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.     

A generalized longitudinal mixture IRT model for measuring differential growth in learning environments

This article describes a generalized longitudinal mixture item response theory (IRT) model that allows for detecting latent group differences in item response data obtained from electronic learning (e-learning) environments or other learning environments that result in large numbers of items. The described model can be viewed as a combination of a longitudinal Rasch model, a mixture Rasch model, and a random-item IRT model, and it includes some features of the explanatory IRT modeling framework. The model assumes the possible presence of latent classes in item response patterns, due to initial person-level differences before learning takes place, to latent class-specific learning trajectories, or to a combination of both. Moreover, it allows for differential item functioning over the classes. A Bayesian model estimation procedure is described, and the results of a simulation study are presented that indicate that the parameters are recovered well, particularly for conditions with large item sample sizes. The model is also illustrated with an empirical sample data set from a Web-based e-learning environment.     

多变量追踪研究的模型整合与拓展：考察往复式影响与增长趋势

追踪研究当中, 交叉滞后模型可以探究多变量之间往复式影响, 潜增长模型可以探究个体增长趋势。对两类模型进行整合, 例如同时关注往复式影响与个体增长趋势, 同时可以定义测量误差、随机截距等变异成分, 衍生出随机截距交叉滞后模型、特质-状态-误差模型、自回归潜增长模型、结构化残差潜增长模型等。以交叉滞后模型和潜增长模型分别作为基础模型, 从个体间/个体内变异分解的角度对上述各类模型梳理, 整合出此类模型的分析框架, 并拓展建立“因子结构化潜增长模型(factor latent curve model with structured reciprocals)”作为统合框架。通过实证研究(早期儿童的追踪研究-幼儿园版, ECLS-K), 建立21049名儿童的阅读和数学能力的往复式影响与增长趋势。研究发现, 分离了稳定特质的模型拟合最优。研究也对模型建模思路和模型选择提供了建议。     

A longitudinal examination of the relationship between teacher burnout and depression

This study investigates the longitudinal relationships between burnout and depression among teachers. Middle and high school teachers participated in a 3‐wave survey. The results of the latent growth modeling analysis revealed that there was a significant relationship between the initial status of burnout and the initial status of depression. Results also indicated a significant relationship between the change rate of burnout and the change rate of depression. Moreover, the autoregressive cross‐lagged modeling revealed a causal relationship such that teacher's burnout leads to subsequent depression symptoms, not vice versa.     

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.     

Analysis of multivariate mixed longitudinal data: A flexible latent process approach

Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random‐effect models, this approach handles individually varying and outcome‐specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome‐specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non‐linear families of transformations for Gaussian and non‐Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini‐Mental State Examination, a 20‐item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non‐linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two‐parameter longitudinal item response models that are specific cases of the approach.     

A Hierarchical Rater Model for Longitudinal Data

Research studies in psychology and education often seek to detect changes or growth in an outcome over a duration of time. This research provides a solution to those interested in estimating latent traits from psychological measures that rely on human raters. Rater effects potentially degrade the quality of scores in constructed response and performance assessments. We develop an extension of the hierarchical rater model (HRM), which yields estimates of latent traits that have been corrected for individual rater bias and variability, for ratings that come from longitudinal designs. The parameterization, called the longitudinal HRM (L-HRM), includes an autoregressive time series process to permit serial dependence between latent traits at adjacent timepoints, as well as a parameter for overall growth. We evaluate and demonstrate the feasibility and performance of the L-HRM using simulation studies. Parameter recovery results reveal predictable amounts and patterns of bias and error for most parameters across conditions. An application to ratings from a study of character strength demonstrates the model. We discuss limitations and future research directions to improve the L-HRM.     

Estimating latent correlations between repeated testings

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.     

Assessing Growth in a Diagnostic Classification Model Framework

A common assessment research design is the single-group pre-test/post-test design in which examinees are administered an assessment before instruction and then another assessment after instruction. In this type of study, the primary objective is to measure growth in examinees, individually and collectively. In an item response theory (IRT) framework, longitudinal IRT models can be used to assess growth in examinee ability over time. In a diagnostic classification model (DCM) framework, assessing growth translates to measuring changes in attribute mastery status over time, thereby providing a categorical, criterion-referenced interpretation of growth. This study introduces the Transition Diagnostic Classification Model (TDCM), which combines latent transition analysis with the log-linear cognitive diagnosis model to provide methodology for analyzing growth in a general DCM framework. Simulation study results indicate that the proposed model is flexible, provides accurate and reliable classifications, and is quite robust to violations to measurement invariance over time. The TDCM is used to analyze pre-test/post-test data from a diagnostic mathematics assessment.     

Constant latent odds-ratios models and the mantel-haenszel null hypothesis

In the present paper, a new family of item response theory (IRT) models for dichotomous item scores is proposed. Two basic assumptions define the most general model of this family. The first assumption is local independence of the item scores given a unidimensional latent trait. The second assumption is that the odds-ratios for all item-pairs are constant functions of the latent trait. Since the latter assumption is characteristic of the whole family, the models are called constant latent odds-ratios (CLORs) models. One nonparametric special case and three parametric special cases of the general CLORs model are shown to be generalizations of the one-parameter logistic Rasch model. For all CLORs models, the total score (the unweighted sum of the item scores) is shown to be a sufficient statistic for the latent trait. In addition, conditions under the general CLORs model are studied for the investigation of differential item functioning (DIF) by means of the Mantel-Haenszel procedure. This research was supported by the Dutch Organization for Scientific Research (NWO), grant number 400-20-026.     

(Re)evaluating the Implications of the Autoregressive Latent Trajectory Model Through Likelihood Ratio Tests of Its Initial Conditions

The autoregressive latent trajectory (ALT) model synthesizes the autoregressive model and the latent growth curve model. The ALT model is flexible enough to produce a variety of discrepant model-implied change trajectories. While some researchers consider this a virtue, others have cautioned that this may confound interpretations of the model's parameters. In this article, we show that some—but not all—of these interpretational difficulties may be clarified mathematically and tested explicitly via likelihood ratio tests (LRTs) imposed on the initial conditions of the model. We show analytically the nested relations among three variants of the ALT model and the constraints needed to establish equivalences. A Monte Carlo simulation study indicated that LRTs, particularly when used in combination with information criterion measures, can allow researchers to test targeted hypotheses about the functional forms of the change process under study. We further demonstrate when and how such tests may justifiably be used to facilitate our understanding of the underlying process of change using a subsample (N = 3,995) of longitudinal family income data from the National Longitudinal Survey of Youth.     

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.     

Thurstonian Scaling of Compositional Questionnaire Data

To prevent response bias, personality questionnaires may use comparative response formats. These include forced choice, where respondents choose among a number of items, and quantitative comparisons, where respondents indicate the extent to which items are preferred to each other. The present article extends Thurstonian modeling of binary choice data to “proportion-of-total” (compositional) formats. Following the seminal work of Aitchison, compositional item data are transformed into log ratios, conceptualized as differences of latent item utilities. The mean and covariance structure of the log ratios is modeled using confirmatory factor analysis (CFA), where the item utilities are first-order factors, and personal attributes measured by a questionnaire are second-order factors. A simulation study with two sample sizes, N = 300 and N = 1,000, shows that the method provides very good recovery of true parameters and near-nominal rejection rates. The approach is illustrated with empirical data from N = 317 students, comparing model parameters obtained with compositional and Likert-scale versions of a Big Five measure. The results show that the proposed model successfully captures the latent structures and person scores on the measured traits.     

Improving measurement precision of test batteries using multidimensional item response models

A conventional way to analyze item responses in multiple tests is to apply unidimensional item response models separately, one test at a time. This unidimensional approach, which ignores the correlations between latent traits, yields imprecise measures when tests are short. To resolve this problem, one can use multidimensional item response models that use correlations between latent traits to improve measurement precision of individual latent traits. The improvements are demonstrated using 2 empirical examples. It appears that the multidimensional approach improves measurement precision substantially, especially when tests are short and the number of tests is large. To achieve the same measurement precision, the multidimensional approach needs less than half of the comparable items required for the unidimensional approach.     

Dynamic GSCA (Generalized Structured Component Analysis) with Applications to the Analysis of Effective Connectivity in Functional Neuroimaging Data

We propose a new method of structural equation modeling (SEM) for longitudinal and time series data, named Dynamic GSCA (Generalized Structured Component Analysis). The proposed method extends the original GSCA by incorporating a multivariate autoregressive model to account for the dynamic nature of data taken over time. Dynamic GSCA also incorporates direct and modulating effects of input variables on specific latent variables and on connections between latent variables, respectively. An alternating least square (ALS) algorithm is developed for parameter estimation. An improved bootstrap method called a modified moving block bootstrap method is used to assess reliability of parameter estimates, which deals with time dependence between consecutive observations effectively. We analyze synthetic and real data to illustrate the feasibility of the proposed method.     

Using mixture models with known class membership to address incomplete covariance structures in multiple‐group growth models

Multi‐group latent growth modelling in the structural equation modelling framework has been widely utilized for examining differences in growth trajectories across multiple manifest groups. Despite its usefulness, the traditional maximum likelihood estimation for multi‐group latent growth modelling is not feasible when one of the groups has no response at any given data collection point, or when all participants within a group have the same response at one of the time points. In other words, multi‐group latent growth modelling requires a complete covariance structure for each observed group. The primary purpose of the present study is to show how to circumvent these data problems by developing a simple but creative approach using an existing estimation procedure for growth mixture modelling. A Monte Carlo simulation study was carried out to see whether the modified estimation approach provided tangible results and to see how these results were comparable to the standard multi‐group results. The proposed approach produced results that were valid and reliable under the mentioned problematic data conditions. We also present a real data example and demonstrate that the proposed estimation approach can be used for the chi‐square difference test to check various types of measurement invariance as conducted in a standard multi‐group analysis.     

Locally dependent latent trait model and the dutch identity revisited

In this paper, we propose a class of locally dependent latent trait models for responses to psychological and educational tests. Typically, item response models treat an individual's multiple response to stimuli as conditional independent given the individual's latent trait. In this paper, instead the focus is on models based on a family of conditional distributions, or kernel, that describes joint multiple item responses as a function of student latent trait, not assuming conditional independence. Specifically, we examine a hybrid kernel which comprises a component for one-way item response functions and a component for conditional associations between items given latent traits. The class of models allows the extension of item response theory to cover some new and innovative applications in psychological and educational research. An EM algorithm for marginal maximum likelihood of the hybrid kernel model is proposed. Furthermore, we delineate the relationship of the class of locally dependent models and the log-linear model by revisiting the Dutch identity (Holland, 1990). The work is supported by a research grant from the Marshall School of Business, University of Southern California. The author thanks the anonymous referees for their suggestions.     

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.     

A latent topic model with Markov transition for process data

We propose a latent topic model with a Markov transition for process data, which consists of time-stamped events recorded in a log file. Such data are becoming more widely available in computer-based educational assessment with complex problem-solving items. The proposed model can be viewed as an extension of the hierarchical Bayesian topic model with a hidden Markov structure to accommodate the underlying evolution of an examinee's latent state. Using topic transition probabilities along with response times enables us to capture examinees' learning trajectories, making clustering/classification more efficient. A forward-backward variational expectation-maximization (FB-VEM) algorithm is developed to tackle the challenging computational problem. Useful theoretical properties are established under certain asymptotic regimes. The proposed method is applied to a complex problem-solving item in the 2012 version of the Programme for International Student Assessment (PISA).     

Evaluation on types of invariance in studying extreme response bias with an IRTree approach

In recent years, item response tree (IRTree) approaches have received increasing attention in the response style literature for their ability to partial out response style latent variables as well as associated item parameters. When an IRTree approach is adopted to measure extreme response styles, directional and content invariance could be assumed at the latent variable and item parameter levels. In this study, we propose to evaluate the empirical validity of these invariance assumptions by employing a general IRTree model with relaxed invariance assumptions. This would allow us to examine extreme response biases, beyond extreme response styles. With three empirical applications of the proposed evaluation, we find that relaxing some of the invariance assumptions improves the model fit, which suggests that not all assumed invariances are empirically supported. Specifically, at the latent variable level, we find reasonable evidence for directional invariance but mixed evidence for content invariance, although we also find that estimated correlations between content-specific extreme response latent variables are high, hinting at the potential presence of a general extreme response tendency. At the item parameter level, we find no directional or content invariance for thresholds and no content invariance for slopes. We discuss how the variant item parameter estimates obtained from a general IRTree model can offer useful insight to help us understand response bias related to extreme responding measured within the IRTree framework.