Quantifying the Bias of Non-linear Equating and Score Transformations

Abstract

This paper shows that using non-linear functions for equating and score transformations leads to consequences that are not commensurable with classical test theory (CTT). More specifically, a well-known theorem from calculus shows that the expected value of a non-linearly transformed variable does not equal the transformed expected value of this variable. Translated to CTT this implies that the transformed observed test score does not have an unbiased expectation, i.e., is different from the transformed true score. In order to quantify the bias, second-order Taylor expansions are used in this work to show that non-linear equating and scale transformations do not only lead to variability of SEMs but also to predictable bias in the expected values of the transformed observed scores. In line with Lord’s finding that is often described as “Equating is either unnecessary or impossible,” this bias due to non-linear equating vanishes either for perfectly reliable tests, or if the equating function is indeed linear, i.e., the tests are congeneric.