CONDITIONAL COVARIANCE THEORY AND DETECT FOR POLYTOMOUS ITEMS

Abstract

This paper extends the theory of conditional covariances to polytomous items. It has been mathematically proven that under some mild conditions, commonly assumed in the analysis of response data, the conditional covariance of two items, dichotomously or polytomously scored, is positive if the two items are dimensionally homogeneous and negative otherwise. The theory provides a theoretical foundation for dimensionality assessment procedures based on conditional covariances or correlations, such as DETECT and DIMTEST, so that the performance of these procedures is theoretically justified when applied to response data with polytomous items. Various estimators of conditional covariances are constructed, and special attention is paid to the case of complex sampling data, such as NAEP data. As such, the new version of DETECT can be applied to response data sets not only with polytomous items but also with missing values, either by design or by random. DETECT is then applied to analyze the dimensional structure of the 2002 NAEP reading samples of grades 4 and 8. The DETECT results show that the substantive test structure based on the purposes for reading is consistent with the statistical dimensional structure for either grade. The results also indicate that the degree of multidimensionality of the NAEP reading data is weak.