  Citation: Fernández, D.; McMillan, L.; Arnold, R.; Spiess, M.; Liu, I. Goodness-of-Fit and Generalized Estimating Equation Methods for Ordinal Responses Based on the Stereotype Model. Stats 2022, 5, 507–520. https://doi.org/ 10.3390/stats5020030 Academic Editor: Wei Zhu Received: 31 March 2022 Accepted: 30 May 2022 Published: 1 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article Goodness-of-Fit and Generalized Estimating Equation Methods for Ordinal Responses Based on the Stereotype Model Daniel Fernández 1,2,3,† , Louise McMillan 4 , Richard Arnold 4 , Martin Spiess 5 and Ivy Liu 4, * 1 Serra Húnter Fellow, Department of Statistics and Operations Research (DEIO), Universitat Politècnica de Catalunya · BarcelonaTech (UPC), 08028 Barcelona, Spain; daniel.fernandez.martinez@upc.edu 2 Institute of Mathematics of UPC-BarcelonaTech (IMTech), 08028 Barcelona, Spain 3 Centro de Investigación Biomédica en Red de Salud Mental, CIBERSAM, Instituto de Salud Carlos III, Monforte de Lemos 3-5, Pabellón 11, 28029 Madrid, Spain 4 School of Mathematics and Statistics, Victoria University of Wellington, Cotton Building 356, Gate 7, Kelburn Parade, Wellington 6012, New Zealand; louise.mcmillan@vuw.ac.nz (L.M.); richard.arnold@vuw.ac.nz (R.A.) 5 Psychological Methods and Statistics, Hamburg University, 20146 Hamburg, Germany; martin.spiess@uni-hamburg.de * Correspondence: ivy.liu@vuw.ac.nz; Tel.: +64-44635648 † These authors contributed equally to this work. Abstract: Background: Data with ordinal categories occur in many diverse areas, but methodologies for modeling ordinal data lag severely behind equivalent methodologies for continuous data. There are advantages to using a model specifically developed for ordinal data, such as making fewer assumptions and having greater power for inference. Methods: The ordered stereotype model (OSM) is an ordinal regression model that is more flexible than the popular proportional odds ordinal model. The primary benefit of the OSM is that it uses numeric encoding of the ordinal response categories without assuming the categories are equally-spaced. Results: This article summarizes two recent advances in the OSM: (1) three novel tests to assess goodness-of-fit; (2) a new Generalized Estimating Equations approach to estimate the model for longitudinal studies. These methods use the new spacing of the ordinal categories indicated by the estimated score parameters of the OSM. Conclusions: The recent advances presented can be applied to several fields. We illustrate their use with the well-known arthritis clinical trial dataset. These advances fill a gap in methodologies available for ordinal responses and may be useful for practitioners in many applied fields. Keywords: goodness-of-fit; longitudinal data; ordinal data; stereotype model 1. Introduction 1.1. Ordinal Responses Many studies use data with ordinal categories (see e.g., [1–5]). For instance, in a questionnaire, Likert scale responses might be “strongly disagree”, “disagree”, “neutral”, “agree”, and “strongly agree” [6,7]. It may be easier for participants to provide rankings than absolute scores. In ecological studies, the ordinal Braun–Blanquet scale is used to collect species abundance data as it reduces sampling time compared with obtaining precise numerical estimates of abundance [3,8,9]. An ordinal variable indicates inherent order [10]. It differs from a nominal variable which has categories without any ordering information. Another defining distinction between a nominal and an ordinal variable is the effect of covariates on the outcome. As a covariate changes value in a particular direction, the distribution of the response consistently moves to higher categories, or consistently moves to lower categories, whereas for a nominal response the covariate may have different effects on different categories. Ordinal responses are often collected and coded as numbers—for example, in the Likert scale above, the levels of agreement might be coded 1, 2, 3, 4, or 5. However, Stats 2022, 5, 507–520. https://doi.org/10.3390/stats5020030 https://www.mdpi.com/journal/stats