A Comparison of Bivariate Smoothing Methods in Common-Item Equipercentile Equating

Abstract

The effectiveness of smoothing the bivariate distributions of common and noncommon item scores in the frequency estimation method of common-item equipercentile equating was exam ined. The mean squared error of equating was computed for several equating methods and sample sizes, for two sets of population bivariate distribu tions of equating and nonequating item scores defined using data from a professional licensure exam. Eight equating methods were compared: five equipercentile methods and three linear methods. One of the equipercentile methods was unsmoothed equipercentile equating. Four methods of smoothed equipercentile (SEP) equating were con sidered : two based on log-linear models, one based on the four-parameter beta binomial model, and one based on the four-parameter beta compound binomial model. The three linear equating methods were the Tucker method, the Levine Equally Reliable method, and the Levine Unequally Reliable method. The results indicated that smoothed distributions produced more accurate equating functions than the unsmoothed distributions, even for the largest sample size. Tucker linear equating produced more accurate results than SEP equating when the systematic error introduced by assuming a linear equating function was small relative to the random error of the methods of SEP equating.