Marginal True-Score Measures and Reliability for Binary Items as a Function of Their IRT Parameters

Abstract

This article provides analytic evaluations of population true-score measures for binary items given their item response theory (IRT) calibration. Under the assumption of normal trait distribution, the expected values of marginalized true scores, error variance, true-score variance, and reliability for norm-referenced and criterion-referenced interpretations are presented as a function of the item parameters. The proposed formulas have methodological and computational value in bridging concepts of IRT and true-score theory. They provide information about the individual contribution of IRT calibrated items to marginal true-score measures and may have valuable applications in test development and analysis. For example, given a bank of IRT calibrated items, one can select binary items to develop a test with known true-score characteristics prior to administering the test (without information about raw scores or trait scores). Calculations with the proposed formulas are easy to perform using basic statistical programs, spreadsheet programs, or even handheld calculators.