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.     

A multidimensional latent trait model for measuring learning and change

A latent trait model is presented for the repeated measurement of ability based on a multidimensional conceptualization of the change process. A simplex structure is postulated to link item performance under a given measurement condition or occasion to initial ability and to one or more modifiabilities that represent individual differences in change. Since item discriminations are constrained to be equal within a measurement condition, the model belongs to the family of multidimensional Rasch models. Maximum likelihood estimators of the item parameters and abilities are derived, and an example provided that shows good recovery of both item and ability parameters. Properties of the model are explored, particularly for several classical issues in measuring change.     

Equivalence classes of functions of finite Markov chains

A matrical representation of a Markov chain consists of the initial vector and transition matrix of the chain, along with matrices that specify which observable response occurs for each state. The likelihood function based on a Markov model can be stated in a general way using the components of the model's matrical representation. It follows directly from that statement that two models are equivalent in likelihood if they are related through matrix operations that constitute a change of basis of the matrical representation. Two necessary properties of a change matrix associating two Markov models that are members of the same equivalence class with respect to likelihood are derived. Examples are provided, involving use of the results in analyzing identifiability of Markov models, including a useful application of diagonalization that provides a connection between the problem of identifiability and the eigenvalue problem.     

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.     

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.     

Quantifying latent social motivation and its associations with joint attention and language in infants at high and low likelihood for autism spectrum disorder

Social motivation—the psychobiological predisposition for social orienting, seeking social contact, and maintaining social interaction—manifests in early infancy and is hypothesized to be foundational for social communication development in typical and atypical populations. However, the lack of infant social-motivation measures has hindered delineation of associations between infant social motivation, other early-arising social abilities such as joint attention, and language outcomes. To investigate how infant social motivation contributes to joint attention and language, this study utilizes a mixed longitudinal sample of 741 infants at high (HL = 515) and low (LL = 226) likelihood for ASD. Using moderated nonlinear factor analysis (MNLFA), we incorporated items from parent-report measures to establish a novel latent factor model of infant social motivation that exhibits measurement invariance by age, sex, and familial ASD likelihood. We then examined developmental associations between 6- and 12-month social motivation, joint attention at 12–15 months, and language at 24 months of age. On average, greater social-motivation growth from 6–12 months was associated with greater initiating joint attention (IJA) and trend-level increases in sophistication of responding to joint attention (RJA). IJA and RJA were both positively associated with 24-month language abilities. There were no additional associations between social motivation and future language in our path model. These findings substantiate a novel, theoretically driven approach to modeling social motivation and suggest a developmental cascade through which social motivation impacts other foundational skills. These findings have implications for the timing and nature of intervention targets to support social communication development in infancy.

Highlights

• We describe a novel, theoretically based model of infant social motivation wherein multiple parent-reported indicators contribute to a unitary latent social-motivation factor.
• Analyses revealed social-motivation factor scores exhibited measurement invariance for a longitudinal sample of infants at high and low familial ASD likelihood.
• Social-motivation growth from ages 6–12 months is associated with better 12−15-month joint attention abilities, which in turn are associated with greater 24-month language skills.
• Findings inform timing and targets of potential interventions to support healthy social communication in the first year of life.

     

A mixture model for responses and response times with a higher-order ability structure to detect rapid guessing behaviour

Many educational and psychological assessments focus on multidimensional latent traits that often have a hierarchical structure to provide both overall-level information and fine-grained diagnostic information. A test will usually have either separate time limits for each subtest or an overall time limit for administrative convenience and test fairness. In order to complete the items within the allocated time, examinees frequently adopt different test-taking behaviours during the test, such as solution behaviour and rapid guessing behaviour. In this paper we propose a new mixture model for responses and response times with a hierarchical ability structure, which incorporates auxiliary information from other subtests and the correlation structure of the abilities to detect rapid guessing behaviour. A Markov chain Monte Carlo method is proposed for model estimation. Simulation studies reveal that all model parameters could be recovered well, and the parameter estimates had smaller absolute bias and mean squared error than the mixture unidimensional item response theory (UIRT) model. Moreover, the true positive rate of detecting rapid guessing behaviour is also higher than when using the mixture UIRT model separately for each subscale, whereas the false detection rate is much lower than the mixture UIRT model. The deviance information criterion and the logarithm of the pseudo-marginal likelihood are employed to evaluate the model fit. Finally, a real data analysis is presented to demonstrate the practical value of the proposed model.     

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).     

The latent variable approach as applied to transitive reasoning

We discuss the limitations of hypothesis testing using (quasi-) experiments in the study of cognitive development and suggest latent variable modeling as a viable alternative to experimentation. Latent variable models allow testing a theory as a whole, incorporating individual differences with respect to developmental processes or abilities in the model. Experiments, in contrast, aim at testing hypotheses that refer to a specific part of a theory; also they ignore individual differences or model the individual differences using age group as a proxy for developmental stage. Drawing on a sample of 409 5–13-year olds, we demonstrate the advantages of latent variable models in the area of transitive reasoning. A comparison of three models showed that the latent variable model that represented fuzzy trace theory had a better fit than the models representing Piaget's theory or linear ordering theory.     

项目间多维测验作答时间数据分析：潜在特质速度间效应建模

随着计算机测验使用的普及化,被试在心理与教育测验上的作答反应时的获取也越发便利。为了充分利用项目反应时信息,单维与多维的反应时模型相继被提出。然后,在项目间多维反应时数据中,潜在特质速度之间可能存在共同关系（比如,层阶关系）,此时现有的反应时模型并不能适用。基于此,本研究提出了高阶对数正态反应时模型与双因子对数正态反应时模型。在模拟研究中,高阶对数正态反应时模型与双因子对数正态反应时模型的各参数都能被准确估计。在瑞文标准推理测验的三组测验项目的反应时数据中,双因子对数正态反应时模型表现出更为优秀的拟合效果,同时基于多个统计量说明了局部与全局潜在特质速度同时存在的必要性。因此,在项目间多维测验反应时数据分析中,非常有必要考虑多维潜在特质速度之间的共同效应。     

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。     

Differential ability antecedents of general and specific dimensions of declarative knowledge: More than g

The purpose of the current study is to test the proposition that the relative contribution of narrow abilities (but not of g) may have been obscured in prior research due to a failure to employ fully multidimensional latent variable analyses. The current study corrects for these deficiencies and examines the relationships between cognitive abilities and domain-specific declarative knowledge. Results show that when observed criterion test scores were individually regressed on abilities, only g was consistently related to the criteria. However, when a latent variable analysis was applied to the same data, both g and narrow ability factors accounted for substantial portions of variance in different latent criterion constructs. Implications 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.     

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.     

Latent trait models in the study of intelligence

This article examines the potential contribution of latent trait models to the study of intelligence. Nontechnical introductions to both unidimensional and multidimensional latent trait models are given, and possible research applications are considered. Latent trait models are shown to resolve several measurement problems in studies of intellectual change, including ability modification studies and life-span development studies. Furthermore, under certain conditions, latent trait models are found useful for construct validation research, since they can represent an individual differences model of cognitive processing on ability test items. Multidimensional latent trait models are shown to be especially useful as processing models, because they can be used to test alternative multiple component theories of test item processing. Furthermore, multidimensional models can be used to decompose test item difficulty into component contributions and estimate individual differences in processing abilities.     

The stability of disruptive childhood behaviors

The stability of child conduct and oppositional defiant behaviors during the period from 7 to 15 years was studied in a birth cohort of New Zealand children. These data were analyzed using two methods. In the first method the observed state to state changes in childhood behavioral tendencies were analyzed using empirical transition matrices. These results suggested that children classified as cases showed high rates of symptom remission, with approximately 50% of cases being classified as noncases 2 years later. In the second approach the data were analyzed using a latent Markov model which took account of errors of measurement in the classification of children. This analysis suggested the presence of strong continuities in childhood problem behaviors, with only 14% of children showing remission of behavioral problems within a 2-year period. The differences in the estimates yielded by the empirical transition matrices and the latent analyses were explained by the fact that there were relatively high probabilities that children who were cases were misclassified as a result of measurement errors.This research was funded by grants from the Health Research Council of New Zealand, the National Child Health Research Foundation, and the Canterbury Medical Research Foundation.     

Assessing the Within‐Trial Treatment Integrity of Discrete‐Trial Teaching Programs Using Sequential Analysis

Discrete‐trial teaching is a strategy frequently used to teach functional skills to individuals with developmental and intellectual disabilities. Research has shown that the within‐trial components of the procedure should be administered with ≥90% treatment integrity to facilitate optimal learning. Usually within‐trial treatment integrity is measured using whole‐session methods such as percentage of trials correctly administered. This study demonstrated one‐step Markov transition matrices as a method of assessing within‐trial treatment integrity. All components of discrete trials were coded and time‐stamped from video recordings of therapist–learner dyads in their typical setting (home or school). Several types of within‐trial treatment integrity errors were identified using the Markov transition matrices, error sequences that could not be identified using a percentage correct analysis. Better identification of errors has the potential both to enhance treatment integrity and to gain efficiency by targeted retraining of therapists. Copyright © 2016 John Wiley & Sons, Ltd.     

Acquiescent Responding in Balanced Multidimensional Scales and Exploratory Factor Analysis

Personality tests often consist of a set of dichotomous or Likert items. These response formats are known to be susceptible to an agreeing-response bias called acquiescence. The common assumption in balanced scales is that the sum of appropriately reversed responses should be reasonably free of acquiescence. However, inter-item correlation (or covariance) matrices can still be affected by the presence of variance due to acquiescence. To analyse these correlation matrices, we propose a method that is based on an unrestricted factor analysis and can be applied to multidimensional scales. This method obtains a factor solution in which acquiescence response variance is isolated in an independent factor. It is therefore possible, without the potentially confounding effect of acquiescence, to: (a) examine the dominant factors related to content latent variables; and (b) estimate participants’ factor scores on content latent variables. This method, which is illustrated by two empirical data examples, has proved to be useful for improving the simplicity of the factor structure. This research was partially supported by a grant from the Spanish Ministry of Science and Technology (SEJ2005-09170-C04-04/PSIC), and a grant from the Catalan Ministry of Universities, the Research and Information Society (2005SGR00017). The authors are obliged to the team of reviewers for helpful comments on an earlier version of this paper.     

On the validity of Raven’s matrices test: Does spatial ability contribute to performance?

The paper reports on an investigation of the convergent and discriminant validity of Raven’s matrices in considering reasoning and spatial abilities. Raven’s advanced progressive matrices (APM) and four scales representing reasoning, visualization, mental rotation and closure were applied to a sample of N = 280 university students. The data were investigated by means of structural equation modeling. The results demonstrated convergent validity of APM with respect to reasoning. In contrast, in investigating discriminant validity substantial correlations of moderate size between APM and the scales representing spatial abilities were observed. However, the investigation of the structure of the prediction of APM revealed that the scales representing spatial abilities did not improve the prediction based on the reasoning scale alone at the latent level. Consequently, it is suggested that Raven’s matrices show good convergent validity and slightly impaired discriminant validity.