Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-08T08:33:10.588Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 January 2025

Francesco Bartolucci  and
Ivonne L. Solis-Trapala
Francesco Bartolucci*
Affiliation:
University of Perugia
Ivonne L. Solis-Trapala
Affiliation:
Lancaster University
*
Requests for reprints should be sent to Francesco Bartolucci, Department of Economics, Finance, and Statistics, University of Perugia, Via A. Pascoli 20, 06123 Perugia, Italy. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

dimensionality assessmentexecutive functionitem response theorylatent Markov modelRasch modeltwo-parameter logistic parameterisation
Type
Original Paper
Information
Psychometrika , Volume 75 , Issue 4 , December 2010 , pp. 725 - 743
Copyright
Copyright © 2010 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartolucci, F. (2006). Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities. Journal of the Royal Statistical Society, Series B, 68, 155–178.CrossRefGoogle Scholar
Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72, 141–157.CrossRefGoogle Scholar
Bartolucci, F., Pennoni, F., Francis, B. (2007). A latent Markov model for detecting patterns of criminal activity. Journal of the Royal Statistical Society, Series A, 170, 115–132.CrossRefGoogle Scholar
Bartolucci, F., Pennoni, F., Lupparelli, M. (2008). Likelihood inference for the latent Markov Rasch model. In Huber, C., Limnios, N., Mesbah, M., Nikulin, M. (Eds.), Mathematical methods for survival analysis, reliability and quality of life (pp. 239–254). London: Wiley.Google Scholar
Baum, L.E., Petrie, T., Soules, G., Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Annals of Mathematical Statistics, 41, 164–171.CrossRefGoogle Scholar
Berchtold, A. (2004). Optimization of mixture models: Comparison of different strategies. Computational statistics, 19, 385–406.CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In Lord, F.M., Novick, M.R. (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading: Addison-Wesley.Google Scholar
Boucheron, S., Gassiat, E. (2007). An information-theoretic perspective on order estimation. In Cappé, O., Moulines, E., Rydén, T. (Eds.), Inference in hidden Markov models (pp. 565–602). New York: Springer.Google Scholar
Christensen, K.B., Bjorner, J.B., Kreiner, S., Petersen, J.H. (2002). Testing unidimensionality in polytomous Rasch models. Psychometrika, 67(4), 563–574.CrossRefGoogle Scholar
Connell, A., Frye, A. (2006). Growth mixture modelling in developmental psychology: Overview and demonstration of heterogeneity in developmental trajectories of adolescent antisocial behaviour. Infant and Child Development, 15, 609–621.CrossRefGoogle Scholar
Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.CrossRefGoogle Scholar
Donohoe, G., Reilly, R., Clarke, S., Meredith, S., Green, B., Morris, D., Gill, M., Corvin, A., Garavan, H., Robertson, I.H. (2006). Do antisaccade deficits in schizophrenia provide evidence of a specific inhibitory function?. Journal of the International Neuropsychological Society, 12, 901–906.CrossRefGoogle ScholarPubMed
Frye, D., Zelazo, P.D., Burack, J.A. (1998). Cognitive complexity and control: I. Implications for theory of mind in typical and atypical development. Current Directions in Psychological Science, 7, 116–121.CrossRefGoogle Scholar
Gerstadt, C.L., Hong, Y.J., Diamond, A. (1994). The relationship between cognition and action: Performance of children 3.5-7 years old on a stroop-like day-night test. Cognition, 53, 129–153.CrossRefGoogle Scholar
Glas, C.A.W., Verhelst, N.D. (1995). Testing the Rasch model. In Fischer, G.H., Molenaar, I.W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 69–95). New York: Springer.CrossRefGoogle Scholar
Happaney, K., Zelazo, D. (2003). Commentaries: Inhibition as a problem in the psychology of behavior. Developmental Science, 6, 468–470.CrossRefGoogle Scholar
Juang, B., Rabiner, L. (1991). Hidden Markov models for speech recognition. Technometrics, 33, 251–272.CrossRefGoogle Scholar
Kamata, A. (2001). Item analysis by hierarchical generalized linear models. Journal of Educational Measurement, 38, 79–93.CrossRefGoogle Scholar
Keribin, C. (2000). Consistent estimation of the order of mixture models. Sankhyā, Series A, 62, 49–66.Google Scholar
Kimberg, D.Y., Farah, M.J. (2000). Is there an inhibitory module in the frontal cortex? Working memory and the mechanisms underlying cognitive control. In Monsell, S., Driver, J. (Eds.), Attention and performance XVIII: Control of cognitive processes (pp. 740–751). Cambridge: MIT Press.Google Scholar
Kirkham, N.Z., Cruess, L., Diamond, A. (2003). Helping children apply their knowledge to their behavior on a dimension-switching task. Developmental Science, 6, 449–476.CrossRefGoogle Scholar
Langeheine, R., van de Pol, F. (2002). Latent Markov chains. In Hagenaars, J.A., McCutcheon, A.L. (Eds.), Advances in latent class analysis (pp. 304–341). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Lazarsfeld, P.F., Henry, N.W. (1968). Latent structure analysis, Boston: Houghton Mifflin.Google Scholar
MacDonald, I.L., Zucchini, W. (1997). Hidden Markov and other models for discrete-valued time series, London: Chapman and Hall.Google Scholar
Martin-Löf, P. (1973). Statistiska modeller. Stockholm: Institütet för Försäkringsmatemetik och Matematisk Statistisk vid Stockholms Universitet.Google Scholar
McLachlan, G.J., Peel, D. (2000). Finite mixture models, New York: Wiley.CrossRefGoogle Scholar
Muthén, B.O. (1983). Latent variable structural equation modelling with categorical data. Journal of Econometrics, 22, 43–65.CrossRefGoogle Scholar
Muthén, B.O. (2001). Second-generation structural equation modeling with a combination of categorical and continuous latent variables: New opportunities for latent class/latent growth modeling. In Collins, L., Sayer, A. (Eds.), New methods for the analysis of change (pp. 291–322). Washington: American Psychological Association.CrossRefGoogle Scholar
Muthén, B.O., Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics, 55, 463–469.CrossRefGoogle ScholarPubMed
Nagin, D. (1999). Analyzing developmental trajectories: A semi-parametric, group-based approach. Psychological Methods, 4, 139–157.CrossRefGoogle Scholar
Nagin, D., Tremblay, R. (2001). Analyzing developmental trajectories of distinct but related behaviors: A group-based method. Psychological Methods, 6, 18–34.CrossRefGoogle ScholarPubMed
Paas, L.J., Vermunt, J.K., Bijmolt, T.H.A. (2007). Discrete-time, discrete-state latent Markov modelling for assessing and predicting household acquisitions of financial products. Journal of the Royal Statisical Society, Series A, 170, 955–974.CrossRefGoogle Scholar
Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the IV Berkeley symposium on mathematical statistics and probability (Vol. 4, pp. 321–333).Google Scholar
Schneider, W., Schumann-Hengsteler, R., Sodian, B. (2005). Young children’s cognitive development: Interrelationships among executive functioning, working memory, verbal ability, and theory of mind, Mahwah: Lawrence Erlbaum Associates.Google Scholar
Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Self, S.G., Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605–610.CrossRefGoogle Scholar
Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review, 56, 49–62.CrossRefGoogle Scholar
Shimmon, K.L. (2004). The development of executive control in young children and its relationship with mental-state understanding: A longitudinal study. Ph.D. thesis, Lancaster University, UK.Google Scholar
Silvapulle, M.J., Sen, P.K. (2004). Constrained statistical inference: Inequality, order, and shape restrictions, New York: Wiley.Google Scholar
Takane, Y., de Leeuw, J. (1987). On the relationships between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408.CrossRefGoogle Scholar
Towse, J.N., Redbond, J., Houston-Price, C.M.T., Cook, S. (2000). Understanding the dimensional change card sort perspectives from task success and failure. Cognitive Development, 15, 347–365.CrossRefGoogle Scholar
van den Wollenberg, A.L. (1982). A simple and effective method to test the dimensionality axiom of the Rasch model. Applied Psychological Measurement, 6, 83–91.CrossRefGoogle Scholar
van den Wollenberg, A.L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar
Verhelst, N.D. (2001). Testing the unidimensionality assumption of the Rasch model. Methods of Psychological Research Online, 6, 231–271.Google Scholar
Vermunt, J.K., Langeheine, R., Böckenholt, U. (1999). Discrete-time discrete-state latent Markov models with time-constant and time-varying covariates. Journal of Educational and Behavioral Statistics, 24, 179–207.CrossRefGoogle Scholar
Vermunt, J.K., Magidson, J. (2007). LGSyntax users guide: Manual for Latent Gold 4.5 and Latent Gold Choice 4.5 Syntax Module, Belmont: Statistical Innovations.Google Scholar
Viterbi, A. (1967). Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory, 13, 260–269.CrossRefGoogle Scholar
Wiggins, L.M. (1973). Panel analysis: Latent probability models for attitude and behavior processes, Amsterdam: Elsevier.Google Scholar
Zelazo, P.D., Frye, D. (1998). Cognitive complexity and control: II. The development of executive function in childhood. Current Directions in Psychological Science, 7, 121–126.CrossRefGoogle Scholar
Zelazo, P.D., Frye, D., Rapus, T. (1996). An age-related dissociation between knowing rules and using them. Cognitive Development, 11, 37–63.CrossRefGoogle Scholar