The bias function of the maximum likelihood estimate of ability for the dichotomous response level

Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

A numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

Temporal perception in closed-skill sports: An experimental study on expert swimmers and runners

The cognitive benefits of closed-skill sports practice have so far been scantily investigated. Here, we thus focused on the potential impact of swimming and running - two sports that highly rely on a precise control of timing - on time processing. To investigate the impact of these closed-skill sports on time perception and estimation, three groups of participants (for a total of eighty-four young adults) took part in the present study: expert swimmers, expert runners, and non-athletes. The ability to process temporal information in the milliseconds and seconds range was assessed through a time reproduction and a finger-tapping tasks, while a motor imagery paradigm was adopted to assess temporal estimation of sport performance in a wider interval range. We also employed the Vividness of Movement Imagery Questionnaire to assess the individual's ability of motor imagery. Results showed that closed-skill sports, specifically time-related disciplines, enhance motor imagery and time perception abilities. Swimmers were more accurate and consistent in perceiving time when compared to runners, probably thanks to the sensory muffled environment that leads these athletes to be more focused on the perception of their internal rhythm.     

Mixture-modelling-based Bayesian MH-RM algorithm for the multidimensional 4PLM

Several recent works have tackled the estimation issue for the unidimensional four-parameter logistic model (4PLM). Despite these efforts, the issue remains a challenge for the multidimensional 4PLM (M4PLM). Fu et al. (2021) proposed a Gibbs sampler for the M4PLM, but it is time-consuming. In this paper, a mixture-modelling-based Bayesian MH-RM (MM-MH-RM) algorithm is proposed for the M4PLM to obtain the maximum a posteriori (MAP) estimates. In a comparison of the MM-MH-RM algorithm to the original MH-RM algorithm, two simulation studies and an empirical example demonstrated that the MM-MH-RM algorithm possessed the benefits of the mixture-modelling approach and could produce more robust estimates with guaranteed convergence rates and fast computation. The MATLAB codes for the MM-MH-RM algorithm are available in the online appendix.     

A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy

Cognitive diagnostic models provide a framework for classifying individuals into latent proficiency classes, also known as attribute profiles. Recent research has examined the implementation of a Pólya-gamma data augmentation strategy binary response model using logistic item response functions within a Bayesian Gibbs sampling procedure. In this paper, we propose a sequential exploratory diagnostic model for ordinal response data using a logit-link parameterization at the category level and extend the Pólya-gamma data augmentation strategy to ordinal response processes. A Gibbs sampling procedure is presented for efficient Markov chain Monte Carlo (MCMC) estimation methods. We provide results from a Monte Carlo study for model performance and present an application of the model.     

On the Bock-Aitkin procedure—From an EM algorithm perspective

The relationship between the EM algorithm and the Bock-Aitkin procedure is described with a continuous distribution of ability (latent trait) from an EM-algorithm perspective. Previous work has been restricted to the discrete case from a probit-analysis perspective. The author is grateful to Bradley A. Hanson for valuable discussion and comments. Thanks also go to Terry A. Ackerman, Meichu Fan, Subrata Kundu, and Robert K. Tsutakawa for their help and encouragement in this study.     

On the loss of information in conditional maximum likelihood estimation of item parameters

In item response models of the Rasch type (Fischer & Molenaar, 1995), item parameters are often estimated by the conditional maximum likelihood (CML) method. This paper addresses the loss of information in CML estimation by using the information concept of F-information (Liang, 1983). This concept makes it possible to specify the conditions for no loss of information and to define a quantification of information loss. For the dichotomous Rasch model, the derivations will be given in detail to show the use of the F-information concept for making comparisons for different estimation methods. It is shown that by using CML for item parameter estimation, some information is almost always lost. But compared to JML (joint maximum likelihood) as well as to MML (marginal maximum likelihood) the loss is very small. The reported efficiency in the use of information of CML to JML and to MML in several comparisons is always larger than 93%, and in tests with a length of 20 items or more, larger than 99%.     

Guess Who Buys Cheap? The Effect of Consumers’ Goal Orientation on Product Preference

The current research examines how goal orientation affects consumer preference among products with different prices. We argue that a less expensive product may have not only lower perceived quality but also greater perceived quality variability. This greater perceived variability provides the opportunity for optimistic, promotion‐oriented consumers to overestimate the quality of the less expensive product. This effect is weaker, however, for a more expensive product that consumers perceive to have less quality variability and which thus provides less room for quality overestimation. As a result, we hypothesize that promotion‐oriented consumers, as compared to prevention‐oriented consumers, will demonstrate a stronger preference for a less expensive product. In a field study and two laboratory experiments, we obtain empirical support both for the hypothesized effect of consumers’ goal orientation on their product preference and for its underlying process. We conclude with a discussion of the theoretical and practical implications of our results.     

Bootstrap区间估计在认知诊断模型误设中的应用

认知诊断模型选择是认知诊断评估中重要研究问题之一。在实际应用中实践者并不知道真正拟合数据的模型,通常会用模型拟合指标检验模型与数据的拟合程度。从测量结果质量来看,除保证模型与数据拟合之外,还需要重点评价模型诊断结果的信度和效度等。考虑到以往研究大都采用基于信息量的拟合指标去判定模型与数据的匹配性,本研究提出综合考虑模型拟合指标与信度指标用于模型选择或评价模型误设。考虑实验因素为真实模型或分析模型（DINA模型、G-DINA模型、R-RUM模型）、样本量、题量和属性个数,在五因素（3×3×2×2×2）实验设计条件下,比较Bootstrap区间估计的属性分类一致性信度平均数与标准误和常用的拟合统计量-2LL、AIC、BIC对正确模型的选择率。结果表明：-2LL在题目数量多的情况下表现较好,而AIC、BIC在被试量较大的情况下表现较好,在不同的研究条件下,-2LL、AIC、BIC的模型选择率很不稳定,而用Bootstrap法估计的属性分类一致性信度平均数和标准误在不同研究条件的模型选择率较稳定,总体表现较好。