Gaussian variational estimation for multidimensional item response theory

Multidimensional item response theory (MIRT) is widely used in assessment and evaluation of educational and psychological tests. It models the individual response patterns by specifying a functional relationship between individuals' multiple latent traits and their responses to test items. One major challenge in parameter estimation in MIRT is that the likelihood involves intractable multidimensional integrals due to the latent variable structure. Various methods have been proposed that involve either direct numerical approximations to the integrals or Monte Carlo simulations. However, these methods are known to be computationally demanding in high dimensions and rely on sampling data points from a posterior distribution. We propose a new Gaussian variational expectation--maximization (GVEM) algorithm which adopts variational inference to approximate the intractable marginal likelihood by a computationally feasible lower bound. In addition, the proposed algorithm can be applied to assess the dimensionality of the latent traits in an exploratory analysis. Simulation studies are conducted to demonstrate the computational efficiency and estimation precision of the new GVEM algorithm compared to the popular alternative Metropolis–Hastings Robbins–Monro algorithm. In addition, theoretical results are presented to establish the consistency of the estimator from the new GVEM algorithm.     

MCMC estimation and some model-fit analysis of multidimensional IRT models

A Bayesian procedure to estimate the three-parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented. The procedure is a generalization of a procedure by Albert (1992) for estimating the two-parameter normal ogive model. The procedure supports analyzing data from multiple populations and incomplete designs. It is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure. The technique is illustrated using simulated and real data. The authors would like to thank Norman Verhelst for his valuable comments and ACT, CITO group and SweSAT for the use of their data.     

Estimating Optimal Weights for Compound Scores: A Multidimensional IRT Approach

A method is proposed for constructing indices as linear functions of variables such that the reliability of the compound score is maximized. Reliability is defined in the framework of latent variable modeling [i.e., item response theory (IRT)] and optimal weights of the components of the index are found by maximizing the posterior variance relative to the total latent variable variance. Three methods for estimating the weights are proposed. The first is a likelihood-based approach, that is, marginal maximum likelihood (MML). The other two are Bayesian approaches based on Markov chain Monte Carlo (MCMC) computational methods. One is based on an augmented Gibbs sampler specifically targeted at IRT, and the other is based on a general purpose Gibbs sampler such as implemented in OpenBugs and Jags. Simulation studies are presented to demonstrate the procedure and to compare the three methods. Results are very similar, so practitioners may be suggested the use of the easily accessible latter method. A real-data set pertaining to the 28-joint Disease Activity Score is used to show how the methods can be applied in a complex measurement situation with multiple time points and mixed data formats.     

Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm

Maximum likelihood estimation of item parameters in the marginal distribution, integrating over the distribution of ability, becomes practical when computing procedures based on an EM algorithm are used. By characterizing the ability distribution empirically, arbitrary assumptions about its form are avoided. The Em procedure is shown to apply to general item-response models lacking simple sufficient statistics for ability. This includes models with more than one latent dimension.Supported in part by NSF grant BNS 7912417 to the University of Chicago and by SSRC (UK) grant HR6132 to the University of Lancaster.We are indebted to Mark Reiser and Robert Gibbons for computer programming. David Thissen clarified a number of points in an earlier draft.     

On local influence analysis of full information item factor models

The full information item factor (FIIF) model is very useful for analyzing relations of dichotomous variables. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the FIIF model. The development is based on a Q-displacement function which is closely related with the Monte Carlo EM algorithm in the ML estimation. In the E-step of this algorithm, the conditional expectations are approximated by sample means of observations simulated by the Gibbs sampler from the appropriate conditional distributions. It turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures. The diagnoses are based on the conformal normal curvature that can be computed easily. A number of interesting perturbation schemes are considered. The methodology is illustrated with two real examples.The research is fully supported by a grant (CUHK 4356/00H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the Editor, Associate Editor, anonymous reviewers, and W.Y. Poon for valuable comments for improving the paper, and to ICPSR and the relevant founding agency for allowing us to use of their data. The assistance of Michael Leung and Esther Tam is gratefully acknowledged.     

High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature

Although the Bock–Aitkin likelihood-based estimation method for factor analysis of dichotomous item response data has important advantages over classical analysis of item tetrachoric correlations, a serious limitation of the method is its reliance on fixed-point Gauss-Hermite (G-H) quadrature in the solution of the likelihood equations and likelihood-ratio tests. When the number of latent dimensions is large, computational considerations require that the number of quadrature points per dimension be few. But with large numbers of items, the dispersion of the likelihood, given the response pattern, becomes so small that the likelihood cannot be accurately evaluated with the sparse fixed points in the latent space. In this paper, we demonstrate that substantial improvement in accuracy can be obtained by adapting the quadrature points to the location and dispersion of the likelihood surfaces corresponding to each distinct pattern in the data. In particular, we show that adaptive G-H quadrature, combined with mean and covariance adjustments at each iteration of an EM algorithm, produces an accurate fast-converging solution with as few as two points per dimension. Evaluations of this method with simulated data are shown to yield accurate recovery of the generating factor loadings for models of upto eight dimensions. Unlike an earlier application of adaptive Gibbs sampling to this problem by Meng and Schilling, the simulations also confirm the validity of the present method in calculating likelihood-ratio chi-square statistics for determining the number of factors required in the model. Finally, we apply the method to a sample of real data from a test of teacher qualifications.     

A multidimensional item response model: Constrained latent class analysis using the gibbs sampler and posterior predictive checks

In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.This paper is supported by grant S40-645 of the Dutch Organization for Scientific Research (NWO).     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.     

A Gibbs sampler for the multidimensional four-parameter logistic item response model via a data augmentation scheme

The four-parameter logistic (4PL) item response model, which includes an upper asymptote for the correct response probability, has drawn increasing interest due to its suitability for many practical scenarios. This paper proposes a new Gibbs sampling algorithm for estimation of the multidimensional 4PL model based on an efficient data augmentation scheme (DAGS). With the introduction of three continuous latent variables, the full conditional distributions are tractable, allowing easy implementation of a Gibbs sampler. Simulation studies are conducted to evaluate the proposed method and several popular alternatives. An empirical data set was analysed using the 4PL model to show its improved performance over the three-parameter and two-parameter logistic models. The proposed estimation scheme is easily accessible to practitioners through the open-source IRTlogit package.     

EM algorithms for ML factor analysis

The details of EM algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models. The algorithm is essentially the same for both cases and involves only simple least squares regression operations; the largest matrix inversion required is for aq ×q symmetric matrix whereq is the matrix of factors. The example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other; such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.     

The EM algorithm for latent class analysis with equality constraints

The EM algorithm is a popular iterative method for estimating parameters in the latent class model where at each step the unknown parameters can be estimated simply as weighted sums of some latent proportions. The algorithm may also be used when some parameters are constrained to equal given constants or each other. It is shown that in the general case with equality constraints, the EM algorithm is not simple to apply because a nonlinear equation has to be solved. This problem arises, mainly, when equality constrints are defined over probabilities indifferent combinations of variables and latent classes. A simple condition is given in which, although probabilities in different variable-latent class combinations are constrained to be equal, the EM algorithm is still simple to apply.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper. C. C. Clogg and R. Luijkx are also acknowledged for verifying our results with their computer programs MLLSA and LCAG, respectively.     

Latent variable selection in multidimensional item response theory models using the expectation model selection algorithm

The aim of latent variable selection in multidimensional item response theory (MIRT) models is to identify latent traits probed by test items of a multidimensional test. In this paper the expectation model selection (EMS) algorithm proposed by Jiang et al. (2015) is applied to minimize the Bayesian information criterion (BIC) for latent variable selection in MIRT models with a known number of latent traits. Under mild assumptions, we prove the numerical convergence of the EMS algorithm for model selection by minimizing the BIC of observed data in the presence of missing data. For the identification of MIRT models, we assume that the variances of all latent traits are unity and each latent trait has an item that is only related to it. Under this identifiability assumption, the convergence of the EMS algorithm for latent variable selection in the multidimensional two-parameter logistic (M2PL) models can be verified. We give an efficient implementation of the EMS for the M2PL models. Simulation studies show that the EMS outperforms the EM-based L₁ regularization in terms of correctly selected latent variables and computation time. The EMS algorithm is applied to a real data set related to the Eysenck Personality Questionnaire.     

More on EM for ML factor analysis

We address several issues that are raised by Bentler and Tanaka's [1983] discussion of Rubin and Thayer [1982]. Our conclusions are: standard methods do not completely monitor the possible existence of multiple local maxima; summarizing inferential precision by the standard output based on second derivatives of the log likelihood at a maximum can be inappropriate, even if there exists a unique local maximum; EM and LISREL can be viewed as complementary, albeit not entirely adequate, tools for factor analysis.This work was partially supported by the Program Statistics Research Project at Educational Testing Service.     

Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.     

Bayesian factor analysis for multilevel binary observations

Multilevel covariance structure models have become increasingly popular in the psychometric literature in the past few years to account for population heterogeneity and complex study designs. We develop practical simulation based procedures for Bayesian inference of multilevel binary factor analysis models. We illustrate how Markov Chain Monte Carlo procedures such as Gibbs sampling and Metropolis-Hastings methods can be used to perform Bayesian inference, model checking and model comparison without the need for multidimensional numerical integration. We illustrate the proposed estimation methods using three simulation studies and an application involving student's achievement results in different areas of mathematics. The authors thank Ian Westbury, University of Illinois at Urbana Champaign for kindly providing the SIMS data for the application.     

An item response theory/confirmatory factor analysis of the Autobiographical Memory Test

The Autobiographical Memory Test (AMT) is used to assess the degree of specificity of autobiographical memory. The AMT usually contains cue words of both positive and negative valence, but it is unclear whether these valences form separate factors or not. Accordingly, confirmatory factor analysis assessed whether the AMT measures one overall factor, or whether different cue types are related to different factors. Results were consistent across three datasets (N = 333, N = 405, and N = 336). A one-factor model fitted each dataset well, which suggests that responses to positive and negative cues are related to the one construct. In addition, item response theory analyses showed that the AMT is most precise for people who score low on memory specificity. Implications for using the AMT with high-functioning samples are discussed.     

A new algorithm for the least-squares solution in factor analysis

A new algorithm to obtain the least-squares or MINRES solution in common factor analysis is presented. It is based on the up-and-down Marquardt algorithm developed by the present authors for a general nonlinear least-squares problem. Experiments with some numerical models and some empirical data sets showed that the algorithm worked nicely and that SMC (Squared Multiple Correlation) performed best among four sets of initial values for common variances but that the solution might sometimes be very sensitive to fluctuations in the sample covariance matrix.Numerical computation was made on a NEAC S-1000 computer in the Computer Center, Osaka University.     

Model specification search using a genetic algorithm with factor reordering for a simple structure factor analysis model

Abstract:  Many techniques for automated model specification search based on numerical indices have been proposed, but no single decisive method has yet been determined. In the present article, the performance and features of the model specification search method using a genetic algorithm (GA) were verified. A GA is a robust and simple metaheuristic algorithm with great searching power. While there has already been some research applying metaheuristics to the model fitting task, we focus here on the search for a simple structure factor analysis model and propose a customized algorithm for dealing with certain problems specific to that situation. First, implementation of model specification search using a GA with factor reordering for a simple structure factor analysis is proposed. Then, through a simulation study using generated data with a known true structure and through example analysis using real data, the effectiveness and applicability of the proposed method were demonstrated.