The Dutch Identity: A new tool for the study of item response models

The Dutch Identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions (IRFs) and the latent trait distribution. The identity may be exploited in several ways. For example: (a) to suggest how item response models behave for large numbers of items—they are approximate submodels of second-order loglinear models for 2 ^J tables; (b) to suggest new ways to assess the dimensionality of the latent trait—principle components analysis of matrices composed of second-order interactions from loglinear models; (c) to give insight into the structure of latent class models; and (d) to illuminate the problem of identifying the IRFs and the latent trait distribution from sample data.This research was supported in part by contract number N00014-87-K-0730 from the Cognitive Science Program of the Office of Naval Research. I realized the usefulness of the identity in Theorem 1 while lecturing in the Netherlands during October, 1986. Because this was in no small part due to the stimulating psychometric atmosphere there, I call the result the Dutch Identity.     

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.     

An Introduction to Network Psychometrics: Relating Ising Network Models to Item Response Theory Models

In recent years, network models have been proposed as an alternative representation of psychometric constructs such as depression. In such models, the covariance between observables (e.g., symptoms like depressed mood, feelings of worthlessness, and guilt) is explained in terms of a pattern of causal interactions between these observables, which contrasts with classical interpretations in which the observables are conceptualized as the effects of a reflective latent variable. However, few investigations have been directed at the question how these different models relate to each other. To shed light on this issue, the current paper explores the relation between one of the most important network models—the Ising model from physics—and one of the most important latent variable models—the Item Response Theory (IRT) model from psychometrics. The Ising model describes the interaction between states of particles that are connected in a network, whereas the IRT model describes the probability distribution associated with item responses in a psychometric test as a function of a latent variable. Despite the divergent backgrounds of the models, we show a broad equivalence between them and also illustrate several opportunities that arise from this connection.     

Conjunctive item response theory kernels

Conjunctive item response models are introduced such that (a) sufficient statistics for latent traits are not necessarily additive in item scores; (b) items are not necessarily locally independent; and (c) existing compensatory (additive) item response models including the binomial, Rasch, logistic, and general locally independent model are special cases. Simple estimates and hypothesis tests for conjunctive models are introduced and evaluated as well. Conjunctive models are also identified with cognitive models that assume the existence of several individually necessary component processes for a global ability. It is concluded that conjunctive models and methods may show promise for constructing improved tests and uncovering conjunctive cognitive structure. It is also concluded that conjunctive item response theory may help to clarify the relationships between local dependence, multidimensionality, and item response function form.I appreciate the many helpful suggestions that were given by the reviewers and Ivo Molenaar.     

Specifying Ability Growth Models Using a Multidimensional Item Response Model for Repeated Measures Categorical Ordinal Item Response Data

When categorical ordinal item response data are collected over multiple timepoints from a repeated measures design, an item response theory (IRT) modeling approach whose unit of analysis is an item response is suitable. This study proposes a few longitudinal IRT models and illustrates how a popular compensatory multidimensional IRT model can be utilized to formulate such longitudinal IRT models, which permits an investigation of ability growth at both individual and population levels. The equivalence of an existing multidimensional IRT model and those longitudinal IRT models is also elaborated so that one can make use of an existing multidimensional IRT model to implement the longitudinal IRT models.     

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.     

多维题组效应Rasch模型

首先, 本文诠释了“题组”的本质即一个存在共同刺激的项目集合。并基于此, 将题组效应划分为项目内单维题组效应和项目内多维题组效应。其次, 本文基于Rasch模型开发了二级评分和多级评分的多维题组效应Rasch模型, 以期较好地处理项目内多维题组效应。最后, 模拟研究结果显示新模型有效合理, 与Rasch题组模型、分部评分模型对比研究后表明：(1)测验存在项目内多维题组效应时, 仅把明显的捆绑式题组效应进行分离而忽略其他潜在的题组效应, 仍会导致参数的偏差估计甚或高估测验信度; (2)新模型更具普适性, 即便当被试作答数据不存在题组效应或只存在项目内单维题组效应, 采用新模型进行测验分析也能得到较好的参数估计结果。     

计算机化多维测验中作答时间和作答精度数据的联合分析

随着心理与教育测量研究的发展和科技的进步,计算机化(大规模)测验逐渐受到人们的关注。为探究在计算机化多维测验中如何利用作答时间数据来辅助评估多维潜在能力,以及为我国义务教育阶段教育质量监测提供数据分析方法上的理论支持。本研究以2012年和2015年国际学生能力评估(PISA)计算机化数学测验数据为例,提出了一种可同时利用作答时间和作答精度数据的联合作答与时间的多维Rasch模型。根据新模型对PISA数据的分析结果,表明引入作答时间数据,不仅有助于提高模型参数的估计精度,还有助于数据分析者利用被试的作答时间信息来做进一步的决策和干预(e.g., 对异常作答行为或预备知识的诊断)。     

The equivalence of two methods of parameter estimation for the rasch model

Two methods of estimating parameters in the Rasch model are compared. It is shown that estimates for a certain loglinear model for the score × item × response table are equivalent to the unconditional maximum likelihood estimates for the Rasch model.     

Loglinear Rasch models for the analysis of stability and change

Loglinear unidimensional and multidimensional Rasch models are considered for the analysis of repeated observations of polytomous indicators with ordered response categories. Reparameterizations and parameter restrictions are provided which facilitate specification of a variety of hypotheses about latent processes of change. Models of purely quantitative change in latent traits are proposed as well as models including structural change. A conditional likelihood ratio test is presented for the comparison of unidimensional and multiple scales Rasch models. In the context of longitudinal research, this renders possible the statistical test of homogeneity of change against subject-specific change in latent traits. Applications to two empirical data sets illustrate the use of the models.The author is greatly indebted to Ulf Böckenholt, Rolf Langeheine, and several anonymous reviewers for many helpful suggestions.     

Log-Multiplicative Association Models as Item Response Models

Log-multiplicative association (LMA) models, which are special cases of log-linear models, have interpretations in terms of latent continuous variables. Two theoretical derivations of LMA models based on item response theory (IRT) arguments are presented. First, we show that Anderson and colleagues (Anderson &; Vermunt, 2000; Anderson &; Böckenholt, 2000; Anderson, 2002), who derived LMA models from statistical graphical models, made the equivalent assumptions as Holland (1990) when deriving models for the manifest probabilities of response patterns based on an IRT approach. We also present a second derivation of LMA models where item response functions are specified as functions of rest-scores. These various connections provide insights into the behavior of LMA models as item response models and point out philosophical issues with the use of LMA models as item response models. We show that even for short tests, LMA and standard IRT models yield very similar to nearly identical results when data arise from standard IRT models. Log-multiplicative association models can be used as item response models and do not require numerical integration for estimation.     

多水平IRT的发展与应用述评

阶层线性模型是处理阶层结构数据的高级统计方法, 项目反应理论是精确测量被试能力的现代测量理论。多水平项目反应理论将阶层线性模型和项目反应理论相结合, 将项目反应模型嵌套在阶层线性模型内, 实现了项目参数和不同水平能力参数的估计, 对回归系数和误差项变异的估计也更加精确。作者概述了多水平项目反应理论的发展历程, 并从项目功能差异、测验等值、学校效能研究等方面评述了多水平项目反应理论在心理与教育测量中的应用, 总结了多水平项目反应理论的价值, 同时展望了今后的研究趋势。     

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 nonlinear mixed model framework for item response theory

Mixed models take the dependency between observations based on the same cluster into account by introducing 1 or more random effects. Common item response theory (IRT) models introduce latent person variables to model the dependence between responses of the same participant. Assuming a distribution for the latent variables, these IRT models are formally equivalent with nonlinear mixed models. It is shown how a variety of IRT models can be formulated as particular instances of nonlinear mixed models. The unifying framework offers the advantage that relations between different IRT models become explicit and that it is rather straightforward to see how existing IRT models can be adapted and extended. The approach is illustrated with a self-report study on anger.     

A person-fit index for polytomous rasch models,latent class models,and their mixture generalizations

A normally distributed person-fit index is proposed for detecting aberrant response patterns in latent class models and mixture distribution IRT models for dichotomous and polytomous data.This article extends previous work on the null distribution of person-fit indices for the dichotomous Rasch model to a number of models for categorical data. A comparison of two different approaches to handle the skewness of the person-fit index distribution is included.Major parts of this paper were written while the first author worked at the Institute for Science Education, Kiel, Germany. Any opinions expressed in this paper are those of the authors and not necessarily of Educational Testing Service. The results presented in this paper were improved by valuable comments from J. Rost, K. Yamamoto, N.D. Verhelst, E. Bedrick and two anonymous reviewers.     

A dual process item response theory model for polytomous multidimensional forced-choice items

The use of multidimensional forced-choice (MFC) items to assess non-cognitive traits such as personality, interests and values in psychological tests has a long history, because MFC items show strengths in preventing response bias. Recently, there has been a surge of interest in developing item response theory (IRT) models for MFC items. However, nearly all of the existing IRT models have been developed for MFC items with binary scores. Real tests use MFC items with more than two categories; such items are more informative than their binary counterparts. This study developed a new IRT model for polytomous MFC items based on the cognitive model of choice, which describes the cognitive processes underlying humans' preferential choice behaviours. The new model is unique in its ability to account for the ipsative nature of polytomous MFC items, to assess individual psychological differentiation in interests, values and emotions, and to compare the differentiation levels of latent traits between individuals. Simulation studies were conducted to examine the parameter recovery of the new model with existing computer programs. The results showed that both statement parameters and person parameters were well recovered when the sample size was sufficient. The more complete the linking of the statements was, the more accurate the parameter estimation was. This paper provides an empirical example of a career interest test using four-category MFC items. Although some aspects of the model (e.g., the nature of the person parameters) require additional validation, our approach appears promising.     

Invariance Properties for General Diagnostic Classification Models

In item response theory (IRT), the invariance property states that item parameter estimates are independent of the examinee sample, and examinee ability estimates are independent of the test items. While this property has long been established and understood by the measurement community for IRT models, the same cannot be said for diagnostic classification models (DCMs). DCMs are a newer class of psychometric models that are designed to classify examinees according to levels of categorical latent traits. We examined the invariance property for general DCMs using the log-linear cognitive diagnosis model (LCDM) framework. We conducted a simulation study to examine the degree to which theoretical invariance of LCDM classifications and item parameter estimates can be observed under various sample and test characteristics. Results illustrated that LCDM classifications and item parameter estimates show clear invariance when adequate model data fit is present. To demonstrate the implications of this important property, we conducted additional analyses to show that using pre-calibrated tests to classify examinees provided consistent classifications across calibration samples with varying mastery profile distributions and across tests with varying difficulties.     

Non‐ignorable missingness item response theory models for choice effects in examinee‐selected items

Examinee‐selected item (ESI) design, in which examinees are required to respond to a fixed number of items in a given set, always yields incomplete data (i.e., when only the selected items are answered, data are missing for the others) that are likely non‐ignorable in likelihood inference. Standard item response theory (IRT) models become infeasible when ESI data are missing not at random (MNAR). To solve this problem, the authors propose a two‐dimensional IRT model that posits one unidimensional IRT model for observed data and another for nominal selection patterns. The two latent variables are assumed to follow a bivariate normal distribution. In this study, the mirt freeware package was adopted to estimate parameters. The authors conduct an experiment to demonstrate that ESI data are often non‐ignorable and to determine how to apply the new model to the data collected. Two follow‐up simulation studies are conducted to assess the parameter recovery of the new model and the consequences for parameter estimation of ignoring MNAR data. The results of the two simulation studies indicate good parameter recovery of the new model and poor parameter recovery when non‐ignorable missing data were mistakenly treated as ignorable.     

Local homogeneity in latent trait models. A characterization of the homogeneous monotone irt model

The stochastic subject formulation of latent trait models contends that, within a given subject, the event of obtaining a certain response pattern may be probabilistic. Ordinary latent trait models do not imply that these within-subject probabilities are identical to the conditional probabilities specified by the model. The latter condition is called local homogeneity. It is shown that local homgeneity is equivalent to subpopulation invariance of the model. In case of the monotone IRT model, local homogeneity implies absence of item bias, absence of item specific traits, and the possibility to join overlapping subtests. The following characterization theorem is proved: the homogeneous monotone IRT model holds for a finite or countable item pool if and only if the pool is experimentally independent and pairwise nonnegative association holds in every positive subpopulation.This research was supported by the Dutch Interuniversity Graduate School of Psychometrics and Sociometrics. The authors wish to thank two reviewers for their thorough comments.     

Robust Measurement via A Fused Latent and Graphical Item Response Theory Model

Item response theory (IRT) plays an important role in psychological and educational measurement. Unlike the classical testing theory, IRT models aggregate the item level information, yielding more accurate measurements. Most IRT models assume local independence, an assumption not likely to be satisfied in practice, especially when the number of items is large. Results in the literature and simulation studies in this paper reveal that misspecifying the local independence assumption may result in inaccurate measurements and differential item functioning. To provide more robust measurements, we propose an integrated approach by adding a graphical component to a multidimensional IRT model that can offset the effect of unknown local dependence. The new model contains a confirmatory latent variable component, which measures the targeted latent traits, and a graphical component, which captures the local dependence. An efficient proximal algorithm is proposed for the parameter estimation and structure learning of the local dependence. This approach can substantially improve the measurement, given no prior information on the local dependence structure. The model can be applied to measure both a unidimensional latent trait and multidimensional latent traits.