Spurious Latent Class Problem in the Mixed Rasch Model: A Comparison of Three Maximum Likelihood Estimation Methods under Different Ability Distributions

Recent research has shown that over-extraction of latent classes can be observed in the Bayesian estimation of the mixed Rasch model when the distribution of ability is non-normal. This study examined the effect of non-normal ability distributions on the number of latent classes in the mixed Rasch model when estimated with maximum likelihood estimation methods (conditional, marginal, and joint). Three information criteria fit indices (Akaike information criterion, Bayesian information criterion, and sample size adjusted BIC) were used in a simulation study and an empirical study. Findings of this study showed that the spurious latent class problem was observed with marginal maximum likelihood and joint maximum likelihood estimations. However, conditional maximum likelihood estimation showed no overextraction problem with non-normal ability distributions.     

A note on the asymptotic variance-covariance matrix of item parameter estimates in the rasch model

Lord and Wingersky have developed a method for computing the asymptotic variance-covariance matrix of maximum likelihood estimates for item and person parameters under some restrictions on the estimates which are needed in order to fix the latent scale. The method is tedious, but can be simplified for the Rasch model when one is only interested in the item parameters. This is demonstrated here under a suitable restriction on the item parameter estimates.     

Addressing Item-Level Missing Data: A Comparison of Proration and Full Information Maximum Likelihood Estimation

Often when participants have missing scores on one or more of the items comprising a scale, researchers compute prorated scale scores by averaging the available items. Methodologists have cautioned that proration may make strict assumptions about the mean and covariance structures of the items comprising the scale (Schafer &; Graham, 2002 Schafer, J.L., &; Graham, J.W. (2002). Missing data: Our view of the state of the art. Psychological Methods, 7, 147–177.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Graham, 2009 Graham, J.W. (2009). Missing data analysis: Making it work in the real world. Annual Review of Psychology, 60, 549–576.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Enders, 2010 Enders, C.K. (2010). Applied missing data analysis. New York, NY: Guilford Press. [Google Scholar]). We investigated proration empirically and found that it resulted in bias even under a missing completely at random (MCAR) mechanism. To encourage researchers to forgo proration, we describe a full information maximum likelihood (FIML) approach to item-level missing data handling that mitigates the loss in power due to missing scale scores and utilizes the available item-level data without altering the substantive analysis. Specifically, we propose treating the scale score as missing whenever one or more of the items are missing and incorporating items as auxiliary variables. Our simulations suggest that item-level missing data handling drastically increases power relative to scale-level missing data handling. These results have important practical implications, especially when recruiting more participants is prohibitively difficult or expensive. Finally, we illustrate the proposed method with data from an online chronic pain management program.     

On the Relationships Between Jeffreys Modal and Weighted Likelihood Estimation of Ability Under Logistic IRT Models

This paper focuses on two estimators of ability with logistic item response theory models: the Bayesian modal (BM) estimator and the weighted likelihood (WL) estimator. For the BM estimator, Jeffreys’ prior distribution is considered, and the corresponding estimator is referred to as the Jeffreys modal (JM) estimator. It is established that under the three-parameter logistic model, the JM estimator returns larger estimates than the WL estimator. Several implications of this result are outlined.     

A relation between a between-item multidimensional IRT model and the mixture rasch model

Two generalizations of the Rasch model are compared: the between-item multidimensional model (Adams, Wilson, and Wang, 1997), and the mixture Rasch model (Mislevy & Verhelst, 1990; Rost, 1990). It is shown that the between-item multidimensional model is formally equivalent with a continuous mixture of Rasch models for which, within each class of the mixture, the item parameters are equal to the item parameters of the multidimensional model up to a shift parameter that is specific for the dimension an item belongs to in the multidimensional model. In a simulation study, the relation between both types of models also holds when the number of classes of the mixture is as small as two. The relation is illustrated with a study on verbal aggression. Frank Rijmen was supported by the Fund for Scientific Research Flanders (FWO). This research is also funded by the GOA/2000/02 granted from the KU Leuven. We would like to thank Kristof Vansteelandt for providing the data of the study on verbal aggression.     

A note on the computation of the second-order derivatives of the elementary symmetric functions in the rasch model

Jansen (1984) gave explicit formulas for the computation of the second-order derivatives of the elementary symmetric functions. But they are only applicable to those pairs of items which have unequal parameters. It is shown here that for equal parameters similar explicit formulas do exist, too, facilitating the application of the Newton-Raphson procedure to estimate the parameters in the Rasch model and related models according to the conditional maximum likelihood principle.The author wishes to thank the reviewers for their helpful comments on an earlier draft of this paper.     

On equivariance and invariance of standard errors in three exploratory factor models

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standardz-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect onz-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis.We gratefully acknowledge the help of the Associate Editor and three referees whose constructive comments lead to an improved version of the paper. This work was supported by National Institute on Drug Abuse Grants DA01070 and DA00017 and by the University of North Texas Faculty Research Grant Program.     

使用似然比D2统计量的题目属性定义方法

题目属性的定义是实施认知诊断评价的关键步骤, 通过有丰富经验的领域专家对题目的属性进行定义是当前的主要方法, 然而该方法受到许多主观经验因素的影响。寻找客观的题目属性定义或验证方法可以为主观定义过程提供策略支持或对结果进行改进, 因此已经引起研究者们的关注。本研究构建了一种简单高效的题目属性定义方法, 研究使用似然比D2统计量从作答数据中估计题目属性的方法, 实现属性掌握模式、题目参数和题目属性向量的联合估计。模拟研究结果表明, 使用似然比D2统计量可以有效地识别题目的属性向量, 该方法一方面可以实现新编制题目属性向量的在线估计, 另一方面可以验证已经定义的题目属性向量的准确性。     

基于作答数据的模型参数和Q矩阵联合估计

Q矩阵在认知诊断的模型参数估计和诊断分类中起着重要作用。本文通过研究Liu等人的方法, 设计了同时估计项目参数和Q矩阵的联合估计算法。在DINA模型下, 对项目参数未知时开展模拟研究。研究假设项目为20个, 考察的属性个数分别是3、4和5, 初始Q矩阵中分别存在3、4和5个属性界定错误的项目。结果表明, 联合估计算法能在错误的初始Q矩阵基础上以很高的概率得到正确的Q矩阵。另外, 当专家认定测验的属性个数存在错误时, 该方法推导的Q矩阵和模型参数能提供很好的鉴别Q矩阵错误的信息。     

On the existence and uniqueness of maximum-likelihood estimates in the Rasch model

Necessary and sufficient conditions for the existence and uniqueness of a solution of the so-called unconditional (UML) and the conditional (CML) maximum-likelihood estimation equations in the dichotomous Rasch model are given. The basic critical condition is essentially the same for UML and CML estimation. For complete data matricesA, it is formulated both as a structural property ofA and in terms of the sufficient marginal sums. In case of incomplete data, the condition is equivalent to complete connectedness of a certain directed graph. It is shown how to apply the results in practical uses of the Rasch model.Paper read at the European Meeting of the Psychometric Society, Groningen, June 19–21, 1980.Part of the research reported herein was done while the author was staying at the Pulmologisches Zentrum der Stadt Wien; he is indebted to Professor Dr. F. Muhar and Dr. R. Mutschlechner for providing excellent working conditions.     

On equivalence between a partial credit item and a set of independent Rasch binary items

Given a Masters partial credit item withn known step difficulties, conditions are stated for the existence of a set of (locally) independent Rasch binary items such that their raw score and the partial credit raw score have identical probability density functions. The conditions are those for the existence ofn positive values with predetermined elementary symmetric functions and include the requirement that then step difficulties form an increasing sequence.     

Application of the Within- and Between-Series Estimators to Non-normal Multiple-Baseline Data: Maximum Likelihood and Bayesian Approaches

In single-case research, multiple-baseline (MB) design provides the opportunity to estimate the treatment effect based on not only within-series comparisons of treatment phase to baseline phase observations, but also time-specific between-series comparisons of observations from those that have started treatment to those that are still in the baseline. For analyzing MB studies, two types of linear mixed modeling methods have been proposed: the within- and between-series models. In principle, those models were developed based on normality assumptions, however, normality may not always be found in practical settings. Therefore, this study aimed to investigate the robustness of the within- and between-series models when data were non-normal. A Monte Carlo study was conducted with four statistical approaches. The approaches were defined by the crossing of two analytic decisions: (a) whether to use a within- or between-series estimate of effect and (b) whether to use restricted maximum likelihood or Markov chain Monte Carlo estimations. The results showed the treatment effect estimates of the four approaches had minimal bias, that within-series estimates were more precise than between-series estimates, and that confidence interval coverage was frequently acceptable, but varied across conditions and methods of estimation. Applications and implications were discussed based on the findings.     

On the Mathematical Relationship Between Latent Change Score and Autoregressive Cross-Lagged Factor Approaches: Cautions for Inferring Causal Relationship Between Variables

The present paper focuses on the relationship between latent change score (LCS) and autoregressive cross-lagged (ARCL) factor models in longitudinal designs. These models originated from different theoretical traditions for different analytic purposes, yet they share similar mathematical forms. In this paper, we elucidate the mathematical relationship between these models and show that the LCS model is reduced to the ARCL model when fixed effects are assumed in the slope factor scores. Additionally, we provide an applied example using height and weight data from a gerontological study. Throughout the example, we emphasize caution in choosing which model (ARCL or LCS) to apply due to the risk of obtaining misleading results concerning the presence and direction of causal precedence between two variables. We suggest approaching model specification not only by comparing estimates and fit indices between the LCS and ARCL models (as well as other models) but also by giving appropriate weight to substantive and theoretical considerations, such as assessing the justifiability of the assumption of random effects in the slope factor scores.     

Natural Realism or Transactionalism: On the Relationships between Putnam and Two Pragmatists,James and Dewey

The relationships between Hilary Putnam and the pragmatists (especially William James and John Dewey) are obvious but subtle.To shed some light on this issue,the author will explore a key issue that not only stands as Putnam's main inheritance from the pragmatists,but that also illuminates the relationships between them more clearly than any other issues.This key issue is the understanding of perception and the philosophical position that arises from this understanding.The author argues that in adopting Dewey's transactionalism (or interactionalism),Putnam advances from James' insight to Dewey's,a shift that is particularly manifest in Putnam's attempt to add another layer of meaning to what he refers to as the second na(i)veté that he detects and appreciates in James' natural realism.     

On the equivalence between elimination-by-aspects and generalised extreme value models of choice behaviour

Elimination-by-aspects and generalised extreme value offer competing paradigms for the representation of a common behaviour, that of individual discrete choice. Observing certain consistencies in their mathematical structure, several eminent authors have commented on the degree of equivalence between the two paradigms. Most contributions to this debate have, however, been less than definitive. More fundamentally, the contributions lack consensus. We advance the debate by establishing formal mathematical conditions under which three-alternative tree models from the two paradigms are exactly equivalent. We then extend our analysis to consider more general models, showing that equivalence can be established for general tree models, but not for cross-nested models.     

On the relationship between sources of within- and between-group differences and measurement invariance in the common factor model

Investigating sources of within- and between-group differences and measurement invariance (MI) across groups is fundamental to any meaningful group comparison based on observed test scores. It is shown that by placing certain restrictions on the multigroup confirmatory factor model, it is possible to investigate the hypothesis that within- and between-group differences are due to the same factors. Moreover, the modeling approach clarifies that absence of measurement bias implies common sources of within- and between-group variation. It is shown how the influence of background variables can be incorporated in the model. The advantages of the modeling approach as compared with other commonly used methods for group comparisons is discussed and illustrated by means of an analysis of empirical data.     

成就目标定向在学业自我概念与学业成绩关系中的调节作用:基于内/外参照模型

为了探究成就目标定向在学业自我概念与学业成绩关系中的调节作用,采用语文学业自我概念量表、数学学业自我概念量表和成就目标定向量表对1515名初中生进行测查。结果显示：(1)初中生语文-数学学业自我概念建构过程符合内/外参照模型(Internal/External Frame of Reference Model, I/EM);(2)掌握回避对I/EM的维度比较路径存在调节作用,掌握回避得分越高越易受维度比较效应的影响,该现象对于语文成绩和数学成绩相对差的学生更明显;(3)成绩回避对I/EM的社会比较路径存在调节作用,成绩回避得分越高越不易受社会比较效应的影响,该现象对于语文成绩和数学成绩相对好的学生更明显。本研究揭示了持有不同成就目标定向的初中生在学业自我概念建构过程中的差异,对引导初中生建构积极学业自我概念具有重要启示。