# 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 model to better analyze and simulate indifference point data by accounting for non-constant variance and natural data boundaries.

## Contribution

The paper introduces a beta regression framework with a scale-location-truncation trick to handle non-normal indifference point data with natural boundaries.

## Key findings

- Beta regression models improve simulation-based approaches by respecting natural data boundaries.
- The proposed model captures non-constant variance as a function of delay.
- Estimated discounting rates from beta regression closely match those from standard nonlinear regression.

## 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 0 and 1. A comparison between beta regression and standard nonlinear regression reveals close agreement in the estimated discounting rate k obtained from both methods.