A Hybrid Model for Orthogonal Regression

Abstract

Linear functional relationships are intended to be symmetric and therefore cannot generally be accurately estimated using ordinary least squares regression equations. Orthogonal regression (OR) models allow for errors in both Y and X and therefore can provide symmetric estimates of these relationships. The most well-established OR model, the errors-in-variables (EIV) model, assumes that the observed scatter around the line is due entirely to errors of measurement in Y and X and that the ratio of the error variances is known. If most of the variance around the line is known to be due to the errors of measurement in Y and X, the EIV model can provide an unbiased maximum likelihood estimate for a functional relationship. However, if a substantial part of the variability around the line is due to natural variability, which is not attributable to errors of measurement in Y or X, the ratio of the measurement error variances is not well defined and the EIV model is not directly applicable. The main contribution of this report is the development of a hybrid model that provides plausible estimates for linear functional relationships in cases with substantial natural variability and substantial errors of measurement. An analysis of female and male differential test functioning between an essay test and an objective test used as parts of a licensure examination provides an illustration of the use of the hybrid model.