Expanding Beta Regression: The Scale Location Truncated Beta Model and Its Applications

Abstract

Many scientific data are bounded between zero and one, such as proportions or probabilities, yet common regression methods assume normal errors and constant variance-assumptions that are unsuitable for such data. This dissertation extends the beta regression framework by introducing the Scale-Location-Truncated Beta (SLTB) model, which naturally accommodates both boundary values and heteroskedasticity. The SLTB model applies a scale–location transformation and truncation to the beta distribution, allowing finite probability at 0 and 1 while preserving the interpretability of standard beta regression. Implemented under both maximum likelihood and Bayesian frameworks, the model accommodates linear and nonlinear relationships while remaining computationally efficient. In addition, a Bayesian model selection approach is developed for mixed-effects SLTB models, using Laplace approximations to efficiently compute marginal likelihoods for variable selection. Simulation studies and real data analyses show that SLTB avoids invalid predictions, captures realistic variance patterns, and performs comparably to or better than Zero-One Inflated Beta (ZOIB) and XBX regression models, with fewer parameters and improved computational efficiency. The proposed model selection framework further demonstrates strong performance in identifying relevant predictors, particularly as sample size increases. Applications to behavioral delay-discounting data highlight the practical utility of the approach for analyzing bounded responses. Overall, the SLTB framework provides a flexible, theoretically grounded, and computationally efficient approach for modeling and selecting models for data confined to the [0, 1] interval.