On completing a measurement model by symmetry

TL;DR

This paper advocates for incorporating symmetry principles into measurement models, linking classical regression, factor analysis, and errors-in-variables frameworks with a focus on both linear and nonlinear associations.

Contribution

It introduces a symmetry-based approach to enhance measurement models, providing a unified interpretation of correlation and error components in regression and factor analysis.

Findings

Symmetry principles can unify measurement model components.

A novel interpretation of correlation includes nonlinear associations.

Connections between regression, factor analysis, and errors-in-variables are established.

Abstract

An appeal for symmetry is made to build established notions of specific representation and specific nonlinearity of measurement (often called model error) into a canonical linear regression model. Additive components are derived from the trivially complete model M = m. Factor analysis and equation error motivate corresponding notions of representation and nonlinearity in an errors-in-variables framework, with a novel interpretation of terms. It is suggested that a modern interpretation of correlation involves both linear and nonlinear association.