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.     

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 Note on Comparing the Estimates of Models for Cluster-Correlated or Longitudinal Data with Binary or Ordinal Outcomes

When using linear models for cluster-correlated or longitudinal data, a common modeling practice is to begin by fitting a relatively simple model and then to increase the model complexity in steps. New predictors might be added to the model, or a more complex covariance structure might be specified for the observations. When fitting models for binary or ordered-categorical outcomes, however, comparisons between such models are impeded by the implicit rescaling of the model estimates that takes place with the inclusion of new predictors and/or random effects. This paper presents an approach for putting the estimates on a common scale to facilitate relative comparisons between models fit to binary or ordinal outcomes. The approach is developed for both population-average and unit-specific models.     

Some remarks on Scheiblechner's treatment of isop models

Scheiblechner (1995) proposes a probabilistic axiomatization of measurement called ISOP (isotonic ordinal probabilistic models) that replaces Rasch's (1980) specific objectivity assumptions with two interesting ordinal assumptions. Special cases of Scheiblechner's model include standard unidimensional factor analysis models in which the loadings are held constant, and the Rasch model for binary item responses. Closely related are the doubly-monotone item response models of Mokken (1971; see also Mokken & Lewis, 1982; Sijtsma, 1988; Molenaar, 1991; Sijtsma & Junker, 1996; and Sijtsma & Hemker, in press). More generally, strictly unidimensional latent variable models have been considered in some detail by Holland and Rosenbaum (1986), Ellis and van den Wollenberg (1993), and Junker (1990, 1993). The purpose of this note is to provide connections with current research in foundations and nonparametric latent variable and item response modeling that are missing from Scheiblechner's (1995) paper, and to point out important related work by Hemker, Sijtsma, Molenaar, & Junker (1996, 1997), Ellis and Junker (in press) and Junker and Ellis (1997). We also discuss counterexamples to three major theorems in the paper. By carrying out these three tasks, we hope to provide researchers interested in the foundations of measurement and item response modeling the opportunity to give the ISOP approach the careful attention it deserves. This research was supported by the National Science Foundation, Grant DMS-94.04438. I thank the Editor and three anonymous referees for careful readings that improved the completeness of this note.     

A Comparison of Imputation Strategies for Ordinal Missing Data on Likert Scale Variables

This article compares a variety of imputation strategies for ordinal missing data on Likert scale variables (number of categories = 2, 3, 5, or 7) in recovering reliability coefficients, mean scale scores, and regression coefficients of predicting one scale score from another. The examined strategies include imputing using normal data models with naïve rounding/without rounding, using latent variable models, and using categorical data models such as discriminant analysis and binary logistic regression (for dichotomous data only), multinomial and proportional odds logistic regression (for polytomous data only). The result suggests that both the normal model approach without rounding and the latent variable model approach perform well for either dichotomous or polytomous data regardless of sample size, missing data proportion, and asymmetry of item distributions. The discriminant analysis approach also performs well for dichotomous data. Naïvely rounding normal imputations or using logistic regression models to impute ordinal data are not recommended as they can potentially lead to substantial bias in all or some of the parameters.     

The linear transformation model with frailties for the analysis of item response times

The item response times (RTs) collected from computerized testing represent an underutilized source of information about items and examinees. In addition to knowing the examinees’ responses to each item, we can investigate the amount of time examinees spend on each item. In this paper, we propose a semi‐parametric model for RTs, the linear transformation model with a latent speed covariate, which combines the flexibility of non‐parametric modelling and the brevity as well as interpretability of parametric modelling. In this new model, the RTs, after some non‐parametric monotone transformation, become a linear model with latent speed as covariate plus an error term. The distribution of the error term implicitly defines the relationship between the RT and examinees’ latent speeds; whereas the non‐parametric transformation is able to describe various shapes of RT distributions. The linear transformation model represents a rich family of models that includes the Cox proportional hazards model, the Box–Cox normal model, and many other models as special cases. This new model is embedded in a hierarchical framework so that both RTs and responses are modelled simultaneously. A two‐stage estimation method is proposed. In the first stage, the Markov chain Monte Carlo method is employed to estimate the parametric part of the model. In the second stage, an estimating equation method with a recursive algorithm is adopted to estimate the non‐parametric transformation. Applicability of the new model is demonstrated with a simulation study and a real data application. Finally, methods to evaluate the model fit are suggested.     

Specifying Ability Growth Models Using a Multidimensional Item Response Model for Repeated Measures Categorical Ordinal Item Response Data

When categorical ordinal item response data are collected over multiple timepoints from a repeated measures design, an item response theory (IRT) modeling approach whose unit of analysis is an item response is suitable. This study proposes a few longitudinal IRT models and illustrates how a popular compensatory multidimensional IRT model can be utilized to formulate such longitudinal IRT models, which permits an investigation of ability growth at both individual and population levels. The equivalence of an existing multidimensional IRT model and those longitudinal IRT models is also elaborated so that one can make use of an existing multidimensional IRT model to implement the longitudinal IRT models.     

Bayesian analysis of longitudinal multitrait–multimethod data with ordinal response variables

A new multilevel latent state graded response model for longitudinal multitrait–multimethod (MTMM) measurement designs combining structurally different and interchangeable methods is proposed. The model allows researchers to examine construct validity over time and to study the change and stability of constructs and method effects based on ordinal response variables. We show how Bayesian estimation techniques can address a number of important issues that typically arise in longitudinal multilevel MTMM studies and facilitates the estimation of the model presented. Estimation accuracy and the impact of between‐ and within‐level sample sizes as well as different prior specifications on parameter recovery were investigated in a Monte Carlo simulation study. Findings indicate that the parameters of the model presented can be accurately estimated with Bayesian estimation methods in the case of low convergent validity with as few as 250 clusters and more than two observations within each cluster. The model was applied to well‐being data from a longitudinal MTMM study, assessing the change and stability of life satisfaction and subjective happiness in young adults after high‐school graduation. Guidelines for empirical applications are provided and advantages and limitations of a Bayesian approach to estimating longitudinal multilevel MTMM models are discussed.     

A Bayesian Semiparametric Latent Variable Model for Mixed Responses

In this paper we introduce a latent variable model (LVM) for mixed ordinal and continuous responses, where covariate effects on the continuous latent variables are modelled through a flexible semiparametric Gaussian regression model. We extend existing LVMs with the usual linear covariate effects by including nonparametric components for nonlinear effects of continuous covariates and interactions with other covariates as well as spatial effects. Full Bayesian modelling is based on penalized spline and Markov random field priors and is performed by computationally efficient Markov chain Monte Carlo (MCMC) methods. We apply our approach to a German social science survey which motivated our methodological development. We thank the editor and the referees for their constructive and helpful comments, leading to substantial improvements of a first version, and Sven Steinert for computational assistance. Partial financial support from the SFB 386 “Statistical Analysis of Discrete Structures” is also acknowledged.     

An autoregressive growth model for longitudinal item analysis

A first-order autoregressive growth model is proposed for longitudinal binary item analysis where responses to the same items are conditionally dependent across time given the latent traits. Specifically, the item response probability for a given item at a given time depends on the latent trait as well as the response to the same item at the previous time, or the lagged response. An initial conditions problem arises because there is no lagged response at the initial time period. We handle this problem by adapting solutions proposed for dynamic models in panel data econometrics. Asymptotic and finite sample power for the autoregressive parameters are investigated. The consequences of ignoring local dependence and the initial conditions problem are also examined for data simulated from a first-order autoregressive growth model. The proposed methods are applied to longitudinal data on Korean students’ self-esteem.     

Measuring change for a multidimensional test using a generalized explanatory longitudinal item response model

Even though many educational and psychological tests are known to be multidimensional, little research has been done to address how to measure individual differences in change within an item response theory framework. In this paper, we suggest a generalized explanatory longitudinal item response model to measure individual differences in change. New longitudinal models for multidimensional tests and existing models for unidimensional tests are presented within this framework and implemented with software developed for generalized linear models. In addition to the measurement of change, the longitudinal models we present can also be used to explain individual differences in change scores for person groups (e.g., learning disabled students versus non‐learning disabled students) and to model differences in item difficulties across item groups (e.g., number operation, measurement, and representation item groups in a mathematics test). An empirical example illustrates the use of the various models for measuring individual differences in change when there are person groups and multiple skill domains which lead to multidimensionality at a time point.     

The effect of multiple indicators on the power to detect inter‐individual differences in change

Hertzog et al. evaluated the statistical power of linear latent growth curve models (LGCMs) to detect individual differences in change, i.e., variances of latent slopes, as a function of sample size, number of longitudinal measurement occasions, and growth curve reliability. We extend this work by investigating the effect of the number of indicators per measurement occasion on power. We analytically demonstrate that the positive effect of multiple indicators on statistical power is inversely related to the relative magnitude of occasion‐specific latent residual variance and is independent of the specific model that constitutes the observed variables, in particular of other parameters in the LGCM. When designing a study, researchers have to consider trade‐offs of costs and benefits of different design features. We demonstrate how knowledge about power equivalent transformations between indicator measurement designs allows researchers to identify the most cost‐efficient research design for detecting parameters of interest. Finally, we integrate different formal results to exhibit the trade‐off between the number of measurement occasions and number of indicators per occasion for constant power in LGCMs.     

A sequential exploratory diagnostic model using a Pólya-gamma data augmentation strategy

Cognitive diagnostic models provide a framework for classifying individuals into latent proficiency classes, also known as attribute profiles. Recent research has examined the implementation of a Pólya-gamma data augmentation strategy binary response model using logistic item response functions within a Bayesian Gibbs sampling procedure. In this paper, we propose a sequential exploratory diagnostic model for ordinal response data using a logit-link parameterization at the category level and extend the Pólya-gamma data augmentation strategy to ordinal response processes. A Gibbs sampling procedure is presented for efficient Markov chain Monte Carlo (MCMC) estimation methods. We provide results from a Monte Carlo study for model performance and present an application of the model.     

On the explaining‐away phenomenon in multivariate latent variable models

Many probabilistic models for psychological and educational measurements contain latent variables. Well‐known examples are factor analysis, item response theory, and latent class model families. We discuss what is referred to as the ‘explaining‐away’ phenomenon in the context of such latent variable models. This phenomenon can occur when multiple latent variables are related to the same observed variable, and can elicit seemingly counterintuitive conditional dependencies between latent variables given observed variables. We illustrate the implications of explaining away for a number of well‐known latent variable models by using both theoretical and real data examples.     

Fitting multilevel models with ordinal outcomes: performance of alternative specifications and methods of estimation

Previous research has compared methods of estimation for fitting multilevel models to binary data, but there are reasons to believe that the results will not always generalize to the ordinal case. This article thus evaluates (a) whether and when fitting multilevel linear models to ordinal outcome data is justified and (b) which estimator to employ when instead fitting multilevel cumulative logit models to ordinal data, maximum likelihood (ML), or penalized quasi-likelihood (PQL). ML and PQL are compared across variations in sample size, magnitude of variance components, number of outcome categories, and distribution shape. Fitting a multilevel linear model to ordinal outcomes is shown to be inferior in virtually all circumstances. PQL performance improves markedly with the number of ordinal categories, regardless of distribution shape. In contrast to binary data, PQL often performs as well as ML when used with ordinal data. Further, the performance of PQL is typically superior to ML when the data include a small to moderate number of clusters (i.e., ≤ 50 clusters).     

Response style analysis with threshold and multi‐process IRT models: A review and tutorial

Two different item response theory model frameworks have been proposed for the assessment and control of response styles in rating data. According to one framework, response styles can be assessed by analysing threshold parameters in Rasch models for ordinal data and in mixture‐distribution extensions of such models. A different framework is provided by multi‐process item response tree models, which can be used to disentangle response processes that are related to the substantive traits and response tendencies elicited by the response scale. In this tutorial, the two approaches are reviewed, illustrated with an empirical data set of the two‐dimensional ‘Personal Need for Structure’ construct, and compared in terms of multiple criteria. Mplus is used as a software framework for (mixed) polytomous Rasch models and item response tree models as well as for demonstrating how parsimonious model variants can be specified to test assumptions on the structure of response styles and attitude strength. Although both frameworks are shown to account for response styles, they differ on the quantitative criteria of model selection, practical aspects of model estimation, and conceptual issues of representing response styles as continuous and multidimensional sources of individual differences in psychological assessment.     

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.     

Analyzing Longitudinal Item Response Data via the Pairwise Fitting Method

Multidimensional item response theory (MIRT) models can be applied to longitudinal educational surveys where a group of individuals are administered different tests over time with some common items. However, computational problems typically arise as the dimension of the latent variables increases. This is especially true when the latent variable distribution cannot be integrated out analytically, as with MIRT models for binary data. In this article, based on the pseudolikelihood theory, we propose a pairwise modeling strategy to estimate item and population parameters in longitudinal studies. Our pairwise method effectively reduces the dimensionality of the problem and hence is applicable to longitudinal IRT data with high-dimensional latent variables, which are challenging for classical methods. And in the low-dimensional case, our simulation study shows that it performs comparably with the classical methods. We further illustrate the implementation of the pairwise method using a development study of mathematics levels of junior high school students in which the response data are collected from 65 classes of 8 schools from 4 different school districts in China.     

A note on monotone likelihood ratio of the total score variable in unidimensional item response theory

This note provides a direct, elementary proof of the fundamental result on monotone likelihood ratio of the total score variable in unidimensional item response theory (IRT). This result is very important for practical measurement in IRT, because it justifies the use of the total score variable to order participants on the latent trait. The proof relies on a basic inequality for elementary symmetric functions which is proved by means of few purely algebraic, straightforward transformations. In particular, flaws in a proof of this result by Huynh [(1994) . A new proof for monotone likelihood ratio for the sum of independent Bernoulli random variables. Psychometrika, 59, 77–79] are pointed out and corrected, and a natural generalization of the fundamental result to non‐linear (quasi‐ordered) latent trait spaces is presented. This may be useful for multidimensional IRT or knowledge space theory, in which the latent ‘ability’ spaces are partially ordered with respect to, for instance, coordinate‐wise vector‐ordering or set‐inclusion, respectively.     

IRTree models with ordinal and multidimensional decision nodes for response styles and trait-based rating responses

IRTree models decompose observed rating responses into sequences of theory-based decision nodes, and they provide a flexible framework for analysing trait-related judgements and response styles. However, most previous applications of IRTree models have been limited to binary decision nodes that reflect qualitatively distinct and unidimensional judgement processes. The present research extends the family of IRTree models for the analysis of response styles to ordinal judgement processes for polytomous decisions and to multidimensional parametrizations of decision nodes. The integration of ordinal judgement processes overcomes the limitation to binary nodes, and it allows researchers to test whether decisions reflect qualitatively distinct response processes or gradual steps on a joint latent continuum. The extension to multidimensional node models enables researchers to specify multiple judgement processes that simultaneously affect the decision between competing response options. Empirical applications highlight the roles of extreme and midpoint response style in rating judgements and show that judgement processes are moderated by different response formats. Model applications with multidimensional decision nodes reveal that decisions among rating categories are jointly informed by trait-related processes and response styles.