An investigation of the accuracy of the estimates of standard errors for the kernel equating functions

This simulation study investigated the accuracy of the estimates of standard errors of kernel equating (SEE) for various sample sizes in the context of the random groups design. Empirical score distributions of four sets of test forms from large-scale standardized achievement testing programs were used as a basis to define the population distributions. Factors studied included sample size, degree of smoothing, choice of bandwidth, and characteristics of the score distributions. Situations in which there was no systematic error introduced by pre-smoothing and in which there was such systematic error were both studied. The criteria were the standard deviations of the equated scores at each score level over the replications, also referred to as empirical standard errors.;The results of the study show that empirical standard errors of kernel equating had regular patterns, and they decreased as sample size increased at each score point. The magnitude of standard errors tended to be larger for tests smoothed with polynomial log linear models of higher degrees or when higher degrees of smoothing were used for the sample. The two small bandwidths produced very similar results in terms of magnitude and pattern of standard errors, as did the two large bandwidths. The characteristics of the score distributions did not influence the above conclusions.;For similar conditions, the accuracy of the estimates of SEE increased as sample size increased, and they tended to be more accurate at score points with higher density than those with lower density. When there was no systematic error introduced by pre smoothing, estimates of SEE were reasonably accurate for sample sizes of 300 or more if the bandwidth was around 0.6 or larger. Using lower degrees of smoothing than for the population distributions resulted in relatively large bias of estimates of SEE for score points about one standard deviation away from the mean. Patterns of bias, standard error and RMSE of SEE were similar for the two small bandwidths, and also for the two large bandwidths. Large bandwidths tended to produce more accurate estimates of SEE than small bandwidths. A bandwidth of 0.33 tended to produce unstable estimates of SEE. These conclusions were the same for tests with different characteristics of score distributions.