A constrained regression model for an ordinal response with ordinal predictors

TL;DR

This paper introduces a new regression model for ordinal response variables with ordinal predictors, using constrained maximum likelihood estimation to enforce monotonic relationships and improve interpretability.

Contribution

It proposes a novel constrained regression approach for ordinal predictors, including a decision rule for monotonicity and a monotonicity test, with methods validated through simulations and real data.

Findings

Constrained model improves parameter estimation accuracy.

Decision rule effectively identifies monotonicity directions.

Model performs well in real-world Likert scale data analysis.

Abstract

A regression model is proposed for the analysis of an ordinal response variable depending on a set of multiple covariates containing ordinal and potentially other variables. The proportional odds model (McCullagh (1980)) is used for the ordinal response, and constrained maximum likelihood estimation is used to account for the ordinality of covariates. Ordinal predictors are coded by dummy variables. The parameters associated to the categories of the ordinal predictor(s) are constrained, enforcing them to be monotonic (isotonic or antitonic). A decision rule is introduced for classifying the ordinal predictors' monotonicity directions, also providing information whether observations are compatible with both or no monotonicity direction. In addition, a monotonicity test for the parameters of any ordinal predictor is proposed. The monotonicity constrained model is proposed together with three estimation methods and compared to the unconstrained one based on simulations. The model is applied to real data explaining a 10-Points Likert scale quality of life self-assessment variable from ordinal and other predictors.