Cross-Validation

The development of the kernel equating (KE) method enhanced the theory of observed-score equating. In KE, discrete test score distributions are converted into continuous distributions through the use of a Gaussian kernel. Traditionally, the optimal bandwidth for a Gaussian kernel was obtained by minimizing a penalty function. In this article, an alternative bandwidth-selection approach for KE was adopted that uses cross-validation (CV) techniques. The method is illustrated through simulations that were conducted with 188 conditions by varying three factors known to influence equating results; these include sample sizes, score distributions, and methods that involve both equating and bandwidth-selection methods. Four equating procedures were considered: traditional equipercentile equating, which uses linear interpolation to make the test distributions continuous; KE with penalty functions; and KE with two newly proposed CV methods. The results were evaluated based on four criteria: bias in continuizing the distributions (i.e., the difference between the estimated and underlying score distributions), the standard error of equating (SEE), the difference between equated scores, and percent relative error (PRE). Overall, the results demonstrate that KE with the two CV methods outperformed the others—the estimated density functions were less biased and the SEEs and PREs were smaller. Equating differences between the different methods were produced, although they were not large. In addition, the bias issues surrounding kernel methods on sample sizes and the shapes of the distributions were addressed and discussed.