This module describes and extends X-to-Y regression measures that have been proposed for use in the assessment of X-to-Y scaling and equating results. Measures are developed that are similar to those based on prediction error in regression analyses but that are directly suited to interests in scaling and equating evaluations. The regression and scaling function measures are compared in terms of their uncertainty reductions, error variances, and the contribution of true score and measurement error variances to the total error variances. The measures are also demonstrated as applied to an assessment of scaling results for a math test and a reading test. The results of these analyses illustrate the similarity of the regression and scaling measures for scaling situations when the tests have a correlation of at least .80, and also show the extent to which the measures can be adequate summaries of nonlinear regression and nonlinear scaling functions, and of heteroscedastic errors. After reading this module, readers will have a comprehensive understanding of the purposes, uses, and differences of regression and scaling functions.

Keywords: concordance, equating, heteroscedastic, regression, prediction error, scaling, scaling error, X-to-Y, Y-toX