Conditional Covariance Theory and Detect for Polytomous Items

This paper extends the theory of conditional covariances to polytomous items. It has been proven that under some mild conditions, commonly assumed in the analysis of response data, the conditional covariance of two items, dichotomously or polytomously scored, given an appropriately chosen composite is positive if, and only if, the two items measure similar constructs besides the composite. 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 those from the National Assessment of Educational Progress (NAEP). 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 at 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. This research was supported by the Educational Testing Service and the National Assessment of Educational Progress (Grant R902F980001), US Department of Education. The opinions expressed herein are solely those of the author and do not necessarily represent those of the Educational Testing Service. The author would like to thank Ting Lu, Paul Holland, Shelby Haberman, and Feng Yu for their comments and suggestions. Requests for reprints should be sent to Jinming Zhang, Educational Testing Service, MS 02-T, Rosedale Road, Princeton, NJ 08541, USA. E-mail: jzhang@ets.org