Delay discounting rates are temporally stable in an equivalent present value procedure using theoretical and area under the curve analyses.
Abstract: Temporal discounting rates have become a popular dependent variable in social science research. While choice procedures are commonly employed to measure discounting rates, equivalent present value (EPV) procedures may be more sensitive to experimental manipulation. However, their use has been impeded by the absence of test retest reliability data. Staff and students at a regional Australian university (n = 53) participated in a matching EPV temporal discounting procedure in 2 sessions, 2 weeks apart, completing 30 one-shot, second-price auctions for two amounts, with delays ranging from 1-43 days. Discounting rates were estimated using hyperbolic and exponential models, as well as atheoretical area under the curve (Au C) analyses. Test retest (relative) stability of the EPV procedure compared favorably with choice procedures (r = .75). Where discounting rates are used as a dependent variable, brief EPV procedures combined with atheoretical analyses of discounting rates are a more sensitive means to detect subtle experimental effects.

Key words: temporal discounting, delay discounting, retest reliability, area under the curve
Article Type: Report
Subject: Discounts (Sales) (Psychological aspects)
Consumer preferences (Research)
Social science research
Authors: Harrison, Justin
McKay, Ryan
Pub Date: 03/22/2012
Publication: Name: The Psychological Record Publisher: The Psychological Record Audience: Academic Format: Magazine/Journal Subject: Psychology and mental health Copyright: COPYRIGHT 2012 The Psychological Record ISSN: 0033-2933
Issue: Date: Spring, 2012 Source Volume: 62 Source Issue: 2
Topic: Event Code: 310 Science & research
Geographic: Geographic Scope: Australia Geographic Code: 8AUST Australia
Accession Number: 288689702
Full Text: As a rule, when an individual wants something (like a new bicycle or a cigarette), he or she prefers to get it sooner rather than later, while for aversive events like dental procedures, the individual is less concerned when the event is still four weeks away than he or she is right before the appointment. The finding that the subjective value (1) of future outcomes reduces in inverse proportion to the delay until the event occurs (Kirby, 1997) is referred to as delay or temporal discounting. This effect is most frequently described by exponential or hyperbolic models (Critchfield & Kollins, 2001; Kirby, 2006; Schweighofer et al., 2006).

Hyperbolic and Exponential Discounting

Expected utility theory requires that an organism will tend toward behaviors that maximize expected gains for effort expended (von Neumann & Morgenstern, 1953). For discounting theory, this led to the assumption by economic theorists that human discounting rates are exponential (Bickel & Johnson, 2003):

V = [Ae.sup.-KD] (1)

(Kirby, 1997, p. 54)

Here V represents the discounted value of the reward A, and D represents units of delay. Solving for the free parameter k provides the discounting rate for a single trial (values in the unit interval, assuming values necessarily decrease as a function of delay, but see van der Pol & Cairns, 2000, for an exception). Exponential discounting applies a fixed rate of discounting per unit of delay, predicting stable long-term preferences. For example, if an individual prefers $25 in 92 days over $20 in 90 days, she should still prefer $25 when there are only two days left to wait, rather than changing her preference and taking the $20 without further delay (Ainslie, 2002, 2009; Thaler, 1981). While the exponential model cannot account for such commonly observed reversals of preference (Green, Myerson, & McFadden, 1997), (2) the hyperbolic model in the following equation does (Ainslie, 2001, 2002):

V = A / 1 + (k* D) (2)

Mazur (1987)

While individual discounting rates are highly variable (Frederick, Loewenstein, & O'Donoghue, 2003, p. 14), observations of discounting rates in humans demonstrate, inter alia, inverse relationships with age (Green, Fry, & Myerson, 1994; Green, Myerson, Lichtman, Rosen, & Fry, 1996), education (Green et al., 1994; Green et al., 1996; Reimers, Maylor, Stewart, & Chater, 2009), and reward size (Green et al., 1997; Kirby & Marakovic, 1996). Increased scientific interest in temporal discounting rates has generated a need for reliable and sensitive measures. In the following section, we describe two common methods.

Choice Procedures Versus Equivalent Present Value

Temporal discounting rates are normally estimated using either a choice or an equivalent present value (EPV) procedure (3) (Loewenstein & Prelec, 1992, p. 576). Choice procedures involve participants having to choose between a small, early reward and a larger, later reward (e.g., "Would you prefer $55 now or $75 in 19 days?"). Amounts and/or delays are adjusted on each trial. The point at which participants switch from the small, early reward to the larger, delayed reward provides an estimate of the temporal discounting rate. This method has been used to infer rates of discounting for health programs (Chapman & Elstein, 1995), environmental gains/losses (Hardisty & Weber, 2009), and money (Kirby, 2009; Whelan & McHugh, 2009).

The widely used Monetary Choice Questionnaire (MCQ; Kirby & Marakovic, 1996) contains 27 dichotomous items (nine questions each for small, medium, and large amounts of money) and a manualized procedure for the computation of discounting rates (Kirby, 2000; Wileyto, 2004). For each of the three amount sizes, the point at which participants cease to select the immediate reward in favor of the later, larger one provides an approximation of the individual's temporal discounting rate (solving for k in Equation 2). The MCQ has allowed nonspecialists to compute temporal discounting rates with relative ease. Another advantage of a standardized procedure for computing discounting rates is ease of comparison between studies.

EPV procedures, on the other hand, require participants to nominate a present value that is subjectively equal to the delayed amount, less what the specified delay is worth to them (e.g., "How much money would you be prepared to pay now to receive $29 in 15 days?"). These procedures do not restrict participants to forced choices between predetermined values. However, EPV procedures are less convenient to administer than choice procedures. The necessary instructions may require the sustained concentration of participants, resulting in some variability in the "orderliness" of responses. Nevertheless, there are circumstances in which the EPV procedure is preferable to the choice procedure. Where researchers are interested in comparing the fit of theoretical models (e.g., hyperbolic versus exponential), a number of "indifference points" are required to provide discounting curves over a number of delays. The 27 items in the MCQ yield a single approximation of an individual's discounting rate, whereas every item in an EPV procedure represents an indifference point. Using such a procedure, Kirby, Winston, and Santiesteban (2005) and Kirby and Santiesteban (2003) obtained 30 indifference points per participant (15 each in relation to two delayed amounts, i.e., $9.90 and $29.90), sufficient to allow estimates of fit that are able to distinguish between theoretical models and provide a more accurate estimate. Choice procedures require many more items to yield the same data, and in doing so cede the advantage of brevity to EPV procedures.

Kirby and Santiesteban's (2003) study was also unusual in that it was one of very few studies that have required participants to commit their own money. This is an important methodological feature, as decisions concerning delayed consequences are influenced by the manner in which they are presented (Highhouse, Mohammed, & Hoffman, 2002). For example, gains tend to be discounted more steeply than losses (Kahneman & Tversky; 1979; LeBoeuf, 2006; Loewenstein & Prelec, 1992; Read, Frederick, Orsel, & Rahman, 2005). Bidding with money that has been personally budgeted elsewhere may be perceived as a greater opportunity cost (a loss) than bidding the same amount with hypothetical funds or with a stake provided by the experimenter (a gain), possibly improving the ecological validity of the results where consumer behavior is concerned. Kirby and Santiesteban's (2003) method is therefore employed in the present study.

Choice procedures may also be subject to ceiling effects. For example, the MCQ questionnaire allows for the estimation of discounting rates in the range of .00016-.25. As delay discounting varies dramatically between individuals, limiting reportable discounting rates may omit participants with higher or lower rates of discounting. In a recent hypothetical study, Harrison and McKay (2010) used the MCQ to infer discounting rates for 200 participants, finding that more than 10% of respondents reported discounting rates of .25. No other single discounting rate was reported by so many participants; thus it is likely that the discounting rate of the sample was distorted by ceiling effects.

Brief choice procedures like the MCQ have made it convenient to measure temporal discounting rates with no participant training or supervision; thus they are particularly well suited to online studies or studies with very large numbers of participants. This is a beneficial development for the study of this important variable, where a single approximation of discounting rates is required from large groups.

However, for experimental studies, participants are often relatively few in number. Moreover, researchers may be concerned with detecting what may be very subtle effects on temporal discounting. In these cases, brief EPV procedures, such as the paper-and-pencil task employed by Chapman (1996) are preferable because in a brief instrument (e.g., 25-35 items) participants provide finer grained estimates of discounting rates, allowing the detection of subtle changes to individual discounting rates as well as the requisite indifference points to facilitate the comparison of theoretical models where desired. The fact that EPV procedures require more instruction and supervision is not necessarily an impediment in the experimental context, as these procedures most frequently take place in a supervised setting where additional tasks are being performed.

Even where model fitting is not required, estimation of discounting rates using theoretical models may not be the most appropriate method in the experimental context. While hyperbolic models consistently fit discounting data better than exponential models, this is a relative finding; frequently, hyperbolic models represent the best of two poorly fit models, such that some participants' data must be omitted (Ohmura, Takahashi, Kitamura, & Wehr, 2006), or the simple hyperbolic equation must be altered, raising the denominator to a power to improve the fit (Critchfield & Rollins, 2001; Myerson, Green, & Warusawitharana, 2001; Ohmura et al., 2006; Whelan & McHugh, 2009). The ubiquitous equation type--dependent error introduces "noise" in the calculation of discounting rates, problematic when trying to detect subtle experimental effects on discounting rates. In addition, estimates of model parameters are usually skewed and kurtotic, making them inappropriate for parametric analyses without transformation (Keene, 1995). This reduction in sensitivity and the increased difficulty of analysis are unnecessary. With sufficient indifference points one may compute discounting rates atheoretically by calculating the area under the curve (AuC; Myerson et al., 2001). AuC analyses do not rely on estimates of model parameters, eliminating equation type dependent error. For researchers interested in a sensitive measure of discounting rates that is amenable to parametric analysis, AuC analysis is a helpful technique. Brief EPV procedures are able to provide the necessary data, while choice procedures of comparable length are not.

The temporal stability of delay discounting rates measured using choice procedures has been established. However, the stability of EPV procedures has not. Simpson and Vuchinich (2000) reported a 1-week test-retest coefficient (n = 15) of .90 for a hypothetical choice procedure. Baker, Johnson, and Bickel (2003) and Johnson, Bickel, and Baker (2007) used an EPV-style questionnaire to obtain subjective equivalence estimates between monetary rewards, periods of health improvement, and nicotine. Equivalencies obtained in the pilot phases of these studies were subsequently used in choice procedures administered in two sessions, 1 week apart. Discounting for money, health gains/losses, and nicotine gains/losses were assessed. Test-retest data for these studies ranged from .71 .90 and .55 .90, respectively. Most recently, Reed and Martens (2011, p. 10) presented elementary school students with hypothetical choices between small, immediate and larger, delayed monetary amounts, reporting a 1-week test-retest reliability coefficient of .88. The obtained discounting rates were also found to predict student behavioral change in response to immediate versus delayed rewards in the classroom, a helpful addition to the literature supporting the ecological validity of temporal discounting rate estimates.

Longer periods between test-retest sessions produce lower reliability estimates. Beck and Triplett (2009) obtained a particularly large sample (n = 299) of undergraduate students for a pencil-and-paper-based procedure in which participants were required to choose the present value of a delayed, hypothetical amount of money from a supplied list. Six-week test-retest coefficients of .64 for hyperbolic model-based estimates and .70 for AuC estimates were observed. Similarly, Ohmura et al. (2006) used a hypothetical choice procedure to collect 3-month test-retest data with 22 undergraduates who were presented with an immediate reward and a set delay period (i.e., 7 days). The authors reported test-retest correlations of .61, assuming a hyperbolic model. Using the MCQ, Kirby (2009) reported test-retest coefficients of .71 at 1 year (n = 56). These stability estimates therefore account for approximately half of the observed variance in temporal discounting rates between individuals, comparable to some measures of personality (Kirby, 2009). However, rates of temporal discounting would be of little use as a dependent variable if they were impervious to context. As it happens, temporal discounting rates appear to be sensitive to sign (gains vs. loss), working memory load, social context (i.e., gambling environment), size of the reward, and relative states of deprivation (Odum & Baumann, 2010). However, while a group of participants may display higher discounting rates for food when they are hungry, it is equally likely that the group would broadly retain their "predeprivation" rank order (Odum & Baumann, 2010, p. 41). As such, the high relative stability of discounting rates does not preclude the impact of situational influences or negate its utility as a dependent variable.

Nevertheless, Kirby (2009) suggested that although situational contexts do affect discounting rates, the particularly high stability rates obtained by Baker et al. (2003), Johnson et al. (2007), Simpson and Vuchinich (2000), and (one may presume) Reed and Martens (2011) may have been inflated by participants being able to recall their previous selections after 1 week. Kirby argued that longer intervals should control for memory effects. However, this may be difficult for repeated-measures research designs where discounting rates are a dependent variable. Longer intervals not only increase logistical difficulties and the cost of experimental studies but also increase the potential effects of confounding variables (i.e., changes in financial situation) on temporal discounting rates during the interval. Experimental manipulations of discounting rates necessarily require as short an interval as possible. Therefore, memory effects need to be controlled using methodological processes. Memory effects could be exacerbated by an artifact of the choice procedures used in these studies, which provide participants with two-option, forced-choice items and at retest present participants with the options they chose earlier. This method lends itself to recognition, which in most contexts is easier than free or even cued recall (Perlmutter, 1979). Brief choice procedures such as the MCQ would be particularly prone to recognition memory effects. EPV procedures provide cues (via presentation of the delay period and amount) to recall but do not present the participants' actual responses, making memory effects less likely at retesting. While the present study also employed a short test-retest interval (2 weeks), it was expected that memory effect would be controlled by the methodology employed. Thus, the resultant reliability coefficients would be more congruent with studies that employed a retest interval of 6 weeks (i.e., Beck & Triplett, 2009) or greater than studies that used a retest interval of 2 weeks or less (i.e., Simpson & Vuchinich, 2000).

Baker et al.'s (2003) pilot studies notwithstanding, there is no published data, as far as we are aware, on the temporal stability of EPV procedures. Such data is overdue since the technique has been in use for some time. For example, Chapman (1996) used a 32-item fill-in-the-blank EPV procedure to measure discounting rates for money and health outcomes; the study's finding that discounting is likely to be domain specific could only be strengthened by test-retest reliability data for either domain. Given that choice and EPV procedures produce different total rates of discounting, as well as different discounting curves as a function of successive delays (Ahlbrecht & Weber, 1997; Read & Roelofsma, 2003), the temporal stability of EPV procedures cannot be taken for granted. Moreover, demonstrating the stability of individual discounting using an EPV procedure would rule out the potential ceiling and recognition memory effects inherent in choice procedures, lending support to earlier findings that delay discounting rates are relatively stable (Critchfield & Kollins, 2001; Kirby, 2009).

The lack of test-retest reliability data for EPV procedures is an impediment to their widespread use, especially for experimental paradigms where sensitivity of the measure may mean the difference between detecting an effect and a Type II error. The present study sought to address this gap in the literature and provide test retest reliability for an EPV procedure used by Kirby and Santiesteban (2003); the method employs a second-price auction where participants are required to bid on delayed sums of money such that they would "break even." Vickrey's (1961) second-price auctions are designed such that it is in the participants' interest to bid exactly what the delayed money is worth to them. Thus each bid provides an equivalent present value. In the current study, 15 such indifference points were obtained for both a large and a small amount of money, and data were fitted to exponential and hyperbolic discounting models for comparison to the extant literature.

It was predicted that discounting rates obtained in the present study would demonstrate inverse relationships with age, education, and reward magnitude, replicating common findings (Kirby & Santiesteban, 2003) and contributing to the evidence base for the validity of this brief EPV procedure. Moreover, to avoid equation type--dependent systematic error, rates of discounting for delayed rewards were quantified in an atheoretical procedure by computing the area under a normalized matching curve (AuC) as suggested by Myerson, Green, and Warusawitharana (2001).

Method

Participants

Participants were 67 staff members and students (53 females, 14 males; mean age = 28.75 years, SD = 12.73) at two campuses of a regional Australian university. Forty-four participants had completed secondary school only; eight also held undergraduate degrees and 15 had completed post-graduate qualifications. Fifty-three participants (42 females, 11 males; mean age 28.53 years, SD = 12.36) returned for the second session. Participants were recruited via the university e-mail and research participation system. All participants were paid AUD$10 for participation in both sessions, while psychology students also received course credit.

Materials

Participants completed all trials on IBM computers. The auctions were run on a Web-based instrument created using the Dreamweaver application. Auction data was collated using the Microsoft Access database application.

Procedure

Sessions 1 and 2 were conducted 2 weeks apart, in groups of between three and eight participants. Prior to the first session participants were contacted by e-mail and informed that they were required to bring AUD$30 with them in order to bid in a series of auctions. No volunteer declined to participate as a result of this requirement. In a series of 30 "one-shot" second-price auctions, participants were asked to nominate an amount they were prepared to pay for delayed sums of money such that they believed they would just break even (Vickrey, 1961). The auctions were preceded by a brief training session and three practice auctions, during which participants had the opportunity to ask questions. Two amounts of money were presented ($9.90 & $29.90), with the amount alternating on each trial. Fifteen delays (of 1, 2, 3, 4, 5, 7, 11, 15, 19, 23, 27, 31, 35, 39, and 43 days) were presented for each amount in randomized order, such that there was no correlation between delay amount and the order of the auctions.

After entering a bid, participants were directed to a second screen where they were asked whether they would like to keep their money or wait for the delayed money, or whether those options "feel about the same to me." This measure was designed to reinforce the instruction that participants should bid to break even and provide an opportunity to fine-tune their bid, raising the accuracy of obtained indifference points (Kirby & Santiesteban, 2003). If the "keep money" option was selected, the participants were instructed to adjust their bid downward; if they selected the "wait" option they were instructed to raise their bid (by $0.10 increments). This process was repeated until the participant selected the "feel about the same" option, at which point the final bid was submitted.

At the conclusion of the auctions, a number between 1 and 30 was selected at random, determining which auction was the "real" auction. The highest bidder was invited to complete the transaction by paying the amount placed by the second highest bidder (Vickrey, 1961). To control transaction costs regardless of delay time, the full amount was received by presenting the receipt at the Psychology School office once the nominated time delay (1-43 days) had elapsed. Participants were paid AUD$10 for attending the second session. (4)

Results

Bidding data were analyzed separately for all participants to avoid artefactual biases introduced by drawing parameter estimates from aggregate data (Kirby, 1997). Discounting rates were calculated separately for the smaller ($9.90) and larger ($29.90) amounts, before taking the geometric mean of both (total) as an estimate of discounting for each participant. Discounting estimates (k) and goodness-of-fit measures against hyperbolic and exponential models were estimated using iterative, nonlinear regression.

We used root mean squared error (RMSE), expressed as a proportion of the total undelayed reward amounts ($9.90 & $29.90) as an index of the equation's goodness of fit; lower RMSE scores indicate smaller error terms and therefore a better model fit. Discounting rates were also estimated atheoretically by measuring AuC using the trapezoid summation method based on the following equation:

[GAMMAR]([x.sub.2]-[x.sub.1])[([y.sub.2]+[y.sub.1])/2] (3)

(Myerson et al., 2001, p. 240)

These estimates of individual discounting rates do not include the residual error (RMSE) inherent in model-based iterations.

Consistent with previous work, theoretical k parameter estimates for the model equations were skewed and kurtotic, while AuC estimates did not violate the assumptions required for parametric analysis. Aggregated RMSE coefficients for hyperbolic discounting were significantly lower than those for the exponential equation (see Table 1). This was also the case for individual repeated-measure comparisons of the hyperbolic and exponential equation, as demonstrated by the frequencies reported in the right column of Table 1. The hyperbolic estimates produced a superior fit over the exponential equation.

Parametric (Pearson) and nonparametric (Spearman) correlation analyses were performed with untransformed equation parameter (k) estimate data. The k estimate data were then subjected to log10 transformation and parametric correlations were performed again. The log10 correlation coefficients were very similar to the nonparametric results, while parametric analysis of the untransformed data produced much higher correlations. We assume that the latter were artificially inflated and therefore do not report them. For the model-based estimates, nonparametric coefficients along with the untransformed data are presented, while parametric correlations are presented for the AuC analyses (see Table 2).

As AuC analyses introduce the least amount of error and are suitable for parametric analyses without further manipulation, analyses using AuC are presented here. Consistent with previous work, correlations between patience (5) and age, r(51) = .41, p = .002, and patience and education, r(51) = .40, p = .003, were detected at both sessions. Reward-magnitude effects were also detected, with the larger reward discounted significantly less than the smaller amount at Time 1, t(51) = 8.90, p < .001, and Time 2, t(65) = 9.40, p < .001.

Test-retest correlations differed for analysis type, with the highest correlation resulting from AuC analyses, [r.sub.total](5l)= .75, p < .001, [r.sub.$9.90](51) = .70, p < .001, and [r.sub.$29.90](51) = .72, p < .001. The difference is presumably a result of the fact that AuC analysis does not include equation type dependent error. Similar effects were noted by Ohmura et al. (2006). Temporal discounting rates are highly variable between participants, meaning a few extreme cases could influence the analyses. However, Cook's D values for all cases were below 0.20. Therefore it is unlikely that the observed correlations are the result of any individual cases producing excessive influence on the data (Belsley, Kuh, & Welsch, 1980; Muller & Mok, 1977).

Discussion

Differences in the temporal discounting estimates produced by choice versus equivalent present value (EPV) procedures have meant that in the absence of test-retest data for the latter, the stability of EPV procedures could not be assumed to match that of choice procedures. This has represented an impediment to their use, which is regrettable since temporal discounting rates are of increasing interest as a dependent variable in experimental research (Benjamin, Choi, & Fisher, 2010; Cox, 2005; Wilson & Daly, 2003), necessitating the most sensitive methods of measurement and analysis. The present study begins to address this gap in the literature by demonstrating that an EPV procedure used in a number of empirical works (Kirby & Marakovic, 1996; Kirby & Santiesteban, 2003; Kirby et al., 2005) is temporally stable. The test-retest coefficients obtained are comparable to those obtained with studies using retest intervals of 6 weeks or greater, such as Beck and Triplett (2009), Kirby (2009), and Ohmura et al. (2006). However, they are lower than those obtained by Baker et al. (2003), Johnson et al. (2007), Reed and Martens (2011), and Simpson and Vuchinich (2000)--these studies all employing a retest interval of less than 2 weeks.

The finding that the present test-retest reliability coefficients are lower than those obtained for choice procedures using intervals of less than 2 weeks (Baker et al., 2003; Johnson et al., 2007; Simpson & Vuchinich, 2000) lends support to Kirby's suggestion that the very high retest correlations (up to r = .91) obtained by these studies may be the result of recognition memory effects. It is reasonable to suggest that briefer choice instruments (like the MCQ) would be even more susceptible to recognition memory effects. The retest period in the present study was also quite short (2 weeks); however, it was expected that, since recognition memory is not possible in this procedure, memory effects could be controlled for methodologically. The lower retest coefficients observed in the present study could be interpreted as a more conservative estimate of the stability of discounting rates and encouraging evidence that EPV procedures are preferable to very short retest periods, such as repeated-measures experimental designs where discounting is a dependent variable. An even more direct test of this would be achieved by replicating the present study using brief choice and EPV procedures in a between-subjects manipulation over a 1-week retest interval.

In addition to being the only study, as far as we are aware, to examine the test-retest stability of delay discounting rates using an EPV procedure, the present study is also one of few studies to examine temporal discounting rates using decisions concerning the participants' own money (Kirby & Santiesteban, 2003; Kirby et al., 2005). The majority of discounting studies use either hypothetical measures (Simpson & Vuchinich, 2000) or provide the participant with a stake (Curry, Price, & Price, 2008; Kirby, 2009). As discussed earlier, the prospect of committing one's own money for up to 43 days presents a more vivid opportunity cost, as it is more likely that the participants' money was already budgeted (formally or otherwise) toward a personally salient goal. Opportunity cost is an important issue for decision making (Baker et al., 2003); thus it might be argued that the inclusion of such an aspect increases the ecological validity of the observations.

The present study is also congruent with previous work in that the AuC analysis produced higher reliability estimates than model parameter estimates (Beck & Triplett, 2009; Ohmura et al., 2006). Such consistent results across studies are encouraging evidence that AuC analysis is as reliable as conventional model fitting. In addition, it is much less likely to require data transformation and, as it eliminates systematic equation type dependent error, may be more sensitive to experimental effects. Thus, researchers have cause to argue that in rare cases where significant changes to discounting rates are not detected by model fitting but are detected by AuC analysis (as suggested by Beck & Triplett, 2009, p. 354), accepting the AuC analysis is justified.

While temporal discounting rates appear to demonstrate similar levels of stability to some personality constructs (Kirby, 2009), they are not impervious to external influence (Odum & Baumann, 2010; Wilson & Daly, 2003). This raises a limitation of the present study. The first session took place during the teaching session at the university, while the second session took place during a mid-semester break. This period may have been associated with greater spending for the holidays or the opportunity to commit more hours to paid work. Ohmura et al. (2006) described a similar situation. As income and financial security are known to affect temporal discounting rates (Green et al., 1996; Ostaszewski, Green, & Myerson, 1998), the test-retest coefficients observed in the present study may be conservative. However, Kirby (2009) obtained discounting rates of .71 over a 57-week delay, during which period one might presume that the financial and social situations of participants could change to a greater extent. Stability of this order accounts for approximately half of the observed variability in discounting rates. It may be that this represents the more stable proportion of temporal discounting rates, while the remaining variability observed at any given time may be better accounted for by situational variables and influences (see Odum & Baumann, 2010, for a review of state vs. trait influences on discounting rates).

A second limitation of the study concerns the relatively short delays (a maximum of 43 days) compared to other studies (Beck & Triplett, 2009, for example, measured delays up to 5 years). While this limitation was a necessary logistical restraint as participants were asked to wait for their own funds to be returned, it does mean that the stability of this brief measure has been established only where short delays are concerned. Caution therefore should be exercised in generalizing these results to longer term discounting, especially when using AuC analysis, as a longer delay interval (i.e., 1 to 5 years, compared to 3 to 6 months) would have a disproportionate impact on the area under the curve.

It is also possible that participants with higher temporal discounting rates were less likely to return for the second testing session, meaning that the study takes account of the stability of participants with relatively low discounting rates. However, such self-selection may be ruled out, as observed discounting rates did not differ between those who participated in the first session only and those who participated in both.

The present study is a useful addition to the literature. The brief EPV procedure under investigation compares favorably with brief choice procedures in terms of relative (test-retest) stability. The detection of age, education, and reward size effects also lends confidence with regard to the convergent validity of the measure with choice procedures.

Moreover, brief EPV procedures such as those used in the present study (or Chapman, 1996) may avoid the problem of recognition memory and ceiling effects, making them a more sensitive measure for use in repeated-measures experimental designs than brief choice procedures. While brief choice procedures like the MCQ are extremely valuable for their ease of administration and scoring, experimenters measuring the effects of independent variables on discounting rates would do well to consider brief EPV procedures combined with AuC analyses. Moreover, it has become increasingly clear that discounting is not domain general (Odum & Baumann, 2010). Individuals may have discounting rates for cigarettes (Baker et al., 2003; Mitchell, 1999; Odum, Madden, & Bickel, 2002), good health (Chapman, 1996), or environmental outcomes (Hardisty & Weber, 2009) that differ from those for money. Ideally, any technique for measuring discounting rates should be modifiable such that these differing domains can be studied. For example, one might use Vickrey auctions to have participants (hypothetically) bid present inconvenience for future convenience as a means to determine a community's tolerance for civil engineering works (i.e., "In order to have a new bypass in my local area that would reduce travel time by 20 minutes, I am prepared to take a detour that would make my commute 40 minutes longer for X days").

Temporal discounting is a rapidly growing area of psychological research (Madden & Bickel, 2010). The exciting creativity that accompanies such fertile research interests should be complemented by rigorous psychometric study of the means used to measure it. To this end, the results of the current study mitigate an obstacle to the use of EPV procedures, thus increasing the methodological options available to researchers interested in this important behavioral variable.

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(1.) The term value as used in the temporal discounting literature does not refer to "attractiveness" of an outcome as it does in the colloquial sense. Rather, value describes the extent to which the outcome is salient, or the emotive force the outcome has for the decision maker. Thus, the value of an aversive outcome increases as it becomes more aversive.

(2.) Green et al. (1997) argued that if discounting rates were dependent on amount of reward, an exponential model could potentially account for preference reversals. However, preference reversals are observed in animals (Green et al., 1997; Rachlin & Green, 1972), whereas "amount-dependent" discounting rates (the so-called "magnitude effect") have not been detected in animals (Grace, 1999; Ong & White. 2004). Thus, the hyperbolic model may be regarded as the more parsimonious of the two.

(3.) EPV procedures are elsewhere referred to as "matching procedures" (Read & Roelofsma, 2003, p. 140). However, to avoid confusion with procedures related to Herrnstein's (1958) matching law, we have adopted Loewenstein and Prelec's (1992, p. 576) term.

(4.) Participants were also asked to provide data for a supraliminal priming study. To this end, participants completed a 20-item sentence-unscrambling task before completing the auctions in the first, but not the second, session. Cox (2005) attempted to manipulate temporal discounting rates using a similar text-editing task and found that the task had no detectable effect. In any case, any impact in this research context would make retest estimates more conservative rather than inflating them.

(5.) AuC supplies a "patience" estimate, in that the greater the area under the curve, the lower the participant's discounting rate. Therefore, positive correlations between AuC estimates and demographic variables should be interpreted as representing an inverse relationship with discounting rates.

Justin Harrison

School of Psychology, Charles Sturt University

Ryan McKay

ARC Centre of Excellence in Cognition and its Disorders, Department of Psychology, Royal Holloway, University of London

Correspondence concerning this article should be addressed to Justin Harrison, School of Psychology, Charles Sturt University, Bathurst, New South Wales, Australia 2975. E-mail: jharrison@csu.edu.au
Means, Standard Errors, t Coefficients, and Individual Comparisons
of Root Mean Squared Error (RMSE) Coefficients for Hyperbolic (H)
and Exponential (E) Models

                   Hyperbolic   Exponential
                    model        model
      n              M    SE     M    SE      t   sig  RMSE H < E

Time  67   $9.90  .124  .009  .162  .012  -7.72  .000      66/1
1
          $29.90  .119  .008  .148  .017  -9.28  .000      67/0

Time  53   $9.92  .132  .014  .167  .015 -11.22  .000      51/2
2
          $29.90  .101  .013  .122  .010  -6.45  .000      48/5


Mean, Standard Errors, and Test-Retest Correlation Coefficients
for Temporal Discounting Estimates at Times 1 and 2

                         Time 1  (n =  67)

                $9.90      $29.90      Total      $9.90
               M    SE     M    SE     M    SE     M   SE

Hyperbolic  .482  .105  .208  .043  .298  .063  .352  .082

Exponenuat  .252  .046  .121  .022  .164  .029  .187  .037

AUC         .383  .030  .506  .031  .434  .030  .423  .030

                     Time 2 (n=53)  [r.sub.total]
              $29.90       Total
               M    SE     M    SE

Hyperbolic  .211  .071  .258  .073        .728**

Exponenuat  .118  .038  .139  .035        .716**

AUC         .542  .032  .476  .031        .745**

** p<.001.
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