Some neglected problems in IRT

The paper addresses three neglected questions from IRT. In section 1, the properties of the “measurement” of ability or trait parameters and item difficulty parameters in the Rasch model are discussed. It is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe. Section 2 deals with the measurement of individual change or “modifiability” based on a Rasch test. A conditional likelihood approach is presented that yields (a) an ML estimator of modifiability for given item parameters, (b) allows one to test hypotheses about change by means of a Clopper-Pearson confidence interval for the modifiability parameter, or (c) to estimate modifiability jointly with the item parameters. Uniqueness results for all three methods are also presented. In section 3, the Mantel-Haenszel method for detecting DIF is discussed under a novel perspective: What is the most general framework within which the Mantel-Haenszel method correctly detects DIF of a studied item? The answer is that this is a 2PL model where, however, all discrimination parameters are known and the studied item has the same discrimination in both populations. Since these requirements would hardly be satisfied in practical applications, the case of constant discrimination parameters, that is, the Rasch model, is the only realistic framework. A simple Pearsonx ² test for DIF of one studied item is proposed as an alternative to the Mantel-Haenszel test; moreover, this test is generalized to the case of two items simultaneously studied for DIF.     

Item-focussed Trees for the Identification of Items in Differential Item Functioning

A novel method for the identification of differential item functioning (DIF) by means of recursive partitioning techniques is proposed. We assume an extension of the Rasch model that allows for DIF being induced by an arbitrary number of covariates for each item. Recursive partitioning on the item level results in one tree for each item and leads to simultaneous selection of items and variables that induce DIF. For each item, it is possible to detect groups of subjects with different item difficulties, defined by combinations of characteristics that are not pre-specified. The way a DIF item is determined by covariates is visualized in a small tree and therefore easily accessible. An algorithm is proposed that is based on permutation tests. Various simulation studies, including the comparison with traditional approaches to identify items with DIF, show the applicability and the competitive performance of the method. Two applications illustrate the usefulness and the advantages of the new method.     

题组项目功能差异（DIF）检验方法的应用探索

篇章形式的阅读测验是一种典型的题组测验,在进行项目功能差异（DIF）检验时需要采用与之匹配的DIF检验方法.基于题组反应模型的DIF检验方法是真正能够处理题组效应的DIF检验方法,能够提供题组中每个项目的DIF效应测量,是题组DIF检验方法中较有理论优势的一种,主要使用的方法是Rasch题组DIF检验方法.该研究将Rasch题组DIF检验方法引入篇章阅读测验的DIF检验中,对某阅读成就测验进行题组DIF检验,结果显示,该测验在内容维度和能力维度的部分子维度上出现了具有显著DIF效应的项目,研究从测验公平的角度对该测验的进一步修改及编制提出了一定的建议.研究中进一步将Rasch题组DIF检验方法与基于传统Rasch模型的DIF检验方法以及变通的题组DIF检验方法的结果进行比较,研究结果体现了进行题组DIF检验的必要性与优越性.研究结果表明,在篇章阅读测验中,能够真正处理题组效应的题组DIF检验方法更加具有理论优势且对于阅读测验的编制与质量的提高具有更重要的意义.     

Explanatory Multidimensional Multilevel Random Item Response Model: An Application to Simultaneous Investigation of Word and Person Contributions to Multidimensional Lexical Representations

This paper presents an explanatory multidimensional multilevel random item response model and its application to reading data with multilevel item structure. The model includes multilevel random item parameters that allow consideration of variability in item parameters at both item and item group levels. Item-level random item parameters were included to model unexplained variance remaining when item related covariates were used to explain variation in item difficulties. Item group-level random item parameters were included to model dependency in item responses among items having the same item stem. Using the model, this study examined the dimensionality of a person’s word knowledge, termed lexical representation, and how aspects of morphological knowledge contributed to lexical representations for different persons, items, and item groups.     

Sufficiency and Conditional Estimation of Person Parameters in the Polytomous Rasch Model

Rasch models are characterised by sufficient statistics for all parameters. In the Rasch unidimensional model for two ordered categories, the parameterisation of the person and item is symmetrical and it is readily established that the total scores of a person and item are sufficient statistics for their respective parameters. In contrast, in the unidimensional polytomous Rasch model for more than two ordered categories, the parameterisation is not symmetrical. Specifically, each item has a vector of item parameters, one for each category, and each person only one person parameter. In addition, different items can have different numbers of categories and, therefore, different numbers of parameters. The sufficient statistic for the parameters of an item is itself a vector. In estimating the person parameters in presently available software, these sufficient statistics are not used to condition out the item parameters. This paper derives a conditional, pairwise, pseudo-likelihood and constructs estimates of the parameters of any number of persons which are independent of all item parameters and of the maximum scores of all items. It also shows that these estimates are consistent. Although Rasch’s original work began with equating tests using test scores, and not with items of a test, the polytomous Rasch model has not been applied in this way. Operationally, this is because the current approaches, in which item parameters are estimated first, cannot handle test data where there may be many scores with zero frequencies. A small simulation study shows that, when using the estimation equations derived in this paper, such a property of the data is no impediment to the application of the model at the level of tests. This opens up the possibility of using the polytomous Rasch model directly in equating test scores.     

Improvement in Detection of Differential Item Functioning Using a Mixture Item Response Theory Model

Usually, methods for detection of differential item functioning (DIF) compare the functioning of items across manifest groups. However, the manifest groups with respect to which the items function differentially may not necessarily coincide with the true source of the bias. It is expected that DIF detection under a model that includes a latent DIF variable is more sensitive to this source of bias. In a simulation study, it is shown that a mixture item response theory model, which includes a latent grouping variable, performs better in identifying DIF items than DIF detection methods using manifest variables only. The difference between manifest and latent DIF detection increases as the correlation between the manifest variable and the true source of the DIF becomes smaller. Different sample sizes, relative group sizes, and significance levels are studied. Finally, an empirical example demonstrates the detection of heterogeneity in a minority sample using a latent grouping variable. Manifest and latent DIF detection methods are applied to a Vocabulary test of the General Aptitude Test Battery (GATB).     

Detection of differential item functioning in Rasch models by boosting techniques

Methods for the identification of differential item functioning (DIF) in Rasch models are typically restricted to the case of two subgroups. A boosting algorithm is proposed that is able to handle the more general setting where DIF can be induced by several covariates at the same time. The covariates can be both continuous and (multi‐)categorical, and interactions between covariates can also be considered. The method works for a general parametric model for DIF in Rasch models. Since the boosting algorithm selects variables automatically, it is able to detect the items which induce DIF. It is demonstrated that boosting competes well with traditional methods in the case of subgroups. The method is illustrated by an extensive simulation study and an application to real data.     

Optimal Items for Assessing Parental Involvement across Different Groups During Middle Childhood

Various definitions and different approaches for assessing the complex construct of parental involvement (PI) have led to inconsistent findings regarding the impact of PI on child development. To date, limited information is available regarding the measurement invariance of PI measures across time and groups (e.g., children’s gender, ethnicity, and socio-economic status), leaving a concern that group differences in PI might reflect item bias instead of true differences in PI. The present study aimed to obtain a set of optimal items for measuring PI from kindergarten through the elementary school years and investigate whether they could be used for parents from different groups. A Rasch measurement model was implemented to investigate item difficulty, step calibrations, and measurement invariance (differential item functioning; DIF, here). The results from the Early Childhood Longitudinal Study, Kindergarten Class of 1998–1999 data set showed that 20 items can be used to measure three dimensions of PI—namely school/home involvement, family educational investment, and family routines—across four time points. Administrative time, children’s gender, ethnicity, and social economic status showed different levels of effect on item difficulty for half of these items. Practitioners and researchers should be cautious when using these items and are suggested to freely estimate the item parameters of DIF items as well as add more items to the PI scale to improve reliability.     

Item Screening in Graphical Loglinear Rasch Models

In behavioural sciences, local dependence and DIF are common, and purification procedures that eliminate items with these weaknesses often result in short scales with poor reliability. Graphical loglinear Rasch models (Kreiner & Christensen, in Statistical Methods for Quality of Life Studies, ed. by M. Mesbah, F.C. Cole & M.T. Lee, Kluwer Academic, pp. 187–203, 2002) where uniform DIF and uniform local dependence are permitted solve this dilemma by modelling the local dependence and DIF. Identifying loglinear Rasch models by a stepwise model search is often very time consuming, since the initial item analysis may disclose a great deal of spurious and misleading evidence of DIF and local dependence that has to disposed of during the modelling procedure.     

Penalization approaches in the conditional maximum likelihood and Rasch modelling context

Recent detection methods for Differential Item Functioning (DIF) include approaches like Rasch Trees, DIFlasso, GPCMlasso and Item Focussed Trees, all of which - in contrast to well established methods - can handle metric covariates inducing DIF. A new estimation method shall address their downsides by mainly aiming at combining three central virtues: the use of conditional likelihood for estimation, the incorporation of linear influence of metric covariates on item difficulty and the possibility to detect different DIF types: certain items showing DIF, certain covariates inducing DIF, or certain covariates inducing DIF in certain items. Each of the approaches mentioned lacks in two of these aspects. We introduce a method for DIF detection, which firstly utilizes the conditional likelihood for estimation combined with group Lasso-penalization for item or variable selection and L1-penalization for interaction selection, secondly incorporates linear effects instead of approximation through step functions, and thirdly provides the possibility to investigate any of the three DIF types. The method is described theoretically, challenges in implementation are discussed. A dataset is analysed for all DIF types and shows comparable results between methods. Simulation studies per DIF type reveal competitive performance of cmlDIFlasso, particularly when selecting interactions in case of large sample sizes and numbers of parameters. Coupled with low computation times, cmlDIFlasso seems a worthwhile option for applied DIF detection.     

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.     

适用于多维迫选测验的IRT计分模型

迫选(forced-choice, FC)测验由于可以控制传统李克特方法带来的反应偏差, 被广泛应用于非认知测验中, 而迫选测验的传统计分方式会产生自模式数据, 这种数据由于不适合于个体间的比较, 一直备受批评。近年来, 多种迫选IRT模型的发展使研究者能够从迫选测验中获得接近常模性的数据, 再次引起了研究者与实践人员对迫选IRT模型的兴趣。首先, 依据所采纳的决策模型和题目反应模型对6种较为主流的迫选IRT模型进行分类和介绍。然后, 从模型构建思路、参数估计方法两个角度对各模型进行比较与总结。其次, 从参数不变性检验、计算机化自适应测验(computerized adaptive testing, CAT)和效度研究3个应用研究方面进行述评。最后提出未来研究可以在模型拓展、参数不变性检验、迫选CAT测验和效度研究4个方向深入。     

A goodness of fit test for the rasch model

The Rasch model is an item analysis model with logistic item characteristic curves of equal slope,i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties estimated from different scoregroups and over-all estimates. Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is formed. It is shown that—2 times the logarithm of this ratio isx ²-distributed when the Rasch model is true. The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves, but unequal discriminating items from a scholastic aptitude test.     

A generalized longitudinal mixture IRT model for measuring differential growth in learning environments

This article describes a generalized longitudinal mixture item response theory (IRT) model that allows for detecting latent group differences in item response data obtained from electronic learning (e-learning) environments or other learning environments that result in large numbers of items. The described model can be viewed as a combination of a longitudinal Rasch model, a mixture Rasch model, and a random-item IRT model, and it includes some features of the explanatory IRT modeling framework. The model assumes the possible presence of latent classes in item response patterns, due to initial person-level differences before learning takes place, to latent class-specific learning trajectories, or to a combination of both. Moreover, it allows for differential item functioning over the classes. A Bayesian model estimation procedure is described, and the results of a simulation study are presented that indicate that the parameters are recovered well, particularly for conditions with large item sample sizes. The model is also illustrated with an empirical sample data set from a Web-based e-learning environment.     

Rasch analysis of a simplified Beck Depression Inventory

Depression is one of the most clinically relevant mood disorders, and many assessment instruments have been developed to measure it. Probably the most frequently used instrument is Beck’s Depression Inventory (BDI). The simplified BDI (BDI-S) is a more efficient version of the BDI that has been shown to be no less reliable or valid. As the BDI-S has not yet been subjected to rigorous tests of Item Response Theory, it is the aim of the present paper to conduct such an analysis using the Rasch model. This study subjected a simplified version of the BDI consisting of 20 items (BDI-S20) to a Rasch analysis in a sample of N = 5,035 participants. The scale, minus one misfitting item (BDI-S19), yielded a good approximation to Rasch assumptions. Moderate differential item functioning (DIF) was present. It is concluded that the BDI-S19 is an internally valid instrument for assessing depression, although some room for improvement exists.     

Exact person fit indexes for the rasch model for arbitrary alternatives

A new algorithm for obtaining exact person fit indexes for the Rasch model is introduced which realizes most powerful tests for a very general family of alternative hypotheses, including tests concerning DIF as well as model-deviating item correlations. The method is also used as a goodness-of-fit test for whole data sets where the item parameters are assumed to be known. For tests with 30 items at most, exact values are obtained, for longer tests a Monte Carlo-algorithm is proposed. Simulated examples and an empirical investigation demonstrate test power and applicability to item elimination.The author wishes to thank Elisabeth Ponocny-Seliger and the reviewers for many helpful comments. All exact goodness-of-fit tests proposed in this article are implemented in the menu-driven program T-Rasch 1.0 by Ponocny and Ponocny-Seliger (1999) which can be obtained from ProGAMMA (WWW: http://www.gamma.rug.nl) and also performs nonparametric tests.     

Assessing the Robustness of Mixture Models to Measurement Noninvariance

Abstract

Recent work reframes direct effects of covariates on items in mixture models as differential item functioning (DIF) and shows that, when present in the data but omitted from the fitted latent class model, DIF can lead to overextraction of classes. However, less is known about the effects of DIF on model performance—including parameter bias, classification accuracy, and distortion of class-specific response profiles—once the correct number of classes is chosen. First, we replicate and extend prior findings relating DIF to class enumeration using a comprehensive simulation study. In a second simulation study using the same parameters, we show that, while the performance of LCA is robust to the misspecification of DIF effects, it is degraded when DIF is omitted entirely. Moreover, the robustness of LCA to omitted DIF differs widely based on the degree of class separation. Finally, simulation results are contextualized by an empirical example.     

Classification of loneliness using the T‐ILS: Is it as simple as it seems?

Student well‐being is a growing issue in higher education, and assessment of the prevalence of conditions as loneliness is therefore important. In higher education and population surveys the Three‐Item Loneliness Scale (T‐ILS) is used increasingly. The T‐ILS is attractive for large multi‐subject surveys, as it consists of only three items (derived from the UCLA Loneliness Scale). Several ways of classifying persons as lonely based on T‐ILS scores exist: dichotomous and trichotomous classification schemes and use of sum scores with rising levels indicating more loneliness. The question remains whether T‐ILS scores are comparable across the different population groups where they are used or across groups of students in the higher education system. The aim was to investigate whether the T‐ILS suffers from differential item functioning (DIF) that might change the loneliness classification among higher education students, using a large sample just admitted to 22 different academy profession degree programs in Denmark (N = 3,757). DIF was tested relative to degree program, age groups and gender. The framework of graphical loglinear Rasch models was applied, as this allows for adjustment of sum scores for uniform DIF, and thus for assessment of whether DIF might change the classification. Two items showed DIF relative to degree program and gender, and adjusting for this DIF changed the classification for some subgroups. The consequences were negligible when using a dichotomous classification and larger when using a trichotomous classification. Therefore, trichotomous classification should be used with caution unless suitable adjustments for DIF are done prior to classification.     

Cognitive Complexity in the Remote Association Test - Chinese Version

The remote association test (RAT) has been applied in various fields; however, evidence of construct validity for the original version and subsequent extensions of the RAT remains limited. This study aimed to elucidate the dimensionality and the relationship between item features and item difficulties for the RAT—Chinese Version (RAT-C) using the Rasch model and the linear logistic test model (LLTM). The revised 30-item RAT-C was administered to 475 undergraduates (263 women and 212 men) in 8 universities in Taiwan. Item features (including types of associations among stimulus words, and frequency and concreteness of target words) were recoded. The analysis found that the RAT-C measured a single latent construct, with all 30 items conforming to the Rasch model’s expectation. Furthermore, according to the LLTM analysis, most item features predicted Rasch item difficulty, suggesting that these features can explain why some items were more difficult than others and can be used to create new items with known item difficulty to tailor the difficulty level for different groups of participants in the future.     

认知诊断测验中基于信息矩阵的多群组DIF检验

基于改进的Wald统计量,将适用于两群组的DIF检测方法拓展至多群组的项目功能差异（DIF）检验;改进的Wald统计量将分别通过计算观察信息矩阵（Obs）和经验交叉相乘信息矩阵（XPD）而得到。模拟研究探讨了此二者与传统计算方法在多个群组下的DIF检验情况,结果表明：（1）Obs和XPD的一类错误率明显低于传统方法,DINA模型估计下Obs和XPD的一类错误率接近理论水平;（2）样本量和DIF量较大时,Obs和XPD具有与传统Wald统计量大体相同的统计检验力。