Bias of Two-Level Scalability Coefficients and Their Standard Errors

Two-level Mokken scale analysis is a generalization of Mokken scale analysis for multi-rater data. The bias of estimated scalability coefficients for two-level Mokken scale analysis, the bias of their estimated standard errors, and the coverage of the confidence intervals has been investigated, under various testing conditions. It was found that the estimated scalability coefficients were unbiased in all tested conditions. For estimating standard errors, the delta method and the cluster bootstrap were compared. The cluster bootstrap structurally underestimated the standard errors of the scalability coefficients, with low coverage values. Except for unequal numbers of raters across subjects and small sets of items, the delta method standard error estimates had negligible bias and good coverage. Post hoc simulations showed that the cluster bootstrap does not correctly reproduce the sampling distribution of the scalability coefficients, and an adapted procedure was suggested. In addition, the delta method standard errors can be slightly improved if the harmonic mean is used for unequal numbers of raters per subject rather than the arithmetic mean.