We demonstrate the use of a multidimensional extension of the latent Markov model to analyse data from studies with correlated 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 and attentional flexibility 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 tests outcomes given the latent process depends on these abilities through a multidimensional two-parameter logistic parameterisation. We outline an EM algorithm to conduct 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 inhibitory control and attentional flexibility 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.
Deep Dive into 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 correlated 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 and attentional flexibility 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 tests outcomes given the latent process depends on these abilities through a multidimensional two-parameter logistic parameterisation. We outline an EM algorithm to conduct 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
A fundamental scientific challenge in neuropsychology and developmental psychology is the search for evidence that two or more postulated mechanisms, thought to underlie the performance of participants on a set of tests, are separable. For example, Donohoe et al. (2006) and Kimberg and Farah (2000) provide an instance where two studies yield contradicting conclusions regarding the separability of the psychological constructs inhibitory control and working memory. Whilst Donohoe et al. 's findings suggest that inhibitory control is a construct that overlaps with, but is separate to working memory, Kimberg and Farah argue against the theory that there is an inhibitory mechanism above and beyond working memory.
The characterisation of the latent trait underlying the performance of participants on a set of tests in terms of a single unidimensional latent variable would imply that all items measure a common construct. This model can be contested by specifying a multivariate latent trait which would postulate that the mechanisms underlying performance are distinct theoretical concepts.
Key changes in cognitive development take place during childhood. In this paper we assess the separability of the executive functions inhibitory control and attentional flexibility in the context of a study involving young children. Longitudinal studies that aim to study developmental aspects of cognition typically involve the administration of several tasks, to participants, where each task is presented in blocks of trials during a single session. This session is subsequently replicated over fixed periods of time, for example every 6 months, when it is believed that key changes in child cognition might have occurred. A common protocol approach involves the recording of participants’ success or failure on every single trial performed during the length of the study. This yields a sequence of correlated binary responses for each participant.
In the presence of dichotomously-scored items, the problem above may be translated into that of assessing the dimensionality of an Item Response Theory (IRT) model, such as the Rasch model (Rasch, 1961), also known as the one-parameter (1PL) logistic model, or the twoparameter logistic (2PL) model (Birnbaum, 1968). This problem has been debated since long time in the statistical and psychometric literature. Relevant contributions in this sense are those of Martin-Löf (1973), van den Wollenberg (1982a,b), Glas and Verhelst (1995), Verhelst (2001), Christensen et al. (2002) and, more recently, Bartolucci (2007). In the latter paper, in particular, a class of multidimensional IRT model is proposed which is based on a 2PL parameterisation of the probability of responding correctly to each item and on the assumption that the latent traits follow a multivariate discrete distribution with an arbitrary number of support points. Bartolucci (2007) showed how to test for undimensionality by a Wald statistic, or equivalently a likelihood ratio (LR) statistic between a bidimensional and a unidimensional model, which are formulated on the basis of the above assumptions. He also showed how this procedure may be extended to assess the number of distinct latent traits measured by the test items and, consequently, to cluster these items in homogeneous groups.
The above approaches, and in particular that of Bartolucci (2007), can be directly applied to assess the dimensionality of the psychological response process described above, provided that responses are collected at a single occasion and that the ability of each subject in responding correctly to each item remain constant during the test. This excludes the cases in which the same set of items is repeatedly administered to the same subjects during a single testing session, which is subsequently replicated at a certain number of occasions separated by a suitable interval of time. This scheme is very common in psychological applications as the one which motivates this paper. The particular feature of this scheme is that a subject may evolve in his/her latent characteristics between occasions and this is not taken into account in the multidimensional IRT models mentioned above. This evolution in time is typically due to tiring effects, learningthrough-training and other developmental phenomena.
In this paper, we propose a multidimensional model for the analysis of data deriving from design protocols that use a repeated measurement scheme at each occasion of a longitudinal study. The basic tool is the latent Markov (LM) model of Wiggins (1973), which may be seen as an extension for longitudinal data of the latent class model Lazarsfeld and Henry (1968) in which each subject is allowed to move between latent classes during the period of observation. For a detailed description of the LM model see Langeheine and van de Pol (2002) and Bartolucci (2006); for a description with an IRT perspective see Bartolucci et al. (2008).
In our formulation: (i) the response v
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