A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.     

Marginal likelihood inference for a model for item responses and response times

Marginal maximum‐likelihood procedures for parameter estimation and testing the fit of a hierarchical model for speed and accuracy on test items are presented. The model is a composition of two first‐level models for dichotomous responses and response times along with multivariate normal models for their item and person parameters. It is shown how the item parameters can easily be estimated using Fisher's identity. To test the fit of the model, Lagrange multiplier tests of the assumptions of subpopulation invariance of the item parameters (i.e., no differential item functioning), the shape of the response functions, and three different types of conditional independence were derived. Simulation studies were used to show the feasibility of the estimation and testing procedures and to estimate the power and Type I error rate of the latter. In addition, the procedures were applied to an empirical data set from a computerized adaptive test of language comprehension.     

Latent variable models for multivariate longitudinal ordinal responses

The paper proposes a full information maximum likelihood estimation method for modelling multivariate longitudinal ordinal variables. Two latent variable models are proposed that account for dependencies among items within time and between time. One model fits item‐specific random effects which account for the between time points correlations and the second model uses a common factor. The relationships between the time‐dependent latent variables are modelled with a non‐stationary autoregressive model. The proposed models are fitted to a real data set.     

Stochastic EM for estimating the parameters of a multilevel IRT model

An item response theory (IRT) model is used as a measurement error model for the dependent variable of a multilevel model. The dependent variable is latent but can be measured indirectly by using tests or questionnaires. The advantage of using latent scores as dependent variables of a multilevel model is that it offers the possibility of modelling response variation and measurement error and separating the influence of item difficulty and ability level. The two‐parameter normal ogive model is used for the IRT model. It is shown that the stochastic EM algorithm can be used to estimate the parameters which are close to the maximum likelihood estimates. This algorithm is easily implemented. The estimation procedure will be compared to an implementation of the Gibbs sampler in a Bayesian framework. Examples using real data are given.     

Robustness of Parameter Estimation to Assumptions of Normality in the Multidimensional Graded Response Model

A central assumption that is implicit in estimating item parameters in item response theory (IRT) models is the normality of the latent trait distribution, whereas a similar assumption made in categorical confirmatory factor analysis (CCFA) models is the multivariate normality of the latent response variables. Violation of the normality assumption can lead to biased parameter estimates. Although previous studies have focused primarily on unidimensional IRT models, this study extended the literature by considering a multidimensional IRT model for polytomous responses, namely the multidimensional graded response model. Moreover, this study is one of few studies that specifically compared the performance of full-information maximum likelihood (FIML) estimation versus robust weighted least squares (WLS) estimation when the normality assumption is violated. The research also manipulated the number of nonnormal latent trait dimensions. Results showed that FIML consistently outperformed WLS when there were one or multiple skewed latent trait distributions. More interestingly, the bias of the discrimination parameters was non-ignorable only when the corresponding factor was skewed. Having other skewed factors did not further exacerbate the bias, whereas biases of boundary parameters increased as more nonnormal factors were added. The item parameter standard errors recovered well with both estimation algorithms regardless of the number of nonnormal dimensions.     

Random effects and extended generalized partial credit models

In this paper it is shown that under the random effects generalized partial credit model for the measurement of a single latent variable by a set of polytomously scored items, the joint marginal probability distribution of the item scores has a closed-form expression in terms of item category location parameters, parameters that characterize the distribution of the latent variable in the subpopulation of examinees with a zero score on all items, and item-scaling parameters. Due to this closed-form expression, all parameters of the random effects generalized partial credit model can be estimated using marginal maximum likelihood estimation without assuming a particular distribution of the latent variable in the population of examinees and without using numerical integration. Also due to this closed-form expression, new special cases of the random effects generalized partial credit model can be identified. In addition to these new special cases, a slightly more general model than the random effects generalized partial credit model is presented. This slightly more general model is called the extended generalized partial credit model. Attention is paid to maximum likelihood estimation of the parameters of the extended generalized partial credit model and to assessing the goodness of fit of the model using generalized likelihood ratio tests. Attention is also paid to person parameter estimation under the random effects generalized partial credit model. It is shown that expected a posteriori estimates can be obtained for all possible score patterns. A simulation study is carried out to show the usefulness of the proposed models compared to the standard models that assume normality of the latent variable in the population of examinees. In an empirical example, some of the procedures proposed are demonstrated.     

Generalized latent variable models with non‐linear effects

Until recently, item response models such as the factor analysis model for metric responses, the two‐parameter logistic model for binary responses and the multinomial model for nominal responses considered only the main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, non‐linear relationships among the latent variables might be necessary in real applications. Methods for fitting models with non‐linear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both non‐linear latent and covariate effects. The model parameters are estimated using full maximum likelihood based on a hybrid integration–maximization algorithm. Finally, a method for obtaining factor scores based on multiple imputation is proposed here for the non‐linear model.     

A general class of latent variable models for ordinal manifest variables with covariate effects on the manifest and latent variables

Previous work on a general class of multidimensional latent variable models for analysing ordinal manifest variables is extended here to allow for direct covariate effects on the manifest ordinal variables and covariate effects on the latent variables. A full maximum likelihood estimation method is used to estimate all the model parameters simultaneously. Goodness‐of‐fit statistics and standard errors are discussed. Two examples from the 1996 British Social Attitudes Survey are used to illustrate the methodology.     

A Multilevel Nonlinear Profile Analysis Model for Dichotomous Data

This study linked nonlinear profile analysis (NPA) of dichotomous responses with an existing family of item response theory models and generalized latent variable models (GLVM). The NPA method offers several benefits over previous internal profile analysis methods: (a) NPA is estimated with maximum likelihood in a GLVM framework rather than relying on the choice of different dissimilarity measures that produce different results, (b) item and person parameters are computed during the same estimation step with an appropriate distribution for dichotomous variables, (c) the model estimates profile coordinate standard errors, and (d) additional individual-level variables can be included to model relationships with the profile parameters. An application examined experimental differences in topographic map comprehension among 288 subjects. The model produced a measure of overall test performance or comprehension in addition to pattern variables that measured the correspondence between subject response profiles and an item difficulty profile and an item-discrimination profile. The findings suggested that subjects who used 3-dimensional maps tended to correctly answer more items in addition to correctly answering items that were more discriminating indicators of map comprehension. The NPA analysis was also compared with results from a multidimensional item response theory model.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

A note on the asymptotic variance-covariance matrix of item parameter estimates in the rasch model

Lord and Wingersky have developed a method for computing the asymptotic variance-covariance matrix of maximum likelihood estimates for item and person parameters under some restrictions on the estimates which are needed in order to fix the latent scale. The method is tedious, but can be simplified for the Rasch model when one is only interested in the item parameters. This is demonstrated here under a suitable restriction on the item parameter estimates.     

A general diagnostic model applied to language testing data

Probabilistic models with one or more latent variables are designed to report on a corresponding number of skills or cognitive attributes. Multidimensional skill profiles offer additional information beyond what a single test score can provide, if the reported skills can be identified and distinguished reliably. Many recent approaches to skill profile models are limited to dichotomous data and have made use of computationally intensive estimation methods such as Markov chain Monte Carlo, since standard maximum likelihood (ML) estimation techniques were deemed infeasible. This paper presents a general diagnostic model (GDM) that can be estimated with standard ML techniques and applies to polytomous response variables as well as to skills with two or more proficiency levels. The paper uses one member of a larger class of diagnostic models, a compensatory diagnostic model for dichotomous and partial credit data. Many well‐known models, such as univariate and multivariate versions of the Rasch model and the two‐parameter logistic item response theory model, the generalized partial credit model, as well as a variety of skill profile models, are special cases of this GDM. In addition to an introduction to this model, the paper presents a parameter recovery study using simulated data and an application to real data from the field test for TOEFL^® Internet‐based testing.     

A rasch model for partial credit scoring

A unidimensional latent trait model for responses scored in two or more ordered categories is developed. This “Partial Credit” model is a member of the family of latent trait models which share the property of parameter separability and so permit “specifically objective” comparisons of persons and items. The model can be viewed as an extension of Andrich's Rating Scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item. The difference between the parameters in this model and the “category boundaries” in Samejima's Graded Response model is demonstrated. An unconditional maximum likelihood procedure for estimating the model parameters is developed. Preparation of this paper was supported by grants from the Spencer Foundation and the National Institute for Justice. I would like to thank Professor Benjamin D. Wright of the University of Chicago for his very kind help with the various drafts of this paper.     

Generalized latent trait models

In this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model. A unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented. We discuss in addition the scoring of individuals on the latent dimensions. The general framework presented allows, not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions. The approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly.     

Logistic response models with item interactions

Items that are clustered according to shared content may violate the principle of conditional independence commonly used in item response theory. This paper investigates the capabilities of a logistic item response model in relation to locally dependent item responses. The model includes main effect and interaction parameters that are computed as linear functions of the latent trait. The paper explains the interpretation of the parameters, the maximum likelihood estimation algorithm, the information matrix and some results concerning parameter identifiability. The problem of over-fitting the data is addressed in a simulation study, and two real data examples are described to illustrate the approach, one from the context of a sample survey and the other from ability testing using testlets.     

Loglinear multidimensional IRT models for polytomously scored items

A loglinear IRT model is proposed that relates polytomously scored item responses to a multidimensional latent space. The analyst may specify a response function for each response, indicating which latent abilities are necessary to arrive at that response. Each item may have a different number of response categories, so that free response items are more easily analyzed. Conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear IRT models.Hank Kelderman is currently affiliated with Vrije Universiteit, Amsterdam.We thank Linda Vodegel-Matzen of the Division of Developmental Psychology of the University of Amsterdam for making available the data used in the example in this article.     

Bayesian estimation of a multilevel IRT model using gibbs sampling

In this article, a two-level regression model is imposed on the ability parameters in an item response theory (IRT) model. The advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error. Another advantage is that, contrary to observed scores, latent scores are test-independent, which offers the possibility of using results from different tests in one analysis where the parameters of the IRT model and the multilevel model can be concurrently estimated. The two-parameter normal ogive model is used for the IRT measurement model. It will be shown that the parameters of the two-parameter normal ogive model and the multilevel model can be estimated in a Bayesian framework using Gibbs sampling. Examples using simulated and real data are given.     

多维题组效应Rasch模型

首先, 本文诠释了“题组”的本质即一个存在共同刺激的项目集合。并基于此, 将题组效应划分为项目内单维题组效应和项目内多维题组效应。其次, 本文基于Rasch模型开发了二级评分和多级评分的多维题组效应Rasch模型, 以期较好地处理项目内多维题组效应。最后, 模拟研究结果显示新模型有效合理, 与Rasch题组模型、分部评分模型对比研究后表明：(1)测验存在项目内多维题组效应时, 仅把明显的捆绑式题组效应进行分离而忽略其他潜在的题组效应, 仍会导致参数的偏差估计甚或高估测验信度; (2)新模型更具普适性, 即便当被试作答数据不存在题组效应或只存在项目内单维题组效应, 采用新模型进行测验分析也能得到较好的参数估计结果。     

Linear structural equations with latent variables

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.     

Inferring the structure of latent class

Present optimization techniques in latent class analysis apply the expectation maximization algorithm or the Newton-Raphson algorithm for optimizing the parameter values of a prespecified model. These techniques can be used to find maximum likelihood estimates of the parameters, given the specified structure of the model, which is defined by the number of classes and, possibly, fixation and equality constraints. The model structure is usually chosen on theoretical grounds. A large variety of structurally different latent class models can be compared using goodness-of-fit indices of the chi-square family, Akaike’s information criterion, the Bayesian information criterion, and various other statistics. However, finding the optimal structure for a given goodness-of-fit index often requires a lengthy search in which all kinds of model structures are tested. Moreover, solutions may depend on the choice of initial values for the parameters. This article presents a new method by which one can simultaneously infer the model structure from the data and optimize the parameter values. The method consists of a genetic algorithm in which any goodness-of-fit index can be used as a fitness criterion. In a number of test cases in which data sets from the literature were used, it is shown that this method provides models that fit equally well as or better than the models suggested in the original articles.