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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Modelling Interval Data with Random Intercepts: A Beta Regression Approach for Clustered and Longitudinal Structures

by
Olga Usuga-Manco
1,*,
Freddy Hernández-Barajas
2 and
Viviana Giampaoli
3
1
Departamento de Ingeniería Industrial, Universidad de Antioquia, Medellín 050010, Colombia
2
Departamento de Estadística, Universidad Nacional de Colombia sede Medellín, Medellín 050034, Colombia
3
Departamento de Estatística, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo 05508-090, Brazil
*
Author to whom correspondence should be addressed.
Modelling 2025, 6(4), 128; https://doi.org/10.3390/modelling6040128
Submission received: 30 August 2025 / Revised: 7 October 2025 / Accepted: 10 October 2025 / Published: 14 October 2025
Versions Notes

Abstract

Beta regression models are a class of models used frequently to model response variables in the interval (0,1). Although there are articles in which these models are used to model clustered and longitudinal data, the prediction of random effects is limited, and residual analysis has not been implemented. In this paper, a random intercept beta regression model is proposed for the complete analysis of this type of data structure. We proposed some types of residuals and formulate a methodology to obtain the best prediction of random effects. This model is developed through the parameterisation of beta distribution in terms of the mean and dispersion parameters. A log-likelihood function is approximated by the Gauss–Hermite quadrature to numerically integrate the distribution of random intercepts. A simulation study is used to investigate the performance of the estimation process and the sampling distributions of the residuals.
Keywords: Gauss–Hermite quadrature; maximum likelihood estimation; beta distribution; beta regression Gauss–Hermite quadrature; maximum likelihood estimation; beta distribution; beta regression

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MDPI and ACS Style

Usuga-Manco, O.; Hernández-Barajas, F.; Giampaoli, V. Modelling Interval Data with Random Intercepts: A Beta Regression Approach for Clustered and Longitudinal Structures. Modelling 2025, 6, 128. https://doi.org/10.3390/modelling6040128

AMA Style

Usuga-Manco O, Hernández-Barajas F, Giampaoli V. Modelling Interval Data with Random Intercepts: A Beta Regression Approach for Clustered and Longitudinal Structures. Modelling. 2025; 6(4):128. https://doi.org/10.3390/modelling6040128

Chicago/Turabian Style

Usuga-Manco, Olga, Freddy Hernández-Barajas, and Viviana Giampaoli. 2025. "Modelling Interval Data with Random Intercepts: A Beta Regression Approach for Clustered and Longitudinal Structures" Modelling 6, no. 4: 128. https://doi.org/10.3390/modelling6040128

APA Style

Usuga-Manco, O., Hernández-Barajas, F., & Giampaoli, V. (2025). Modelling Interval Data with Random Intercepts: A Beta Regression Approach for Clustered and Longitudinal Structures. Modelling, 6(4), 128. https://doi.org/10.3390/modelling6040128

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