Limits on Log Odds Ratios for Unidimensional Item Response Theory Models

Bounds are established for log odds ratios (log cross-product ratios) involving pairs of items for item response models. First, expressions for bounds on log odds ratios are provided for one-dimensional item response models in general. Then, explicit bounds are obtained for the Rasch model and the two-parameter logistic (2PL) model. Results are also illustrated through an example from a study of model-checking procedures. The bounds obtained can provide an elementary basis for assessment of goodness of fit of these models. Any opinions expressed in this publication are those of the authors and not necessarily those of the Educational Testing Service. The authors thank Dan Eignor, Matthias von Davier, Lydia Gladkova, Brian Junker, and the three anonymous reviewers for their invaluable advice. The authors gratefully acknowledge the help of Kim Fryer with proofreading.     

Geometric representation of association between categories

Categories can be counted, rated, or ranked, but they cannot be measured. Likewise, persons or individuals can be counted, rated, or ranked, but they cannot be measured either. Nevertheless, psychology has realized early on that it can take an indirect road to measurement: What can be measured is the strength of association between categories in samples or populations, and what can be quantitatively compared are counts, ratings, or rankings made under different circumstances, or originating from different persons. The strong demand for quantitative analysis of categorical data has thus created a variety of statistical methods, with substantial contributions from psychometrics and sociometrics. What is the common basis of these methods dealing with categories? The basic element they share is that the sample space has a special geometry, in which categories (or persons) are point masses forming a simplex, while distributions of counts or profiles of ratings are centers of gravity, which are also point masses. Rankings form a discrete subset in the interior of the simplex, known as the permutation polytope, and paired comparisons form another subset on the edges of the simplex. Distances between point masses form the basic tool of analysis. The paper gives some history of major concepts, which naturally leads to a new concept: the shadow point. It is then shown how loglinear models, Luce and Rasch models, unfolding models, correspondence analysis and homogeneity analysis, forced classification and classification trees, as well as other models and methods, fit into this particular geometrical framework.     

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.     

Constant latent odds-ratios models and the mantel-haenszel null hypothesis

In the present paper, a new family of item response theory (IRT) models for dichotomous item scores is proposed. Two basic assumptions define the most general model of this family. The first assumption is local independence of the item scores given a unidimensional latent trait. The second assumption is that the odds-ratios for all item-pairs are constant functions of the latent trait. Since the latter assumption is characteristic of the whole family, the models are called constant latent odds-ratios (CLORs) models. One nonparametric special case and three parametric special cases of the general CLORs model are shown to be generalizations of the one-parameter logistic Rasch model. For all CLORs models, the total score (the unweighted sum of the item scores) is shown to be a sufficient statistic for the latent trait. In addition, conditions under the general CLORs model are studied for the investigation of differential item functioning (DIF) by means of the Mantel-Haenszel procedure. This research was supported by the Dutch Organization for Scientific Research (NWO), grant number 400-20-026.     

IRT_Δb法和修正LR法对矩阵取样DIF检验的有效性

矩阵取样测验包含多个题册,单个题册的总分不能直接作为匹配变量用于 DIF 检测。本研究首先基于模拟数据,同时采用 I RT_Δb法,以及用 I RT模型估计的考生能力作为匹配变量修订后的 L R法对矩阵取样测验进行DIF检测,分析二者进行DIF检测的有效性及其相关影响因素;并根据已有的LR法DIF判断标准划定出I RT_Δb法分类标准;最后使用实证数据加以验证。结果显示：矩阵取样测验中, I RT_Δb法和修正LR法均能较好地区分DIF量不同的题目;样本量、题册中DIF题目的比例和考生群体间真实能力的差异对两种方法的检验力、犯I类错误的概率和分类结果都有较大影响。     

Psychometric evaluation of the Basic Traits Inventory in the multilingual South African environment

The objective of this study was to evaluate the psychometric properties of the BTI for all the official language groups in South Africa. Item bias analysis was performed by means of Rasch analysis for a sample of 105 342 respondents with representation of each of the 11 language groups. Reliability values were good for all the language groups and bias was revealed in only 12 of the 193 items (6.2%) through item response modelling. Based on these results, the BTI may be an appropriate measure of personality for all the different language groups in South Africa.     

Reliability measures in item response theory: Manifest versus latent correlation functions

For item response theory (IRT) models, which belong to the class of generalized linear or non‐linear mixed models, reliability at the scale of observed scores (i.e., manifest correlation) is more difficult to calculate than latent correlation based reliability, but usually of greater scientific interest. This is not least because it cannot be calculated explicitly when the logit link is used in conjunction with normal random effects. As such, approximations such as Fisher's information coefficient, Cronbach's α, or the latent correlation are calculated, allegedly because it is easy to do so. Cronbach's α has well‐known and serious drawbacks, Fisher's information is not meaningful under certain circumstances, and there is an important but often overlooked difference between latent and manifest correlations. Here, manifest correlation refers to correlation between observed scores, while latent correlation refers to correlation between scores at the latent (e.g., logit or probit) scale. Thus, using one in place of the other can lead to erroneous conclusions. Taylor series based reliability measures, which are based on manifest correlation functions, are derived and a careful comparison of reliability measures based on latent correlations, Fisher's information, and exact reliability is carried out. The latent correlations are virtually always considerably higher than their manifest counterparts, Fisher's information measure shows no coherent behaviour (it is even negative in some cases), while the newly introduced Taylor series based approximations reflect the exact reliability very closely. Comparisons among the various types of correlations, for various IRT models, are made using algebraic expressions, Monte Carlo simulations, and data analysis. Given the light computational burden and the performance of Taylor series based reliability measures, their use is recommended.     

The relevance of security: A latent domain of attachment relationships

In the present study an adult attachment dimension, latent to the constructs of security, anxiety, and avoidance, was hypothesized, wherein security was expected to occupy the most relevant position. Furthermore, the reciprocal functioning of attachment constructs and their interactions with self‐esteem were explored. Four hundreds and thirty‐four Italian university students responded to two adult attachment questionnaires (Attachment Style Questionnaire and Adult Attachment Questionnaire) and to the Rosenberg Self‐esteem scale. A Many‐Facet Rasch Measurement modeling approach was adopted. The main results can be summarized as follows: (a) security, anxiety, and avoidance are nested under one latent attachment dimension; (b) security occupies the most prominent position on the dimension; (c) security is positively associated with a moderate level of attachment anxiety and negatively related to avoidance; and (d) a positive interaction between self‐esteem and security, and a negative relation between self‐esteem and anxiety, were detected. Theoretical, clinical, and empirical implications of the results are further discussed.     

K-means、潜在类别模型和混合Rasch模型的比较

基于模拟研究比较了K-means方法、潜在类别模型和混合Rasch模型在二分外显变量情境下的聚类效果.结果表明：（1）潜在类别数量、变量数量、样本量、样本平衡和变量间相关对K-means方法、潜在类别模型和混合Rasch模型的分类准确性均有影响且因素间的交互作用存在;（2）除了在2个潜在类别的样本不平衡条件下K-means方法表现较差外,在其他条件下与潜在类别模型和混合Rasch模型的表现相当;（3）混合Rasch模型的分类一致性在2个潜在类别的情境下要好于潜在类别模型,但是在4个潜在类别的情境下要差于潜在类别模型.