Applying Kaplan-Meier to Item Response Data

Some IRT models can be equivalently modeled in alternative frameworks such as logistic regression. Logistic regression can also model time-to-event data, which concerns the probability of an event occurring over time. Using the relation between time-to-event models and logistic regression and the relation between logistic regression and IRT, this article outlines how the nonparametric Kaplan-Meier estimator for time-to-event data can be applied to IRT data. Established Kaplan-Meier computational formulas are shown to aid in better approximating “parametric-type” item difficulty compared to methods from existing nonparametric methods, particularly for the less-well-defined scenario wherein the response function is monotonic but invariant item ordering is unreasonable. Limitations and the potential for Kaplan-Meier within differential item functioning are also discussed.     

The Stability of IRT b Values

This study investigated possible explanations for an observed change in Rasch item parameters (b values) obtained from consecutive administrations of a professional licensure examination. Considered in this investigation were variables related to item position, item type, item content, and elapsed time between administrations of the item. An analysis of covariance methodology was used to assess the relations between these variables and change in item b values, with the elapsed time index serving to control for differences that could be attributed to average or pool changes in b values over time. A series of analysis of covariance models were fitted to the data in an attempt to identify item characteristics that were significantly related to the change in b values after the time elapsed between item administrations had been controlled. The findings indicated that the change in item b values was not related either to item position or to item type. A small, positive relationship between this change and elapsed time indicated that the pool b values were increasing over time. A test of simple effects suggested the presence of greater change for one of the content categories analyzed. These findings are interpreted, and suggestions for future research are provided.     

A Response Time Process Model for Not-Reached and Omitted Items

Item nonresponses are prevalent in standardized testing. They happen either when students fail to reach the end of a test due to a time limit or quitting, or when students choose to omit some items strategically. Oftentimes, item nonresponses are nonrandom, and hence, the missing data mechanism needs to be properly modeled. In this paper, we proposed to use an innovative item response time model as a cohesive missing data model to account for the two most common item nonresponses: not-reached items and omitted items. In particular, the new model builds on a behavior process interpretation: a person chooses to skip an item if the required effort exceeds the implicit time the person allocates to the item (Lee & Ying, 2015; Wolf, Smith, & Birnbaum, 1995), whereas a person fails to reach the end of the test due to lack of time. This assumption was verified by analyzing the 2015 PISA computer-based mathematics data. Simulation studies were conducted to further evaluate the performance of the proposed Bayesian estimation algorithm for the new model and to compare the new model with a recently proposed “speed-accuracy + omission” model (Ulitzsch, von Davier, & Pohl, 2019). Results revealed that all model parameters could recover properly, and inadequately accounting for missing data caused biased item and person parameter estimates.     

Item Position and Item Difficulty Change in an IRT-Based Common Item Equating Design

In operational testing programs using item response theory (IRT), item parameter invariance is threatened when an item appears in a different location on the live test than it did when it was field tested. This study utilizes data from a large state's assessments to model change in Rasch item difficulty (RID) as a function of item position change, test level, test content, and item format. As a follow-up to the real data analysis, a simulation study was performed to assess the effect of item position change on equating. Results from this study indicate that item position change significantly affects change in RID. In addition, although the test construction procedures used in the investigated state seem to somewhat mitigate the impact of item position change, equating results might be impacted in testing programs where other test construction practices or equating methods are utilized.     

A Unified Latent Growth Curve Model

Applying item response theory models to repeated observations has demonstrated great promise in developmental research. By allowing the researcher to take account of the characteristics of both item response and measurement error in longitudinal trajectory analysis, it improves the reliability and validity of latent growth curve analysis. This has enabled the study, to differentially weigh individual items and examine developmental stability and change over time, to propose a comprehensive modeling framework, combining a measurement model with a structural model. Despite a large number of components requiring attention, this study focuses on model formulation, evaluates the performance of the estimators of model parameters, incorporates prior knowledge from Bayesian analysis, and applies the model using an illustrative example. It is hoped that this fundamental study can demonstrate the breadth of this unified latent growth curve model.     

Asymptotic Standard Errors for Item Response Theory True Score Equating of Polytomous Items

Building on previous works by Lord and Ogasawara for dichotomous items, this article proposes an approach to derive the asymptotic standard errors of item response theory true score equating involving polytomous items, for equivalent and nonequivalent groups of examinees. This analytical approach could be used in place of empirical methods like the bootstrap method, to obtain standard errors of equated scores. Formulas are introduced to obtain the derivatives for computing the asymptotic standard errors. The approach was validated using mean‐mean, mean‐sigma, random‐groups, or concurrent calibration equating of simulated samples, for tests modeled using the generalized partial credit model or the graded response model.     

Mokken Scale Analysis: Theoretical Considerations and an Application to Transitivity Tasks

This study provides a discussion and an application of Mokken scale analysis. Mokken scale analysis can be characterized as a nonparametric item response theory approach. The Mokken approach to scaling consists of two different item response models, the model of monotone homogeneity and the more restrictive model of double monotonicity. Methods for empirical data analysis using the two Mokken model versions are discussed. Both dichotomous and polytomous item scores can be analyzed by means of Mokken scale analysis. Three empirical data sets pertaining to transitive inference items were analyzed using the Mokken approach. The results are compared with the results obtained from a Rasch analysis.     

Achievement and Growth on English Language Proficiency and Content Assessments for English Learners in Elementary Grades

English language proficiency (ELP) assessment scores are used by states to make high-stakes decisions related to linguistic support in instruction and assessment for English learner (EL) students and for EL student reclassification. Changes to both academic content standards and ELP academic standards within the last decade have resulted in increased academic rigor and language demands. In this study, we explored the association between EL student performance over time on content (English language arts and mathematics) and ELP assessments, generally finding evidence of positive associations. Modeling the simultaneous association between changes over time in both content and ELP assessment performance contributes empirical evidence about the role of language in ELA and mathematics development and provides contextual information to serve as validity evidence for score inferences for EL students.     

An Investigation of the Power of the Likelihood Ratio Goodness-of-Fit Statistic in Detecting Differential Item Functioning

The purpose of this study was to investigate the power and Type I error rate of the likelihood ratio goodness-of-fit (LR) statistic in detecting differential item functioning (DIF) under Samejima's (1969, 1972) graded response model. A multiple-replication Monte Carlo study was utilized in which DIF was modeled in simulated data sets which were then calibrated with MULTILOG (Thissen, 1991) using hierarchically nested item response models. In addition, the power and Type I error rate of the Mantel (1963) approach for detecting DIF in ordered response categories were investigated using the same simulated data, for comparative purposes. The power of both the Mantel and LR procedures was affected by sample size, as expected. The LR procedure lacked the power to consistently detect DIF when it existed in reference/focal groups with sample sizes as small as 500/500. The Mantel procedure maintained control of its Type I error rate and was more powerful than the LR procedure when the comparison group ability distributions were identical and there was a constant DIF pattern. On the other hand, the Mantel procedure lost control of its Type I error rate, whereas the LR procedure did not, when the comparison groups differed in mean ability; and the LR procedure demonstrated a profound power advantage over the Mantel procedure under conditions of balanced DIF in which the comparison group ability distributions were identical. The choice and subsequent use of any procedure requires a thorough understanding of the power and Type I error rates of the procedure under varying conditions of DIF pattern, comparison group ability distributions.–or as a surrogate, observed score distributions–and item characteristics.     

Exploring the Utility of Background and Cognitive Variables in Explaining Latent Differential Item Functioning: An Example of the PISA 2009 Reading Assessment

Differential item functioning (DIF) may be caused by an interaction of multiple manifest grouping variables or unexplored manifest variables, which cannot be detected by conventional DIF detection methods that are based on a single manifest grouping variable. Such DIF may be detected by a latent approach using the mixture item response theory model and subsequently explained by multiple manifest variables. This study facilitates the interpretation of latent DIF with the use of background and cognitive variables. The PISA 2009 reading assessment and student survey are analyzed. Results show that members in manifest groups were not homogenously advantaged or disadvantaged and that a single manifest grouping variable did not suffice to be a proxy of latent DIF. This study also demonstrates that DIF items arising from the interaction of multiple variables can be effectively screened by the latent DIF analysis approach. Background and cognitive variables jointly well predicted latent class membership.     

Computerized Adaptive Testing in Early Education: Exploring the Impact of Item Position Effects on Ability Estimation

Studies have shown that item difficulty can vary significantly based on the context of an item within a test form. In particular, item position may be associated with practice and fatigue effects that influence item parameter estimation. The purpose of this research was to examine the relevance of item position specifically for assessments used in early education, an area of testing that has received relatively limited psychometric attention. In an initial study, multilevel item response models fit to data from an early literacy measure revealed statistically significant increases in difficulty for items appearing later in a 20‐item form. The estimated linear change in logits for an increase of 1 in position was .024, resulting in a predicted change of .46 logits for a shift from the beginning to the end of the form. A subsequent simulation study examined impacts of item position effects on person ability estimation within computerized adaptive testing. Implications and recommendations for practice are discussed.     

Reliably Assessing Growth with Longitudinal Diagnostic Classification Models

Recent advances have enabled diagnostic classification models (DCMs) to accommodate longitudinal data. These longitudinal DCMs were developed to study how examinees change, or transition, between different attribute mastery statuses over time. This study examines using longitudinal DCMs as an approach to assessing growth and serves three purposes: (1) to define and evaluate two reliability measures to be used in the application of longitudinal DCMs; (2) through simulation, demonstrate that longitudinal DCM growth estimates have increased reliability compared to longitudinal item response theory models; and (3) through an empirical analysis, illustrate the practical and interpretive benefits of longitudinal DCMs. A discussion describes how longitudinal DCMs can be used as practical and reliable psychometric models when categorical and criterion‐referenced interpretations of growth are desired.     

Parameter Estimation in Rasch Models for Examinee‐Selected Items

The examinee‐selected‐item (ESI) design, in which examinees are required to respond to a fixed number of items in a given set of items (e.g., choose one item to respond from a pair of items), always yields incomplete data (i.e., only the selected items are answered and the others have missing data) that are likely nonignorable. Therefore, using standard item response theory models, which assume ignorable missing data, can yield biased parameter estimates so that examinees taking different sets of items to answer cannot be compared. To solve this fundamental problem, in this study the researchers utilized the specific objectivity of Rasch models by adopting the conditional maximum likelihood estimation (CMLE) and pairwise estimation (PE) methods to analyze ESI data, and conducted a series of simulations to demonstrate the advantages of the CMLE and PE methods over traditional estimation methods in recovering item parameters in ESI data. An empirical data set obtained from an experiment on the ESI design was analyzed to illustrate the implications and applications of the proposed approach to ESI data.     

The Use of Hierarchical Generalized Linear Model for Item Dimensionality Assessment

To assess item dimensionality, the following two approaches are described and compared: hierarchical generalized linear model (HGLM) and multidimensional item response theory (MIRT) model. Two generating models are used to simulate dichotomous responses to a 17-item test: the unidimensional and compensatory two-dimensional (C2D) models. For C2D data, seven items are modeled to load on the first and second factors, θ₁ and θ₂, with the remaining 10 items modeled unidimensionally emulating a mathematics test with seven items requiring an additional reading ability dimension. For both types of generated data, the multidimensionality of item responses is investigated using HGLM and MIRT. Comparison of HGLM and MIRT's results are possible through a transformation of items' difficulty estimates into probabilities of a correct response for a hypothetical examinee at the mean on θ and θ₂. HGLM and MIRT performed similarly. The benefits of HGLM for item dimensionality analyses are discussed.     

Comparing the Difficulty of Examination Subjects with Item Response Theory

Methods are presented for comparing grades obtained in a situation where students can choose between different subjects. It must be expected that the comparison between the grades is complicated by the interaction between the students' pattern and level of proficiency on one hand, and the choice of the subjects on the other hand. Three methods based on item response theory (IRT) for the estimation of proficiency measures that are comparable over students and subjects are discussed: a method based on a model with a unidimensional representation of proficiency, a method based on a model with a multidimensional representation of proficiency, and a method based on a multidimensional representation of proficiency where the stochastic nature of the choice of examination subjects is explicitly modeled. The methods are compared using the data from the Central Examinations in Secondary Education in the Netherlands. The results show that the unidimensional IRT model produces unrealistic results, which do not appear when using the two multidimensional IRT models. Further, it is shown that both the multidimensional models produce acceptable model fit. However, the model that explicitly takes the choice process into account produces the best model fit.     

Modeling Partial Knowledge on Multiple‐Choice Items Using Elimination Testing

Single‐best answers to multiple‐choice items are commonly dichotomized into correct and incorrect responses, and modeled using either a dichotomous item response theory (IRT) model or a polytomous one if differences among all response options are to be retained. The current study presents an alternative IRT‐based modeling approach to multiple‐choice items administered with the procedure of elimination testing, which asks test‐takers to eliminate all the response options they consider to be incorrect. The partial credit model is derived for the obtained responses. By extracting more information pertaining to test‐takers’ partial knowledge on the items, the proposed approach has the advantage of providing more accurate estimation of the latent ability. In addition, it may shed some light on the possible answering processes of test‐takers on the items. As an illustration, the proposed approach is applied to a classroom examination of an undergraduate course in engineering science.     

An Expected Response Function Approach to Graphical Differential Item Functioning

In this paper a new approach to graphical differential item functioning (DIF) is offered. The methodology is based on a sampling-theory approach to expected response functions (Lewis, 1985; Mislevy, Wingersky, & Sheehan, 1994). Essentially error in item calibrations is modeled explicitly, and repeated samples are taken from the posterior distributions of the item parameters. Sampled parameter values are used to estimate the posterior distribution of the difference in item characteristic curves (ICCs)for two groups. A point-wise expectation is taken as an estimate of the true difference between the ICCs, and the sampled-difference functions indicate uncertainty in the estimate. Tbe approach is applied to a set of pretest items, and the results are compared to traditional Mantel-Haenszel DIF statistics. The expected-response-function approach is contrasted with Pashley's (1992) graphical DIF approach.     

Investigating the Effect of Item Position in Computer‐Based Tests

Computer‐based tests (CBTs) often use random ordering of items in order to minimize item exposure and reduce the potential for answer copying. Little research has been done, however, to examine item position effects for these tests. In this study, different versions of a Rasch model and different response time models were examined and applied to data from a CBT administration of a medical licensure examination. The models specifically were used to investigate whether item position affected item difficulty and item intensity estimates. Results indicated that the position effect was negligible.     

Latent variable modeling in the hierarchical modeling framework in longitudinal studies: a fully bayesian approach

This paper presents a strategy for specifying latent variable regressions in the hierarchical modeling framework (LVR-HM). This model takes advantage of the Structural Equation Modeling (SEM) approach in terms of modeling flexibility—regression among latent variables—and of the HM approach in terms of allowing for more general data structures. A fully Bayesian approach via Markov Chain Monte Carlo (MCMC) techniques is applied to the LVR-HM. Through analyzing the data from a longitudinal study of educational achievement, gender difference are explored in the growth of mathematical achievement across grade 7 through grade 10. Allowing for the fact that initial status effect to rates of change may differ for girls and boys, the LVR-HM is specified in a way that rates of change parameters are modeled as a function of initial status parameters and the interaction between initial status and gender.     

Examining the Precision of Cut Scores Within a Generalizability Theory Framework: A Closer Look at the Item Effect

An Angoff standard setting study generally yields judgments on a number of items by a number of judges (who may or may not be nested in panels). Variability associated with judges (and possibly panels) contributes error to the resulting cut score. The variability associated with items plays a more complicated role. To the extent that the mean item judgments directly reflect empirical item difficulties, the variability in Angoff judgments over items would not add error to the cut score, but to the extent that the mean item judgments do not correspond to the empirical item difficulties, variability in mean judgments over items would add error to the cut score. In this article, we present two generalizability-theory–based analyses of the proportion of the item variance that contributes to error in the cut score. For one approach, variance components are estimated on the probability (or proportion-correct) scale of the Angoff judgments, and for the other, the judgments are transferred to the theta scale of an item response theory model before estimating the variance components. The two analyses yield somewhat different results but both indicate that it is not appropriate to simply ignore the item variance component in estimating the error variance.