A Multilevel Mixture IRT Framework for Modeling Response Times as Predictors or Indicators of Response Engagement in IRT Models

Disengaged item responses pose a threat to the validity of the results provided by large-scale assessments. Several procedures for identifying disengaged responses on the basis of observed response times have been suggested, and item response theory (IRT) models for response engagement have been proposed. We outline that response time-based procedures for classifying response engagement and IRT models for response engagement are based on common ideas, and we propose the distinction between independent and dependent latent class IRT models. In all IRT models considered, response engagement is represented by an item-level latent class variable, but the models assume that response times either reflect or predict engagement. We summarize existing IRT models that belong to each group and extend them to increase their flexibility. Furthermore, we propose a flexible multilevel mixture IRT framework in which all IRT models can be estimated by means of marginal maximum likelihood. The framework is based on the widespread Mplus software, thereby making the procedure accessible to a broad audience. The procedures are illustrated on the basis of publicly available large-scale data. Our results show that the different IRT models for response engagement provided slightly different adjustments of item parameters of individuals’ proficiency estimates relative to a conventional IRT model.     

Multilevel Modeling of Item Position Effects

In many testing programs it is assumed that the context or position in which an item is administered does not have a differential effect on examinee responses to the item. Violations of this assumption may bias item response theory estimates of item and person parameters. This study examines the potentially biasing effects of item position. A hierarchical generalized linear model is formulated for estimating item‐position effects. The model is demonstrated using data from a pilot administration of the GRE wherein the same items appeared in different positions across the test form. Methods for detecting and assessing position effects are discussed, as are applications of the model in the contexts of test development and item analysis.     

A Comparison Between Linear IRT Observed‐Score Equating and Levine Observed‐Score Equating Under the Generalized Kernel Equating Framework

In this article, linear item response theory (IRT) observed‐score equating is compared under a generalized kernel equating framework with Levine observed‐score equating for nonequivalent groups with anchor test design. Interestingly, these two equating methods are closely related despite being based on different methodologies. Specifically, when using data from IRT models, linear IRT observed‐score equating is virtually identical to Levine observed‐score equating. This leads to the conclusion that poststratification equating based on true anchor scores can be viewed as the curvilinear Levine observed‐score equating.     

Investigating Experimental Effects Within the Framework of Structural Equation Modeling: An Example With Effects on Both Error Scores and Reaction Times

Structural equation modeling provides the framework for investigating experimental effects on the basis of variances and covariances in repeated measurements. A special type of confirmatory factor analysis as part of this framework enables the appropriate representation of the experimental effect and the separation of experimental and nonexperimental parts of variance. The constraint of the matrix of loadings is essential for the representation of the effect. Appropriate constraints of loadings are achievable with the aid of the polynomial function. The representation can even bear on several response modes. The usefulness of this method is demonstrated in data obtained by an experimental task with 3 treatment levels with respect to reaction times and error scores. A model with latent variables representing constancy and increase in reaction times and one latent variable representing increase in error scores serves best in these data. Both reaction times and error scores show experimental effects.     

An NCME Instructional Module on Item‐Fit Statistics for Item Response Theory Models

Drawing valid inferences from item response theory (IRT) models is contingent upon a good fit of the data to the model. Violations of model‐data fit have numerous consequences, limiting the usefulness and applicability of the model. This instructional module provides an overview of methods used for evaluating the fit of IRT models. Upon completing this module, the reader will have an understanding of traditional and Bayesian approaches for evaluating model‐data fit of IRT models, the relative advantages of each approach, and the software available to implement each method.     

The Philosophical Aspects of IRT Equating: Modeling Drift to Evaluate Cohort Growth in Large‐Scale Assessments

Calibration and equating is the quintessential necessity for most large‐scale educational assessments. However, there are instances when no consideration is given to the equating process in terms of context and substantive realization, and the methods used in its execution. In the view of the authors, equating is not merely an exhibit of the statistical methodology, but it is also a reflection of the thought process undertaken in its execution. For example, there is hardly any discussion in literature of the ideological differences in the selection of an equating method. Furthermore, there is little evidence of modeling cohort growth through an identification and use of construct‐relevant linking items’ drift, using the common item nonequivalent group equating design. In this article, the authors philosophically justify the use of Huynh's statistical method for the identification of construct‐relevant outliers in the linking pool. The article also dispels the perception of scale instability associated with the inclusion of construct‐relevant outliers in the linking item pool and concludes that an appreciation of the rationale used in the selection of the equating method, together with the use of linking items in modeling cohort growth, can be beneficial to the practitioners.     

An Item‐Level Expected Classification Accuracy and Its Applications in Cognitive Diagnostic Assessment

Most of the existing classification accuracy indices of attribute patterns lose effectiveness when the response data is absent in diagnostic testing. To handle this issue, this article proposes new indices to predict the correct classification rate of a diagnostic test before administering the test under the deterministic noise input “and” gate (DINA) model. The new indices include an item‐level expected classification accuracy (ECA) for attributes and a test‐level ECA for attributes and attribute patterns, and both of them are calculated based solely on the known item parameters and Q ‐matrix. Theoretical analysis showed that the item‐level ECA could be regarded as a measure of correct classification rates of attributes contributed by an item. This article also illustrates how to apply the item‐level ECA for attributes to estimate the correct classification rate of attributes patterns at the test level. Simulation results showed that two test‐level ECA indices, ECA_I_W (an index based on the independence assumption and the weighted sum of the item‐level ECAs) and ECA_C_M (an index based on Gaussian Copula function that incorporates the dependence structure of the events of attribute classification and the simple average of the item‐level ECAs), could make an accurate prediction for correct classification rates of attribute patterns.     

The Long‐Term Sustainability of IRT Scaling Methods in Mixed‐Format Tests

Due to recent research in equating methodologies indicating that some methods may be more susceptible to the accumulation of equating error over multiple administrations, the sustainability of several item response theory methods of equating over time was investigated. In particular, the paper is focused on two equating methodologies: fixed common item parameter scaling (with two variations, FCIP‐1 and FCIP‐2) and the Stocking and Lord characteristic curve scaling technique in the presence of nonequivalent groups. Results indicated that the improvements made to fixed common item parameter scaling in the FCIP‐2 method were sustained over time. FCIP‐2 and Stocking and Lord characteristic curve scaling performed similarly in many instances and produced more accurate results than FCIP‐1. The relative performance of FCIP‐2 and Stocking and Lord characteristic curve scaling depended on the nature of the change in the ability distribution: Stocking and Lord characteristic curve scaling captured the change in the distribution more accurately than FCIP‐2 when the change was different across the ability distribution; FCIP‐2 captured the changes more accurately when the change was consistent across the ability distribution.     

The Impact of Model Misspecification on Parameter Estimation and Item‐Fit Assessment in Log‐Linear Diagnostic Classification Models

Using a complex simulation study we investigated parameter recovery, classification accuracy, and performance of two item‐fit statistics for correct and misspecified diagnostic classification models within a log‐linear modeling framework. The basic manipulated test design factors included the number of respondents (1,000 vs. 10,000), attributes (3 vs. 5), and items (25 vs. 50) as well as different attribute correlations (.50 vs. .80) and marginal attribute difficulties (equal vs. different). We investigated misspecifications of interaction effect parameters under correct Q‐matrix specification and two types of Q‐matrix misspecification. While the misspecification of interaction effects had little impact on classification accuracy, invalid Q‐matrix specifications led to notably decreased classification accuracy. Two proposed item‐fit indexes were more strongly sensitive to overspecification of Q‐matrix entries for items than to underspecification. Information‐based fit indexes AIC and BIC were sensitive to both over‐ and underspecification.     

Evaluating the Wald Test for Item‐Level Comparison of Saturated and Reduced Models in Cognitive Diagnosis

This article used the Wald test to evaluate the item‐level fit of a saturated cognitive diagnosis model (CDM) relative to the fits of the reduced models it subsumes. A simulation study was carried out to examine the Type I error and power of the Wald test in the context of the G‐DINA model. Results show that when the sample size is small and a larger number of attributes are required, the Type I error rate of the Wald test for the DINA and DINO models can be higher than the nominal significance levels, while the Type I error rate of the A‐CDM is closer to the nominal significance levels. However, with larger sample sizes, the Type I error rates for the three models are closer to the nominal significance levels. In addition, the Wald test has excellent statistical power to detect when the true underlying model is none of the reduced models examined even for relatively small sample sizes. The performance of the Wald test was also examined with real data. With an increasing number of CDMs from which to choose, this article provides an important contribution toward advancing the use of CDMs in practical educational settings.     

Asymptotic Standard Errors of Observed‐Score Equating With Polytomous IRT Models

In observed‐score equipercentile equating, the goal is to make scores on two scales or tests measuring the same construct comparable by matching the percentiles of the respective score distributions. If the tests consist of different items with multiple categories for each item, a suitable model for the responses is a polytomous item response theory (IRT) model. The parameters from such a model can be utilized to derive the score probabilities for the tests and these score probabilities may then be used in observed‐score equating. In this study, the asymptotic standard errors of observed‐score equating using score probability vectors from polytomous IRT models are derived using the delta method. The results are applied to the equivalent groups design and the nonequivalent groups design with either chain equating or poststratification equating within the framework of kernel equating. The derivations are presented in a general form and specific formulas for the graded response model and the generalized partial credit model are provided. The asymptotic standard errors are accurate under several simulation conditions relating to sample size, distributional misspecification and, for the nonequivalent groups design, anchor test length.     

Performance of the Generalized S‐X2 Item Fit Index for Polytomous IRT Models

Orlando and Thissen's S‐X ² item fit index has performed better than traditional item fit statistics such as Yen's Q₁ and McKinley and Mill's G² for dichotomous item response theory (IRT) models. This study extends the utility of S‐X ² to polytomous IRT models, including the generalized partial credit model, partial credit model, and rating scale model. The performance of the generalized S‐X ² in assessing item model fit was studied in terms of empirical Type I error rates and power and compared to G². The results suggest that the generalized S‐X ² is promising for polytomous items in educational and psychological testing programs.     

Predicting Freshman Grade‐Point Average from Test Scores: Effects of Variation Within and Between High Schools

Most studies predicting college performance from high‐school grade point average (HSGPA) and college admissions test scores use single‐level regression models that conflate relationships within and between high schools. Because grading standards vary among high schools, these relationships are likely to differ within and between schools. We used two‐level regression models to predict freshman grade point average from HSGPA and scores on both college admissions and state tests. When HSGPA and scores are considered together, HSGPA predicts more strongly within high schools than between, as expected in the light of variations in grading standards. In contrast, test scores, particularly mathematics scores, predict more strongly between schools than within. Within‐school variation in mathematics scores has no net predictive value, but between‐school variation is substantially predictive. Whereas other studies have shown that adding test scores to HSGPA yields only a minor improvement in aggregate prediction, our findings suggest that a potentially more important effect of admissions tests is statistical moderation, that is, partially offsetting differences in grading standards across high schools.     

Agent‐Based Modeling of Growth Processes

Growth processes abound in nature, and are frequently the target of modeling exercises in the sciences. In this article we illustrate an agent‐based approach to modeling, in the case of a single example from the social sciences: bullying.     

Development of Information Functions and Indices for the GGUM‐RANK Multidimensional Forced Choice IRT Model

This research derived information functions and proposed new scalar information indices to examine the quality of multidimensional forced choice (MFC) items based on the RANK model. We also explored how GGUM‐RANK information, latent trait recovery, and reliability varied across three MFC formats: pairs (two response alternatives), triplets (three alternatives), and tetrads (four alternatives). As expected, tetrad and triplet measures provided substantially more information than pairs, and MFC items composed of statements with high discrimination parameters were most informative. The methods and findings of this study will help practitioners to construct better MFC items, make informed projections about reliability with different MFC formats, and facilitate the development of MFC triplet‐ and tetrad‐based computerized adaptive tests.     

Examining the Value of a Scaffolded Critique Framework to Promote Argumentative and Explanatory Writings Within an Argument-Based Inquiry Approach

This study investigated the value of using a scaffolded critique framework to promote two different types of writing—argumentative writing and explanatory writing—with different purposes within an argument-based inquiry approach known as the Science Writing Heuristic (SWH) approach. A quasi-experimental design with sixth and seventh grade students taught by two teachers was used. A total of 170 students participated in the study, with 87 in the control group (four classes) and 83 in the treatment group (four classes). All students used the SWH templates as an argumentative writing to guide their written work and completed these templates during the SWH investigations of each unit. After completing the SWH investigations, both groups of students were asked to complete the summary writing task as an explanatory writing at the end of each unit. All students’ writing samples were scored using analytical frameworks developed for the study. The results indicated that the treatment group performed significantly better on the explanatory writing task than the control group. In addition, the results of the partial correlation suggested that there is a very strong significantly positive relationship between the argumentative writing and the explanatory writing.     

Diagnosing Teachers’ Understandings of Rational Numbers: Building a Multidimensional Test Within the Diagnostic Classification Framework

We report a multidimensional test that examines middle grades teachers’ understanding of fraction arithmetic, especially multiplication and division. The test is based on four attributes identified through an analysis of the extensive mathematics education research literature on teachers’ and students’ reasoning in this content area. We administered the test to a national sample of 990 in‐service middle grades teachers and analyzed the item responses using the log‐linear cognitive diagnosis model. We report the diagnostic quality of the test at the item level, mastery classifications for teachers, and attribute relationships. Our results demonstrate that, when a test is grounded in research on cognition and is designed to be multidimensional from the onset, it is possible to use diagnostic classification models to detect distinct patterns of attribute mastery.     

Modeling Student Test‐Taking Motivation in the Context of an Adaptive Achievement Test

This study examined the utility of response time‐based analyses in understanding the behavior of unmotivated test takers. For the data from an adaptive achievement test, patterns of observed rapid‐guessing behavior and item response accuracy were compared to the behavior expected under several types of models that have been proposed to represent unmotivated test taking behavior. Test taker behavior was found to be inconsistent with these models, with the exception of the effort‐moderated model. Effort‐moderated scoring was found to both yield scores that were more accurate than those found under traditional scoring, and exhibit improved person fit statistics. In addition, an effort‐guided adaptive test was proposed and shown by a simulation study to alleviate item difficulty mistargeting caused by unmotivated test taking.     

Bayesian Model Selection Methods for Multilevel IRT Models: A Comparison of Five DIC‐Based Indices

Model selection is important in any statistical analysis, and the primary goal is to find the preferred (or most parsimonious) model, based on certain criteria, from a set of candidate models given data. Several recent publications have employed the deviance information criterion (DIC) to do model selection among different forms of multilevel item response theory models (MLIRT). The majority of the practitioners use WinBUGS for implementing MCMC algorithms for MLIRT models, and the default version of DIC provided by WinBUGS focused on the measurement‐level parameters only. The results herein show that this version of DIC is inappropriate. This study introduces five variants of DIC as a model selection index for MLIRT models with dichotomous outcomes. Considering a multilevel IRT model with three levels, five forms of DIC are formed: first‐level conditional DIC computed from the measurement model only, which is the index given by many software packages such as WinBUGS; second‐level marginalized DIC and second‐level joint DIC computed from the second‐level model; and top‐level marginalized DIC and top‐level joint DIC computed from the entire model. We evaluate the performance of the five model selection indices via simulation studies. The manipulated factors include the number of groups, the number of second‐level covariates, the number of top‐level covariates, and the types of measurement models (one‐parameter vs. two‐parameter). Considering the computational viability and interpretability, the second‐level joint DIC is recommended for MLIRT models under our simulated conditions.     

Orienting Attention Within Visual Short‐Term Memory: Development and Mechanisms

How does developing attentional control operate within visual short‐term memory (VSTM)? Seven‐year‐olds, 11‐year‐olds, and adults (total n = 205) were asked to report whether probe items were part of preceding visual arrays. In Experiment 1, central or peripheral cues oriented attention to the location of to‐be‐probed items either prior to encoding or during maintenance. Cues improved memory regardless of their position, but younger children benefited less from cues presented during maintenance, and these benefits related to VSTM span over and above basic memory in uncued trials. In Experiment 2, cues of low validity eliminated benefits, suggesting that even the youngest children use cues voluntarily, rather than automatically. These findings elucidate the close coupling between developing visuospatial attentional control and VSTM.