Bootstrap-Calibrated Interval Estimates for Latent Variable Scores in Item Response Theory

In most item response theory applications, model parameters need to be first calibrated from sample data. Latent variable (LV) scores calculated using estimated parameters are thus subject to sampling error inherited from the calibration stage. In this article, we propose a resampling-based method, namely bootstrap calibration (BC), to reduce the impact of the carryover sampling error on the interval estimates of LV scores. BC modifies the quantile of the plug-in posterior, i.e., the posterior distribution of the LV evaluated at the estimated model parameters, to better match the corresponding quantile of the true posterior, i.e., the posterior distribution evaluated at the true model parameters, over repeated sampling of calibration data. Furthermore, to achieve better coverage of the fixed true LV score, we explore the use of BC in conjunction with Jeffreys’ prior. We investigate the finite-sample performance of BC via Monte Carlo simulations and apply it to two empirical data examples.     

被试作答的偶然性对IRT能力估计的影响研究

被试能力参数估计是项目反应理论应用研究最重要的技术之一。本文在理想的测验情境下,研究被试作答的偶然性对被试能力值估计的影响。研究设计了被试作答的两种偶然性情况：一是偶然做对了一道项目难度高于其能力值的试题,二是偶然做错了一道或几道项目难度低于其能力值的试题．然后分别探讨了这两种情况下对被试的能力估计所带来的影响,并且就如何消除这些偶然性所带来的影响提出了相应的方法。     

A Note on the Linearly and Quadratically Weighted Kappa Coefficients

The linearly and quadratically weighted kappa coefficients are popular statistics in measuring inter-rater agreement on an ordinal scale. It has been recently demonstrated that the linearly weighted kappa is a weighted average of the kappa coefficients of the embedded 2 by 2 agreement matrices, while the quadratically weighted kappa is insensitive to the agreement matrices that are row or column reflection symmetric. A rank-one matrix decomposition approach to the weighting schemes is presented in this note such that these phenomena can be demonstrated in a concise manner.     

Assessing Item Fit for Unidimensional Item Response Theory Models Using Residuals from Estimated Item Response Functions

Residual analysis (e.g. Hambleton & Swaminathan, Item response theory: principles and applications, Kluwer Academic, Boston, 1985; Hambleton, Swaminathan, & Rogers, Fundamentals of item response theory, Sage, Newbury Park, 1991) is a popular method to assess fit of item response theory (IRT) models. We suggest a form of residual analysis that may be applied to assess item fit for unidimensional IRT models. The residual analysis consists of a comparison of the maximum-likelihood estimate of the item characteristic curve with an alternative ratio estimate of the item characteristic curve. The large sample distribution of the residual is proved to be standardized normal when the IRT model fits the data. We compare the performance of our suggested residual to the standardized residual of Hambleton et al. (Fundamentals of item response theory, Sage, Newbury Park, 1991) in a detailed simulation study. We then calculate our suggested residuals using data from an operational test. The residuals appear to be useful in assessing the item fit for unidimensional IRT models.     

A Note on Item Analysis of Family Relations Test Data

Two statistical problems prevent meaningful item analysis of Family Relations Test (FRT) data: (a) multiple assignment invalidates the assumption of independence and (b) low frequencies prevent computation of probabilities. It is suggested that: (a) multiple assignment be disallowed and (b) items be made more representative of family interactions.     

A Truncated-Probit Item Response Model for Estimating Psychophysical Thresholds

Human abilities in perceptual domains have conventionally been described with reference to a threshold that may be defined as the maximum amount of stimulation which leads to baseline performance. Traditional psychometric links, such as the probit, logit, and t, are incompatible with a threshold as there are no true scores corresponding to baseline performance. We introduce a truncated probit link for modeling thresholds and develop a two-parameter IRT model based on this link. The model is Bayesian and analysis is performed with MCMC sampling. Through simulation, we show that the model provides for accurate measurement of performance with thresholds. The model is applied to a digit-classification experiment in which digits are briefly flashed and then subsequently masked. Using parameter estimates from the model, individuals’ thresholds for flashed-digit discrimination is estimated.     

Combining Item Response Theory and Diagnostic Classification Models: A Psychometric Model for Scaling Ability and Diagnosing Misconceptions

Traditional testing procedures typically utilize unidimensional item response theory (IRT) models to provide a single, continuous estimate of a student’s overall ability. Advances in psychometrics have focused on measuring multiple dimensions of ability to provide more detailed feedback for students, teachers, and other stakeholders. Diagnostic classification models (DCMs) provide multidimensional feedback by using categorical latent variables that represent distinct skills underlying a test that students may or may not have mastered. The Scaling Individuals and Classifying Misconceptions (SICM) model is presented as a combination of a unidimensional IRT model and a DCM where the categorical latent variables represent misconceptions instead of skills. In addition to an estimate of ability along a latent continuum, the SICM model provides multidimensional, diagnostic feedback in the form of statistical estimates of probabilities that students have certain misconceptions. Through an empirical data analysis, we show how this additional feedback can be used by stakeholders to tailor instruction for students’ needs. We also provide results from a simulation study that demonstrate that the SICM MCMC estimation algorithm yields reasonably accurate estimates under large-scale testing conditions.     

Multilevel Reliability Measures of Latent Scores Within an Item Response Theory Framework

Abstract

This paper evaluated multilevel reliability measures in two-level nested designs (e.g., students nested within teachers) within an item response theory framework. A simulation study was implemented to investigate the behavior of the multilevel reliability measures and the uncertainty associated with the measures in various multilevel designs regarding the number of clusters, cluster sizes, and intraclass correlations (ICCs), and in different test lengths, for two parameterizations of multilevel item response models with separate item discriminations or the same item discrimination over levels. Marginal maximum likelihood estimation (MMLE)-multiple imputation and Bayesian analysis were employed to evaluate the accuracy of the multilevel reliability measures and the empirical coverage rates of Monte Carlo (MC) confidence or credible intervals. Considering the accuracy of the multilevel reliability measures and the empirical coverage rate of the intervals, the results lead us to generally recommend MMLE-multiple imputation. In the model with separate item discriminations over levels, marginally acceptable accuracy of the multilevel reliability measures and empirical coverage rate of the MC confidence intervals were found in a limited condition, 200 clusters, 30 cluster size, .2 ICC, and 40 items, in MMLE-multiple imputation. In the model with the same item discrimination over levels, the accuracy of the multilevel reliability measures and the empirical coverage rate of the MC confidence intervals were acceptable in all multilevel designs we considered with 40 items under MMLE-multiple imputation. We discuss these findings and provide guidelines for reporting multilevel reliability measures.     

Model-Based Measures for Detecting and Quantifying Response Bias

This paper proposes a model-based family of detection and quantification statistics to evaluate response bias in item bundles of any size. Compensatory (CDRF) and non-compensatory (NCDRF) response bias measures are proposed, along with their sample realizations and large-sample variability when models are fitted using multiple-group estimation. Based on the underlying connection to item response theory estimation methodology, it is argued that these new statistics provide a powerful and flexible approach to studying response bias for categorical response data over and above methods that have previously appeared in the literature. To evaluate their practical utility, CDRF and NCDRF are compared to the closely related SIBTEST family of statistics and likelihood-based detection methods through a series of Monte Carlo simulations. Results indicate that the new statistics are more optimal effect size estimates of marginal response bias than the SIBTEST family, are competitive with a selection of likelihood-based methods when studying item-level bias, and are the most optimal when studying differential bundle and test bias.     

应用项目反应理论对瑞文测验联合型的分析

使用BILOG-MG3.0软件,边际极大似然估计,3参数Logistic模型对354名不同能力水平的男性青年的瑞文测验联合型数据进行了分析。结果显示:大多数瑞文测验联合型的题目都适合3参数Logistic模型(有6道题不适合)。整个测验的信息函数峰值的位置在难度量表的-3到-2之间,其值为16.82。共有18道题的信息函数峰值在0.2以下。从区分度来看,72道题目的区分度均大于0.5,比较理想。难度参数显示所有题目均较低,绝大部分都在0以下,最高的只有1.01。题目的难度主要由所需的操作水平决定。伪猜测参数在0.07-0.24之间。综合分析表明瑞文测验联合型对正常青年的智力评价精度较差。     

A Doubly Latent Space Joint Model for Local Item and Person Dependence in the Analysis of Item Response Data

Item response theory (IRT) is one of the most widely utilized tools for item response analysis; however, local item and person independence, which is a critical assumption for IRT, is often violated in real testing situations. In this article, we propose a new type of analytical approach for item response data that does not require standard local independence assumptions. By adapting a latent space joint modeling approach, our proposed model can estimate pairwise distances to represent the item and person dependence structures, from which item and person clusters in latent spaces can be identified. We provide an empirical data analysis to illustrate an application of the proposed method. A simulation study is provided to evaluate the performance of the proposed method in comparison with existing methods.

     

Bifactor Modeling and the Estimation of Model-Based Reliability in the WAIS-IV

Previous confirmatory factor analytic research that has examined the factor structure of the Wechsler Adult Intelligence Scale–Fourth Edition (WAIS-IV) has endorsed either higher order models or oblique factor models that tend to amalgamate both general factor and index factor sources of systematic variance. An alternative model that has not yet been examined for the WAIS-IV is the bifactor model. Bifactor models allow all subtests to load onto both the general factor and their respective index factor directly. Bifactor models are also particularly amenable to the estimation of model-based reliabilities for both global composite scores (ω _h ) and subscale/index scores (ω _s ). Based on the WAIS-IV normative sample correlation matrices, a bifactor model that did not include any index factor cross loadings or correlated residuals was found to be better fitting than the conventional higher order and oblique factor models. Although the ω _h estimate associated with the full scale intelligence quotient (FSIQ) scores was respectably high (.86), the ω _s estimates associated with the WAIS-IV index scores were very low (.13 to .47). The results are interpreted in the context of the benefits of a bifactor modeling approach. Additionally, in light of the very low levels of unique internal consistency reliabilities associated with the index scores, it is contended that clinical index score interpretations are probably not justifiable.     

The Impact of Fallible Item Parameter Estimates on Latent Trait Recovery

In this paper we propose an upward correction to the standard error (SE) estimation of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} , the maximum likelihood (ML) estimate of the latent trait in item response theory (IRT). More specifically, the upward correction is provided for the SE of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} when item parameter estimates obtained from an independent pretest sample are used in IRT scoring. When item parameter estimates are employed, the resulting latent trait estimate is called pseudo maximum likelihood (PML) estimate. Traditionally, the SE of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} is obtained on the basis of test information only, as if the item parameters are known. The upward correction takes into account the error that is carried over from the estimation of item parameters, in addition to the error in latent trait recovery itself. Our simulation study shows that both types of SE estimates are very good when θ is in the middle range of the latent trait distribution, but the upward-corrected SEs are more accurate than the traditional ones when θ takes more extreme values.     

Nested Logit Models for Multiple-Choice Item Response Data

Nested logit item response models for multiple-choice data are presented. Relative to previous models, the new models are suggested to provide a better approximation to multiple-choice items where the application of a solution strategy precedes consideration of response options. In practice, the models also accommodate collapsibility across all distractor categories, making it easier to allow decisions about including distractor information to occur on an item-by-item or application-by-application basis without altering the statistical form of the correct response curves. Marginal maximum likelihood estimation algorithms for the models are presented along with simulation and real data analyses.     

A Speeded Item Response Model with Gradual Process Change

An item response theory model for dealing with test speededness is proposed. The model consists of two random processes, a problem solving process and a random guessing process, with the random guessing gradually taking over from the problem solving process. The involved change point and change rate are considered random parameters in order to model examinee differences in both respects. The proposed model is evaluated on simulated data and in a case study. The research reported in this paper was supported by IAP P5/24 and GOA/2005/04, both awarded to Paul De Boeck and Iven Van Mechelen, and by IAP P6/03, awarded to Iven Van Mechelen. Yuri Goegebeur’s research was supported by a grant of the Danish Natural Science Research Council.     

A Speeded Item Response Model: Leave the Harder till Later

A speeded item response model is proposed. We consider the situation where examinees may retain the harder items to a later test period in a time limit test. With such a strategy, examinees may not finish answering some of the harder items within the allocated time. In the proposed model, we try to describe such a mechanism by incorporating a speeded-effect term into the two-parameter logistic item response model. A Bayesian estimation procedure of the current model using Markov chain Monte Carlo is presented, and its performance over the two-parameter logistic item response model in a speeded test is demonstrated through simulations. The methodology is applied to physics examination data of the Department Required Test for college entrance in Taiwan for illustration.