CADEM: A conditional augmented data EM algorithm for fitting one parameter probit models

Joint maximum likelihood (JML) estimation is one of the earliest approaches to fitting item response theory (IRT) models. This procedure treats both the item and person parameters as unknown but fixed model parameters and estimates them simultaneously by solving an optimization problem. However, the JML estimator is known to be asymptotically inconsistent for many IRT models, when the sample size goes to infinity and the number of items keeps fixed. Consequently, in the psychometrics literature, this estimator is less preferred to the marginal maximum likelihood (MML) estimator. In this paper, we re-investigate the JML estimator for high-dimensional exploratory item factor analysis, from both statistical and computational perspectives. In particular, we establish a notion of statistical consistency for a constrained JML estimator, under an asymptotic setting that both the numbers of items and people grow to infinity and that many responses may be missing. A parallel computing algorithm is proposed for this estimator that can scale to very large datasets. Via simulation studies, we show that when the dimensionality is high, the proposed estimator yields similar or even better results than those from the MML estimator, but can be obtained computationally much more efficiently. An illustrative real data example is provided based on the revised version of Eysenck's Personality Questionnaire (EPQ-R).

The estimation of high-dimensional latent regression item response theory (IRT) models is difficult because of the need to approximate integrals in the likelihood function. Proposed solutions in the literature include using stochastic approximations, adaptive quadrature, and Laplace approximations. We propose using a second-order Laplace approximation of the likelihood to estimate IRT latent regression models with categorical observed variables and fixed covariates where all parameters are estimated simultaneously. The method applies when the IRT model has a simple structure, meaning that each observed variable loads on only one latent variable. Through simulations using a latent regression model with binary and ordinal observed variables, we show that the proposed method is a substantial improvement over the first-order Laplace approximation with respect to the bias. In addition, the approach is equally or more precise to alternative methods for estimation of multidimensional IRT models when the number of items per dimension is moderately high. Simultaneously, the method is highly computationally efficient in the high-dimensional settings investigated. The results imply that estimation of simple-structure IRT models with very high dimensions is feasible in practice and that the direct estimation of high-dimensional latent regression IRT models is tractable even with large sample sizes and large numbers of items.

Two common aspects of symptom measurement are 1) the occurrence or presence of symptoms, and 2) the intensity or severity of symptoms when they occur. We adopt a latent trait perspective based on item response theory (IRT), using both unidimensional and multidimensional IRT models. We demonstrate how to (a) prepare data for analysis, (b) specify, estimate, and compare models, (c) interpret model parameters, (d) compare scores from models, and (e) visualize analysis results. We develop the relevant sequential IRT model, noting its flexibility, congruence with the theorized data generating process for symptom measures, and its promise for facilitating additional research and practical applications. The sequential model is less frequently used than other IRT models for polytomous data such as the generalized partial credit or graded response models. However, estimation of the sequential model can be readily accomplished with standard latent variable modeling and IRT software for binary indicators that allows constraints on the discrimination parameters. We compare the sequential model to other modeling options. We provide discussion of future research directions.

Questionnaires are widely used in clinical research and mental health studies, but their responses are usually more subjective than clinical biomarkers. Researchers may use latent variable models such as the item response theory (IRT) model to characterize individual ability, but response instability and small sample sizes can lead to increased uncertainty in estimation. Historical data from similar questionnaires may help reduce variation, but they must be incorporated adaptively. The challenge is further compounded by the need to ensure computational efficiency in IRT models with a large number of parameters. In this work, we develop a Bayesian framework for adaptive borrowing in IRT models based on an approximated normalized power prior (NPP) that treats the borrowing weight as a random parameter. The proposed method makes NPP feasible for general IRT models and is coupled with a full Bayesian data augmentation strategy that enables joint estimation of ability and item parameters through an efficient Gibbs sampler. In simulations, the approach adaptively increases borrowing when historical and current data are compatible and automatically downweights conflicting information. Relative to analyzes without borrowing, the method reduces variance and mean squared error while maintaining coverage across a range of test lengths, item discrimination profiles, and historical-current concordance scenarios. We illustrate the method by integrating historical mental health surveys into current assessments, yielding more precise ability estimates with preserved calibration. An efficient implementation is provided in our updated package NPP available on the Comprehensive R Archive Network.

Standard 3.9 of the Standards for Educational and Psychological Testing () demands evidence of model fit when item response theory (IRT) models are employed to data from tests. Hambleton and Han () and Sinharay () recommended the assessment of practical significance of misfit of IRT models, but few examples of such assessment can be found in the literature concerning IRT model fit. In this article, practical significance of misfit of IRT models was assessed using data from several tests that employ IRT models to report scores. The IRT model did not fit any data set considered in this article. However, the extent of practical significance of misfit varied over the data sets.

在心理与教育测量中，项目反应理论（Item Response Theory，IRT）模型的参数估计方法是理论研究与实践应用的基本工具。最近，由于IRT模型的不断扩展与EM（expectation-maximization）算法自身的固有问题，参数估计方法的改进与发展显得尤为重要。这里介绍了IRT模型中边际极大似然估计的发展，提出了它的阶段性特征，即联合极大似然估计阶段、确定性潜在心理特质“填补”阶段、随机潜在心理特质“填补”阶段，重点阐述了它的潜在心理特质“填补”（dataaugmentation）思想。EM算法与Metropolis-Hastings Robbins—Monro（MH-RM）算法作为不同的潜在心理特质“填补”方法，都是边际极大似然估计的思想跨越。目前，潜在心理特质“填补”的参数估计方法仍在不断发展与完善。

Although item response theory (IRT) models such as the bifactor, two-tier, and between-item-dimensionality IRT models have been devised to confirm complex dimensional structures in educational and psychological data, they can be challenging to use in practice. The reason is that these models are multidimensional IRT (MIRT) models and thus are highly parameterized, making them only suitable for data provided by large samples. Unfortunately, many educational and psychological studies are conducted on a small scale, leaving the researchers without the necessary MIRT models to confirm the hypothesized structures in their data. To address the lack of modeling options for these researchers, we present a general Bayesian MIRT model based on adaptive informative priors. Simulations demonstrated that our MIRT model could be used to confirm a two-tier structure (with two general and six specific dimensions), a bifactor structure (with one general and six specific dimensions), and a between-item six-dimensional structure in rating scale data representing sample sizes as small as 100. Although our goal was to provide a general MIRT model suitable for smaller samples, the simulations further revealed that our model was applicable to larger samples. We also analyzed real data from 121 individuals to illustrate that the findings of our simulations are relevant to real situations.

The novelty of this study is to propose multidimensional item response theory (IRT) models for the investigation of heterogeneity in residents’ perceptions and attitudes toward the tourism industry. By following the social exchange theory, a sample of residents in the Romagna area (Italy) were invited to declare their agreement to some aspects related to benefits and costs of the tourism industry in 2010. The analysis suggests that the additive IRT model with two kind of latent variables (viz. the general latent trait measuring attitude toward tourism and two specific latent traits measuring perceptions on benefits and costs of tourism) fits the data adequately. Moreover, the estimated scores show that the tourism features are perceived differently by the residents, supporting the hypothesis that perceptions and attitudes change with regard to the individuals’ characteristics. Finally, the analysis proposes how the parameters of the IRT models can be interpreted in this new context.

A ranking pattern approach is proposed to build item response theory (IRT) models for forced-choice (FC) items. This new approach is an addition to the two existing approaches, sequential selection and Thurstone's law of pairwise comparison. A new dominance IRT model, the multidimensional generalized partial preference model (MGPPM), is proposed for FC items with any number (greater than 1) of statements. The maximum marginal likelihood estimation using an expectation-maximization algorithm (MML-EM) and Markov chain Monte Carlo (MCMC) estimation are developed. A simulation study is conducted to show satisfactory parameter recovery on triplet and tetrad data. The relationships between the newly proposed approach/model and the existing approaches/models are described, and the MGPPM, Thurstonian IRT (TIRT) model, and Triplet-2PLM are compared when applied to simulated and real triplet data. The new approach offers more flexible IRT modeling than the other two approaches under different assumptions, and the MGPPM is more statistically elegant than the TIRT and Triple-2PLM.

Quality of life studies must be concerned with the accuracy, or reliability (as it is usually called in psychometrics) of measurement. A reliability coefficient gives an evaluation of a questionnaire’s ability to yield interpretable statements about the construct being measured. There are two classes of models for analyzing unidimensional scales: classical models and Item Response Theory (IRT) models. Classical models are based on a linear decomposition of the score, and reliability is estimated using Cronbach’s alpha coefficient. IRT models focus on the relationship between the probability of a correct answer and a latent variable. The Rasch model is a special IRT model which having good measurement properties. In the context of the Rasch model, or IRT models in general, no reliability coefficient similar to Cronbach Alpha is clearly defined. Some authors have studied the Fisher information of the latent parameter, which provides a measure of accuracy of the estimator. In this chapter, we present how reliability is estimated for classical models, and we propose a reliability coefficient for the Rasch model. Simulated data and data derived from the communication subscale of the Sickness Impact Profile are used to illustrate the methods.

Item response theory (IRT) models provide an appropriate alternative to the classical ordinal confirmatory factor analysis (CFA) during the development of patient-reported outcome measures (PROMs). Current literature has identified the assessment of IRT model fit as both challenging and underdeveloped (Sinharay & Johnson, 2003; Sinharay, Johnson, & Stern, 2006). This study evaluates the performance of Ordinal Bayesian Instrument Development (OBID), a Bayesian IRT model with a probit link function approach, through applications in two breast cancer-related instrument development studies. The primary focus is to investigate an appropriate method for comparing Bayesian IRT models in PROMs development. An exact Bayesian leave-one-out cross-validation (LOO-CV) approach (Vehtari & Lampinen, 2002) is implemented to assess prior selection for the item discrimination parameter in the IRT model and subject content experts’ bias (in a statistical sense and not to be confused with psychometric bias as in differential item functioning) toward the estimation of item-to-domain correlations. Results support the utilization of content subject experts’ information in establishing evidence for construct validity when sample size is small. However, the incorporation of subject experts’ content information in the OBID approach can be sensitive to the level of expertise of the recruited experts. More stringent efforts need to be invested in the appropriate selection of subject experts to efficiently use the OBID approach and reduce potential bias during PROMs development.

ABSTRACTMany standardized educational tests include groups of items based on a common stimulus, known as testlets. Standard unidimensional item response theory (IRT) models are commonly used to model examinees' responses to testlet items. However, it is known that local dependence among testlet items can lead to biased item parameter estimates when using standard IRT models, and to overestimated reliability. In this study, a general polytomous testlet model was proposed to account for local dependence in testlet‐based tests that contain both dichotomously and polytomously scored items. The proposed model and a standard IRT model were fit to simulated data and several real data sets from the reading sections of a large‐scale English‐language test, and model fit was evaluated. Item parameters and test information obtained from the two models were compared to check the impact of local item dependence. In addition, a multidimensional IRT model with simple structure was also fit to the real data sets. Results based on both simulated and real data suggested that local dependence had a small impact on item parameter estimates and a relatively larger impact on test information and reliability. It was also found that the multidimensional IRT model with simple structure fit the real data sets better than the general polytomous testlet model and the standard IRT model did.

Item response theory (IRT) is the statistical paradigm underlying a dominant family of generative probabilistic models for test responses, used to quantify traits in individuals relative to target populations. The graded response model (GRM) is a particular IRT model that is used for ordered polytomous test responses. Both the development and the application of the GRM and other IRT models require statistical decisions. For formulating these models (calibration), one needs to decide on methodologies for item selection, inference, and regularization. For applying these models (test scoring), one needs to make similar decisions, often prioritizing computational tractability and/or interpretability. In many applications, such as in the Work Disability Functional Assessment Battery (WD-FAB), tractability implies approximating an individual’s score distribution using estimates of mean and variance, and obtaining that score conditional on only point estimates of the calibrated model. In this manuscript, we evaluate the calibration and scoring of models under this common use-case using Bayesian cross-validation. Applied to the WD-FAB responses collected for the National Institutes of Health, we assess the predictive power of implementations of the GRM based on their ability to yield, on validation sets of respondents, ability estimates that are most predictive of patterns of item responses. Our main finding indicates that regularized Bayesian calibration of the GRM outperforms the regularization-free empirical Bayesian procedure of marginal maximum likelihood. We also motivate the use of compactly supported priors in test scoring.

This study investigates the impact of item parameter drift (IPD) on parameter and ability estimation when the underlying measurement model fits a mixture distribution, thereby violating the item invariance property of unidimensional item response theory (IRT) models. An empirical study was conducted to demonstrate the occurrence of both IPD and an underlying mixture distribution using real-world data. Twenty-one trended anchor items from the 1999, 2003, and 2007 administrations of Trends in International Mathematics and Science Study (TIMSS) were analyzed using unidimensional and mixture IRT models. TIMSS treats trended anchor items as invariant over testing administrations and uses pre-calibrated item parameters based on unidimensional IRT. However, empirical results showed evidence of two latent subgroups with IPD. Results also showed changes in the distribution of examinee ability between latent classes over the three administrations. A simulation study was conducted to examine the impact of IPD on the estimation of ability and item parameters, when data have underlying mixture distributions. Simulations used data generated from a mixture IRT model and estimated using unidimensional IRT. Results showed that data reflecting IPD using mixture IRT model led to IPD in the unidimensional IRT model. Changes in the distribution of examinee ability also affected item parameters. Moreover, drift with respect to item discrimination and distribution of examinee ability affected estimates of examinee ability. These findings demonstrate the need to caution and evaluate IPD using a mixture IRT framework to understand its effects on item parameters and examinee ability.

Vision-Specific Quality-of-Life Research: A Need to Improve the Quality