Statistical equivalence and test-retest reliability of delay and probability discounting using real and hypothetical rewards

Abstract

Delay discounting (DD) and probability discounting (PD) refer to the reduction in the subjective value of outcomes as a function of delay and uncertainty, respectively. Elevated measures of discounting are associated with a variety of maladaptive behaviors, and confidence in the validity of these measures is imperative. The present research examined (1) the statistical equivalence of discounting measures when rewards were hypothetical or real, and (2) their 1-week reliability. While previous research has partially explored these issues using the low threshold of nonsignificant difference, the present study fully addressed this issue using the more-compelling threshold of statistical equivalence. DD and PD measures were collected from 28 healthy adults using real and hypothetical $50 rewards during each of two experimental sessions, one week apart. Analyses using area-under-the-curve measures revealed a general pattern of statistical equivalence, indicating equivalence of real/hypothetical conditions as well as 1-week reliability. Exceptions are identified and discussed.

Keywords: Delay discounting; Hypothetical outcomes; Probability discounting; Real outcomes; Statistical equivalence; Test–retest reliability.

Conflict of interest statement

The authors report no potential conflicts of interest.

Copyright © 2013 Elsevier B.V. All rights reserved.

Figures

Figure 1. Equivalence of discounting of real and hypothetical rewards as a function of type of discounting and session

(A) Observed discounting expressed as a ratio of real/hypothetical rewards, jittered vertically to better distinguish individual points. S1 and S2 represent session 1 and session 2, respectively. The vertical dashed lines represent the equivalence region (0.8, 1.25). Filled points fall within the equivalence region, and open points do not. (B) Observed discounting expressed as a ratio of real/hypothetical rewards. S1 and S2 represent session 1 and session 2, respectively. Each line is marked at the the median ratio of real/hypothetical rewards. Thick bars represent the 90% confidence intervals for the median ratios, and thin bars represent the 95% confidence intervals for the median ratios. The nonparametric nature of the 90% and 95% confidence intervals is such that an upper or lower bound can be common to both. Two measurements are considered statistically equivalent if (i) the endpoints of the thick bar fall entirely within the equivalence region, (0.8, 1.25), and (ii) the thin bar covers 1.0. If the thin bar does not cross 1.0, the two measures are statistically different, and therefore cannot be statistically equivalent.

Figure 2. Equivalence of two measures of discounting taken one week apart, as a function of type of discounting and reward type (real, hypothetical [hyp])

(A) Observed discounting expressed as a ratio of session 1/session 2, jittered vertically to better distinguish individual points. The vertical dashed lines represent the equivalence region (0.8, 1.25). Filled points fall within the predeterimined equivalence region (0.8, 1.25); open do not. (B) Observed discounting expressed as a ratio of session 1/session 2. Each line is marked at the median ratio of session 1/session 2 discounting. Thick bars represent the 90% confidence intervals for the median ratios, and thin bars represent the 95% confidence intervals for the median ratios. The nonparametric nature of the 90% and 95% confidence intervals is such that an upper or lower bound can be common to both. Two measurements are considered statistically equivalent if (i) the endpoints of the thick bar fall entirely within the equivalence region, (0.8, 1.25), and (ii) the thin bar covers 1.0. If the thin bar does not cross 1.0, the two measures are statistically different, and therefore cannot be statistically equivalent.