The Continuous Parameter Estimation Model is applied to develop individual genomic-level heritabilities for the latent hierarchical structure and developmental dynamics of Life History (LH) strategy LH strategies relate to the allocations of bioenergetic resources into different domains of fitness. LH has moderate to high population-level heritability in humans, both at the level of the high-order Super-K Factor and the lower-order factors, the K-Factor, Covitality Factor, and General Factor of Personality (GFP). Several important questions remain unexplored. We developed measures of (...) genome-level heritabilities employing an American sample of 316 monozygotic (MZ) and 274 dizygotic (DZ) twin dyads and a Swedish sample of 863 MZ and 475 DZ twin dyads. This novel heritability index measures individual genetic transmissibility, therefore opening new avenues for analyzing complex interactions among heritable traits inaccessible to standard structural equations methods. For these samples: (1) moderate to high heritability of factor loadings of Super-K on its lower-order factors is demonstrated, evidencing biological preparedness, genetic accommodation, and the gene-culture coevolution of biased epigenetic rules of development; (2) moderate to high heritability of the magnitudes of the effect of the higher-order factors upon their loadings on their constituent factors, evidencing genetic constraints upon phenotypic plasticity; and (3) that heritability of the LH factors, of factor loadings, and of the magnitudes of the correlations among factors are weaker among those with slower LH speeds, demonstrating that inter-individual variation in transmissibility is a function of individual socioecological selection pressures. (shrink)

After writing my response to the commentaries, I sat back and reflected on the fascination and frustration of work on this topic. There is the ancient fascination of trying to understand the nature of the sensory bridge linking us to the external world. Also, discussing the measurability of sensation brings to the surface concepts we use and take for granted when we are working in other areas of psychology; and it holds them before us for critical examination. The frustration lies (...) chiefly in the difficulty of formulating concepts clearly, in a way understood by all. At least when constructing the original Tower of Babel, workers hearing strange words knew that communication was not occurring. The psychophysical Tower of Babel sometimes has workers using the same words, but with different meanings. Much of my response is based on attempts to clarify concepts and clear up confusions. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single (...) events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

The reliability of a test score is discussed from the viewpoint of underestimation of and, specifically, deflation in estimates or reliability. Many widely used estimators are known to underestimate reliability. Empirical cases have shown that estimates by widely used estimators such as alpha, theta, omega, and rho may be deflated by up to 0.60 units of reliability or even more, with certain types of datasets. The reason for this radical deflation lies in the item–score correlation embedded in the estimators: because (...) the estimates by Rit are deflated when the number of categories in scales are far from each other, as is always the case with item and score, the estimates of reliability are deflated as well. A short-cut method to reach estimates closer to the true magnitude, new types of estimators, and deflation-corrected estimators of reliability, are studied in the article. The empirical section is a study on the characteristics of combinations of DCERs formed by different bases for estimators, different alternative estimators of correlation as the linking factor between item and the score variable, and different conditions. Based on the simulation, an initial typology of the families of DCERs is presented: some estimators are better with binary items and some with polytomous items; some are better with small sample sizes and some with larger ones. (shrink)

Being able to estimate quantity is important in everyday life and for success in the STEM disciplines. However, people have difficulty reasoning about magnitudes outside of human perception. This study examines patterns of estimation errors across temporal and spatial magnitudes at large scales. We evaluated the effectiveness of hierarchical alignment in improving estimations, and transfer across dimensions. The activity was successful in increasing accuracy for temporal and spatial magnitudes, and learning transferred to the estimation of numeric magnitudes associated (...) with events and objects. However, there were also a number of informative differences in performance on temporal, spatial, and numeric magnitude measures, suggesting that participants possess different categorical information for these scales. Educational implications are discussed. (shrink)

This chapter presents a re-understanding of the contents of our analog magnitude representations (e.g., approximate duration, distance, number). The approximate number system (ANS) is considered, which supports numerical representations that are widely described as fuzzy, noisy, and limited in their representational power. The contention is made that these characterizations are largely based on misunderstandings—that what has been called “noise” and “fuzziness” is actually an important epistemic signal of confidence in one’s estimate of the value. Rather than the ANS having (...) noisy or fuzzy numerical content, it is suggested that the ANS has exquisitely precise numerical content that is subject to epistemic limitations. Similar considerations will arise for other analog representations. The chapter discusses how this new understanding of ANS representations recasts the learnability problem for number and the conceptual changes that children must accomplish in the number domain. (shrink)

This month the American Journal of Bioethics examines the intersectionality of medicine, ethics, and race. In “Race, Power, and COVID-19: A Call for Advocacy Within Bioethics,” Mithani and colleagu...

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the magnitude judgments or (...) results from changes in precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and (...) response time are expected near the boundaries between categories. In contrast to previous work that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers. (shrink)

Continuous magnitude estimation and continuous cross-modality matching with line length can efficiently track the momentary loudness of time-varying sounds in behavioural experiments. These methods are known to be prone to systematic biases but may be checked for consistency using their counterpart, magnitude production. Thus, in Experiment 1, we performed such an evaluation for time-varying sounds. Twenty participants produced continuous cross-modality matches to assess the momentary loudness of fourteen songs by continuously adjusting the length of a line. In (...) Experiment 2, the resulting temporal line length profile for each excerpt was played back like a video together with the given song and participants were asked to continuously adjust the volume to match the momentary line length. The recorded temporal line length profile, however, was manipulated for segments with durations between 7 to 12 s by eight factors between 0.5 and 2, corresponding to expected differences in adjusted level of −10, −6, −3, −1, 1, 3, 6, and 10 dB according to Stevens’s power law for loudness. The average adjustments 5 s after the onset of the change were −3.3, −2.4, −1.0, −0.2, 0.2, 1.4, 2.4, and 4.4 dB. Smaller adjustments than predicted by the power law are in line with magnitude-production results by Stevens and co-workers due to “regression effects.” Continuous cross-modality matches of line length turned out to be consistent with current loudness models, and by passing the consistency check with cross-modal productions, demonstrate that the method is suited to track the momentary loudness of time-varying sounds. (shrink)

It is well-documented that mathematics achievement is an important predictor of many positive life outcomes like college graduation, career opportunities, salary, and even citizenship. As such, it is important for researchers and educators to help students succeed in mathematics. Although there are undoubtedly many factors that contribute to students' success in mathematics, much of the research and intervention development has focused on variations in instructional techniques. Indeed, even a cursory glance at many educational journals and granting agencies reveals that there (...) is a large amount of time, energy, and resources being spent on determining the best way to convey information through direct, declarative instruction. The proposed project is motivated by recent calls to expand the focus of research in mathematics education beyond direct, declarative instruction. The overarching goal of the presented experiment is to evaluate the efficacy of a novel mathematics intervention designed using principles taken from the literature on non-declarative learning. The intervention combines errorless learning and structured cue fading to help second grade students improve their understanding of symbolic magnitude. Results indicate that students who learned about symbolic magnitude using the novel intervention did better than students who were provided with extensive declarative support. These findings offer preliminary evidence in favor of using learning combination of errorless learning and cue fading techniques in the mathematics classroom. (shrink)

BackgroundThe amount and value of researchers’ peer review work is critical for academia and journal publishing. However, this labor is under-recognized, its magnitude is unknown, and alternative ways of organizing peer review labor are rarely considered.MethodsUsing publicly available data, we provide an estimate of researchers’ time and the salary-based contribution to the journal peer review system.ResultsWe found that the total time reviewers globally worked on peer reviews was over 100 million hours in 2020, equivalent to over 15 thousand years. (...) The estimated monetary value of the time US-based reviewers spent on reviews was over 1.5 billion USD in 2020. For China-based reviewers, the estimate is over 600 million USD, and for UK-based, close to 400 million USD.ConclusionsBy design, our results are very likely to be under-estimates as they reflect only a portion of the total number of journals worldwide. The numbers highlight the enormous amount of work and time that researchers provide to the publication system, and the importance of considering alternative ways of structuring, and paying for, peer review. We foster this process by discussing some alternative models that aim to boost the benefits of peer review, thus improving its cost-benefit ratio. (shrink)

The anchoring heuristic refers to phenomena when an arbitrary number affects subsequent numerical estimations. Oppenheimer, LeBoeuf and Brewer showed that it is not necessary for the anchor to be a numerical value, yet current models describing the anchoring heuristic do not fully account for the mechanism of non-numerical anchoring. However, this effect shows similarity to the basic anchoring effect - obtained without the comparative question and based on the availability of the given number in working memory. In this study, we (...) attempt to verify whether those two effect share the same psychological mechanism. In Experiment 1, we show that non-numerical anchoring based on magnitude priming cannot be obtained when the lines are just observed. The examined mechanism proves to be dependent on the act of drawing, displaying limitations similar to the basic anchoring effect, previously pointed out by Brewer and Chapman. By using the same numerical anchors in different size formats, in Experiment 2 we showed that anchoring based on magnitude priming occurs even when the numerical values do not affect the estimations. The results are discussed in the light of a possible mechanism that underlies the investigated effect. (shrink)

Adjusting for covariates is a well-established method to estimate the total causal effect of an exposure variable on an outcome of interest. Depending on the causal structure of the mechanism under study, there may be different adjustment sets, equally valid from a theoretical perspective, leading to identical causal effects. However, in practice, with finite data, estimators built on different sets may display different precisions. To investigate the extent of this variability, we consider the simplest non-trivial non-linear model of a v-structure (...) on three nodes for binary data. We explicitly compute and compare the variance of the two possible different causal estimators. Further, by going beyond leading-order asymptotics, we show that there are parameter regimes where the set with the asymptotically optimal variance does depend on the edge coefficients, a result that is not captured by the recent leading-order developments for general causal models. As a practical consequence, the adjustment set selection needs to account for the relative magnitude of the relationships between variables with respect to the sample size and cannot rely on purely graphical criteria. (shrink)

The disambiguation of a syntactically ambiguous sentence in favor of a less preferred parse can lead to slower reading at the disambiguation point. This phenomenon, referred to as a garden‐path effect, has motivated models in which readers initially maintain only a subset of the possible parses of the sentence, and subsequently require time‐consuming reanalysis to reconstruct a discarded parse. A more recent proposal argues that the garden‐path effect can be reduced to surprisal arising in a fully parallel parser: words consistent (...) with the initially dispreferred but ultimately correct parse are simply less predictable than those consistent with the incorrect parse. Since predictability has pervasive effects in reading far beyond garden‐path sentences, this account, which dispenses with reanalysis mechanisms, is more parsimonious. Crucially, it predicts a linear effect of surprisal: the garden‐path effect is expected to be proportional to the difference in word surprisal between the ultimately correct and ultimately incorrect interpretations. To test this prediction, we used recurrent neural network language models to estimate word‐by‐word surprisal for three temporarily ambiguous constructions. We then estimated the slowdown attributed to each bit of surprisal from human self‐paced reading times, and used that quantity to predict syntactic disambiguation difficulty. Surprisal successfully predicted the existence of garden‐path effects, but drastically underpredicted their magnitude, and failed to predict their relative severity across constructions. We conclude that a full explanation of syntactic disambiguation difficulty may require recovery mechanisms beyond predictability. (shrink)

Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log-to-linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power-shaped function (Barth & Paladino, ; Siegler & Opfer, ). While (...) about half of people did estimate numbers linearly over this range, nearly all the remaining participants placed 1 million approximately halfway between 1 thousand and 1 billion, but placed numbers linearly across each half, as though they believed that the number words “thousand, million, billion, trillion” constitute a uniformly spaced count list. Participants in this group also tended to be optimistic in evaluations of largely ineffective political strategies, relative to linear number-line placers. The results indicate that the surface structure of number words can heavily influence processes for dealing with numbers in this range, and it can amplify the possibility that analogous surface regularities are partially responsible for parallel phenomena in children. In addition, these results have direct implications for lawmakers and scientists hoping to communicate effectively with the public. (shrink)

According to Kant’s Critique of the Power of Judgement, in the end all estimation of magnitude is sensible, or ‘aesthetic’, and the absolutely great in aesthetic estimation is called ‘the mathematical sublime’. This article identifies the relevant sensible element with an inner sensation of a temporal tension: in aesthetic comprehension, the imagination encounters an inevitable tension between the successive reproduction of a magnitude’s individual parts and the simultaneous unification of these parts. The sensation of this tension (...) varies in degree and facilitates aesthetic estimation in general. But in the special case where it comes to involve a sense of exceeding our imagination’s limit, we judge a magnitude to be mathematically sublime. Pace Kant, I argue that this limit is private and contingent rather than transcendental, such that the judgement of the mathematical sublime is neither universal a priori nor necessary. (shrink)

ABSTRACTAdults use a variety of strategies to reason about fraction magnitudes, and this variability is adaptive. In two studies, we examined the relationships between mathematics anxiety, working memory, strategy variability and performance on two fraction tasks: fraction magnitude comparison and estimation. Adults with higher mathematics anxiety had lower accuracy on the comparison task and greater percentage absolute error on the estimation task. Unexpectedly, mathematics anxiety was not related to variable strategy use. However, variable strategy use was linked (...) to more accurate magnitude comparisons, especially among adults with lower working memory performance or those who use mathematics less frequently, as well as lower PAE on the estimation task. These findings shed light on the role of strategy variability in fraction problem solving and demonstrate a link between mathematics anxiety and fraction magnitude reasoning, a key predictor of general mathematics achievement. (shrink)

The measurement of sensory intensity has had a long history, attracting the attention of investigators from many disciplines including physiology, psychology, physics, mathematics, philosophy, and even chemistry. While there has been a continuing doubt by some that sensation has the properties necessary for measurement, experiments designed to obtain estimates of sensory intensity have found that a general rule applies: Equal stimulus ratios produce equal sensory ratios. Theories concerning the basis for this simple psychophysical rule are discussed, with emphasis given to (...) the physical correlate theory, which considers judgments of subjective magnitudes to be based upon estimates of physical dimensions that vary regularly with changes in degree of stimulation. For the most thoroughly investigated sensory scales, brightness and loudness, the physical correlate is considered to be distance. Our “tacit knowledge” of the sensory effects of changing distance plays an essential role in matching motor activities to environmental conditions and in ensuring accurate perceptual evaluations through brightness and loudness constancies. In psychophysical experiments, subjects apparently use this same tacit knowledge when required to estimate relative subjective magnitudes. The evidence related to the physical correlate theory is summarized, and it is concluded that, while under appropriate conditions we demonstrate considerable skill in evaluating environmental relationships, we are quite unable to estimate the neurophysiological nature or quantity of sensory response. A psychophysics devoted to studying conditions required for accuracy and conditions producing errors in the perception of environmental relationships would seem to be more valuable than one devoted to subjective magnitudes. (shrink)

The “erasing race” effect is the reduction of the salience of “race” as an alliance cue when recalling coalition membership, once more accurate information about coalition structure is presented. We conducted a random-effects model meta-analysis of this effect using five United States studies (containing nine independent effect sizes). The effect was found (ρ = 0.137, K = 9, 95% CI = 0.085 to 0.188). However, no decline effect or moderation effects were found (a “decline effect” in this context would be (...) a decrease in the effect size over time). Furthermore, we found little evidence of publication bias. Synthetically correcting the effect size for bias stemming from the use of an older method for calculating error base rates reduced the magnitude of the effect, but the it remained significant. Taken together, these findings indicate that the “erasing race” effect generalizes quite well across experimental contexts and would, therefore, appear to be quite robust. We reinterpret the theoretical basis for these effects in line with Brunswikian evolutionary-developmental theory and present a series of predictions to guide future research in this area. (shrink)

This paper offers an analysis of Kant’s account of the mathematical sublime with reference to his claim that ‘Nature is thus sublime in those of its appearances the intuition of which brings with them the idea of its infinity’. In undertaking this analysis I challenge Paul Crowther’s interpretation of this species of aesthetic experience, and I reject his interpretation as not being reflective of Kant’s actual position. I go on to show that the experience of the mathematical sublime is necessarily (...) connected with the progression of the imagination in its move towards the infinite. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Kant’s Mathematical Sublime and the Role of the Infinite: Reply to CrowtherVolume 20, Issue 1Simon D. Smith DOI: https://doi.org/10.1017/S1369415414000302Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation. (shrink)