Figure 1.
Trial structure of the task in Experiment 1.
A: sequence of two Choice Trials, demonstrating the display of outcome options and outcome phases. The dotted arrow denotes how choices on previous trials determine expected shock rates on the future trials referred to by those choices. B: An example No Choice Trial.
Figure 2.
Observed time preference: Experiment 1.
Mean probability across participants (N = 25) of choosing the more delayed shock outcome (S2) over the sooner shock outcome (S1) [referred to as p(Choose S2)] as a function of difference in delay between delivery of S2 and S1 (D2 – D1), expressed in units of trials. Delay difference (D2 - D1) is binned into tertiles, corresponding to short (1–10 trials), medium (11–20 trials) and long (>20 trials) delay differences. A: choice probabilities for all choices. At delay difference of zero, S1 and S2 would occur at the same time-point; since there are equal numbers of trials in which S1>S2 as in which S2>S1, this plot is theoretically bounded to cross the probability axis at (Choose S2) = 0.5, represented by the blue and red square. Blue circles represent choice probabilities for the relief frame, red circles choice probabilities for the pain frame. Error bars represent one standard error from the between subject mean. A 2-way repeated ANOVA revealed a significant main effect of both frame [F(1,24) = 9.505; p = 0.005)] and delay [F(3,72) = 8.156; p = 0.002)], as well as a significant delay by frame interaction [F(3,72) = 4.169; p = 0.023)]. B: Choice probabilities for choices in which the more delayed option was a smaller number of shocks. At delay difference of zero, S1 and S2 would occur at the same time-point, under which circumstance it might be assumed that participants would show preference for the smaller number of shocks, denoted by the blue and red square.
Figure 3.
Observed time preference in individual participants categorized by time preference.
p(Choose S2) as a function of delay difference, expressed in units of trials, for all 25 participants included in the modeling analysis. Choice probabilities shown are the mean of those on the two frames. Delay difference scaling is identical that in Figure 2. Time preference is approximated by the slope of the choice probability lines. A: participants with no significant time preference at any delay. B: participants who show positive time preference, but no significant negative time preference at any delay. C: participants who show negative time preference, but no significant positive time preference at any delay. D: participants with initial negative time preference followed by significant positive time preference at longer delays. Data are plotted as solid lines to assist visualization of the choice patterns. Each gray line represents data from a single participant. The bold purple lines represent the between-subject means in each category. At delay difference of zero, S1 and S2 would occur at the same time-point; since there are equal numbers of trials in which S1>S2 as in which S2>S1, the plots are theoretically bounded to cross the probability axis at p(Choose S2) = 0.5.
Figure 4.
Model predictions on the task: Experiment 1.
p(Choose S2) as a function of delay difference according to alternative models of dread. Choice probabilities shown are the mean of those on the two frames. Delay difference scaling is identical that in Figure 2. The fine gray lines represent mean p(choose S2) for the four participant subgroups shown in Figure 3. Data points marked by blue squares, joined with lines for illustrative purposes, represent model data simulated at the parameter values denoted in each panel. These do not represent the results of model fitting, but serve to illustrate the basic form of the alternative model predictions. Notably different parameterizations of the more complex models can produce diverse shapes of choice frequency plot (see Figures S2, S3, S4, S5). Error bars represent one standard deviation of the binomial distribution. In each case the softmax inverse temperature parameter, , is set to 0.25, a representative value.
Figure 5.
Model comparison and framing effects: Experiment 1.
A: Bayesian Information Criterion (), summed across participants (N = 25) for the alternative models. Lower values of
indicate better fits of the model. Exponential Dread outperformed other models, with Undiscounted Exponential Dread providing the most parsimonious fit at the group level, indicated by the red circle. B: Mean frequency of choosing sooner pain across all choices by all participants in either frame. Error bars show one standard error from the mean in each direction. Two-tailed paired t-test showed significant difference between the two frames t(32) = 2.84, p = 0.0077. This result was confirmed with non-parametric testing for differences between paired samples using the Wilcoxon Signed Rank test, which revealed significant differences between the two medians (N = 33, Z = −2.6, p = 0.0093). C: Results of fitting the general form Exponential Dread model to both pain and relief frames, whilst restricting which parameters were allowed to vary between frames. In the unrestricted framing model (All-Framing) all four parameters, the inverse softmax temperature,
, the discount parameters,
and
, and the anticipation parameter,
, were applied separately to each frame, yielding an eight parameter model. In the fully restricted framing model (No-Framing) all parameters were constrained to be equal across frames, yielding a four-parameter model. The best fit, indicated by the red circle, was provided by a four-parameter model in which
,
and
were fixed across frames, leaving between-frame differences explained by differences in
(
-Framing). Likelihood ratios are displayed at both the group level and the individual level, strongly favoring the
-Framing model over the No-Framing model at both the group (fixed effects) (LR = 10108∶1, χ2 = 497.3, p<0.001, d. f. = 25) and individual levels (Mean individual LR = 2088∶1, χ2 = 19.9, p<0.001, d. f. = 1) .
Figure 6.
Model comparison with non-linear utility: Experiment 1.
The results of an identical model comparison, performed with subject-specific non-linear utility functions for pain, derived from subjective ratings of stimuli with differing shock rates, as shown in Figure S6. Blue bars represent summed values for linear utility models, gray bars the
values for non-linear utility models. For each alternative model using linear utility provided better model fits, as indicated by lower
values. Importantly, the rank order of the models was largely unchanged using non-linear utility, the only exception being that the general form Exponential Dread model outperformed the Restricted Discounted version with non-linear utility, but not with linear utility. The green bar labelled “Fit Utility Only” represents the result of implementing the Null model with a freely fitted three-parameter Weibull utility function, showing that a variable utility function alone was unable to account competitively for the observed data.
Figure 7.
Time preference of sample participants on Experiment 1 and fits of the (discounted) Exponential Dread model.
Observed p(Choose S2), combined across both frames, as a function of delay difference, expressed in units of trials, is displayed for a single participant from each of the four subgroups shown in Figure 3, indicated by the purple circles. Delay difference scaling is identical that in Figure 2. Data simulated from the general form Exponential Dread model at the maximum likelihood parameter estimates for each participant, subsequently combined across frames, are plotted as cyan squares. Error bars represent one standard deviation of the binomial distribution. A: a participant with zero time preference B: a participant with positive time preference (left hand column). C: a participant with negative time. D: a participant with reversing time preference: showing initial negative time preference reverting to positive time preference at longer delay differences. The general form of the Exponential Dread model adequately captures all four patterns of time preference.
Figure 8.
Time preference for a hypothetical painful dental appointment: Experiment 2.
Observed p(Choose A2) is plotted as a function of the delay to the later appointment; the sooner appointment was always at 0 days, i.e. “today”. Error bars represent one standard error from the group mean. A: group mean p(Choose A2) for all participants (N = 30). B: mean p(Choose A2) in the zero time preference group (N = 12). C: mean p(Choose A2) in the positive time preference group (N = 3). D: mean p(Choose A2) in the negative time preference group (N = 15).
Figure 9.
Model comparison for Experiment 2.
Bayesian Information Criterion (), summed across participants (N = 30) for the alternative models. Lower values of
indicate better fits of the model. Undiscounted Exponential Dread provided the most parsimonious fit at the group level, indicated by the red circle.