Thinking inside the bounds: Improved error distributions for indifference point data analysis and simulation via beta regression using common discounting functions

TL;DR

This paper introduces a beta regression framework for indifference point data that better captures variability, respects data boundaries, and improves modeling of discounting functions compared to traditional nonlinear regression.

Contribution

The work develops a novel beta regression approach for indifference points, addressing limitations of normality assumptions and constant variance in standard methods.

Findings

Beta regression provides a better fit to discounting data.

The model captures non-constant variance as a function of delay.

Estimated discounting rates are consistent with traditional methods.

Abstract

Standard nonlinear regression is commonly used when modeling indifference points due to its ability to closely follow observed data, resulting in a good model fit. However, standard nonlinear regression currently lacks a reasonable distribution-based framework for indifference points, which limits its ability to adequately describe the inherent variability in the data. Software commonly assumes data follow a normal distribution with constant variance. However, typical indifference points do not follow a normal distribution or exhibit constant variance. To address these limitations, this paper introduces a class of nonlinear beta regression models that offers excellent fit to discounting data and enhances simulation-based approaches. This beta regression model can accommodate popular discounting functions. This work proposes three specific advances. First, our model automatically captures non-constant variance as a function of delay. Second, our model improves simulation-based approaches since it obeys the natural boundaries of observable data, unlike the ordinary assumption of normal residuals and constant variance. Finally, we introduce a scale-location-truncation trick that allows beta regression to accommodate observed values of zero and one. A comparison between beta regression and standard nonlinear regression reveals close agreement in the estimated discounting rate k obtained from both methods.