The role of feedback and dot presentation format in younger and older adults’ number estimation

Numerosity estimation, the rapid assessment of the number of items in a visual scene, is historically inaccurate. We assessed whether providing feedback regarding the correct numerosity on either 0%, 50%, or 100% of the trials would affect younger and older adults’ estimation accuracy for randomized, clustered (i.e., groups of 3 or 7 dots), and stacked (i.e., column) dot formats. Participants provided estimates and confidence ratings in six blocks, each containing 48 trials (16 numerosities shown in each format). Feedback frequency was manipulated between participants during blocks 1–4; no feedback was provided during blocks 5 and 6, which contained old and new numerosities and previously estimated presentations rotated 90°. Estimation accuracy was age equivalent across blocks despite younger adults initially being more accurate than older adults. Feedback improved both age groups’ accuracy. Stacked presentations were most accurately estimated but were more likely to be over-estimated than clustered and randomized presentations. Older adults gave lower confidence ratings than younger adults despite both age groups showing increased confidence across blocks, for more structured presentation formats, and as feedback frequency increased. These results expand our understanding of the role of presentation format and feedback in producing age equivalence or age-related differences in numerosity estimation.     

题组反应理论及其在中学英语考试中的应用研究

题组作为众多测验中的一种常见题型,由于项目间存在一定程度的依赖性而违背了局部独立性假设,若用项目反应模型进行参数估计将会出现较大的偏差.题组反应理论将被试与题组的交互作用纳入到模型中,解决了项目间相依性的问题.笔者对题组反应理论的发展、基本原理及其相关研究进行了综述,并将其应用在中学英语考试中.与项目反应理论相对比,结果发现：（1）题组反应模型与项目反应模型在各参数估计值的相关系数较强,尤其是能力参数和难度参数;（2）在置信区间宽度的比较上,题组反应模型在各个参数上均窄于项目反应模型,即题组反应模型的估计精度优于项目反应模型.     

The development of numerosity estimation: Evidence for a linear number representation early in life

Several studies investigating the development of approximate number representations used the number-to-position task and reported evidence for a shift from a logarithmic to a linear representation of numerical magnitude with increasing age. However, this interpretation as well as the number-to-position method itself has been questioned recently. The current study tested 5- and 8-year-old children on a newly established numerosity production task to examine developmental changes in number representations and to test the idea of a representational shift. Modelling of the children's numerical estimations revealed that responses of the 8-year-old children approximate a simple positive linear relation between estimated and actual numbers. Interestingly, however, the estimations of the 5-year-old children were best described by a bilinear model reflecting a relatively accurate linear representation of small numbers and no apparent magnitude knowledge for large numbers. Taken together, our findings provide no support for a shift of mental representations from a logarithmic to a linear metric but rather suggest that the range of number words which are appropriately conceptualised and represented by linear analogue magnitude codes expands during development.     

四参数Logistic模型潜在特质参数的Warm加权极大似然估计

本文以四参数Logistic(4-parameter Logistic,4PL)模型为研究对象,根据Warm的加权极大似然估计技巧,提出了4PL模型潜在特质参数的加权极大似然估计方法,并借助模拟研究对加权极大似然估计的性质进行验证。研究结果表明,与通常的极大似然估计和后验期望估计相比,加权极大似然估计的偏差(bias)明显减小,并且具有良好的返真性能。此外,在测试的长度较短和项目的区分度较小的情况下,加权极大似然估计依然保持了良好的统计性质,表现出更加显著的优势。     

Item Calibration Samples and the Stability of Achievement Estimates and System Rankings: Another Look at the PISA Model

Using an empirically-based simulation study, we show that typically used methods of choosing an item calibration sample have significant impacts on achievement bias and system rankings. We examine whether recent PISA accommodations, especially for lower performing participants, can mitigate some of this bias. Our findings indicate that standard operational methods, while not ideal, recover underlying proficiency reasonably well and generally outperform methods that more completely include all participants. Translating results onto the PISA scale, the calibration sample can induce bias of up to 12.49 points, which is important given that standard errors are around three points. Although ranking correlations are at least.95, we note the policy implications of slight ranking changes. Our findings indicate that limited accommodations targeted at low achieving educational systems do not outperform either of the other methods considered. Research that further explores accommodations for heterogeneous populations is recommended.     

Further evidence of the invalidity of the magnitude estimation method

IntroductionThe magnitude estimation method is widely applied in diverse areas of psychology. However, Weiss (1972) found that this method may produce judgments that inaccurately represent subjective magnitudes: constructing the stimuli according to a factorial design allowed the bias caused by magnitude estimation to be exposed as a spurious divergence of factorial curves.ObjectiveThe present study experimentally compared magnitude estimation with graphic rating and ratio estimation. The comparison of magnitude estimation with graphic rating retested Weiss's finding. The comparison of magnitude estimation with ratio estimation explored the hypothesis that magnitude estimation may cause a divergence of factorial curves due to the requirement of magnitude estimation to multiply and divide numbers—both ratio estimation and magnitude estimation required participants to judge ratios, but only magnitude estimation required mental multiplication and division.MethodUsing a 3 × 4 factorial experimental design, an achromatic disk was presented concentrically with a larger achromatic square. The brightness of the disk varied with the luminance of the disk and, due to simultaneous contrast, with the luminance of the square. Three different groups of participants judged the brightness of the disk: one group by graphic rating, another by ratio estimation, and the third by magnitude estimation.ResultsIn accord with Weiss's results, graphic rating yielded roughly parallel factorial curves whereas magnitude estimation yielded diverging factorial curves. Ratio estimation also yielded roughly parallel factorial curves, supporting the hypothesis that the magnitude estimation task itself may cause divergence of factorial curves due to the requirement of this method to mentally multiply and divide numbers.ConclusionCaution should be taken in using magnitude estimation, especially in factorial experiments, because this method is likely to cause the researcher to make incorrect inferences about the functional relations between the variables involved in mental information integration. Ratio estimation yields a pattern of factorial curves essentially equivalent to that obtained by graphic rating.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

On Judgments of Approximately Equal

Although precision is often important in quantitative judgment, sometimes, it is valuable to recognize that two quantities are roughly the same. Fuzzy‐trace theory suggests that approximately equal judgments rely on gist representations (i.e., meaningful fuzzy categories of quantity). We conducted three experiments to investigate approximately equal judgments with number pairs presented in different formats, both with and without semantic content (breast cancer statistics). In each study, the ratio of the smaller divided by larger number predicted approximately equal judgments. Experiment 1 also examined how knowledge of breast cancer, presentation format (frequencies vs. percentages), and differences in gist comprehension of breast cancer information influence fuzzy equality judgments. As predicted by the fuzzy‐trace theory concept of denominator neglect, approximately equal judgments were more sensitive, as measured by signal detection theory (SDT) analyses, when presented as percentages. In both experiments with breast cancer statistics, people were more likely to judge number pairs approximately equal when they were embedded in sentences about breast cancer, and breast cancer knowledge predicted increased perception of equality, when appropriately consistent with reliable sources, and increased judgment sensitivity. In Experiment 2, a simple intervention focusing on gist meaning increased source‐consistent approximately equal judgments, increased SDT judgment sensitivity, and decreased SDT response bias. In Experiment 3, using number pairs devoid of semantic context spanning four orders of magnitude, we further examined ratio similarity. Overall, more knowledgeable judges and those who better understood the gist of meaningful numbers were more likely to judge literally different numbers as “approximately equal” rather than make precise discriminations that were meaningless. Copyright © 2017 John Wiley & Sons, Ltd.     

Feedback frequency of graphical countdown timers affects pedestrian estimated waiting time at red lights

Graphical countdown timers (similar to a progress bar) have the potential to shorten pedestrians’ perceived waiting time at the red light. Based on the findings from human-computer interaction (HCI) and time estimation theories, the potential may be affected by feedback frequency. This study tested how the feedback frequency of countdown progress bar in traffic light influenced time estimation in a field experiment, and further explored its mechanism in a laboratory experiment. In both experiments, participants estimated various durations (30 s, 45 s, 60 s) under three levels of feedback frequency: low (0.5 Hz), medium (1 Hz), and high (2 Hz). The results showed that although waiting time was underestimated in all conditions, lower feedback frequency led to shorter estimated duration, with slight changes in different contexts. In the traffic context, the effect of feedback frequency was comparable across different countdown duration, but it became stronger at longer countdown duration in the non-traffic context. Overall, the effect of feedback frequency was accompanied by changes in neither arousal level (measured by the subjective scale and physiological indices) nor internal clock speed, which are two critical determinants of time perception. The findings have practical implications on the display design of red light and theoretical implications on time estimation processes.     

The left digit effect in a complex judgment task: Evaluating hypothetical college applicants

A left digit effect has been broadly observed across judgment and decision-making contexts ranging from product evaluation to medical treatment decisions to number line estimation. For example, $3.00 is judged to be a much greater cost than $2.99, and “801” is estimated strikingly too far to the right of “798” on a number line. Although the consequences of the effects for judgment and decision behavior have been documented, the sources of the effects are not well established. The goal of the current work is to extend investigations of the left digit effect to a new complex judgment activity and to assess whether the magnitude of the effect at the individual level can be predicted from performance on a simpler number skills task on which the left digit effect has also recently been observed. In three experiments (N = 434), adults completed a judgment task in which they rated the strength of hypothetical applicants for college admission and a self-paced number line estimation task. In all experiments, a small or medium left digit effect was found in the college admissions task, and a large effect was found in number line estimation. Individual-level variation was observed, but there was no relationship between the magnitudes of the effects in the two tasks. These findings provide evidence of a left digit effect in a novel multiattribute judgment task but offer no evidence that such performance can be predicted from a simple number skills task such as number line estimation.