An Overall Test of Pairwise Mean Conditional Covariances in IRT

We study how the Conditioning on Added Regression Predictions (CARP) statistics from different item pairs can be aggregated into a single overall test of monotone homogeneity. As a pairwise statistic, we use the mean conditional covariance (MCC) or its standardized value (). We use three different estimates of the covariance matrix of the pairwise test statistics: (1) the covariance matrix of the MCCs, based on the sample moments; (2) the covariance matrix of the MCCs or s, based on bootstrapping; and (3) the covariance matrix of the s, equated to the identity matrix. We consider various aggregation methods, including (a) the chi-bar-square statistic; (b) the preselected standardized partial sum of pairwise statistics; (c) the product of preselected -values; (d) the minimum of preselected -values; and (e–h) the same statistics, but now conditioned on post-selecting only the negative values in the test sample. We study the Type 1 error rate and power of the ensuing 20 tests based on simulations. The tests with the highest power among the tests that control the Type I error rate are based on -statistics with the identity matrix: the conditional likelihood ratio test, the conditionalized product of -values, the conditionalized sum of Z-values, and the preselected product of -values.