The left digit effect in an unbounded number line task.

The left digit effect in number line estimation refers to the phenomenon where numerals with similar magnitudes but different leftmost digits (e.g., 19 and 22) are estimated to be farther apart on a number line than is warranted. The effect has been studied using a bounded number line task, a task in which a line is bounded by two endpoints (e.g., 0 and 100), and where one must indicate the correct location of a target numeral on the line. The goal of the present work is to investigate the left digit effect in an unbounded number line task, a task that involves using the size of one unit to determine a target numeral's location, and that elicits strategies different from those used in the bounded number line task. In a preregistered study, participants (N = 58 college students) completed four blocks of 38 trials each of an unbounded number line task, with target numerals ranging between 0 and 100. We found a medium and statistically reliable left digit effect (d = 0.70). The study offers further evidence that the effect is not driven by response strategies specific to the bounded number line task. We discuss other possible sources of the effect including conversion of symbols to magnitudes in these and other contexts.

Modeling the left digit effect in adult number line estimation.

Number line estimation tasks are frequently used to study numerical cognition skills. In a typical version, the bounded number line task, target numerals must be placed on a bounded line labeled only at its endpoints (e.g., with 0 and 100). Placements by adults, while highly accurate, reveal a cyclical pattern of over- and underestimation of target numerals. The pattern suggests use of proportion judgment strategies and is well-captured by cyclical power models. Another systematic number line bias that has recently been observed, but has not yet been considered in modeling efforts, is the left digit effect. Numerals with different leftmost digits (e.g., 39 and 41) are placed farther apart on a line than is warranted. In the current study (N = 60), adult estimates were obtained for all numerals on a 0-100 number line estimation task, and fit of the standard cyclical power model was compared with two modified versions of the model. One modified version included a parameter that underweights the rightward digit's place value (e.g., the ones digit here), and the other used the same parameter to underweight all digits' place values. We found that both modifications provided a considerably better fit for individual and median data than the standard model, and we discuss their relative merits and cognitive interpretations. The data and models suggest how a left digit bias might impact estimates across the number line.

Summary accuracy feedback and the left digit effect in number line estimation.

A robust left digit effect arises in number line estimation such that adults' estimates for numerals with different hundreds place digits but nearly identical magnitudes are systematically different from one another (e.g., 299 is placed too far to the left of 302). In two experiments, we investigate whether brief feedback interventions designed to increase task effort can reduce or eliminate the left digit effect in a self-paced 0-1,000 number line estimation task. Participants were assigned to complete three blocks of 120 trials each where the middle block contained feedback or no feedback. Feedback was in the form of summary accuracy scores (Experiment 1; N = 153) or competitive (summary) accuracy scores (Experiment 2; N = 145). In both experiments, planned analyses revealed large left digit effects in all blocks regardless of feedback condition. Feedback did not lead to a reduction in the left digit effect in either experiment, but improvements in overall accuracy were observed. We conclude that there are no changes in the left digit effect resulting from either summary accuracy feedback or competitive accuracy feedback. Also reported are exploratory analyses of trial characteristics (e.g., whether 299 is presented before or after 302) and the left digit effect.

Number line estimation and standardized test performance: The left digit effect does not predict SAT math score.

INTRODUCTION: Recent work reveals a new source of error in number line estimation (NLE), the left digit effect (Lai, Zax, et al., 2018), whereby numerals with different leftmost digits but similar magnitudes (e.g., 399, 401) are placed farther apart on a number line (e.g., 0 to 1,000) than is warranted. The goals of the present study were to: (1) replicate the left digit effect, and (2) assess whether it is related to mathematical achievement. METHOD: Participants were all individuals (adult college students) who completed the NLE task in the laboratory between 2014 and 2019 for whom SAT scores were available (n = 227). RESULTS: We replicated the left digit effect but found its size was not correlated with SAT math score, although it was negatively correlated with SAT verbal score for one NLE task version. CONCLUSIONS: These findings provide further evidence that individual digits strongly influence estimation performance and suggest that this effect may have different cognitive contributors, and predict different complex skills, than overall NLE accuracy.

Developmental change in partition dependent resource allocation behavior.

Partition dependence, the tendency to distribute choices differently based on the way options are grouped, has important implications for decision making. This phenomenon, observed in adults across a variety of contexts such as allocating resources or making selections from a menu of items, can bias decision makers toward some choices and away from others. Only one study to date (Reichelson, Zax, Patalano, & Barth, Quarterly Journal of Experimental Psychology, 72, 1029-1036, 2019) has investigated the developmental trajectory of this phenomenon. In the current study we investigate children's and adults' susceptibility to partitioning effects in a child-friendly resource allocation task. In Experiment 1 (N = 80), adults distributed 12 food tokens to animals at the zoo. Based on previous findings that older children show weaker partition dependence in this task, we predicted that adults might exhibit reduced partition dependent behavior: they showed none. In Experiment 2 (N = 272), we used a less transparent task with only five food tokens, predicting that both adults and children (ages 3-10 years) would show partition dependence. Children, but not adults, made partition dependent resource allocations, with younger children exhibiting greater effects than older children. These experiments provide further evidence that children's decisions, like adults' (in other tasks), are influenced by the arbitrary partitioning of the available options. This work supports previous findings that younger children may be more susceptible to these effects, and maps developmental change in partition dependent behavior from early childhood to adulthood on this child-friendly partition dependence task.

Brief report on the relationship between temporal discount rate and error related negativity for immediate versus future choice options.

It is well documented that individuals vary in their economic patience - their willingness to choose delayed larger rewards (e.g., $100 in a month) over immediate smaller rewards (e.g., $25 now) - and that high levels of impatience, or temporal discounting, can be behaviorally problematic. Using event-related potential (ERP) method, we investigated error monitoring, as indexed by the error related negativity (ERN) component, as a function of discounting behavior. This work builds on prior work on risky decision making that revealed that individuals have greater ERNs for trials in which they select a risky option over a certain one (Yu and Zhou 2009), especially individuals not inclined towards risk taking (Martin and Potts 2009). In the present study, participants completed a temporal discounting task (choosing between a fixed immediate reward versus a future reward that varied across trials) while electroencephalogram (EEG) activity was recorded. We found an asymmetric relationship between discounting and the ERN: the greater an individual's overall rate of discounting, the greater the ERN component of the ERP waveform on trials where the future reward was selected, but not on trials in which the immediate reward was selected. The ERN may reflect an early warning signal alerting high discounters to potential negative consequences of future-oriented choices.

Intuitive symbolic magnitude judgments and decision making under risk in adults.

Performance on an intuitive symbolic number skills task-namely the number line estimation task-has previously been found to predict value function curvature in decision making under risk, using a cumulative prospect theory (CPT) model. However there has been no evidence of a similar relationship with the probability weighting function. This is surprising given that both number line estimation and probability weighting can be construed as involving proportion judgment, that is, involving estimating a number on a bounded scale based on its proportional relationship to the whole. In the present work, we re-evaluated the relationship between number line estimation and probability weighting through the lens of proportion judgment. Using a CPT model with a two-parameter probability weighting function, we found a double dissociation: number line estimation bias predicted probability weighting curvature while performance on a different number skills task, number comparison, predicted probability weighting elevation. Interestingly, while degree of bias was correlated across tasks, the direction of bias was not. The findings provide support for proportion judgment as a plausible account of the shape of the probability weighting function, and suggest directions for future work.

Probability range and probability distortion in a gambling task.

In decision making under risk, adults tend to overestimate small and underestimate large probabilities (Tversky & Kahneman, 1992). This inverse S-shaped distortion pattern is similar to that observed in a wide variety of proportion judgment tasks (see Hollands & Dyre, 2000, for review). In proportion judgment tasks, distortion patterns tend not to be fixed but rather to depend on the reference points to which the targets are compared. Here, we tested the novel hypothesis that probability distortion in decision making under risk might also be influenced by reference points-in this case, references implied by the probability range. Adult participants were assigned to either a full-range (probabilities from 0-100%), upper-range (50-100%), or lower-range (0-50%) condition, where they indicated certainty equivalents for 176 hypothetical monetary gambles (e.g., "a 50% chance of $100, otherwise $0"). Using a modified cumulative prospect theory model, we found only minimal differences in probability distortion as a function of condition, suggesting no differences in use of reference points by condition, and broadly demonstrating the robustness of distortion pattern across contexts. However, we also observed deviations from the curve across all conditions that warrant further research.

Partition dependence in development: Are children's decisions shaped by the arbitrary grouping of options?

The grouping of options into arbitrary categories influences adults' decisions about allocating choices or resources among those options; this is called "partition dependence." Partition dependence has been demonstrated in a wide range of contexts in adults and is often presented as a technique for designing choice architectures that nudge people towards better decisions. Whether children also make partition dependent decisions is unknown, as are potential patterns of developmental change. In this experiment ( N = 159), we examined whether children exhibit partition dependence using a novel resource allocation task. This novel task, distributing food tokens to zoo animals, did elicit partition dependence in our developmental sample. Both older children (ages 7-10 years) and younger children (ages 3-6 years) made partition dependent allocations, and younger children exhibited a larger partition dependence effect than did older children. This work provides the first evidence that children's decisions, like adults', are influenced by the arbitrary grouping of the options, and suggests that younger children may be more susceptible to this influence, at least in the context explored here.

Gratitude intervention modulates P3 amplitude in a temporal discounting task.

Gratitude has been shown to reduce economic impatience. In particular, individuals induced to experience heightened gratitude are more willing to choose delayed larger rewards over immediate smaller rewards (i.e., they have lower discounting rates) than those in a neutral condition. Using the event-related potential (ERP) method, we investigated the relation between gratitude level and neurophysiological correlates. Of interest was motivated information processing, as indexed by the P3 component. Participants were administered a gratitude or a neutral mood induction followed by a temporal discounting task (choosing between a fixed immediate reward versus a future reward that varied across trials) while electroencephalogram (EEG) activity was recorded. Individuals in the gratitude condition had greater P3 amplitude, suggesting greater attention to the future-reward option (the choice option that varied across trials), even when this option was not selected, and providing the first evidence of gratitude-induced changes in electrophysiological activity.

Partition dependence in consumer choice: Perceptual groupings do not reliably shape decisions.

The partitioning of options into arbitrary categories has been shown to influence decisions about allocating choices or resources among those options; this phenomenon is called partition dependence. While we do not call into question the validity of the partition dependence phenomenon in the present work, we do examine the robustness of one of the experimental paradigms reported by Fox, Ratner, and Lieb (Journal of Experimental Psychology: General, 134, 538-551, 2005, Study 4). In three experiments (N = 300) conducted here, participants chose from a menu of perceptually partitioned options (varieties of candy distributed across bowls). We found no clear evidence of partition dependent choice in children (Experiment 1) and no evidence at all of partition dependence in adults' choices (Experiments 1-3). This was true even when methods were closely matched to those of Fox et al.'s Study 4 (Experiment 3). We conclude that the candy-bowl choice task does not reliably elicit partition dependence and propose possible explanations for the discrepancy between these findings and prior reports. Future work will explore the conditions under which partition dependence in consumer choice does reliably arise.

Suboptimality in perceptual decision making and beyond.

We concur with the authors' overall approach and suggest that their analysis should be taken even further. First, the same points apply to areas beyond perceptual decision making. Second, the same points apply beyond issues of optimality versus suboptimality.

Early group bias in the Faroe Islands: Cultural variation in children's group-based reasoning.

Recent developmental research demonstrates that group bias emerges early in childhood. However, little is known about the extent to which bias in minimal (i.e., arbitrarily assigned) groups varies with children's environment and experience, and whether such bias is universal across cultures. In this study, the development of group bias was investigated using a minimal groups paradigm with 46 four- to six-year-olds from the Faroe Islands. Children observed in-group and out-group members exhibiting varying degrees of prosocial behaviour (egalitarian or stingy sharing). Children did not prefer their in-group in the pretest, but a pro-in-group and anti-out-group sentiment emerged in both conditions in the posttest. Faroese children's response patterns differ from those of American children [Schug, M. G., Shusterman, A., Barth, H., & Patalano, A. L. (2013). Minimal-group membership influences children's responses to novel experience with group members. Developmental Science, 16(1), 47-55], suggesting that intergroup bias shows cultural variation even in a minimal groups context.

The role of numeracy and approximate number system acuity in predicting value and probability distortion.

It is well documented that individuals distort outcome values and probabilities when making choices from descriptions, and there is evidence of systematic individual differences in distortion. In the present study, we investigated the relationship between individual differences in such distortions and two measures of numerical competence, numeracy and approximate number system (ANS) acuity. Participants indicated certainty equivalents for a series of simple monetary gambles, and data were used to estimate individual-level value and probability distortion, using a cumulative prospect theory framework. We found moderately strong negative correlations between numeracy and value and probability distortion, but only weak and non-statistically reliable correlations between ANS acuity and distortions. We conclude that low numeracy contributes to number distortion in decision making, but that approximate number system acuity might not underlie this relationship.

Minimal-group membership influences children's responses to novel experience with group members.

Children, like adults, tend to prefer ingroup over outgroup individuals, but how this group bias affects children's processing of information about social groups is not well understood. In this study, 5- and 6-year-old children were assigned to artificial groups. They observed instances of ingroup and outgroup members behaving in either a positive (egalitarian) or a negative (stingy) manner. Observations of positive ingroup and negative outgroup behaviors reliably reduced children's liking of novel outgroup members, while observations of negative ingroup and positive outgroup behaviors had little effect on liking ratings. In addition, children successfully identified the more generous group only when the ingroup was egalitarian and the outgroup stingy. These data provide compelling evidence that children treat knowledge of and experiences with ingroups and outgroups differently, and thereby differently interpret identical observations of ingroup versus outgroup members.

The influence of category coherence on inference about cross-classified entities.

A critical function of categories is their use in property inference (Heit, 2000). However, one challenge to using categories in inference is that most entities in the world belong to multiple categories (e.g., Fido could be a dog, a pet, a mammal, or a security system). Building on Patalano, Chin-Parker, and Ross (2006), we tested the hypothesis that category coherence (the extent to which category features go together in light of prior knowledge) influences the selection of categories for use in property inference about cross-classified entities. In two experiments, we directly contrasted coherent and incoherent categories, both of which included cross-classified entities as members, and we found that the coherent categories were used more readily as the source of both property transfer and property extension. We conclude that category coherence, which has been found to be a potent influence on strength of inference for singly classified entities (Rehder & Hastie, 2004), is also central to category use in reasoning about novel cross-classified ones.

The role of category coherence in experience-based prediction.

Both real-world category knowledge and instance-based sample data are often available as sources of inductive inference. In three experiments using natural social categories, we test the influence of generalcategory knowledge on the use of category instances to make property inductions both to other category members and to others in the population. We find that a category's coherence--the extent to which its features are interrelated through prior knowledge (Murphy & Medin, 1985)--influences inductions positively to new category members and negatively to the population. This effect of coherence is strongest with small as compared with large samples of instances. The results are interpreted from both similarity and explanation-based perspectives.

Spontaneous number discrimination of multi-format auditory stimuli in cotton-top tamarins (Saguinus oedipus).

Studies using operant training have demonstrated that laboratory animals can discriminate the number of objects or events based on either auditory or visual stimuli, as well as the integration of both auditory and visual modalities. To date, studies of spontaneous number discrimination in untrained animals have been restricted to the visual modality, leaving open the question of whether such capacities generalize to other modalities such as audition. To explore the capacity to spontaneously discriminate number based on auditory stimuli, and to assess the abstractness of the representation underlying this capacity, a habituation-discrimination procedure involving speech and pure tones was used with a colony of cotton-top tamarins. In the habituation phase, we presented subjects with either two- or three-speech syllable sequences that varied with respect to overall duration, inter-syllable duration, and pitch. In the test phase, we presented subjects with a counterbalanced order of either two- or three-tone sequences that also varied with respect to overall duration, inter-syllable duration, and pitch. The proportion of looking responses to test stimuli differing in number was significantly greater than to test stimuli consisting of the same number. Combined with earlier work, these results show that at least one non-human primate species can spontaneously discriminate number in both the visual and auditory domain, indicating that this capacity is not tied to a particular modality, and within a modality, can accommodate differences in format.