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.     

二参数逻辑斯蒂模型项目参数的估计精度

项目参数的估计精度对于测验的编制尤其是题库的建立十分重要。目前,国内外对项目参数估计精度的研究,大部分是基于在已知项目参数真值的情况下,运用各种参数估计方法产生新的估计值,再和真值进行偏度(BIAS)和均方根差(RMSE)的比较,从而说明该种估计方法的有效性。但是这种方法不能提供不同的参数真值之间的估计误差的变化规律。为了弥补这一缺陷,本文尝试从项目参数估计信息函数的角度出发研究项目参数的估计精度问题。本研究以二参数Logistic模型作为研究对象,首先定义了项目参数的估计信息函数,然后基于完全随机实验设计,通过模拟研究的方法探索影响项目参数的估计精度的因素,实验共设计了(2×3×2)种情形。研究结果表明：(1)项目参数(a,b)的估计精度均随着被试样本量的增大而提高;(2)被试的能力分布对难度参数的估计精度影响较大,对区分度参数的估计精度影响相对较小;(3)难度参数和区分度参数的估计精度都分别受到参数a和参数b的共同作用。     

改进的MCMC算法及其在估计IRT模型参数中的应用

本文首先简要的阐述了MCMC算法的思想及在IRT参数估计中的操作过程;其次,针对该算法存在的一些问题,提出相应的改进建议;然后,分别运用传统的和改进型的MCMC算法进行模拟数据分析和比较,结果显示新的方法表现更好;最后总结新方法的优点所在,并指出下一步的研究方向。     

Benchmark‐based strategies in whole number line estimation

In this study, we used verbal protocols to identify whether adults spontaneously apply quartile‐based strategies or whether they need additional external support to use these strategies when solving a 0–1,000 number line estimation (NLE) task. Participants were assigned to one of three conditions based on the number of external benchmarks provided on the number line. In the bounded condition only the origin and endpoint were indicated, the mid‐point condition included an additional external benchmark at 50%, and in the quartile condition three additional external benchmarks at 25%, 50%, and 75% were specified. Firstly, participants in the bounded condition reported to spontaneously apply quartile‐based strategies to calibrate their estimates. Moreover, participants frequently relied on the external benchmarks for creating internal benchmarks at the mid‐point, quartiles, and even octiles of the number line. Secondly, overall estimation accuracy improved as the number of external benchmarks increased, and target numbers close to external benchmarks were estimated more accurately and with less variability. Thirdly, the use of a larger variety in benchmark‐based strategies was positively related to NLE accuracy. In summary, this study provides evidence that the NLE task induces more sophisticated strategy use in participants than initially anticipated.     

Bayesian Estimation for Item Factor Analysis Models with Sparse Categorical Indicators

Psychometric models for item-level data are broadly useful in psychology. A recurring issue for estimating item factor analysis (IFA) models is low-item endorsement (item sparseness), due to limited sample sizes or extreme items such as rare symptoms or behaviors. In this paper, I demonstrate that under conditions characterized by sparseness, currently available estimation methods, including maximum likelihood (ML), are likely to fail to converge or lead to extreme estimates and low empirical power. Bayesian estimation incorporating prior information is a promising alternative to ML estimation for IFA models with item sparseness. In this article, I use a simulation study to demonstrate that Bayesian estimation incorporating general prior information improves parameter estimate stability, overall variability in estimates, and power for IFA models with sparse, categorical indicators. Importantly, the priors proposed here can be generally applied to many research contexts in psychology, and they do not impact results compared to ML when indicators are not sparse. I then apply this method to examine the relationship between suicide ideation and insomnia in a sample of first-year college students. This provides an important alternative for researchers who may need to model items with sparse endorsement.