Applying Mixed-Effects Modeling to Behavioral Economic Demand: An Introduction

Abstract

Behavioral economic demand methodology is increasingly being used in various fields such as substance use and consumer behavior analysis. Traditional analytical techniques to fitting demand data have proven useful yet some of these approaches require preprocessing of data, ignore dependence in the data, and present statistical limitations. We term these approaches "fit to group" and "two stage" with the former interested in group or population level estimates and the latter interested in individual subject estimates. As an extension to these regression techniques, mixed-effect (or multilevel) modeling can serve as an improvement over these traditional methods. Notable benefits include providing simultaneous group (i.e., population) level estimates (with more accurate standard errors) and individual level predictions while accommodating the inclusion of "nonsystematic" response sets and covariates. These models can also accommodate complex experimental designs including repeated measures. The goal of this article is to introduce and provide a high-level overview of mixed-effects modeling techniques applied to behavioral economic demand data. We compare and contrast results from traditional techniques to that of the mixed-effects models across two datasets differing in species and experimental design. We discuss the relative benefits and drawbacks of these approaches and provide access to statistical code and data to support the analytical replicability of the comparisons.

Applications of operant demand to treatment selection I: Characterizing demand for evidence‐based practices

“…Caregiver consumption of EBPs and ATs across both HTPTs was evaluated using the Zero Bounded Exponential (ZBE) model of demand (Gilroy et al, 2021) in a mixed-effects modeling approach (Kaplan et al, 2021). Briefly, the ZBE model is an extension of the original Exponential model of operant demand (Hursh & Silberberg, 2008) with a modified scale (Inverse Hyperbolic Sine) that optionally supports a true lower bound at zero consumption.…”

“…Second, GEE is similar to multilevel models and is often applied in experiments to account for repeated measurements across individuals (Hardin, 2005;Kaplan et al, 2020). Such an approach avoids issues associated with ordinary least squares regression, for example, nonindependence (DeHart & Kaplan, 2019;Kaplan et al, 2021). Third, similar to the methods proposed in Hursh and Roma, the quantity regressed upon price in the GEE approach captures the direction and rate of changes in consumption as the price to consume EBPs changes.…”

Hidden equivalence in the operant demand framework: A review and evaluation of multiple methods for evaluating nonconsumption

“…For example, this alternative (i.e., Inverse Hyperbolic Sine transformation) replicates the behavior of logarithms across a range of values (e.g., 10, 100), normalizes residual error variance, and supports a true lower bound at zero. This approach is not discussed at length in this report, though interested readers are encouraged to consult Gilroy, Kaplan et al (2021) for a discussion and demonstration of this "log-like" scale and its benefits (e.g., decoupling of α from the span of the curve, separate span parameter not necessary).…”

“…This perspective and this framework currently reflect a range of consumption (and nonconsumption) and efforts are underway to leverage multilevel modeling as a methodological extension (Kaplan et al, 2021). Indeed, various labs are working toward increasing the applicability and generality of this approach.…”

“…The original implementation of the Hursh and Silberberg (2008) framework was modeled from the notion that the prototypical shape of the demand curve was an "S"-type form bounded by upper and lower limits. The original intent of Hursh and Silberberg was to have an upper limit defined at a price of zero and a lower limit reached as prices approached infinity (i.e., lim P!∞ ; Gilroy, Kaplan et al, 2021). The upper limit is interpreted as the intensity of demand for a particular reinforcer (i.e., consumption [Q] at a price of 0 = Q 0 ), and the rate by which demand progresses from the upper to lower limit refers to the rate of change in elasticity (i.e., = α, Δelasticity).…”