A Quadratic Curve Equating Method to Equate the First Three Moments in Equipercentile Equating

ACT Research Report Series94-2A Quadratic Curve Equating Method to Equate the First Three Moments in Equipercentile EquatingTian-You Wang Michael J. KolenJune 1994 For additional copies write: ACT Research Report Series P.O. Box 168 Iowa City, Iowa 52243©1994 by The American College Testing Program. All rights reserved. A Quadratic Curve Equating Method to Equate the First Three Moments in Equipercentile EquatingTianyou Wang Michael J. Kolen I (Abstract)In this paper, a quadratic curve equating method for equating different test forms under a random group data collection design is proposed. Procedures for implementing this method and related issues are described and discussed. The quadratic curve method was evaluated using real test data and simulated data in terms of model fit and equating error, and was compared to several other equating methods. It was found that the quadratic curve method fit many of the real test data examined and that when model fits the population, this method could perform better than other more sophisticated equating methods. Index terms: Equipercentile equating, smoothing procedures, quadratic curve equating, linear equating, random groups equating design.A Quadratic Curve Equating Method to Equatethe First Three Moments in Equipercentile Equating In standardized testing, often multiple test forms are needed because examinees need to take the test at different occasions and one test form can be administered only once to ensure test security. In this situation, it is typically required that test scores derived from different forms are equivalent. Efforts can be made in the test construction process to make different forms as nearly equivalent as possible (e.g., forms can be built based on the same table of specifications; items can be selected to have approximately equal average difficulty level). But often these efforts are not enough to ensure test score equivalency for different forms. So, test equating based on test data is often performed to adjust test scores so that scores on different forms are more nearly equivalent. There are several designs for collecting test equating data. One of the designs is the random groups design, in which different test forms are administered to different but randomly equivalent groups of examinees.Under the random groups equating design, the examinee groups that take different test forms (for simplicity, say, form X and form Y) are regarded as being sampled from the same population. The differences in score distributions for different test forms are attributed to form differences and sampling variations of the examinee groups. Equating form X to form Y involves transforming the X scores so that the transformed X scores have the same distribution as the Y scores. If an assumption can be made that the population distributions for X and Y scores have the same shape and only differ in mean and variance, then the linear equating method will be most appropriate. Linear equating takes the formA Quadratic Curve Equating Method to Equatethe First Three Moments in Equipercentile Equating 2where x is the score on form X, fix and jiY are means for form X and form Y, <JX and oY are standard deviations (s.d.) for form X and form Y, and lY(x) is the equated form Y score for x.If no assumptions can be made about the shape of the population score distributions, equipercentile equating method is the method of choice. Equipercentile equating for a discrete score distribution is given bywhere Pr means probability, p* (x) = Pr(X < x)+.5P r(X = *) , and u*(;c) is the smallest integer such that p*(x) = Pr[Y < «*(*)].Equipercentile equating based on samples may have large sampling error because for any particular score, the equating relationship is based on local frequencies at that score point. Two types of smoothing techniques have been introduced to reduce random errors: pre-smoothing and post-smoothing. Pre-smoothing smoothes the score distributions for form X and form Y separately and equates the smoothed score distributions. Post-smoothing (Kolen, 1984) smoothes the equipercentile equating function directly.Studies have been done to evaluate these methods (see Kolen, 1984, Fairbank, 1987, Cope & Kolen, 1990, Hanson, 1990, Hanson, Zeng, & Colton, 1991). Results from Hanson, Zeng, and Colton (1991) showed that smoothed equating was more accurate than unsmoothed equipercentile and linear methods in terms of mean squared errors. However, linear equating consistently had smaller random error, especially when sample sizes were small. This finding resulted because the linear method uses only means and standard deviations in computing the equating equation and these aggregate statistics typically have small sample variability. However, a fundamental limitation of linear methods is that if the shape of the distribution of X scores is different from that of Y scores in the popul