On the Appropriate Measure to Estimate Hyperbolic Discounting Rate (K) using the Method of Least Squares

Abstract

The estimation of the rate at which value declines with delay requires identifying the correct discounting model, applying the appropriate parameter estimation method, and choosing the dependent measure from which parameters are estimated. The simplest adequate discounting model is the hyperbolic model; the simplest method to estimate its sole free parameter, K, is the method of least squares. Estimates of K based on relative subjective values (RSV), although typical, are not necessarily more reliable than those obtained through other measures. We consider an alternative measure termed immediacy premium (IP): the ratio of value lost over value preserved due to outcome delay. According to hyperbolic discounting, IP is a linear function of delay. As a result, estimates of K obtained from IP circumvent the divergence between individual and aggregate estimates obtained with the RSV method. Moreover, published data suggests that estimates of K based on RSV and IP differ systematically in humans and in pigeons. Regardless of the dependent measure adopted, estimates of K obtained from nonhuman animals, but not from humans, yield residuals that conform with error-distribution assumptions of the method of least squares. Although residuals obtained using the IP method in human data diverged more from normality than those obtained using the RSV method, the sequential dependence over delays of the former was weaker than that of the latter. We therefore recommend adopting both RSV and IP when estimating hyperbolic K using the method of least squares, resorting to more elaborate estimation methods when inferences drawn from these estimates are inconsistent.

Delay discounting, or temporal discounting, is the decrease in the value of an outcome as the delay to that outcome increases (Madden & Bickel, 2010; Odum, 2011; Vanderveldt et al., 2016). The value of an outcome can be positive or negative, depending on whether it is a gain or a loss; its decrease means that it approaches zero (Green et al., 2014). Estimates of delay discounting rate are often used to study impulsivity in humans and animals and to mathematically characterize such abstract psychological constructs as willpower and compulsion (Levitt et al., 2020). The accurate estimation of delay discounting rates has important implications for research on psychopathology, prosocial behavior, and other social concerns (Arbuthnott, 2010; da Matta et al., 2012; Johnson & Bickel, 2008; Lempert et al., 2019).

The accurate estimation of discounting rates involves at least three challenges: (1) identifying the correct discounting model; (2) applying an appropriate method to estimate model parameters, including the discounting rate; and (3) choosing a dependent measure from which parameters may be estimated. We first review challenges (1) and (2) but because these challenges have been extensively discussed (e.g., Cavagnaro et al., 2016), the review is succinct. This article focuses on challenge (3), which has been largely neglected in the delay discounting literature.

Model Uncertainty

The accuracy with which discounting rate is estimated depends on how value is assumed to decline over time. The normative exponential discounting function, which assumes that value declines at a constant rate over time, is substantially inferior in accounting for empirical data than other models (Mazur & Biondi, 2009; McKerchar et al., 2009). A prevalent model of delay discounting, derived from the matching law (Davison & McCarthy, 1988), is the hyperbolic discounting function (Ainslie, 2017; Madden & Bickel, 2010),

$$V\left(D\right)=V\left(0\right){\left(1+\mathit{KD}\right)}^{-1},$$ 1

where V(D) is the value of an outcome with delay D, discounted at a rate K; V(0) is the undiscounted value of the outcome. Equation 1 captures a key feature of empirical discounting data: value typically declines more steeply between shorter delays than between longer delays. The fit of Equation 1 to data improves by allowing the exponent to vary, or by raising D to the power of a free parameter (Green & Myerson, 2004; McKerchar et al., 2009; Rachlin, 2006). However, the complexity of these models has two important drawbacks. First, there is no unambiguous interpretation of the exponent parameter in these models. Second, given the small number of delays on which estimates of K and the exponent parameter are typically based (rarely more than six), there is a substantial risk of overfitting.

Other delay discounting models have been proposed (e.g., Killeen, 2009; van den Bos & McClure, 2013). Inferences on how steeply value is discounted given some empirical data depend on which model is assumed. Although that same data may be used to establish the relative merit of competing models (e.g., Ishii et al., 2018), the range of models considered is necessarily limited. An alternative approach suggests drawing inferences on impulsivity not from estimated discounting rates, but from more directly observed measures, such as the area below the value-delay function (area under the curve [AUC]; Myerson et al. 2001). This approach replaces model uncertainty with uncertainty about the appropriate measures from which impulsivity may be inferred. Measures of AUC, for instance, depend on the particular delays at which the value-delay function is probed (Borges et al., 2016). In the absence of a model of delay discounting, there is no principled rationale for computing AUC using one set of delays or another.

Model uncertainty may be circumvented by considering multiple discounting models and even nontheoretical approaches such as AUC. If one variable (e.g., smoking status) is similarly associated with analogous parameters and measures across multiple models and approaches (e.g., smokers have a higher K and lower AUC), it may be safe to conclude that that variable is associated with impulsivity (e.g., smokers are more impulsive). However, for model-based approaches, model parameters must be estimated, and such estimation entails addressing a second challenge.

Parameter Uncertainty

The accuracy of inferences on discounting rate depends not only on the discounting model that is assumed, but also on the way in which model parameters are estimated. A common and computationally simple method for estimating model parameters is to choose those that minimize the sum of the squared differences between model predictions and empirical observations. For instance, parameter K in Equation 1 can be estimated as the value of K that, across delays, minimizes the squared difference between V(D) and an observable measure of the value of the delayed outcome. Estimates based on this method assume that observable measures include a normally distributed error centered on zero that is not correlated across levels of the independent variable—i.e., across delays (Draper & Smith, 1998). These assumptions are rarely validated in the delay discounting literature.

Alternative assumptions on the variability in model output may be adopted using other estimation methods, such as maximum likelihood estimation (Myung, 2003). Deterministic models like Equation 1, however, are silent about the provenance of variability in the data. Assumptions of output variability may be incorporated as extensions of the hyperbolic discounting model, such as a choice probability component (Hwang et al., 2009). Maximum likelihood estimates are the basis for Bayesian analyses, which provide measures of parameter uncertainty (the dispersion of its posterior distribution) that can be reduced using population estimates in hierarchical models (Farrell & Lewandowsky, 2018). Such analyses have been implemented using delay-discounting data (Franck et al., 2019; Vincent, 2016). Although these methods have demonstrable advantages over those based on squared residuals, they require further assumptions and are computationally demanding.

The estimation of model parameters requires not only a model, but also data. Choosing the most appropriate dependent measure is the third challenge in the estimation of discounting rates. In the remainder of this article, we will consider the conventional measure of relative subjective value (RSV), and will compare its merit against an alternative measure, the immediacy premium (IP).

Relative Subjective Value

In typical delay discounting studies, the subjective value of a delayed outcome is assumed to be equal to the nominal value of an immediate outcome that the subject is just as likely to choose. The relative subjective value (RSV) of a delayed outcome is its subjective value divided by its undiscounted (nominal) value. For instance, for an individual who is indifferent between a $1,000 reward in 12 months and a $625 reward now, the subjective value of $1,000 in 12 months is $625; the RSV of $1000 in 12 months is thus $625 / $1,000 = 0.625.

Although RSV are typically treated as observations, they are actually estimations based on the pattern of choices that subjects make over various combinations of outcomes and delays. Most concerns about the adequacy of RSV as a measure to estimate K are in regard to the empirical methods to elicit choices from subjects and to estimate RSV from those choices (Robles & Vargas, 2007). Here, we set those concerns aside and assume that obtained RSV are accurate, in order to focus on the assumptions necessary to estimate K from RSV.

Based on Equation 1, K may be readily estimated by minimizing the squared differences between predicted and observed RSV. Predicted RSV may be obtained from a simple reparameterization of Equation 1. Both sides of Equation 1 may be divided by V(0), such that RSV(D) = V(D) / V(0), and thus

$$\mathit{RSV}\left(D\right)={\left(1+\mathit{KD}\right)}^{-1}.$$ 2

For simplicity, we will refer to this method of estimating K as the RSV method. For clarity, predicted and theoretical variables are written here in uppercase (e.g., RSV, K), whereas observed and estimated variables are written in lowercase (e.g., rsv, k). The RSV method assumes that observed rsv include error variance ε_RSV(D); i.e.,

$$\mathit{rsv}\left(D\right)={\left(1+\mathit{KD}\right)}^{-1}+{\mathrm{\epsilon }}_{\mathrm{RSV}}\left(D\right).$$ 3

Error variance is assumed to be sampled for each probe D from independent normal distributions with mean of zero. We will refer to the estimates of K obtained from the RSV method as k_RSV, to distinguish these estimates from those obtained using other measures.

A key limitation of the RSV method of is that the delay discounting function derived from the mean of multiple k_RSV estimates may not be representative of the individual empirical discounting functions. This mismatch between individual and aggregated data, a particular case of Simpson’s paradox (Kievit et al., 2013), can lead to serious misinterpretations of empirical findings: even though discounting functions are typically the basis for inferences about how individuals discount, they are often presented in aggregated form.

Figure 1A illustrates the divergence between delay discounting functions across levels of aggregation. The circles represent the mean of the observed rsv of three participants at each of six delays. k_RSV was obtained for each participant using the RSV method. The mean of those k_RSV, $\overline{k}$_RSV, was used to trace Equation 3, assuming K = $\overline{k}$_RSV (continuous line). In this example, that trace generally falls substantially below the mean rsv. That is, the mean rsv, in particular those for rewards delayed by more than 12 months, are much larger than the RSV predicted from $\overline{k}$_RSV.

Fig. 1.

(A) Mean (± SEM) observed relative subjective values (rsv, shown as circles) obtained from individual datasets in the $250 reward condition reported in Figure 3 (top row) of Green et al. (2014). Note. The gray band within dashed curves represents the 95% confidence interval enveloping the discounting functions drawn from averaging individual k_RSV estimates (obtained from fitting Equation 3, assuming that K = $\overline{k}$_RSV = 0.203 months^-1, represented as solid black curve; 95% CI = ± 0.103 months^-1). (B) Representation of the data in panel A as immediacy premiums (ip, shown as circles; individual rsv were transformed to ip using Equation 4 prior to averaging). The mean fit of Equation 6 to individual subjects coincides with the trace of Equation 6 drawn from averaging individual k_IP estimates (assuming that K = $\overline{k}$_IP = 0.061 months^-1, represented as solid black line; 95% CI = ± 0.038 months^-1, represented as gray band within dashed lines)

Immediacy Premium

The discrepancy between individual and aggregated value-delay functions may be circumvented using an alternative method to estimate K. This method relies on the computation of the immediacy premium (IP) instead of the RSV. The IP of a delayed outcome is the ratio of the value lost due to its delay to the value preserved despite its delay; in terms of Equation 1, IP(D) = [V(0) – V(D)] / V(D). For example, for an individual who is indifferent between $1,000 in 12 months and $625 now, the IP of the delayed $1,000 is the ratio of the $1,000 – $625 = $375 lost due to the 12-month delay to the $625 preserved despite that delay: $375 / $625 = 0.600. That is, for every dollar of value preserved over 12 months, 60 cents are lost over that period of time. An alternative description of this indifference relation is that, barring endowment effects (Marzilli Ericson & Fuster, 2014), this individual is willing to pay no more than a 60-cent premium for every delayed dollar that is delivered immediately.

Like RSV, IP must be estimated from observed choices. In fact, IP may be thought of as a mere transformation of RSV,

$$\mathit{IP}\left(D\right)=\left[1–\mathit{RSV}\left(D\right)\right]/\mathit{RSV}\left(D\right).$$ 4

Therefore, similar questions about the adequacy of RSV estimates (rsv) based on choices may be raised about IP estimates (ip). However, just like rsv, we will assume that ip are accurate, in order to focus on the assumptions necessary to estimate K from ip.

If hyperbolic discounting (Equations 1 and 2) is assumed, Equation 4 indicates that IP(D) is a linear function of K,

$$\mathit{IP}\left(D\right)=\mathit{KD}.$$ 5

Discounting rate K may thus be estimated from ip using the method of least squares, assuming that

$$\mathit{ip}\left(D\right)=\mathit{KD}+{\mathrm{\epsilon }}_{\mathrm{IP}},$$ 6

where error ε_IP has the same characteristics as ε_RSV in Equation 3. We will refer to the estimates of K from the IP method as k_IP.

Unlike RSV derived from k_RSV, the IP derived from the mean of multiple k_IP estimates is the mean IP from which each individual k_IP was estimated. Note in Figure 1B that ip means fall close to the range of discounting functions drawn from mean k_IP ± 95% confidence interval (gray band within dashed curves), and that the fit of Equation 6 to those means overlaps with the function drawn from mean k_IP ($\overline{k}$_IP; solid black line).

The coherence between the IP drawn from $\overline{k}$_IP and the mean IP drawn from individual k_IP suggests that this dependent measure should be preferred over RSV to estimate K. Before reaching this conclusion, however, it is important to determine the extent to which k_RSV and k_IP estimates vary systematically. If k_RSV and k_IP do not vary systematically, the choice between dependent measures is moot.

Difference between k_RSV and k_IP Estimates

Estimates of discounting rate K (k_RSV and k_IP) were obtained from empirical studies and compared. Studies were first selected for potential inclusion using the PsycINFO database on January 8, 2018. A search was conducted only within the Journal of the Experimental Analysis of Behavior using the terms “(temporal OR delay) discounting”; this journal was selected because it systematically reports individual-subject data. This search yielded 84 articles. These articles were further examined, keeping only those that contained at least one figure that reported at least five data points for individual subjects that could be converted into rsv. Figures that contained layered data points, such that data could not be reliably extracted, were not considered. This filter narrowed down the initial pool to seven articles. Table 1 lists the seven articles included in the analyses along with key characteristics. Pooled across studies, data was extracted from 15 figures representing data from 27 humans, 12 pigeons, and 19 rats. The type of reward, the range of its magnitude, the range of delays, and the methods for eliciting choices and estimating K varied between studies. Data were extracted from relevant figures using DigitizeIt 2.2.2 (Bormann, 2016). When possible, values on the x-axis (delays) were obtained from the methods section in the corresponding article. These data were then transformed into rsv (if necessary) and ip.

Table 1.

Sources of Data for Analysis

Study	Figures included	Species and number of subjects (and datasets)	Delays	Outcomes	Model used to estimate k	RSV estimation procedure
Human Participants
Green et al. (2013)	Fig. 4	Humans 6 (9)	Months: 1, 3, 6, 12, 72, 144	$20; $250; $3,000; $20,000; $50,000; $100,000; $500,000; $2,000,000; $10,000,000	Hyperboloid, Quasi-Hyperbolic, and Double Exponential	Adjusting-Amount Procedure
Green et al. (2014)	Fig. 3	Humans 3 (9)	Months: 1, 3, 6, 12, 72, 144	-$20; -$250; -$3,000; -$20,000; -$50,000; -$100,000; -$500,000	Hyperboloid	Adjusting-Amount Procedure
Johnson & Bickel (2002)	Fig. 1–4	Humans 6 (48)	Days: 1, 7, 14, 30, 182	$10; $25; $100; $250	Hyperbolic and Exponential Decay	Adjusting-Amount Procedure
Myerson & Green (1995)	Fig. 2 & 3	Humans 12 (24)	Days: 7, 30, 180, 360, 1080, 3600, 9000	$1,000; $10,000.	Hyperbola-like and Exponential Decay	Fixed-Amount Procedure
Non- Human Subjects
Green et al. (2004)	Fig. 1 & 2	Pigeons 4 (16)	Seconds: 1, 2, 4, 8, 16, 32	5, 12, 20, and 32 pellets	Hyperbolic and Hyperbola-like Decay	Adjusting-Amount Procedure
Green et al. (2004)	Fig. 1 & 2	Rats 4 (10)	Seconds: 1, 2, 4, 8, 16, 32	5, 12, and 20 pellets	Hyperbolic and Hyperbola-like Decay	Adjusting-Amount Procedure
Oliveira et al. (2014)	Fig. 1	Pigeons 8 (16)	Seconds: 1, 3, 6, 10, 20	16 and 32 pellets	Hyperbolic Decay	Concurrent-Chain Adjusting-Amount Procedures
Stein et al. (2012)	Fig. 1	Rats [Lewis] 8 (13)	Seconds: 0, 1.25, 2.5, 5, 10	10 food pellets	Hyperbolic Decay	Rapid-Determination and Steady-State Procedures
Stein et al. (2012)	Fig. 1	Rats [Fischer] 8 (13)	Seconds: 0, 1.25, 2.5, 5, 10	10 food pellets	Hyperbolic Decay	Rapid-Determination and Steady-State Procedures

Individual datasets were defined as the set of data points for each individual subject in each condition. So, for instance, if a set of data points was obtained from one participant using a $20 larger-later reward, and another set of data points was obtained from the same individual using a $20,000 larger-later reward, each set counted as an individual dataset. Pooled across studies, 158 datasets were analyzed in total. k_RSV and k_IP were obtained by fitting Equations 3 and 6 to each individual dataset using the method of least squares implemented using the Solver add-in of Microsoft ® Excel for Mac 16.2.

Figure 2 shows the correlation between k_RSV and k_IP in humans (panel A) and nonhuman animals (panel B) in log-log scale. In humans, the difference between k_RSV and k_IP appears to increase as the estimates increase. In addition, k_RSV are systematically higher [mean k_RSV – k_IP = 0.026 ± 0.010 days^-1; t(85) = 2.53, p = .013] and more variable (variance of k_RSV = 0.040 days^-2, variance of k_IP = 0.014 days^-2) than k_IP. In animals , mean k_RSV appear to be systematically lower than mean k_IP estimates. Figure 2B shows this difference as a larger number of data points above the identity line. This difference is visible in pigeons [mean k_RSV – k_IP = -0.263 ± 0.057 s^-1; t(31) = 4.69, p < .001] but less so in rats [mean k_RSV – k_IP = -0.041 ± 0.049 s^-1; t(34) = 0.856 , p = .398]. A systematic difference between the variance of k_RSV and k_IP was observed in pigeons, but not in rats (k_rsv pigeons = 0.285 s^-2, k_IP pigeons = 0.548 s^-2; k_RSV rats = 0.541 s^-2, k_IP rats = 0.582 s^-2).

Fig. 2.

Delay discounting rate (K) estimates obtained from fitting Equations 3 (K = k_RSV, x-axis in log scale) and 6 (K = k_IP, y-axis in log scale) to human (A) and nonhuman animal (B) data. Note. Identity lines are dashed.

This analysis suggests that estimates of K vary systematically depending on whether they were obtained using the RSV or IP method, both in central tendency and dispersion measures, at least in two of three species. These differences highlight the importance of selecting between these two estimation methods. This selection may be based on the extent to which each method conforms with the assumptions of the method of least squares. Residuals obtained from the estimation of k_RSV and k_IP in the preceding analysis were analyzed to determine conformity with assumptions. In particular, we verified, for each method, whether residuals are (1) centered near zero, (2) normally distributed, and (3) sequentially independent over delays.

Central Tendency of Residuals

If the mean of rsv and ip samples are the best estimates of the mean RSV and IP in the population, then the residuals from fitting Equations 3 and 6 to data are the best estimates of error in rsv (ε_RSV in Equation 3) and ip (ε_IP in Equation 6) measurement. Figure 3 shows the correlation between normalized residuals obtained from rsv and ip in humans (panel A) and nonhuman subjects (panel B). Normalized residuals obtained from human rsv (x-axis) were on average positive, and those obtained from ip negative (y-axis; see inset graph of panel A). This suggests that, in general, fitted k_RSV overestimated human rsv, and fitted k_IP underestimated human ip. Nonetheless, squared normalized residuals, which measure average distance from origin, were generally smaller when obtained from human rsv (mean = 1.441 ± 0.085) than when obtained from human ip [2.343 ± 0.229; t(539) = 3.77, p < .001].

Fig. 3.

Normalized residuals (divided by their standard deviation in own dataset) obtained from fitting Equations 3 (rsv; x-axis) and 6 (ip; y-axis) to human (A) and animal (B) data. Note. Inset graphs show the corresponding normalized residuals averaged within each study (gray symbols) and pooled across studies (closed symbols)

Normalized residuals obtained from animals clustered closer to zero than those obtained from humans, regardless of whether they were obtained from rsv or ip (compare Figure 3B to Figure 3A, and corresponding inset graphs). Squared normalized residuals obtained from rsv (mean = 0.849 ± 0.051) were similar to those obtained from ip [mean = 0.932 ± 0.060; t(365) = 0.575, p = .283].

Distribution of Residuals

Normality of the distribution of residuals was assessed using quantile-quantile (Q-Q) plots, which represent the correlation between (empirical) normalized residuals observed in a study and the theoretical (normally distributed) normalized residuals. A theoretical normalized residual is a quantile q of a normal distribution with mean zero and standard deviation of the empirical normalized residuals; the interval between quantiles contains a proportion 1 / (n – 1) of the distribution, where n is the number of empirical residuals in a study. To the extent that normalized residuals are normally distributed and centered on zero, the data points in the Q-Q plot would fall on the identity line. Distance relative to the identity line was quantified as the sum of the square difference between observed and theoretical normalized residuals.

Figure 4 shows the Q-Q plots for normalized residuals obtained for humans (panel A) and nonhuman animals (panel B), from rsv and ip. Human data appear to be more layered than animal data (see inset graphs), suggesting more variability in normality among human studies than among animal studies. Human data also suggest that the distribution of residuals was more negatively skewed (long left tail) when obtained from rsv, and more positively skewed (right long tail) when obtained from ip. Layering and differences in skewness across methods is not evident in animal data.

Fig. 4.

Q-Q plots of normalized residuals for humans (A) and animals (B) obtained from rsv (“○”) and ip (“×”). Note. Empirical normalized residuals are in the y-axis and their normally distributed correlates are in the x-axis; the identity line indicates perfect normality. The inset graphs enlarge the square area between -1 and +1 in both x- and y-axes, to better visualize the layering of Q-Q plots

Table 2 shows the sum of the squared distance to the identity line (and index of nonnormality) in humans and nonhuman animals, arranged by study. In all human studies, residuals were more normally distributed when obtained from rsv than when obtained from ip. Results from animal studies were more balanced, with more normal residuals when obtained from rsv in three of five studies. This is reflected in the overlap among Q-Q plots in Figure 4B.

Table 2.

Sum of Square Distance between Observed and Theoretical Normalized Residuals

Study	rsv	ip
Human Participants
Green et al. (2013)	0.619*	12.556
Green et al. (2014)	3.681*	12.887
Johnson & Bickel (2002)	61.533*	126.650
Myerson & Green (1995)	3.500*	35.569
Nonhuman Subjects
Green et al. (2004) Pigeons	7.630	6.022*
Green et al. (2004) Rats	2.627*	3.658
Oliveira et al. (2014) Pigeons	1.709*	3.086
Stein et al. (2012) Rats [Lewis]	0.525*	1.563
Stein et al. (2012) Rats [Fisher]	2.040	1.959*

Note. *Value closer to zero.

Sequential Independence of Residuals

To determine the sequential independence of residuals, lag-1 autocorrelations were conducted on normalized residuals obtained from rsv and ip in each dataset (Hibbs, 1973). Autocorrelation coefficients were obtained from each dataset by correlating each normalized residual at each delay D with the normalized residual at the next delay. Autocorrelation coefficients were squared (r²) in each dataset, and the difference between r² obtained from rsv and ip (Δr²) served as a measure of relative sequential dependence among residuals. Positive Δr² indicate that residuals obtained from rsv are more sequentially dependent than residuals obtained from ip.

Table 3 reports the Δr² for each study under consideration. Only two studies showed significant Δr²; both were positive, indicating that, in those studies, residuals obtained from rsv were more sequentially dependent than those obtained from ip. Both studies were conducted with human participants; no animal study yielded a significant Δr², suggesting that concerns of sequential dependency among residuals may be restricted to estimates of K based on human rsv.

Table 3.

Relative Sequential Dependence of Normalized Residuals

Study	Mean Δr2 (+/- 95% CI)
Human Participants
Green et al. (2013)	0.01 (0.12)
Green et al. (2014)	-0.07 (0.28)
Johnson et al. (2002)	0.07* (0.06)
Myerson et al. (1995)	0.18* (0.10)
Nonhuman Subjects
Green et al. (2004) Pigeons	0.06 (0.15)
Green et al. (2004) Rats	-0.08 (0.19)
Oliveira et al. (2014) Pigeons	-0.05 (0.12)
Stein et al. (2012) Rats (Lewis)	-0.03 (0.18)
Stein et al. (2012) Rats (Fisher)	-0.13 (0.18)

Note. *Absolute mean Δr² greater than its 95% CI. Positive Δr² indicate that normalized residuals are more sequentially dependent when obtained from rsv than from ip.

Summary and Recommendations

This brief review identified three “uncertainties” in the estimation of delay discounting rates: model uncertainty, parameter uncertainty, and uncertainty about the appropriate dependent measure for parameter estimation. Model and parameter uncertainties have been examined extensively elsewhere and are therefore just summarized here. Nonetheless, they provide an informative context for an inquiry on the third uncertainty.

Although there are multiple models of delay discounting, the best supported single-free-parameter model is the hyperbolic discounting model (Equation 1). Multiple-free-parameter models provide better fits to empirical data but, given the typically few data points collected per subject, they do so at a substantial risk of overfitting. This challenge may be circumvented by collecting more data per subject, implementing hierarchical models, testing for convergence among multiple models, and implementing theoretically meaningful manipulations. Atheoretical approaches (e.g., AUC) dispel model uncertainty, but worsen the uncertainty about dependent measures: without a model of impulsivity, it is impossible to determine the merit of an empirical measure as indicative of impulsivity.

The method of least squares is a simple and common method for estimating the discounting rate parameter K of the hyperbolic discounting model. The method assumes normally distributed error around the dependent measure from which K is estimated. This is certainly not the only method to estimate K, and it is not necessarily the most informative nor the most accurate. Alternative methods, however, typically require extending the (deterministic) hyperbolic discounting model to include sources of variability; incorporating such variability into parameter estimation is neither theoretically nor computationally trivial. Estimates of K based on the method of least squares may be sufficient to detect significant effects on delay discounting rate, as long as those estimates are based on adequately obtained data.

There are, of course, many factors that determine the quality of empirical data. We focused on one factor that is often overlooked: the empirical measure from which K is estimated. In particular, we focused on two measures: relative subjective values (rsv) and immediacy premiums (ip). These measures need not necessarily be collected using different methods, because their expected means (RSV and IP) can be transformed into one another (Equation 4). However, estimates of K obtained through the IP method may be more accurate than those obtained through the RSV method. For instance, whereas average fits of the hyperbolic discounting to ip are representative of individual fits, that is not the case with fits to rsv (Figure 1). We extended this comparison between dependent measures to determine their relative conformity to the assumptions of the method of least squares, based on published data. Selection between these methods is particularly important because estimates of K vary systematically between methods, at least in humans and pigeons (Figure 2).

A close examination of normalized residuals suggests that assumptions of normality and independence of error were likely met in the case of nonhuman animals (Figure 3B and 4B), regardless of whether K was estimated from rsv or ip. In contrast, fits of Equation 3 to rsv systematically overestimated rsv, whereas fits of Equation 6 to ip systematically underestimated ip (Figure 3A), suggesting a nonnormal distribution of error around these measures (Figure 4A). This divergence of error distribution from normality appears to be larger around ip than rsv (Table 2), thus favoring the adoption of the RSV method over the IP method, at least in humans. However, two human studies (Johnson & Bickel, 2002; Myerson & Green, 1995) suggest that error around rsv are substantially more sequentially dependent over delays than error around ip; no study, human or otherwise, show the opposite relation (Table 3). These latter findings favor the adoption of the IP method over the RSV method.

A set of recommendations may be drawn from the observations and analyses included in this article. These recommendations are based on two assumptions that delay discounting studies typically meet. First, our recommendations assume the single-free-parameter (K) hyperbolic discounting function (Equation 1). To adopt more complex models would require testing on a large number of delays to avoid overfitting. Second, our recommendations assume that theoretically justified sources of variability in model output are not identified. If such sources of variance were identified, K may be estimated using methods based on maximum likelihood.

These assumptions entail that K must be estimated using the conventional RSV method or the novel IP method, both based on the method of least squares. Because these methods may yield different estimates of K, but the evidence does not unequivocally favor one method for estimating K over the other, and both methods are equally intuitive and simple to implement, our recommendation is to estimate K using both methods. This solution is analogous to the adoption of multiple models of delay discounting to circumvent model uncertainty. In terms of absolute estimates of K, those obtained from the RSV and IP methods provide a range over which the true K is likely located. In terms of differences in estimates of K, if an independent variable is similarly associated with k_RSV and with k_IP, it may be safe to conclude that, as long as the hyperbolic discounting model is valid, that variable is associated with impulsivity. k_RSV and k_IP that change in different directions across levels of an independent variable may indicate that the relation between that variable and impulsivity is weak. Finally, we recommend that, if average data are displayed, it should be ip. Unlike averaged rsv-delay functions, averaged ip-delay functions are representative of the average estimate of K that generate the individual functions.

Declarations

Conflict of Interest

The authors declare that they have no conflict of interest. This chapter does not contain any studies with human participants or animals performed by any of the authors.