stratified bivariate data.

If the relationship between two ordered categorical variables X and Y is influenced by a third categorical variable with K levels, the Cochran-Mantel-Haenszel (CMH) correlation statistic QC is a useful stratum-adjusted summary statistic for testing the null hypothesis of no association between X and Y. Although motivated by and developed for the case of K I x J contingency tables, the correlation statistic QC is also applicable when X and Y are continuous variables. In this paper we derive a corresponding estimator of the average correlation coefficient for K I x J tables. We also study two estimates of the variance of the average correlation coefficient. The first is a restricted variance based on the variances of the observed cell frequencies under the null hypothesis of no association. The second is an unrestricted variance based on an asymptotic variance derived by Brown and Benedetti. The estimator of the average correlation coefficient works well in tables with balanced and unbalanced margins, for equal and unequal stratum-specific sample sizes, when correlation coefficients are constant over strata, and when correlation coefficients vary across strata. When the correlation coefficients are zero, close to zero, or the cell frequencies are small, the confidence intervals based on the restricted variance are preferred. For larger correlations and larger cell frequencies, the unrestricted confidence intervals give superior performance. We also apply the CMH statistic and proposed estimators to continuous non-normal data sampled from bivariate gamma distributions. We compare our methods to statistics for data sampled from normal distributions. The size and power of the CMH and normal theory statistics are comparable. When the stratum-specific sample sizes are small and the distributions are skewed, the proposed estimator is superior to the normal theory estimator. When the correlation coefficient is zero or close to zero, the restricted confidence intervals provide the best performance. None of the confidence intervals studied provides acceptable performances across all correlation coefficients, sample sizes and non-normal distributions.