A Comprehensive Simulation Study of Estimation Methods for the Rasch Model

The Rasch model is one of the most prominent item response models. In this article, different item parameter estimation methods for the Rasch model are systematically compared through a comprehensive simulation study: Different alternatives of joint maximum likelihood (JML) estimation, different alternatives of marginal maximum likelihood (MML) estimation, conditional maximum likelihood (CML) estimation, and several limited information methods (LIM). The type of ability distribution (i.e., nonnormality), the number of items, sample size, and the distribution of item difficulties were systematically varied. Across different simulation conditions, MML methods with flexible distributional specifications can be at least as efficient as CML. Moreover, in many situations (i.e., for long tests), penalized JML and JML with ε adjustment resulted in very efficient estimates and might be considered alternatives to JML implementations currently used in statistical software. Moreover, minimum chi-square (MINCHI) estimation was the best-performing LIM method. These findings demonstrate that JML estimation and LIM can still prove helpful in applied research.

A Flexible Approach to Modeling Over-, Under- and Equidispersed Count Data in IRT: The Two-Parameter Conway-Maxwell-Poisson Model

Several psychometric tests generate count data, e.g. the number of ideas in divergent thinkingtasks. The most prominent count data IRT model, the Rasch Poisson Counts Model (RPCM)assumes constant discriminations across items as well as the equidispersion assumption of thePoisson distribution (i.e., E(X) = Var(X)), considerably limiting modeling flexibility. Violationsof these assumptions are associated with impaired ability, reliability, and standard error estimates.Models have been proposed to loose the one or the other assumption. The Two-Parameter PoissonCounts Model (2PPCM) allows varying discriminations but retains the equidispersion assumption.The Conway-Maxwell-Poisson Counts Model (CMPCM) that allows for modeling equi- but alsoover- and underdispersion (more or less variance than implied by the mean under the Poisson distribution)but assumes constant discriminations. The present work introduces the Two-ParameterConway-Maxwell-Poisson (2PCMP) model which generalizes the RPCM, the 2PPCM, and the CMPCM(all contained as special cases) to allow for varying discriminations and dispersions withinone model. A marginal maximum likelihood method based on a fixed quadrature Expectation-Maximization (EM) algorithm is derived. Standard errors as well as two methods for latent abilityestimation are provided. An implementation of the 2PCMP model in R and C++ is provided. Twosimulation studies examine the model’s statistical properties and compare the 2PCMP model toestablished methods. Data from divergent thinking tasks are re-analyzed with the 2PCMP modelto illustrate the model’s flexibility and ability to test assumptions of special cases.

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.

A Lasso and a Regression Tree Mixed-Effect Model with Random Effects for the Level, the Residual Variance, and the Autocorrelation

Research in psychology is experiencing a rapid increase in the availability of intensive longitudinal data. To use such data for predicting feelings, beliefs, and behavior, recent methodological work suggested combinations of the longitudinal mixed-effect model with Lasso regression or with regression trees. The present article adds to this literature by suggesting an extension of these models that—in addition to a random effect for the mean level—also includes a random effect for the within-subject variance and a random effect for the autocorrelation. After introducing the extended mixed-effect location scale (E-MELS), the extended mixed-effect location-scale Lasso model (Lasso E-MELS), and the extended mixed-effect location-scale tree model (E-MELS trees), we show how its parameters can be estimated using a marginal maximum likelihood approach. Using real and simulated example data, we illustrate how to use E-MELS, Lasso E-MELS, and E-MELS trees for building prediction models to forecast individuals’ daily nervousness. The article is accompanied by an R package (called ) and functions that support users in the application of the suggested models.

Power analysis for the Wald, LR, score, and gradient tests in a marginal maximum likelihood framework: Applications in IRT

The Wald, likelihood ratio, score and the recently proposed gradient statistics can be used to assess a broad range of hypotheses in item response theory models, for instance, to check the overall model fit or to detect differential item functioning. We introduce new methods for power analysis and sample size planning that can be applied when marginal maximum likelihood estimation is used. This avails the application to a variety of IRT models, which are increasingly used in practice, e.g., in large-scale educational assessments. An analytical method utilizes the asymptotic distributions of the statistics under alternative hypotheses. For a larger number of items, we also provide a sampling-based method, which is necessary due to an exponentially increasing computational load of the analytical approach. We performed extensive simulation studies in two practically relevant settings, i.e., testing a Rasch model against a 2PL model and testing for differential item functioning. The observed distributions of the test statistics and the power of the tests agreed well with the predictions by the proposed methods. We provide an openly accessible R package that implements the methods for user-supplied hypotheses.

Bayesian Modal Estimation for the One-Parameter Logistic Ability-Based Guessing (1PL-AG) Model

The calibration of the one-parameter logistic ability-based guessing (1PL-AG) model in item response theory (IRT) with a modest sample size remains a challenge for its implausible estimates and difficulty in obtaining standard errors of estimates. This article proposes an alternative Bayesian modal estimation (BME) method, the Bayesian Expectation-Maximization-Maximization (BEMM) method, which is developed by combining an augmented variable formulation of the 1PL-AG model and a mixture model conceptualization of the three-parameter logistic model (3PLM). By comparing with marginal maximum likelihood estimation (MMLE) and Markov Chain Monte Carlo (MCMC) in JAGS, the simulation shows that BEMM can produce stable and accurate estimates in the modest sample size. A real data example and the MATLAB codes of BEMM are also provided.

Latent D-Scoring Modeling: Estimation of Item and Person Parameters

This study presents a latent (item response theory–like) framework of a recently developed classical approach to test scoring, equating, and item analysis, referred to as D-scoring method. Specifically, (a) person and item parameters are estimated under an item response function model on the D-scale (from 0 to 1) using marginal maximum-likelihood estimation and (b) analytic expressions are provided for item information function, test information function, and standard error of estimation for D-scores obtained under the proposed latent treatment of the D-scoring method. The results from a simulation study reveal very good recovery of item and person parameters via the marginal maximum-likelihood estimation method. Discussion and recommendations for practice are provided.

Adaptive lasso with weights based on normalized filtering scores in molecular big data

The molecular big data are highly correlated, and numerous genes are not related. The various classification methods performance mainly rely on the selection of significant genes. Sparse regularized regression (SRR) models using the least absolute shrinkage and selection operator (lasso) and adaptive lasso (alasso) are popularly used for gene selection and classification. Nevertheless, it becomes challenging when the genes are highly correlated. Here, we propose a modified adaptive lasso with weights using the ranking-based feature selection (RFS) methods capable of dealing with the highly correlated gene expression data. Firstly, an RFS methods such as Fisher’s score (FS), Chi-square (CS), and information gain (IG) are employed to ignore the unimportant genes and the top significant genes are chosen through sure independence screening (SIS) criteria. The scores of the ranked genes are normalized and assigned as proposed weights to the alasso method to obtain the most significant genes that were proven to be biologically related to the cancer type and helped in attaining higher classification performance. With the synthetic data and real application of microarray data, we demonstrated that the proposed alasso method with RFS methods is a better approach than the other known methods such as alasso with filtering such as ridge and marginal maximum likelihood estimation (MMLE), lasso and alasso without filtering. The metrics of accuracy, area under the receiver operating characteristics curve (AUROC), and geometric mean (GM-mean) are used for evaluating the performance of the models.

A Deep Learning Algorithm for High-Dimensional Exploratory Item Factor Analysis

Marginal maximum likelihood (MML) estimation is the preferred approach to fitting item response theory models in psychometrics due to the MML estimator's consistency, normality, and efficiency as the sample size tends to infinity. However, state-of-the-art MML estimation procedures such as the Metropolis-Hastings Robbins-Monro (MH-RM) algorithm as well as approximate MML estimation procedures such as variational inference (VI) are computationally time-consuming when the sample size and the number of latent factors are very large. In this work, we investigate a deep learning-based VI algorithm for exploratory item factor analysis (IFA) that is computationally fast even in large data sets with many latent factors. The proposed approach applies a deep artificial neural network model called a variational autoencoder for exploratory IFA. An importance sampling technique to help the variational estimator better approximate the MML estimator is explored. We provide a real data application that recovers results aligning with psychological theory across random starts. Via simulation studies, we empirically demonstrate that the variational estimator is consistent (although factor correlation estimates exhibit some bias) and yields similar results to MH-RM in less time. Our simulations also suggest that the proposed approach performs similarly to and is potentially faster than constrained joint maximum likelihood estimation, a fast procedure that is consistent when the sample size and the number of items simultaneously tend to infinity.