A Comparison of Presmoothing and Postsmoothing Methods in Equipercentile Equating. ACT Research Report Series 94-4.

This paper compares various methods of smoothed equipercentile equating and linear equating in the random groups equating design. Three presmoothing methods (based on the beta binomial model, four-parameter beta binomial model and a log-linear model) are compared to postsmoothing using cubic splines, linear equating and unsmoothed equipercentile equating. Performance of these methods is evaluated by specifying several pairs of population distributions and estimating equating error by Monte Carlo methods. The results indicate that both presmoothing and postsmoothing methods can improve estimation of the equipercentile equating function and that presmoothing and postsmoothing methods provide comparable levels of performance in terms of equating error. A Comparison of Presmoothing and Postsmoothing Methods in Equipercentile Equating Two types of smoothing methods have been suggested for improving equipercentile equat­ ing results in the random groups equating design. Presmoothing methods involve smoothing the score distributions for the two test forms to be equated before equipercentile equating is performed. Postsmoothing methods involve smoothing the equipercentile equating function produced from the unsmoothed score distributions. Both presmoothing and postsmoothing methods have been found to reduce equating error as compared with unsmoothed equipercentile equating. Kolen (1984) concluded that a postsmoothing method based on cubic splines is preferred to unsmoothed equiper­ centile equating. Fairbank (1987) studied seven presmoothing methods and seven postsmoothing methods and concluded that of the methods he investigated a presmoothing method based on the negative hypergeometric distribution and a postsmoothing method based on cubic splines were the preferred presmoothing and postsmoothing methods. Cope & Kolen (1990) and Hanson (1990) investigated various methods of smoothing distri­ butions of test scores (these methods could be used for presmoothing in equipercentile equating). Cope & Kolen (1990) found that smoothing based on the four-parameter beta binomial model (Lord, 1965) provided more accurate results than smoothing based on the beta binomial (negative hyper­ geometric) model. Hanson (1990) found the four-parameter beta binomial model and a log-linear model (Holland & Thayer, 1987) provided the most accurate results of the methods examined. The results presented in the papers of Cope & Kolen (1990) and Hanson (1990) suggest that the four-parameter beta binomial model and the log-linear model, when used in presmoothing o f score distributions, may produce less equating error than presmoothing based on the beta binomial model which produced the most accurate results in Fairbank (1987). This paper investigates the relative performance of unsmoothed equipercentile equating, sev­ eral presmoothing methods (using the two and four parameter beta binomial models, and a log-linear model), a postsmoothing method based on cubic splines, and linear equating (Angoff, 1971). The relative performance o f these methods is investigated using several example data sets. Equipercentile Equating The focus o f this paper is on the random groups equating design. In the random groups equating design the new and old forms are each administered to a random sample from a common population. Let the random variables X and Y represent the test scores on the new and old forms