Quantifying the contribution of individual variation in timing to delay-discounting

Delay-discounting studies in neuroscience, psychology, and economics have been mostly focused on concepts of self-control, reward evaluation, and discounting. Another important relationship to consider is the link between intertemporal choice and time perception. We presented 50 college students with timing tasks on the range of seconds to minutes and intertemporal-choice tasks on both the time-scale of seconds and of days. We hypothesized that individual differences in time perception would influence decisions about short experienced delays but not long delays. While we found some evidence that individual differences in internal clock speed account for some unexplained variance between choices across time-horizons, overall our findings suggest a nominal contribution of the altered sense of time in intertemporal choice.

. Each plot is the softmax-hyperbolic fit for each subject in the intertemporal choice task (new group, N = 26). In each panel, the marker and error bar indicate the mean and binomial confidence intervals of the subjects choices for that offer. The smooth ribbon indicates the BHM model fits (at 50, 80, 99% credible intervals). At the top of each subject plot we indicate the mean estimates of log(k) and τ for each task for that subject. We also indicate the Bayesian r 2 for each task. Plots from left to right, row-by-row are ordered by discount factor (as estimated using BHM) for the verbal short delay task (SV).   Plots from left to right, row-by-row are ordered by ICSe. Thus, the first plot is the data of the outlier subject, who is removed from analysis whenever proxies for ICS and ICSError were used.  Figure S5A. n=1500 responses (5 intervals, 3 repeats, 2 tasks, 50 subjects).  Table S2. kfold power and linear model comparison: the 2 nd column shows the difference between the expected log pointwise predictive density (∆ELPD 3 ) of the best (the first row) model with the model on that row.

Regression analysis
We ran linear regressions according to equations 4 and 5 separately for the followup and the new groups (main text, Methods). The remaining regression analysis tables were displayed in this section. Note: * p<.05; * * p<.01 Table S3. Follow-up Group log(k SV ) Regression Results

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Production Estimation n = 50 subjects 1500 trials A B C Figure S5. Subjects performance is consistent with scalar timing. A. The mean and standard deviation across all responses for each interval and task. The standard deviation grows with the interval. These data are also presented as Table S1. Production in red and Estimation in blue. The position of the points for Estimation (blue) on the x-axis were shifted to more easily visualize the standard deviation. The dashed line shows the unity line (y=x Note: * p<.05; * * p<.01 Table S5. Follow-up Group log(k LV ) Regression Results

Demographics and BIS
We used permutation tests to compare timing and discount factors between demographic categories. We did not find any effects of gender or nationality (all p > .1) across discount factors (log(k SV ), log(k LV )) and timing variables (ICSe, ICSp, ICSError). We used the Barratt Impulsiveness Scale (BIS-11 4 ) as a standard measure of impulsivity. The mean total score for the new group was 64.31 (std = 9.57), which was consistent with other reports in the literature 5 . In Lukinova et al. 1 we did not find significant correlations between discount factors and the BIS. We replicated those null results with the new group of subjects: no significant correlations between BIS and (a) log(k SV ) (Pearson r = −.10, p = .640) or (b) log(k LV ) (Pearson r = −.02, p = .909).
Impulsive individuals may be characterized by an impairment in their ability to make temporal judgements. However, studies provide inconsistent and contradictory evidence on the relationship between BIS and timing 6 . In line with Corvi et al. 7 , we did not find significant relationship (for the follow-up and the new group separately and combined, the correlation coefficients were reported for the latter) between BIS and timing variables: (a) ICSe (Pearson r = .11, p = .432), (b) ICSp (Pearson r = .13, p = .375), (c) ICSError (Pearson r = −.10, p = .482).

Post hoc analysis
Although this analysis was not preregistered, we checked whether timing was related to decision noise in the intertemporal choice task and performed a joint group analysis.
The joint group analysis was used to test whether the increase in statistical power would change the null results we got when exploring the association of timing with discounting. In the main text we already listed some of the results of the joint analysis. Here in figures S6B-C, we re-plot correlations of timing variables with discount factors separately for the followup and the new groups. As in the joint analysis, there were no significant correlations between ICS measures and delay discounting coefficients. Also, we re-plot the kernel density estimations of ICSe as proxy for ICS divided by differences of subject's impulsivity in seconds compared to that in days in figures S6D-E separately for two groups. The permutation tests confirmed that there were no significant differences in ICSe between K LV > K SV and K SV > K LV subgroups for the follow-up group, M K LV >K SV = 0.87 and M K SV >K LV = 0.84, p = .624 and for the new group, M K LV >K SV = 0.83 and M K SV >K LV = 0.94, p = .178. Thus, as in the joint analysis in the main text, we did not find any significant difference in internal clocks between subjects who were more impulsive in the seconds task than the days task and those, who were more impulsive in the days task than the seconds task.
In the joint analysis (compared to the new group analysis in the main text), dropping ICSe (a proxy for ICS) no longer resulted in a significant decrease in the likelihood for explaining short delay in Figure S6F. As before, we generated linear regression models of log(k) for each task (short delay and long delay) against the discount factor of the other task, as well as timing variables. In order to test which factors were important, we dropped each factor and tested whether the decrease in likelihood was significant by a χ 2 test. We plot the change in AIC, with significant drops in black (p < .0125, Bonferroni Corrected p < .05/4).
Further, we looked at the possibility of a delay-dependent relationship of timing with discounting. According to Figure  S6G, there was no evidence that people who tend to overestimate 64 seconds (or other relatively longer delays) were more impulsive or more patient for 64 second delays (or other relatively longer delays) compared to the relatively smaller delays that were estimated or produced more accurately.
In addition, we used the following linear models to test the contribution of timing variables to each discount factor while controlling for the group the subjects belonged to: The detailed regression results in Tables S6 and S7 for joint groups data were similar to the respective regression tables here and in the main text. There was no main effect of the group. In the reduced models, we found that log(k LV ) significantly predicted log(k SV ) (Table S6) and log(k SV ) significantly predicted log(k LV ) (Table S7). These reduced models contained the only factors that mattered (according to the joint 'drop 1' analysis in Figure S6F). As in the main text, we also considered the model where two regressors (log(k LV ) and ICSe) predict log(k SV ) in Table S6. With the addition of ICSe we explained 34% (an increase in 5% compared to 29% for reduced model) variance, but the coefficient for ICSe was not significant. Using subjective time, rather than objective time in joint analysis did not improve our ability to predict subjects' choices (Table S8). In fact, when we used subjective time from time production task we significantly decreased the correlation between the delay discounting coefficients. Follow-up D Figure S6. (A-C) Correlation plots (N = 49, separately for the followup and new groups arranged in columns highlighted by a colored border) between noise and discount factors (y-axis, rows) and ICSe, ICSp, and ICSError (x-axis). Each circle is one subject. Pearson's r is reported on the figure ( * * p < .01; for all correlations with discount factors p > .05). Best linear fit line (y ∼ x) is displayed. (D-E) Kernel density estimations of ICSe and ICSp as proxies for ICS and ICSError divided by positive or negative difference between discount factor in short and long delay tasks for both of our subjects' groups. (F) Drop-one regression analysis for joint data no longer resulted in a significant decrease in the likelihood for any variables dropped, except for the discount factor. (G) Relationship between timing ratio (TeRatio(t) and T pRatio(t), x-axis) and the proportion of later choices (y-axis) grouped by a particular delay (or actual time interval, columns), intertemporal (rows) and timing (color) tasks.

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Dependent variable: log(k LV ) (1)  Table S8. Pearson Correlation between log(k SV ) and log(k LV ). 'Significantly Better?' answers whether there was significant increase in correlations from the objective time model to the respective subjective time model tested using R package cocor.