Multidimensional Latent Markov Models in a Developmental Study of Inhibitory Control and Attentional Flexibility in Early Childhood

We demonstrate the use of a multidimensional extension of the latent Markov model to analyse data from studies with repeated binary responses in developmental psychology. In particular, we consider an experiment based on a battery of tests which was administered to pre-school children, at three time periods, in order to measure their inhibitory control (IC) and attentional flexibility (AF) abilities. Our model represents these abilities by two latent traits which are associated to each state of a latent Markov chain. The conditional distribution of the test outcomes given the latent process depends on these abilities through a multidimensional one-parameter or two-parameter logistic parameterisation. We outline an EM algorithm for likelihood inference on the model parameters; we also focus on likelihood ratio testing of hypotheses on the dimensionality of the model and on the transition matrices of the latent process. Through the approach based on the proposed model, we find evidence that supports that IC and AF can be conceptualised as distinct constructs. Furthermore, we outline developmental aspects of participants’ performance on these abilities based on inspection of the estimated transition matrices.     

An overview of Markov chain methods for the study of stage-sequential developmental processes

This article presents an overview of quantitative methodologies for the study of stage-sequential development based on extensions of Markov chain modeling. Four methods are presented that exemplify the flexibility of this approach: the manifest Markov model, the latent Markov model, latent transition analysis, and the mixture latent Markov model. A special case of the mixture latent Markov model, the so-called mover-stayer model, is used in this study. Unconditional and conditional models are estimated for the manifest Markov model and the latent Markov model, where the conditional models include a measure of poverty status. Issues of model specification, estimation, and testing using the Mplus software environment are briefly discussed, and the Mplus input syntax is provided. The author applies these 4 methods to a single example of stage-sequential development in reading competency in the early school years, using data from the Early Childhood Longitudinal Study--Kindergarten Cohort.     

A Latent Transition Model With Logistic Regression

Latent transition models increasingly include covariates that predict prevalence of latent classes at a given time or transition rates among classes over time. In many situations, the covariate of interest may be latent. This paper describes an approach for handling both manifest and latent covariates in a latent transition model. A Bayesian approach via Markov chain Monte Carlo (MCMC) is employed in order to achieve more robust estimates. A case example illustrating the model is provided using data on academic beliefs and achievement in a low-income sample of adolescents in the United States. This research was partially supported by the National Institute on Drug Abuse Grant 1-R03-DA021639. This research was partially supported by the National Institute on Drug Abuse Grant 1-P50-DA10075, The Methodology Center, The Pennsylvania State University. This research was partially supported by the National Institute of Mental Health funds as part of the Studying Diverse Lives research support program at the Henry A. Murray Research Archive, Institute for Quantitative Science, Harvard University.     

Latent Class Models for Diary Method Data: Parameter Estimation by Local Computations

The increasing use of diary methods calls for the development of appropriate statistical methods. For the resulting panel data, latent Markov models can be used to model both individual differences and temporal dynamics. The computational burden associated with these models can be overcome by exploiting the conditional independence relations implied by the model. This is done by associating a probabilistic model with a directed acyclic graph, and applying transformations to the graph. The structure of the transformed graph provides a factorization of the joint probability function of the manifest and latent variables, which is the basis of a modified and more efficient E-step of the EM algorithm. The usefulness of the approach is illustrated by estimating a latent Markov model involving a large number of measurement occasions and, subsequently, a hierarchical extension of the latent Markov model that allows for transitions at different levels. Furthermore, logistic regression techniques are used to incorporate restrictions on the conditional probabilities and to account for the effect of covariates. Throughout, models are illustrated with an experience sampling methodology study on the course of emotions among anorectic patients. Frank Rijmen was partly supported by the Fund for Scientific Research Flanders (FWO).     

Multilevel IRT using dichotomous and polytomous response data

A structural multilevel model is presented where some of the variables cannot be observed directly but are measured using tests or questionnaires. Observed dichotomous or ordinal polytomous response data serve to measure the latent variables using an item response theory model. The latent variables can be defined at any level of the multilevel model. A Bayesian procedure Markov chain Monte Carlo (MCMC), to estimate all parameters simultaneously is presented. It is shown that certain model checks and model comparisons can be done using the MCMC output. The techniques are illustrated using a simulation study and an application involving students' achievements on a mathematics test and test results regarding management characteristics of teachers and principles.     

Bayesian Inferences of Latent Class Models with an Unknown Number of Classes

This paper focuses on analyzing data collected in situations where investigators use multiple discrete indicators as surrogates, for example, a set of questionnaires. A very flexible latent class model is used for analysis. We propose a Bayesian framework to perform the joint estimation of the number of latent classes and model parameters. The proposed approach applies the reversible jump Markov chain Monte Carlo to analyze finite mixtures of multivariate multinomial distributions. In the paper, we also develop a procedure for the unique labeling of the classes. We have carried out a detailed sensitivity analysis for various hyperparameter specifications, which leads us to make standard default recommendations for the choice of priors. The usefulness of the proposed method is demonstrated through computer simulations and a study on subtypes of schizophrenia using the Positive and Negative Syndrome Scale (PANSS).     

Analyzing sequential categorical data: Individual variation in markov chains

Markov chains are probabilistic models for sequences of categorical events, with applications throughout scientific psychology. This paper provides a method for anlayzing data consisting of event sequences and covariate observations. It is assumed that each sequence is a Markov process characterized by a distinct transition probability matrix. The objective is to use the covariate data to explain differences between individuals in the transition probability matrices characterizing their sequential data. The elements of the transition probability matrices are written as functions of a vector of latent variables, with variation in the latent variables explained through a multivariate regression on the covariates. The regression is estimated using the EM algorithm, and requires the numerical calculation of a multivariate integral. An example using simulated cognitive developmental data is presented, which shows that the estimation of individual variation in the parameters of a probability model may have substantial theoretical importance, even when individual differences are not the focus of the investigator's concerns.Research contributing to this article was supported by B.R.S. Subgrant 5-35345 from the University of Virginia. I thank the DADA Group, Bill Fabricius, Don Hartmann, William Griffin, Jack McArdle, Ivo Molenaar, Ronald Schoenberg, Simon Tavaré, and several anonymous reviewers for their discussion of these points.     

Modelling multiple problem-solving strategies and strategy shift in cognitive diagnosis for growth

Problem-solving strategies, defined as actions people select intentionally to achieve desired objectives, are distinguished from skills that are implemented unintentionally. In education, strategy-oriented instructions that guide students to form problem-solving strategies are found to be more effective for low-achieving students than the skill-oriented instructions designed for enhancing their skill implementation ability. Although the existing longitudinal cognitive diagnosis models (CDMs) can model the change in students' dynamic skill mastery status over time, they are not designed to model the shift in students' problem-solving strategies. This study proposes a longitudinal CDM that considers both between-person multiple strategies and within-person strategy shift. The model, separating the strategy choice process from the skill implementation process, is intended to provide diagnostic information on strategy choice as well as skill mastery status. A simulation study is conducted to evaluate the parameter recovery of the proposed model and investigate the consequences of ignoring the presence of multiple strategies or strategy shift. Further, an empirical data analysis is conducted to illustrate the use of the proposed model to measure strategy shift, growth in skill implementation ability and skill mastery status.     

Bayesian Analysis of Structural Equation Models With Nonlinear Covariates and Latent Variables

In this article, we formulate a nonlinear structural equation model (SEM) that can accommodate covariates in the measurement equation and nonlinear terms of covariates and exogenous latent variables in the structural equation. The covariates can come from continuous or discrete distributions. A Bayesian approach is developed to analyze the proposed model. Markov chain Monte Carlo methods for obtaining Bayesian estimates and their standard error estimates, highest posterior density intervals, and a PP p value are developed. Results obtained from two simulation studies are reported to respectively reveal the empirical performance of the proposed Bayesian estimation in analyzing complex nonlinear SEMs, and in analyzing nonlinear SEMs with the normal assumption of the exogenous latent variables violated. The proposed methodology is further illustrated by a real example. Detailed interpretation about the interaction terms is presented.     

Generalized graded unfolding model with structural equation for subject parameters

The generalized graded unfolding model (GGUM) is capable of analyzing polytomous scored, unfolding data such as agree‐disagree responses to attitude statements. In the present study, we proposed a GGUM with structural equation for subject parameters, which enabled us to evaluate the relation between subject parameters and covariates and/or latent variables simultaneously, in order to avoid the influence of attenuation. Additionally, an algorithm for parameter estimation is newly implemented via the Markov Chain Monte Carlo (MCMC) method, based on Bayesian statistics. In the simulation, we compared the accuracy of estimates of regression coefficients between the proposed model and a conventional method using a GGUM (where regression coefficients are estimated using estimates of θ). As a result, the proposed model performed much better than the conventional method in terms of bias and root mean squared errors of estimates of regression coefficients. The study concluded by verifying the efficacy of the proposed model, using an actual data example of attitude measurement.     

Bayesian generalized structured component analysis

Generalized structured component analysis (GSCA) is a component-based approach to structural equation modelling, which adopts components of observed variables as proxies for latent variables and examines directional relationships among latent and observed variables. GSCA has been extended to deal with a wider range of data types, including discrete, multilevel or intensive longitudinal data, as well as to accommodate a greater variety of complex analyses such as latent moderation analysis, the capturing of cluster-level heterogeneity, and regularized analysis. To date, however, there has been no attempt to generalize the scope of GSCA into the Bayesian framework. In this paper, a novel extension of GSCA, called BGSCA, is proposed that estimates parameters within the Bayesian framework. BGSCA can be more attractive than the original GSCA for various reasons. For example, it can infer the probability distributions of random parameters, account for error variances in the measurement model, provide additional fit measures for model assessment and comparison from the Bayesian perspectives, and incorporate external information on parameters, which may be obtainable from past research, expert opinions, subjective beliefs or knowledge on the parameters. We utilize a Markov chain Monte Carlo method, the Gibbs sampler, to update the posterior distributions for the parameters of BGSCA. We conduct a simulation study to evaluate the performance of BGSCA. We also apply BGSCA to real data to demonstrate its empirical usefulness.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

An exploratory analysis of the latent structure of process data via action sequence autoencoders

Computer simulations have become a popular tool for assessing complex skills such as problem-solving. Log files of computer-based items record the human–computer interactive processes for each respondent in full. The response processes are very diverse, noisy, and of non-standard formats. Few generic methods have been developed to exploit the information contained in process data. In this paper we propose a method to extract latent variables from process data. The method utilizes a sequence-to-sequence autoencoder to compress response processes into standard numerical vectors. It does not require prior knowledge of the specific items and human–computer interaction patterns. The proposed method is applied to both simulated and real process data to demonstrate that the resulting latent variables extract useful information from the response processes.     

A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the metropolis-hastings algorithm

Nonlinear latent variable models are specified that include quadratic forms and interactions of latent regressor variables as special cases. To estimate the parameters, the models are put in a Bayesian framework with conjugate priors for the parameters. The posterior distributions of the parameters and the latent variables are estimated using Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis-Hastings algorithm. The proposed estimation methods are illustrated by two simulation studies and by the estimation of a non-linear model for the dependence of performance on task complexity and goal specificity using empirical data.     

Markov Decision Process Measurement Model

Within-task actions can provide additional information on student competencies but are challenging to model. This paper explores the potential of using a cognitive model for decision making, the Markov decision process, to provide a mapping between within-task actions and latent traits of interest. Psychometric properties of the model are explored, and simulation studies report on parameter recovery within the context of a simple strategy game. The model is then applied to empirical data from an educational game. Estimates from the model are found to correlate more strongly with posttest results than a partial-credit IRT model based on outcome data alone.     

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.     

Flexible Computerized Adaptive Tests to Detect Misconceptions and Estimate Ability Simultaneously

The Scaling Individuals and Classifying Misconceptions (SICM) model is an advanced psychometric model that can provide feedback to examinees’ misconceptions and a general ability simultaneously. These two types of feedback are represented by a discrete and a continuous latent variable, respectively, in the SICM model. The complex structure of the SICM model brings difficulties in estimating both misconception profile and ability efficiently in a linear test. To overcome this challenge, this study proposes a flexible computerized adaptive test (FCAT) design as a new test delivery method to increase test efficiency by administering an individualized test to examinees. We propose three item selection methods and two transition criteria to determine adaptive steps based on the needs of estimating one or two latent variables. Through two simulation studies, we demonstrate how to select an appropriate item selection method for an adaptive step and what transition criterion should be used between two adaptive steps. Results reveal the combination of the item selection method and the transition criterion could improve the estimation accuracy of a specific latent variable to a different extent and thus provide further guidance in designing an FCAT.     

Sequential Analyses of Adolescent,Mother, and Father Behaviors in Distressed and Nondistressed Families

The Response Class Matrix was used to compare sequential relations among behaviors in clinic families with a problem adolescent and those in non-clinic families with an adolescent under two experimental conditions, discussing a problem and planning something together. Clinic dyads manifested significantly more negative an les problem-solving behavior overall. When the family was planning something together, clinic mothers responded to their adolescents' negative behavior significantly more often with negative behavior and less often with problem-solving than did their nonclinic counterparts. Clinic adolescents responded to their mothers' negative behavior significantly more often with negative behavior and with less problem-solving than non-clinic adolescents, both when planning something together and when discussing a problem. These patterns were not observed with fathers. When the family was planning something together, clinic adolescents responded to both their mothers' and fathers' problem-solving with significantly more negative and less problem-solving behavior than did non-clinic adolescents. Results are discussed in terms of the aversive nature of the family environment for clinic adolescents, and the differences between mothers' and fathers' interactions with adolescents.     

Bayesian estimation and model selection in ordered latent class models for polytomous items

In a latent class IRT model in which the latent classes are ordered on one dimension, the class specific response probabilities are subject to inequality constraints. The number of these inequality constraints increase dramatically with the number of response categories per item, if assumptions like monotonicity or double monotonicity of the cumulative category response functions are postulated. A Markov chain Monte Carlo method, the Gibbs sampler, can sample from the multivariate posterior distribution of the parameters under the constraints. Bayesian model selection can be done by posterior predictive checks and Bayes factors. A simulation study is done to evaluate results of the application of these methods to ordered latent class models in three realistic situations. Also, an example of the presented methods is given for existing data with polytomous items. It can be concluded that the Bayesian estimation procedure can handle the inequality constraints on the parameters very well. However, the application of Bayesian model selection methods requires more research.