Diagnosis on the Performance of the Delta Method for Estimating Standard Errors of IRT Equating Under the Common-Item Design

Abstract

The purpose of the present study is to use computer simulations to examine the performance of the delta method formulas for estimating standard errors (SEs) of item response theory (IRT) true score and observed score equating that have been presented under the common-item design. Test type (3PL model test, 3PL+GPC model test, and 3PL+GR model test), sample size (1,000 and 4,000), ratio of common items to the total items (20% and 40%), and method for scale linking [mean/mean (MM) and test response function (TRF) methods] were considered as simulation factors. Main results were as follows. First, for the three types of tests, the theoretical SEs of IRT equating estimated by the delta method were slightly larger than the empirical SEs computed using the simulated equated scores, but as the sample size increased, the average differences between the theoretical and empirical SEs decreased to less than 0.01. Second, for the three types of tests, the SEs of IRT equating tended to be inversely proportional to the square root of the sample size. Third, regardless of the test type and sample size, the SEs of IRT equating based on the MM method were greater than those based on the TRF method. Fourth, the SEs of IRT equating for the 3PL model test (30-item test) were, on the average, smaller than those for the mixed-format tests (40-item tests).