Attentional dynamics of evidence accumulation explain why more numerate people make better decisions under risk

In decisions under risk, more numerate people are typically more likely to choose the option with the highest expected value (EV) than less numerate ones. Prior research indicates that this finding cannot be explained by differences in the reliance on explicit EV calculation. The current work uses the attentional Drift Diffusion Model as a unified computational framework to formalize three candidate mechanisms of pre-decisional information search and processing—namely, attention allocation, amount of deliberation, and distorted processing of value—which may differ between more and less numerate people and explain differences in decision quality. Computational modeling of an eye-tracking experiment on risky choice demonstrates that numeracy is linked to how people allocate their attention across the options, how much evidence they require before committing to a choice, and also how strongly they distort currently non-attended information during preference formation. Together, especially the latter two mechanisms largely mediate the effect of numeracy on decision quality. Overall, the current work disentangles and quantifies latent aspects of the dynamics of preference formation, explicates how their interplay may give rise to manifest differences in decision quality, and thereby provides a fully formalized, mechanistic explanation for the link between numeracy and decision quality in risky choice.

one of the options is attended to, evidence regarding the alternative option is accumulated in a distorted, i.e., downscaled, manner.A full formal description of the model is provided in the "Methods" section.Crucially, according to the aDDM, there are at least three facets of pre-decisional information search and processing which may explain why more numerate people tend to maximize more: biases in attention allocation, distorted processing of value information, and criteria for committing to a choice.The next paragraphs briefly summarize prior evidence indicating that more and less numerate people might differ in these processes, outline how they can be formalized and disentangled using the aDDM, and formulate hypotheses on how each mechanism might give rise to the link between numeracy and decision quality.
Prior research indicates that more and less numerate people seem to differ in how they allocate their attention across the available information before making a choice.For instance, the more numerate seem to rely more on alternative-based search 8 and attend more to numbers 15 .Crucially, how people allocate their attention across the available information prior to making a decision is robustly linked to choice behavior 11,[16][17][18][19][20][21][22][23][24] , and more specifically, to decision quality 13,14 .Capturing this link between attention allocation and decision quality, the aDDM predicts that predominantly looking at one option (vs. the other one) increases the probability of choosing this option.In this light, a first potential mechanistic explanation for the link between decision quality and numeracy is that more numerate people may attend more to higher-valued options (relative to lower-valued options) compared to less numerate people, and therefore also choose them more.This hypothesis can be tested by using eye-tracking data and assessing whether option-specific attention allocation differs systematically between the more and less numerate.
Moreover, there is evidence that more (vs.less) numerate people seem to perform a higher number of simple heuristic considerations 10 , engage in more effortful and elaborate information search 8 , deliberate longer 25,26 , and sample more information in decisions from experience 27,28 before choice.In summary, these findings suggest that more numerate people engage in more extensive considerations before committing to an option.In the aDDM-as in other models belonging to the evidence accumulation framework 29,30 -differences in how much evidence decision makers consider before choice can be formalized in terms of the boundary separation parameter α.Decision makers who rely on wider boundaries, captured in higher values of α, deliberate more extensively.Consequently, their choices tend to be slower but also more accurate, in the sense of having an increased probability of choosing the higher-valued option on a choice problem.Therefore, another potential explanation for higher decision quality in more (vs.less) numerate people is that they may elaborate more (vs.less) extensively, captured in higher values of the boundary separation parameter α.
Prior work employing models such as Cumulative Prospect Theory (CPT) 31 has shown that choices of less (vs.more) numerate people can be characterized by more nonlinear value functions and probability-weighting functions [32][33][34] , indicating a stronger distortion of options' outcomes and probabilities.However, neo-Bernoullian models such as CPT are conventionally considered to be "as-if " models, which do not strive to offer a mechanistically plausible perspective on the cognitive processes of preference formation, but rather to account for patterns in choice data in a purely descriptive manner 35,36 .The aDDM captures distortions of options' values during preference formation while operating closer to the level of cognitive processes.Specifically, given lower values of the parameter θ, decision makers tend to update their preferences based on more strongly distorted value representations regarding the currently unattended option, such that merely attending more to an option can make it seem disproportionally more attractive.Therefore, decision makers with lower values of θ are predicted to perform worse at maximizing EV, consistent with empirical evidence 13,14 .The third hypothesis thus posits that less (vs.more) numerate people may maximize less because they process information on non-attended options in a more strongly distorted manner, captured in lower values of θ.
These three hypotheses are not mutually exclusive and, in principle, any combination of the outlined mechanisms could operate concurrently.Yet prior work has, at best, investigated the proposed mechanisms in isolation, often relying on statistical rather than computational models.In contrast, the aDDM offers a unified computational framework that allows one to concurrently formalize, disentangle, and quantify all three mechanisms in terms of distinct building blocks of the model's architecture.Thereby, it accounts for the potential interplay between the mechanisms and allows one to assess the extent to which each mechanism alone, or their combination, can explain the link between numeracy and decision making.
The three mechanistic hypotheses were tested by reanalyzing data from an eye-tracking experiment on risky choice.Each participant made 100 binary choices offering safe and risky options.Each safe option offered one outcome, X s > 0, with p X S = 1 , and each risky option offered two outcomes, X R1 > 0 with p X R1 < 1 and X R2 = 0 with p X R2 = 1 − p X R1 .The values of the non-zero outcomes ranged between 1 and 95.On each trial, the non-zero outcome of each option and its associated probability were displayed on the screen.The choice task, illustrated in Fig. 1a, included both problems where the risky option had a higher EV than the safe option (henceforth, risky.betterproblems, 49% of the data) and problems where the safe option had a higher EV than the risky option (henceforth, safe.betterproblems, 51% of the data).Pre-decisional attention allocation was measured using eye-tracking.Numeracy was measured using the 7-item version of the Berlin Numeracy Test 5 .Further details regarding the experimental procedure are provided in the "Methods" section.
Three process-level measures, each corresponding to one of the outlined mechanisms, were computed: the proportion of time spent attending to the higher-valued option on each trial; participants' posterior mean estimates of the aDDM's boundary separation parameter, α; and participants' posterior mean estimates of the aDDM's distorted processing parameter, θ.To obtain the latter two measures, a Bayesian implementation of the aDDM was fitted to data from each participant (see "Methods" for details).Each hypothesis was tested in a mediation analysis 37 consisting of three Bayesian GLMs-a Total Effect Model, a Mediator Model, and a Direct Effect Model-which allow one to assess the extent to which the link between numeracy and decision quality is mediated by a given process-level measure (further details below).

Descriptive analyses
Figure 1b illustrates the distribution of numeracy scores across the 80 participants (37 male, 43 female, M numeracy = 3.7, M age = 28.9years).On average, participants chose the risky option on 26% of the trials, and the option with the higher EV on 72% of the trials.Choice patterns differed notably between problem types: on average, participants chose the option with the higher EV on 48% of the risky.betterproblems and on 95% of the safe.betterproblems.

Is numeracy linked to decision quality?
Before turning to potential differences in cognitive processing between the more and less numerate, it is important to establish whether numeracy was indeed linked to decision quality.Figure 2 displays the proportion of choices of the option with the higher EV-that is, decision quality-in each participant, depending on their numeracy score.Total Effect Models (TEMs), using choice of the option with the higher EV as the dependent variable and numeracy as the predictor, were estimated to test whether numeracy was credibly linked to decision quality before statistically accounting for any of the process-level measures.All posterior mean β-coefficients and 95% posterior intervals are reported in Table 2 (see also Fig. 2).Higher numeracy was credibly linked to higher decision quality when analyzing data from all choice problems concurrently.Importantly, fitting the TEM to data from the two types of choice problems separately revealed that this association was mainly driven by behavior on risky.betterproblems.Here, less numerate people were credibly more likely to choose the safe option, even though it offered a lower EV than the risky alternative, compared to more numerate people.In safe.betterproblems, where the same option was both safe and superior in terms of EV, decision quality was generally very high and not credibly linked to numeracy (see Table 2 and Fig. 2).

Is numeracy linked to differences in processing mechanisms?
The associations between numeracy and the process-level measures are illustrated in Fig. 3. Three Mediator Models (MMs) were estimated to test whether more and less numerate people differed in each of the processlevel measures.The proportion of time attending to the option with the higher EV (i.e., attention allocation), α, and θ were used as the dependent variables.Numeracy, problem type (safe.bettervs. risky.better),and their interaction were included as fixed predictors.The models included a random intercept for each participant.All posterior mean β-coefficients and 95% posterior intervals are reported in Table 1.Since risky.betterwas used as the reference level for the factor problem type, the coefficient of numeracy reflects the link between numeracy and a given process-level measure in the risky.betterproblems.These coefficients indicate that numeracy was credibly and positively linked to all three process-level measures in the risky.betterproblems (cf.Table 1), consistent with the three mechanistic hypotheses.Specifically, the more numerate attended slightly more to the option with the higher EV than the less numerate.The more numerate also tended to rely on wider choice boundaries α-meaning that they required more evidence before committing to a choice.Moreover, they tended to have higher values on the distorted processing parameter θ, indicating that they processed information in a less distorted manner than the less numerate.The latter finding suggests that the choices of more numerate participants seemed to be driven by comparisons between relatively objective representations of the two options' values, whereas less www.nature.com/scientificreports/numerate people seemed to be more strongly drawn towards whatever option they were currently looking at, regardless of whether its value was higher or lower than that of the other option, and by how much.Overall, these results indicate that in risky.betterchoice problems, where more and less numerate people differed credibly in decision quality, they also differed in all three facets of cognitive processing in the direction expected under the three mechanistic hypotheses.The relationship between numeracy and decision quality separately for risky.betterproblems and safe.betterproblems.Each subplot includes the posterior mean estimate and corresponding 95% posterior interval of the β-coefficient representing the link between numeracy and decision quality, obtained in the Total Effect Model for the respective (sub-)set of data (see also Table 2).Associations between numeracy and each of the process-level measures, and how they depend on the type of choice problem.Solid (dashed) lines represent the associations in risky.better(safe.better)problems.The β-coefficients and 95% posterior intervals for the effects of numeracy, problem type, and their interaction on each of the process-level measures are reported in Table 1.(a) Association between attention allocation, i.e., the proportion of time attending to the option with the higher EV relative to time attending to any option, and numeracy.Each dot (triangle) represents the average attention allocation score in a given participant in the risky.better(safe.better)problems.(b) Association between the aDDM's distorted processing parameter, θ, and numeracy.Each dot (triangle) represents the posterior mean estimate of θ in a given participant in the risky.better(safe.better)problems.(c) Association between the aDDM's boundary separation parameter, α, and numeracy.Each dot (triangle) represents the posterior mean estimate of α in a given participant in the risky.better (safe.better)problems.
Table 1.Posterior mean β-coefficients and 95% posterior intervals for the Mediator Models using either attention allocation, α, or θ as the dependent variable, and numeracy, problem type, as well as their interaction, as fixed predictors.MMs were estimated for different process-level measures as dependent variables (attention allocation, α, and θ).The predictor variable numeracy was z-standardized.Boldface indicates credible effects.MM = Mediator Model.www.nature.com/scientificreports/

MM
The interaction terms between numeracy and problem type (cf.Table 1) reflect if and how these patterns changed in the safe.betterproblems compared to the risky.betterproblems.The interaction term in the MM for α was not credible.However, credible and negative interactions indicated that numeracy was less strongly linked to attention allocation and to θ in safe.betterproblems compared to risky.betterproblems (cf.Table 1).As illustrated in Fig. 3, the link between numeracy and attention allocation even seemed to reverse in safe.betterproblems.That is, in safe.betterproblems, where numeracy was not credibly linked to differences in decision quality, also differences in two facets of cognitive processing between the more and less numerate were less pronounced or even reversed in direction.

Are process-level measures linked to decision quality?
Next, Direct Effect Models (DEMs), using choice of the option with the higher EV as the dependent variable and numeracy as well as one of the three process-level measures as the predictors, were estimated to test whether each process-level measure was linked to decision quality in the expected direction.Since differences in decision quality and in cognitive processing between the more and less numerate hinged on the structure of the choice problem, the DEMs were first fitted to data from all choice problems concurrently and subsequently also to data from risky.better and safe.betterproblems separately.All posterior mean β-coefficients and 95% posterior intervals are reported in Table 2.
Attention allocation was positively and credibly associated with decision quality across all (sub-)sets of data.That is, consistent with the attention-allocation hypothesis, the more people attended to higher-valued options, the more likely they were to choose them.The boundary separation parameter α was credibly and positively linked to decision quality when considering all data concurrently and risky.betterproblems alone, but not when considering safe.betterproblems alone.That is, consistent with the boundary-separation hypothesis, people who relied on wider boundaries were indeed more likely to choose the higher-valued option-but only in risky.betterproblems (see also Fig. 4).The distorted processing parameter θ was credibly and positively linked to decision quality across all (sub-)sets of data.That is, consistent with the distorted-processing hypothesis, people with higher values of θ, who processed information on the options' values in a less distorted manner, were systematically more likely to choose the higher-valued option (see also Fig. 4).Not only did more and less numerate people differ in the three facets of pre-decisional information processing, but such differences in processing also tended to be associated with differences in decision quality in the direction predicted by the three mechanistic hypotheses-at least in the risky.betterproblems.

Do process-level measures mediate the link between numeracy and decision quality?
To assess the extent to which a given process-level measure can explain the link between numeracy and decision quality, one can compare the coefficients of numeracy between the TEMs and the DEMs fitted to the same (sub-)sets of data (Table 2).If the link between numeracy and decision quality is-at least partly-mediated by a given processing mechanism, one would expect the link between numeracy and decision quality to be reduced after (vs.before) statistically accounting for the corresponding process-level measure 37 .This comparison is most meaningful in the analyses considering all data concurrently and risky.betterproblems alone.After all, there was no credible link between numeracy and decision quality to be explained by the process-level measures in safe.betterproblems, even before accounting for any of these measures.Nevertheless, for completeness, Table 2 reports the full set of results, including those for safe.betterproblems.
The link between numeracy and decision quality was slightly reduced after accounting for attention allocation (TEM vs. DEM attention allocation; Table 2) in the analysis of risky.betterproblems, but not in the analysis considering all data.In contrast, statistically accounting for α (TEM vs. DEM α; Table 2) considerably reduced the link between numeracy and decision quality, both when considering all data and the risky.betterproblems alone.Likewise, statistically accounting for θ (TEM vs. DEM θ; Table 2) considerably reduced the link between numeracy and decision quality, both when considering all data and the risky.betterproblems alone.
These findings provide evidence that each of the proposed mechanisms partly mediated the link between numeracy and decision quality in the risky.betterproblems.This pattern carried over to the analysis of the full data regarding the mechanisms captured in the aDDM parameters α and θ, but not regarding attention allocation.

Interplay between the mechanisms
While the link between numeracy and decision quality was reduced after statistically accounting for each of the process-level measures to some extent, it did not vanish entirely.Crucially, however, the proposed mechanisms do not act in isolation.Instead, the aDDM posits that they concurrently shape the dynamics of decision making.To what extent can their combined effects explain the link between numeracy and decision quality?To address this question, additional DEMs, including numeracy as well as a combination of process-level measures as predictors, were estimated.The highest Variance Inflation Factor across these DEMs was 1.2, thus alleviating potential concerns about excessive multicollinearity between the predictors (for details, see Supplementary Information S5).
Accounting for both θ and α concurrently (DEM α, θ in Table 2) reduced the link between numeracy and decision quality to a considerably larger degree than accounting for either parameter in isolation.Accounting for attention allocation in addition to α and θ (DEM α, θ, attention allocation in Table 2) did not reduce the link between numeracy and decision quality further in the analysis of all data, and only slightly in the analysis of risky.betterproblems.That is, accounting for differences in how people allocated their attention across options seemed to add comparably little to explaining the link between numeracy and decision quality, beyond what is achieved by accounting for the parameters capturing latent characteristics of information processing, θ and α.
These analyses underscore that the link between numeracy and decision quality does not stem from a lone determinant.Instead, it seems to result from the interplay of at least two facets of pre-decisional information Figure 4. Associations between the posterior mean estimates of aDDM parameters and the proportion of choices of the option with the higher EV-i.e., decision quality-displayed for data and estimates from all choice problems concurrently (left subplot), as well as separately for risky.betterproblems (middle subplot) and safe.betterproblems (right subplot).Each subplot includes the posterior mean estimate and corresponding 95% posterior interval of the β-coefficient representing the link between a given aDDM parameter and decision quality obtained in the Direct Effect Model for the respective aDDM parameter and (sub-)set of data (see also Table 2).(a) Association between the aDDM's boundary separation parameter α and decision quality.(b) Association between the aDDM's distorted processing parameter θ and decision quality.numerate did not credibly differ in decision quality, they also differed less in terms of attention allocation and distorted processing.
Mediation analyses highlighted that none of the outlined mechanisms alone is sufficient to fully account for the link between numeracy and decision quality.Instead, consistent with the integrative formal framework offered by the aDDM, the different mechanisms seem to concurrently give rise to differences in decision quality, and their combined effects mediate the effect of numeracy on decision quality almost entirely.Notably, differences in overt attention allocation between the more and less numerate added relatively little explanatory power compared to differences in how much evidence they required and also in how objectively they processed the information on the options' values before committing to a choice.Importantly, these findings highlight that studies focusing on any individual facet of pre-decisional search and processing in isolation may be prone to overlooking a crucial part of the explanation for individual differences in decision quality.Thereby, they underscore the merits of relying on an overarching computational framework to concurrently formalize various candidate mechanisms.
One puzzling finding in the literature on decision quality and numeracy is that, although more numerate people tend to behave in a manner highly consistent with EV maximization, they rarely seem to explicitly calculate the options' EVs to arrive at their choices 10 .The formal framework of sequential sampling offers a novel and genuinely mechanistic explanation for this apparent paradox.Specifically, the highly numerate might achieve high levels of EV maximization by relying on a simple mechanism of internal sampling, which frugally bypasses the comparatively elaborate operations of explicit EV calculation.Indeed, the aDDM's decision process can be implemented by simulating sequential samples from each option's outcome distribution, summing up the partly distorted representations of the sampled outcomes within each option, and comparing the thus accumulated evidence between the options 24 .Crucially, depending on parameter settings, this simple process can achieve high levels of EV maximization, although it requires no weighting of outcomes by their probabilities, i.e., EV calculation.Therefore, in line with the notion that probabilistic cognition may often be implemented in terms of approximate sampling mechanisms, rather than precise calculation 38 , the highly numerate might achieve relatively high levels of EV maximization by exploiting such a simple mechanism of internal sampling.
The current results also add novel insights regarding thus far barely studied characteristics of the aDDM's core parameter θ.Previous work has demonstrated that θ can be modulated by features of the choice task 39,40 , suggesting that this parameter may have a state-like component.However, it has also been shown that substantial differences in θ between individuals exist within the same task, that these differences are related to individual differences in decision quality, and that accounting for such differences notably improves predictions of choice behavior 14 .To date, however, it is unclear how such individual differences in θ relate to other aspects of cognitive performance.The current finding that θ is related to numeracy constitutes a first step in this direction.Since decisions of people with lower (higher) values of θ are driven to a higher (lower) extent by how they allocate their attention across the available options, this insight also promises a better understanding of individual differences in susceptibility to attentional manipulations in decision making.Specifically, the current results suggest that more (less) numerate people might be less (more) susceptible to manipulations or choice architectures that aim to bias choice behavior in favor of a target option by making it attentionally salient, for instance, in the domain of marketing 41 .
3][34] provide counterexamples].Emphasizing the immense but largely untapped potential of computational approaches, the current work demonstrates that individual differences in decision quality between the more and less numerate can be precisely characterized and explained by locating individuals in a continuous parameter space of a mechanistic model.In this sense, the current work can be understood as an early step toward deriving a computational phenotype of people high and low in numeracy 42 .By doing so, one obtains insights into latent processes underlying manifest differences in behavior between the more and less numerate, which are difficult or even impossible to measure or observe directly.Crucially, only after understanding the association between numeracy and decision quality mechanistically, one can start to think about how such mechanisms might be manipulated 43,44 .Therefore, deriving a computational phenotype of numeracy is not only theoretically illuminating.It also bears the promise of designing targeted interventions to improve decision making-an approach that has already gained some prominence in other domains 45,46 .Even though the current results constitute only an early step in this direction and are not aimed at immediately deriving practical interventions, they point toward a promising route to enhance decision quality in the less numerate-namely, by designing interventions that allow one to manipulate the aDDM's parameters α and θ 39,40 .Overall, the current work thus showcases that computational modeling in the evidence accumulation framework is a powerful tool for understanding how numeracy shapes the dynamics of preference formation, bearing the promise to explain individual differences in decision making under risk from a genuinely mechanistic perspective.

Data
The hypotheses were tested by reanalyzing existing data from a risky choice experiment 47 investigating attentional foundations of the description-experience gap in risky choice 48 .Data were collected at the Max Planck Institute for Human Development Berlin.The study was approved by the Ethics Committee of the Max Planck Institute for Human Development, Berlin (A 2021-11), and all methods were carried out in accordance with relevant guidelines and regulations.The experiment consisted of two conditions, varied between subjects, one in which participants made decisions from description (DfD, where people consult abstract descriptions of the options' outcomes and their probabilities) and one in which participants made decisions from experience (DfE, where people learn about the options by repeatedly sampling their outcome distributions) 48 .In the DfD condition, eye-tracking data were collected to capture pre-decisional information search.Since such data are necessary to problem determined the probability p of the corresponding outcome displayed to the yoked participant in the DfD condition.Due to this yoked design, some choice problems in the DfD condition also offered a choice between two safe options-such as a choice between a 100% chance to win $5 and a 100% chance to win $15which are trivial and were excluded from the current analyses.Moreover, choice problems where both options had an equal EV were excluded from the current analyses, because decision quality cannot be assessed on such problems.The position of outcomes and probabilities (on top or bottom of the screen) and of options (on the left or right side of the screen) was randomized across choice problems, uniquely within each participant.The option displayed on the left (right) side of the screen could be chosen in a self-paced manner by pressing the key F (J) on each choice problem.After the risky choice task participants completed several cognitive, affective and demographic measures, including the 7-item version of the Berlin Numeracy Test 5 .Each participant's numeracy score was computed as the number of correct responses on the Berlin Numeracy Test.

Preprocessing of eye-tracking data
Individual fixations and their durations were extracted from raw samples of the eye-tracking data using a velocity-based algorithm 49 implemented in the saccades package in R 50 .The location of identified fixations on each choice problem was classified into two areas of interest (AOIs), corresponding to the two options on each choice problem.For each choice problem, the relative dwell time fixating on each option's AOI, relative to the total dwell time fixating on any option's AOI, was calculated.Choice problems on which no fixations in any AOIs were identified were excluded from analyses.The final data set used for analyses included a total of 7051 trials from 80 participants.

Statistical approaches
The statistical analyses rely on Bayesian approaches for data analysis 51,52 .Bayesian Generalized Linear Models (GLMs) are used for regression analyses, implemented using the package brms in R 53 .In general, the posterior mean of the β-coefficients representing the effects of interest in the GLMs and the corresponding two-sided 95% posterior intervals are reported.A given effect is considered statistically credible if the 95% posterior interval excludes 0. All logistic regression analyses using choice of the option with the higher EV as the dependent variable (TEMs and DEMs) were first conducted using data from all choice problems, followed by separate analyses for risky.betterproblems and safe.betterproblems.All continuous predictor variables were z-standardized, rendering the resulting estimates of coefficients comparable in terms of magnitude.Correction for multiple comparisons in the frequentist sense was not applied.However, zero-centered Gaussian distributions with a standard deviation of 1, N (0, 1) , were used as priors for all β-coefficients in these GLMs.Such priors shrink the posterior distributions towards zero (indicating no effect) to some extent, thus guarding against overconfident inferences regarding the existence and magnitudes of effects.To check for multicollinearity in DEMs including more than one processlevel measure as predictors, Variance Inflation Factors (VIFs), reported in detail in Supplementary Information S5, were computed using the package performance in R 54 .Leave-one-out information criteria (loo−IC), reported in Table 2 and computed using the package brms in R 53 , allow one to compare the predictive performance of models predicting decision quality based on different combinations of predictors.Lower values of loo−IC compared to other models estimated using the same data indicate a better predictive performance of the model.

Figure 1 .
Figure 1.(a) Illustration of the risky choice task.Each trial of the task consisted of a choice between two options, displayed on the left and right side of the screen.The position of outcomes and probabilities on the top (vs.bottom) of the screen was randomized across trials.Each trial of the choice task was preceded by a 500 ms fixation period.The choice itself was self-paced.Font size in the illustration was increased compared to that in the real experiment to ensure readability.(b) Distribution of numeracy scores in the sample of participants.

Figure 2 .
Figure2.Proportion of choices of the option with the higher EV, i.e., decision quality, depending on numeracy.(a) The relationship between numeracy and decision quality across all choice problems.(b) The relationship between numeracy and decision quality separately for risky.betterproblems and safe.betterproblems.Each subplot includes the posterior mean estimate and corresponding 95% posterior interval of the β-coefficient representing the link between numeracy and decision quality, obtained in the Total Effect Model for the respective (sub-)set of data (see also Table2).

Figure 3 .
Figure3.Associations between numeracy and each of the process-level measures, and how they depend on the type of choice problem.Solid (dashed) lines represent the associations in risky.better(safe.better)problems.The β-coefficients and 95% posterior intervals for the effects of numeracy, problem type, and their interaction on each of the process-level measures are reported in Table1.(a) Association between attention allocation, i.e., the proportion of time attending to the option with the higher EV relative to time attending to any option, and numeracy.Each dot (triangle) represents the average attention allocation score in a given participant in the risky.better(safe.better)problems.(b) Association between the aDDM's distorted processing parameter, θ, and numeracy.Each dot (triangle) represents the posterior mean estimate of θ in a given participant in the risky.better(safe.better)problems.(c) Association between the aDDM's boundary separation parameter, α, and numeracy.Each dot (triangle) represents the posterior mean estimate of α in a given participant in the risky.better (safe.better)problems.

Table 2 .
Posterior mean β-coefficients and 95% posterior intervals for the Bayesian logistic generalized linear models using choice of the option with the higher EV as the dependent variable.Results are shown separately for analyses across all data (top panel), analyses including only data from risky.better problems (middle panel), and analyses including only data from safe.better problems (bottom panel).The TEMs and DEMs included different combinations of predictor variables (numeracy, attention allocation, α, and/or θ).Continuous predictor variables were z-standardized.Coefficients and 95% Posterior Intervals are reported for all predictors included in a given model.Loo-IC indicates the leave-one-out information criterion.Lower values of loo-IC indicate a better predictive performance of the model compared to other models estimated using the same data.Boldface indicates credible effects.TEM = Total Effect Model.DEM = Direct Effect Model.