Assessing Fit of Unidimensional Item Response Theory Models Using a Bayesian Approach

Even though Bayesian estimation has recently become quite popular in item response theory (IRT), there is a lack of works on model checking from a Bayesian perspective. This paper applies the posterior predictive model checking (PPMC) method ( Guttman, 1967 ; Rubin, 1984 ), a popular Bayesian model checking tool, to a number of real applications of unidimensional IRT models. The applications demonstrate how to exploit the flexibility of the posterior predictive checks to meet the need of the researcher. This paper also examines practical consequences of misfit, an area often ignored in educational measurement literature while assessing model fit.     

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.     

Assessing Fit of Item Response Models Using the Information Matrix Test

The information matrix can equivalently be determined via the expectation of the Hessian matrix or the expectation of the outer product of the score vector. The identity of these two matrices, however, is only valid in case of a correctly specified model. Therefore, differences between the two versions of the observed information matrix indicate model misfit. The equality of both matrices can be tested with the so‐called information matrix test as a general test of misspecification. This test can be adapted to item response models in order to evaluate the fit of single items and the fit of the whole scale. The performance of different versions of the test is compared in a simulation study with existing tests of model fit, among them the test of Orlando and Thissen, the score test of local independence due to Glas and Suarez‐Falcon, and the limited information approach of Maydeu‐Olivares and Joe. In general, the different versions of the information matrix test adhere to the nominal Type I error rate and have high power for detecting misspecified item characteristic curves. Additionally, some versions of the test can be used in order to detect violations of the local independence assumption.     

Comparing the Fit of Item Response Theory and Factor Analysis Models

Linear factor analysis (FA) models can be reliably tested using test statistics based on residual covariances. We show that the same statistics can be used to reliably test the fit of item response theory (IRT) models for ordinal data (under some conditions). Hence, the fit of an FA model and of an IRT model to the same data set can now be compared. When applied to a binary data set, our experience suggests that IRT and FA models yield similar fits. However, when the data are polytomous ordinal, IRT models yield a better fit because they involve a higher number of parameters. But when fit is assessed using the root mean square error of approximation (RMSEA), similar fits are obtained again. We explain why. These test statistics have little power to distinguish between FA and IRT models; they are unable to detect that linear FA is misspecified when applied to ordinal data generated under an IRT model.     

非参数项目反应理论模型及其在教育考试中的应用

非参数项目反应理论模型包括单调均匀性模型和双单调模型。用单调均匀性模型对某英语听力考试结果研究发现,按照顺序选择法,可从16道听力试题中选出11道满足要求的试题,组成单维量表。用考生在这11道试题上的总得分对考生进行排序与按照潜质排序等效。利用双单调模型对11道听力试题组成的单维量表进行试题功能偏差研究发现,有5道试题在女生子群体中的排序与在男生子群体以及整个群体排序不同,显示女生子群体作出正确应答的概率明显高于男生子群体作出正确应答的概率。这种差异至少部分是由两个子群体听力能力上的差异引起的。     

A Comparison of Item Fit Statistics for Mixed IRT Models

In this study we examined procedures for assessing model-data fit of item response theory (IRT) models for mixed format data. The model fit indices used in this study include PARSCALE's G² , Orlando and Thissen's S − X² and S − G² , and Stone's χ²* and G²* . To investigate the relative performance of the fit statistics at the item level, we conducted two simulation studies: Type I error and power studies. We evaluated the performance of the item fit indices for various conditions of test length, sample size, and IRT models. Among the competing measures, the summed score-based indices S − X² and S − G² were found to be the sensible and efficient choice for assessing model fit for mixed format data. These indices performed well, particularly with short tests. The pseudo-observed score indices, χ²* and G²* , showed inflated Type I error rates in some simulation conditions. Consistent with the findings of current literature, the PARSCALE's G² index was rarely useful, although it provided reasonable results for long tests.     

An NCME Instructional Module on Polytomous Item Response Theory Models

A polytomous item is one for which the responses are scored according to three or more categories. Given the increasing use of polytomous items in assessment practices, item response theory (IRT) models specialized for polytomous items are becoming increasingly common. The purpose of this ITEMS module is to provide an accessible overview of polytomous IRT models. The module presents commonly encountered polytomous IRT models, describes their properties, and contrasts their defining principles and assumptions. After completing this module, the reader should have a sound understating of what a polytomous IRT model is, the manner in which the equations of the models are generated from the model's underlying step functions, how widely used polytomous IRT models differ with respect to their definitional properties, and how to interpret the parameters of polytomous IRT models.     

Assessing the Sensitivity of Weighted Least Squares Model Fit Indexes to Local Dependence in Item Response Theory Models

Given the relationships of item response theory (IRT) models to confirmatory factor analysis (CFA) models, IRT model misspecifications might be detectable through model fit indexes commonly used in categorical CFA. The purpose of this study is to investigate the sensitivity of weighted least squares with adjusted means and variance (WLSMV)-based root mean square error of approximation, comparative fit index, and Tucker–Lewis Index model fit indexes to IRT models that are misspecified due to local dependence (LD). It was found that WLSMV-based fit indexes have some functional relationships to parameter estimate bias in 2-parameter logistic models caused by violations of LD. Continued exploration into these functional relationships and development of LD-detection methods based on such relationships could hold much promise for providing IRT practitioners with global information on violations of local independence.     

Assessing Goodness of Fit of Item Response Theory Models: A Comparison of Traditional and Alternative Procedures

Testing the goodness of fit of item response theory (IRT) models is relevant to validating IRT models, and new procedures have been proposed. These alternatives compare observed and expected response frequencies conditional on observed total scores, and use posterior probabilities for responses across θ levels rather than cross-classifying examinees using point estimates of θ and score responses. This research compared these alternatives with regard to their methods, properties (Type 1 error rates and empirical power), available research, and practical issues (computational demands, treatment of missing data, effects of sample size and sparse data, and available computer programs). Different advantages and disadvantages related to these characteristics are discussed. A simulation study provided additional information about empirical power and Type 1 error rates.     

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.     

Investigation of a Nonparametric Procedure for Assessing Goodness-of-Fit in Item Response Theory

There have been many studies of the comparability of computer-administered and paper-administered tests. Not surprisingly (given the variety of measurement and statistical sampling issues that can affect any one study) the results of such studies have not always been consistent. Moreover, the quality of computer-based test administration systems has changed considerably over recent years, as has the computer-experience of students. This study synthesizes the results of 81 studies performed between 1997 and 2007. The estimated effect size across all studies was very small (–.01 weighted, .00 unweighted). Meta-analytic methods were used to ascertain whether grade (elementary, middle, or high school) or subject (English Language Arts, Mathematics, Reading, Science, or Social Studies) had an impact on comparability. Grade appeared to have no affect on comparability. Subject did appear to affect comparability, with computer administration appearing to provide a small advantage for English Language Arts and Social Studies test (effect sizes of .11 and .15, respectively), and paper administration appearing to provide a small advantage for Mathematics tests (effect size of??.06).     

Item Response Theory Models for Performance Decline During Testing

Sometimes, test‐takers may not be able to attempt all items to the best of their ability (with full effort) due to personal factors (e.g., low motivation) or testing conditions (e.g., time limit), resulting in poor performances on certain items, especially those located toward the end of a test. Standard item response theory (IRT) models fail to consider such testing behaviors. In this study, a new class of mixture IRT models was developed to account for such testing behavior in dichotomous and polytomous items, by assuming test‐takers were composed of multiple latent classes and by adding a decrement parameter to each latent class to describe performance decline. Parameter recovery, effect of model misspecification, and robustness of the linearity assumption in performance decline were evaluated using simulations. It was found that the parameters in the new models were recovered fairly well by using the freeware WinBUGS; the failure to account for such behavior by fitting standard IRT models resulted in overestimation of difficulty parameters on items located toward the end of the test and overestimation of test reliability; and the linearity assumption in performance decline was rather robust. An empirical example is provided to illustrate the applications and the implications of the new class of models.     

Item Response Models for Examinee‐Selected Items

In some tests, examinees are required to choose a fixed number of items from a set of given items to answer. This practice creates a challenge to standard item response models, because more capable examinees may have an advantage by making wiser choices. In this study, we developed a new class of item response models to account for the choice effect of examinee‐selected items. The results of a series of simulation studies showed: (1) that the parameters of the new models were recovered well, (2) the parameter estimates were almost unbiased when the new models were fit to data that were simulated from standard item response models, (3) failing to consider the choice effect yielded shrunken parameter estimates for examinee‐selected items, and (4) even when the missingness mechanism in examinee‐selected items did not follow the item response functions specified in the new models, the new models still yielded a better fit than did standard item response models. An empirical example of a college entrance examination supported the use of the new models: in general, the higher the examinee's ability, the better his or her choice of items.     

Item Response Models for Multiple Attempts With Incomplete Data

Allowance for multiple chances to answer constructed response questions is a prevalent feature in computer‐based homework and exams. We consider the use of item response theory in the estimation of item characteristics and student ability when multiple attempts are allowed but no explicit penalty is deducted for extra tries. This is common practice in online formative assessments, where the number of attempts is often unlimited. In these environments, some students may not always answer‐until‐correct, but may rather terminate a response process after one or more incorrect tries. We contrast the cases of graded and sequential item response models, both unidimensional models which do not explicitly account for factors other than ability. These approaches differ not only in terms of log‐odds assumptions but, importantly, in terms of handling incomplete data. We explore the consequences of model misspecification through a simulation study and with four online homework data sets. Our results suggest that model selection is insensitive for complete data, but quite sensitive to whether missing responses are regarded as informative (of inability) or not (e.g., missing at random). Under realistic conditions, a sequential model with similar parametric degrees of freedom to a graded model can account for more response patterns and outperforms the latter in terms of model fit.     

Item Response Models for Local Dependence Among Multiple Ratings

Ratings given to the same item response may have a stronger correlation than those given to different item responses, especially when raters interact with one another before giving ratings. The rater bundle model was developed to account for such local dependence by forming multiple ratings given to an item response as a bundle and assigning fixed‐effect parameters to describe response patterns in the bundle. Unfortunately, this model becomes difficult to manage when a polytomous item is graded by more than two raters. In this study, by adding random‐effect parameters to the facets model, we propose a class of generalized rater models to account for the local dependence among multiple ratings and intrarater variation in severity. A series of simulations was conducted with the freeware WinBUGS to evaluate parameter recovery of the new models and consequences of ignoring the local dependence or intrarater variation in severity. The results revealed a good parameter recovery when the data‐generating models were fit, and a poor estimation of parameters and test reliability when the local dependence or intrarater variation in severity was ignored. An empirical example is provided.     

Use of Restricted Item Response Models for Examining Item Difficulty Ordering and Slope Uniformity

This article demonstrates the utility of restricted item response models for examining item difficulty ordering and slope uniformity for an item set that reflects varying cognitive processes. Twelve sets of paired algebra word problems were developed to systematically reflect various types of cognitive processes required for successful performance. This resulted in a total of 24 items. They reflected distance-rate–time (DRT), interest, and area problems. Hypotheses concerning difficulty ordering and slope uniformity for the items were tested by constraining item difficulty and discrimination parameters in hierarchical item response models. The first set of model comparisons tested the equality of the discrimination and difficulty parameters for each set of paired items. The second set of model comparisons examined slope uniformity within the complex DRT problems. The third set of model comparisons examined whether the familiarity of the story context affected item difficulty for two types of complex DRT problems. The last set of model comparisons tested the hypothesized difficulty ordering of the items.     

Using the Bayes Factors to Evaluate Person Fit in the Item Response Theory

In this article, we propose using the Bayes factors (BF) to evaluate person fit in item response theory models under the framework of Bayesian evaluation of an informative diagnostic hypothesis. We first discuss the theoretical foundation for this application and how to analyze person fit using BF. To demonstrate the feasibility of this approach, we further use it to evaluate person fit in simulated and empirical data, and compare the results with those of H^T and the infit and outfit statistics. We found that overall BF performed as well as H^T statistics and better than the infit and outfit statistics when detecting aberrant responses. Given the BF flexibility in handling data set with a small number of examinees, we suggest that BF can be used as person fit statistics, especially in computerized adaptive tests.     

从项目特征曲线的估算比较参数型及非参数型项目反应理论模型

项目反应理论模型的参数估计一般需要较大样本量,小样本量条件下参数型与非参数型项目反应理论模型的相对优势并无定论。通过计算机模拟数据比较两类模型在小样本量时(n<=200)估计项目特征曲线所产生的偏误及均方根误差。当模拟数据基于3PL模型生成时,参数型与非参数型模型在样本量低于200时估值偏误方面无差别,但前者均方根误差较小。在样本量为200时,两模型估算值类似。当真实数据基于3PL模型且样本量小于200时,参数型Rasch模型比非参数核平滑模型更值得推荐。     

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.     

Use of Restricted Item Response Theory Models for Examining the Stability of Item Parameter Estimates Over Time

Item parameter instability can threaten the validity of inferences about changes in student achievement when using Item Response Theory- (IRT) based test scores obtained on different occasions. This article illustrates a model-testing approach for evaluating the stability of IRT item parameter estimates in a pretest-posttest design. Stability of item parameter estimates was assessed for a random sample of pretest and posttest responses to a 19-item math test. Using MULTILOG (Thissen, 1986), IRT models were estimated in which item parameter estimates were constrained to be equal across samples (reflecting stability) and item parameter estimates were free to vary across samples (reflecting instability). These competing models were then compared statistically in order to test the invariance assumption. The results indicated a moderately high degree of stability in the item parameter estimates for a group of children assessed on two different occasions.