Reversible Conversion Formulas Based on Partial Symmetric Linear Regression Models

Symmetric regression deals with a reversible functional relationship involving a set of variables, where all of them are measured with error and it is not meaningful to consider one as the response and the remaining ones as explanatory. Therefore, it is unsuitable to study any functional (linear) relationship between them by fixing one direction of the regression rather than the other. The scope of the symmetric regression can be expanded by considering a partial symmetric linear regression where the functional relationship is controlled for other variables, which are not assumed to be error-prone. Actually, the word partial in this context, means that we are not interested in a fully symmetric relationship between all the variables but in a symmetric and reversible relationship that holds for some variables of interest, whose functional relationship is of primary concern, for any given value of the control variables. Therefore, a partial symmetric regression modeling strategy is developed within a very general framework that includes different symmetric regression strategies. The finite sample behaviors of the proposed estimators are investigated through numerical studies and illustrated with an application to rheumatology data to find a reversible conversion formula between the Stanford Health Assessment Questionnaire (HAQ) score and the Multi-Dimensional HAQ (MDHAQ) score.