Accurate characterization of delay discounting: A multiple model approach using approximate bayesian model selection and a unified discounting measure

Abstract

The study of delay discounting, or valuation of future rewards as a function of delay, has contributed to understanding the behavioral economics of addiction. Accurate characterization of discounting can be furthered by statistical model selection given that many functions have been proposed to measure future valuation of rewards. The present study provides a convenient Bayesian model selection algorithm that selects the most probable discounting model among a set of candidate models chosen by the researcher. The approach assigns the most probable model for each individual subject. Importantly, effective delay 50 (ED50) functions as a suitable unifying measure that is computable for and comparable between a number of popular functions, including both one- and two-parameter models. The combined model selection/ED50 approach is illustrated using empirical discounting data collected from a sample of 111 undergraduate students with models proposed by Laibson (1997); Mazur (1987); Myerson & Green (1995); Rachlin (2006); and Samuelson (1937). Computer simulation suggests that the proposed Bayesian model selection approach outperforms the single model approach when data truly arise from multiple models. When a single model underlies all participant data, the simulation suggests that the proposed approach fares no worse than the single model approach.

Discounting of qualitatively different delayed health outcomes in current and never smokers.

“…Table 3 includes the parameter estimates for the highest quality model for each group and outcome (see Franck, Koffarnus, House, & Bickel, 2015). The parameters in Table 3 were those used to create the lines of best fit in Figure 1.…”

Adolescent methylmercury exposure affects choice and delay discounting in mice

“…The present study employed a model-comparison analysis (Burnham & Anderson, 2002), a robust statistical approach that does not rely on null-hypothesis testing and has been growing in popularity in the neurobehavioral sciences (Avila et al, 2009; Franck et al, 2015; Sanabria et al, 2008), to examine responding following adolescent MeHg exposure. The model that allowed magnitude sensitivity ( s M ) to vary across all exposure groups and a separate delay sensitivity ( s D ) for the 0.3 ppm MeHg group was the best model (Table 2.2).…”

“…This model-comparison analysis has been growing in popularity in the behavioral and neural sciences to model drug and neurotoxicant effects (Avila et al, 2009; Franck, Koffarnus, House, & Bickel, 2015; Sanabria, Acosta, Killeen, Neisewander, & Bizo, 2008) and it has been used in our laboratory for model construction (Hutsell & Newland, 2013). It does not rely on traditional null-hypothesis testing.…”

The Value of Clean Air: Comparing Discounting of Delayed Air Quality and Money Across Magnitudes

“…First, four models of delay discounting were fit to the median indifference points obtained from each outcome. The models selected were a random noise model (see Franck, Koffarnus, House, & Bickel, 2015; E ( Y ) = c ), the exponential model (Samuelson, 1937; E ( Y ) = e − kD ), the hyperbolic model (Mazur, 1987; E ( Y ) =1/(1 + kD )), and the hyperboloid model (Myerson & Green, 1995; E ( Y ) =1/(1 + kD ) s ). The highest quality model for each outcome was selected using an Akaike information criterion (AIC) process (see Wagenmakers & Farrell, 2004).…”