Multidimensional latent Markov models in a developmental study of inhibitory control and attentional flexibility in early childhood

# Multidimensional latent Markov models in a developmental study of inhibitory control and attentional flexibility in early childhood

Francesco Bartolucci111Department of Economics, Finance and Statistics, University of Perugia, Via A. Pascoli 20, 06123 Perugia, Italy.   and Ivonne L. Solis-Trapala222Division of Medicine, Faraday Building, Lancaster University, Lancaster, LA1 4YB, UK.
###### Abstract

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.

Keywords: dimensionality assessment; executive function; item response theory; latent Markov model; Rasch model; two-parameter logistic parameterisation

## 1 Introduction

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, donohoe06 and kimberg00 provide an instance where two studies yield contradicting conclusions regarding the separability of the psychological constructs inhibitory control and working memory. Whilst donohoe06 ’s findings suggest that inhibitory control is a construct that overlaps with, but is separate to working memory, kimberg00 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 (rasch61), also known as the one-parameter (1PL) logistic model, or the two-parameter logistic (2PL) model (birn68). This problem has been debated since long time in the statistical and psychometric literature. Relevant contributions in this sense are those of martin:1973, van:simp:1982; van:two:1982, glas:1995, Verh:2001, Chri:Bjor:Krei:Pete:test:2002 and, more recently, bart:07. 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. bart:07 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 bart:07, 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, learning-through-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 wig73, which may be seen as an extension for longitudinal data of the latent class model laz-hen68 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 lan-pol02 and bar06; for a description with an IRT perspective see bart:08. In our formulation: (i) the response variables are conditionally independent given a sequence of latent variables that follows a first-order Markov chain which is partially homogeneous; (ii) each state of the chain is associated to a certain level of each latent trait measured by the test items; (iii) the distribution of each response variable depends on these latent states through the same 2PL parameterisation adopted by bart:07. Evolution of the subjects with respect to these latent traits depends on the transition matrices of the model and, on the basis of these matrices, we can test the hypothesis that the subjects always remain in the same state. Obviously, by exploiting the proposed model, we can also perform an LR test for the hypothesis of unidimensionality against that of multidimensionality. In contrast to more standard approaches, this test takes into account the longitudinal structure of the data. We also allow for the inclusion of covariates affecting the probability to belong to each latent state by a parameterisation similar to that of ver-al99.

For the maximum likelihood (ML) estimation of the LM model, we illustrate an EM algorithm (demp-al77) which closely recalls that illustrated by bar06. At the E-step, the algorithm exploits certain recursions developed within the hidden Markov literature (mac-zuc97). We also deal with the asymptotic distribution of the LR statistic for testing hypotheses on the transition probabilities. Note that, under hypotheses of this type, the parameters may be on the boundary of the parameter space and, therefore, an inferential problem under non-standard conditions arises (sel-lia87; sil-sen04). However, as shown by bar06, this LR test statistic has an asymptotic chi-bar-squared distribution under the null hypothesis.

The remainder of this paper is organised as follows. In the next Section we describe the psychological study that motivates this paper. In Section 3 we illustrate the main assumptions of the proposed multidimensional LM model and in Section 4 we illustrate likelihood inference on this model, including testing hypotheses on the dimension of the model and on the transition matrices of the latent process. Results of the application of our modelling approach to the study on executive function are presented in Section 5. We close with a final discussion in Section 6.

## 2 A longitudinal study on executive function

The study conducted by shi04 aims to investigate theoretical questions regarding the relationships between the components of the cognitive construct executive function in young children. The study comprises data collected from the administration of a battery of tests to 115 children during a single testing session. This session was subsequently replicated over two 6-month periods when it was believed that key changes in child cognition might have occurred. At the first period of testing the participants’ age range was 34-55 months. In this paper we restrict our attention to two components of executive function, namely inhibitory control and attentional flexibility. These are two abstract concepts that define two closely related psychological constructs. We address two methodological issues: (i) we investigate the nature of the interrelationship between the two constructs and (ii) we assess developmental trends in task performance and scalability of the tasks at various ages.

The response data collected for each participant consist of a sequence of correlated binary outcomes with each component indicating a success or failure for each trial on four tasks administered at each of three time periods. Inhibitory control was measured by the day/night and the abstract pattern tasks developed by gerstadt94; each of these tasks was administered in blocks of 16 trials. Similarly, attentional flexibility was measured by two versions of the Dimensional change card-sort (DCCS) tasks: the DCCS face-up and the DCCS face-down tests (zelazoetal96). Each of these tests was administered in blocks of 6 trials. Participants were randomly allocated to one of two testing orderings. Half of the group performed the tasks in the following order: day/night, DCCS face-down, abstract pattern, DCCS face-up, whereas the other half of the group followed the order: abstract pattern, DCCS face-up, day/night, DCCS face-down. Thus the length of the response sequence for each participant is 44 (16+6+16+6) at each time period. The maximum number of observations of each subject is then . For an illustration of this experiment see Figure 1.

Note that the abstract pattern task is expected by design to demand less inhibitory control skill than the day/night task. Similarly, the face-up version of the DCCS task has lower working memory demands than the face-down version. The rationale of the experimental design was to administer the harder tasks previous to their easier versions to half of the participants, and the reverse order to the other half in order to control for potential testing ordering effects.

## 3 The multidimensional latent Markov model

### 3.1 Basic assumptions

The proposed model is based on the assumption that the tasks under consideration measure different types of latent traits or, more specifically, abilities; then, we denote by , , the subsets of containing indices of the tasks measuring ability of type . In our application, for instance, it is reasonable to set , with and , so that day/night and abstract pattern tasks measure the ability of type 1 and DCCS tasks measures the ability of type 2 in both face-down and face-up versions. Consequently to each subject is associated an -dimensional vector of abilities which is allowed to depend on time and is denoted by . We also denote by the probability that this subject performs correctly a task of type at occasion .

Following bart:07, we adopt a 2PL parameterisation (birn68) for this conditional probability, so that

 logit[λj(\boldmathθit)]=γj(∑dδjdθitd−βj),j=1,…,J. (1)

where is the discriminant index for the th item and is a dummy variable equal to 1 if and to 0 otherwise. With this notation, and taking into account that each subject faces a specific sequence of tasks, the probability that subject responds to the -th task administered is

 p(yit|\boldmathθit)=λjit(\boldmathθit)yit[1−λjit(\boldmathθit)]1−yit. (2)

Obviously, the simpler one-parameter (1PL) logistic parameterisation (rasch61) could also be used. In this case, we have

 logit[λj(\boldmathθit)]=∑dδjdθitd−βj,j=1,…,J. (3)

This parameterisation is less flexible than the above one since it assumes that all taks have the same discriminant power, i.e. , . On the other hand, the resulting model is simpler to estimate.

In order to model the dynamics of each ability across , we assume that for each subject the latent process follows a first-order Markov chain with state space , where is the number of latent states, with initial probabilities , , and transition probabilities , . Since we consider the abilities as the only explanatory variables of the probability of performing correctly an item, as in the latent Markov model of wig73 we assume that the response variables in are conditional independent given . The resulting model is then an extension of the latent Markov Rasch model described by bart:08 based on a multidimensional 2PL parameterisation for the conditional distribution of the response variables given the latent process and allowing for different sequences of items between subjects. Therefore, experimental design aspects such as testing order can also be taken into consideration in the statistical analysis of the data. Moreover, by assuming the existence of a latent Markov chain, we allow a subject to move between latent classes in a way that depends on the transition probabilities and hence we also generalise the multidimensional IRT model of bart:07 which implicitly assumes that every subject maintains the same level of each ability across time occasions.

A final point concerns how to model in a parsimonious way the initial and transition probabilities of the latent process taking into account the difference between subjects in terms of individual covariates and considering the experimental design. For this aim, and similarly to ver-al99, we adopt the following logistic parameterisation for the initial probabilities:

 logπicπi1=\boldmathx′i\boldmathϕc,c=2,…,k,i=1,…,n, (4)

where is a vector of covariates measured at the first occasion. In our application, we allow these initial probabilities to depend on the age of the subject at beginning of the study, indicated by for subject , and then we let .

For what concerns the transition probabilities of the latent process, we specify those transition matrices that encompass the different transitional stages of substantive interest. For instance, for our application, we use a first type of transition matrix within the sequence of day/night tasks and a second type within the second of abstract pattern tasks and so on; see Section 6 for a detailed description. Note that we could also allow the transition probabilities to depend on the individual covariates through a parameterisation of the probabilities similar to that in (4). This however would complicate the approach and we do not pursue this further here.

Submodels may also be considered. The most interesting is obviously that of unidimensionality that may be formulated by requiring that , with denoting a column vector of ones, so that the elements of are equal to each other. The hypothesis that there is no transition between latent states at a certain occasion is also of interest. This may be formulated by requiring that the corresponding matrix of transition probabilities is diagonal; see bar06 for a detailed description of these constraints. Finally, the assumption that there is no effect of the covariates on the initial probabilities is also of interest. In our application this assumption may be formulated by letting , .

### 3.2 Manifest distribution of the response variables

In order to derive the manifest distribution of the response variables , we consider the vector , with being the state to which subject belongs at occasion , with , . In practice, , for , if and only if .

The assumption of conditional independence between the elements of given the latent process implies that

 p(\boldmathyi|\boldmathci)=∏jλyititcit(1−λitcit)1−yit,

where denotes a realisation of , denotes a realisation of whose elements are indicated by and

 λitc=p(Yit=yit|\boldmathθit=\boldmathξc), (5)

which is defined on the basis of (2). Consequently, the manifest distribution of may be expressed as

 p(\boldmathyi)=∑\scriptsize{\boldmathc}ip(% \boldmathyi|\boldmathci)p(\boldmathci), (6)

where the sum is extended to all the possible configurations of . Clearly, the probability depends on the initial and transition probabilities and .

Avoiding the sum in (6), the manifest probability may be efficiently computed by exploiting certain recursions known in the hidden Markov literature (mac-zuc97) and which may be efficiently implemented by using matrix notation; we refer to bar-al07 for details. Let denote the initial probability vector with element , , and let denote the transition probability matrix with elements , . For a fixed vector , also let denote the column vector with elements for . For , this vector may be computed by using the recursion

 \boldmathlit={diag(% \boldmathλi1)\boldmathπi if j=1,diag(\boldmathλit)[\boldmathΠ(t)]′\boldmathli,t−1 otherwise, (7)

where denotes the column vector with elements , . Once (7) has been computed for , we obtain as the sum of the elements of .

## 4 Likelihood inference on the model parameters

Let denote the vector of all model parameters, i.e. the item parameters and , the support points of the latent distribution, the parameters for the initial probabilities of the latent process and the transition probabilities .

As usual, we assume that the subjects in the sample have response patterns independent from one another, so that the log-likelihood of the LMR model may be expressed as

 ℓ(\boldmathη)=∑ilog[p(\boldmathyi)],

where is computed as a function of by using recursion (7). We can maximise by means of the EM algorithm (demp-al77) implemented as in bar-al07 and which we briefly illustrate in the following.

### 4.1 Log-likelihood maximisation

First of all, we need to consider the so-called complete data log-likelihood that may be expressed as

 ℓ∗(\boldmathη)=∑i[∑cwi1clog(πic)+∑Tit=2∑c∑dzicdlog(π(t)icd)++∑Tit=1∑cwitc{yitlog(λitc)+(1−yit)log(1−λitc)}], (8)

where is a dummy variable equal to 1 if subject is in latent state at occasion (i.e. ), is a dummy variable equal to 1 if subject moves from the -th to the -th latent state at occasion (i.e. and i.e. ) and the probabilities in are defined in (5).

Obviously, the dummy variables in are not known and then this log-likelihood is exploited within the EM algorithm to maximize the incomplete data log-likelihood by alternating the following two steps until convergence:

• E-step: compute the conditional expected value of the dummy variables and given the observed data , the covariates and the current estimate of ;

• M-step: update the estimate of by maximizing the expected value of the complete log-likelihood ; this is defined as in (8) with each dummy variable involved in this expression substituted by the corresponding expected value computed at the E-step.

The E-step may be performed by using certain recursions similar to (7). In order to udpate the parameters , and , the M-step requires to run an algorithm to maximize the weighted likelihood of a logistic model, which is easily available. A similar algorithm is necessary to update the parameters for the initial probabilities, whereas an explicit expression is available to update the transition probabilities . For a detailed description on how to implement these steps, see bar06, bar-al07 and bart:08.

The algorithm requires to be initialized by choosing suitable starting values for the parameters in . Trying different sets of starting values is useful to detect multimodality of the likelihood, which often arises for latent variable models. These starting values may be generated by a random rule and then we take, as ML estimate of the parameters, the value of which at convergence gives the highest value of . This estimate is denoted by .

### 4.2 Model seleciton and testing hypotheses on the parameters

Obviously, analysing a dataset through the model described above requires to choose the number of latent states . Following the current literature on latent variable models, and the related literature on finite mixture models (see mcla:peel:00), we suggest to rely on the Bayesian Information Criterion (BIC) which was proposed by schw78. According to this criterion, we choose the value of corresponding to the minimum of the index

 BICk=−2^ℓk+gklog(n), (9)

where denotes the maximised log-likelihood and is the corresponding number of parameters.

Note that the penalization term in (9) only takes the sample size into account. In order to take into account the overall number of observations, corresponding to , we can alternatively use the criterion BIC based on the minimisation of the index

 BIC∗k=−2^ℓk+gklog(∑iTi).

Once the number of latent states has been chosen, a hypothesis on the parameters may be tested by the likelihood ratio (LR) statistic

 D=−2[ℓ(^\boldmathη0)−ℓ(^\boldmathη)],

where is the ML estimate of under ; this estimate may also be computed by using the EM algorithm illustrated above.

When standard regularity conditions hold, the asymptotic null distribution of the statistic is a chi-squared distribution with a number of degrees of freedom equal to the number of independent constraints used to express . In particular, through this procedure we may test hypotheses on the transition probabilities of the latent process. As already mentioned, standard asymptotic results on the distribution of LR test statistic are not ensured to hold in this case. This happens, for instance, for the hypothesis that a transition matrix is diagonal and thus transitions between latent states is not possible. However, by using certain results on constrained statistical inference (sel-lia87; sil-sen04), it was shown by bar06 that the null asymptotic distribution of is of chi-bar-squared type (shap88; sil-sen04). This is a mixture of standard chi-squared distribution with weights that may be computed by a simple rule.

## 5 Results

In this section we investigate models that describe the developmental and dimensional aspects of the data collected to measure inhibitory control and attentional flexibility, described in Section 2. The tasks that measure the above abilities correspond to types of item, namely: (i) day/night, (ii) abstract pattern, (iii) DCCS face-down and (iv) DCCS face-up. As we described in detail in Section 3, our modelling approach relies on a latent Markov process. The parameters of the distribution of this process are specified as follows:

• the initial probabilities of the latent process depend on age according to (4);

• there are 8 different transition matrices of the latent process that model participants’ ability transition during the period of experimentation. The experimental stages of substantive interest and their corresponding transition probability matrices are depicted in Figure 1. Each testing session in the study involves two transition matrices. The first matrix measures changes in performance within the inhibitory control sequence of trials and transition to the attentional flexibility task, whereas the second one measures transitions within the attentional flexibility sequence of trials. Thus, changes in performance during the two types of testing sessions yield four transition matrices, and for the experimental design where harder tasks precede easier tasks, and and for the alternative experimental design. Transition matrices and represent changes in abilities between sequences over a week period of time, and similarly, matrices and model changes over six-month time periods for the two experimental designs.

To complete our model specification we initially assume that the probability of responding correctly to each item is unconstrained, so that we have a specific parameter for the probability of responding correctly to each item of type , , given each possible latent state , . Subsequently we test whether the 2PL parameterisation based on (1) is reasonable.

We next turn to the problem of choosing an LM model that provides an appropriate representation of the dependence structure. We first address the choice of the number of latent classes. We consider four models with 1 to 4 latent classes and select the fitted model with smallest values of the Bayesian information criteria BIC and BIC* where, as shown earlier, BIC is penalised by a term equal to the sample size 115 and BIC* is penalised by the total number of single trials equal to 12798. Table 1 displays the values of the log-likelihood function evaluated at the maximum likelihood estimate, the BIC and BIC* indices for model comparison.

On the basis of the results in Table 1, we select a model with latent states. This model may be simplified by testing hypotheses on the equality between some of the transition matrices deriving from the experimental stages depicted in Figure 1. For instance, it is of interest to test whether patterns of participants’ performance within each of the two tasks of inhibitory control are comparable. Formally, this involves testing the hypothesis of equality between the transition matrices and . Similarly, comparison between matrices and and matrices and is important because rejection of the hypothesis of equality of these matrices would suggest the presence of task order effects.

Table 2 summarises the results of testing the above and other relevant hypotheses on the transition matrices. We highlight in boldface those constrained models for which the BIC and BIC indices are minimum among fitted models with comparable number of parameters.

Based on the results presented in Table 2, we adopt a constrained model where the matrices and are equal and so are matrices and , and and . Further reduction is not plausible as can be seen from the fact that constraining the transition matrices to be identity matrices yields larger values of BIC and BIC. Hence, we can represent the dependence structure through four transition matrices. The first two transition matrices comprise patterns of performance within the inhibitory control and within the attention flexibility items respectively. The third transition matrix yields information on changes in performance after a week’s time; the fourth transition matrix describes developmental changes that take place in 6 months’ time.

Furthermore, we tested the dependence of the initial probabilities on participant’s age. We observe that the value of the BIC increases from 7334.2 to 7446.4 when the dependence is removed. Thus we conclude that age has a significant effect on the initial probabilities.

Another important aspect in the search of an appropriate latent Markov model concerns the parameterisation of the conditional probabilities of success at performing the task given the latent state. We explore the 1PL and 2PL parameterisations, contrasting between the unidimensional and the bidimensional cases; see equations (1) and (3). We adopt a bidimensional 2PL model which yields the smallest values of BIC compared with its competing models, as can be seen from Table 3. The first ability underlies the inhibitory control tasks, whereas the second ability is the basis of the attentional flexibility tasks. This completes our model choice.

In summary, we adopt a latent Markov model with three latent variables. The initial probabilities of the latent process depend on participant’s age and there are four transition probability matrices whose interpretation was discussed earlier. For this model, the conditional probability of responding correctly to each item given the latent process is parameterised as in a 2PL bidimensional model bart:07. Here, the first dimension represents inhibitory control and the second one attentional flexibility. Under the selected model the estimated ability parameters are summarised in Table 4.

The latent states are ordered to reflect lowest to highest ability for both dimensions. Note that the estimated ability parameters corresponding to attentional flexibility are large in absolute value due to the fact that a large number of participants tended to score either zero or one during the full 6-item sequences. As noted by the experimenter, this is not a particular characteristic of this data set; but other studies (e.g. zelazoetal96) report a similar phenomenon.

Table 5 displays estimates of the discriminant and difficulty parameters associated with each item. These estimates reveal that the abstract pattern task is considerably easier and less discriminant than the day/night task. On the other hand, the two DCCS tasks are comparable in level of difficultly, but the DCCS face-up task discriminates better than the face-down version of this task. These results confirm the experimenter’s hypotheses (see shi04), namely that participants would perform better at the abstract pattern and the DCCS face-up tasks than at their counterparts. Furthermore, in the case of the inhibitory control tasks, the abstract pattern task was regarded as a control for the day/night task, in the sense that it is expected to make less inhibitory demands; therefore, our finding that the day/night task is a better discriminator is reassuring.

Based on the estimates reported in Tables 4 and 5 we can calculate the conditional probabilities of success to respond to an item for a given participant given his/her latent state; see equation (1). These probabilities are presented in Table 6.

A noteworthy fact is that the participants who fall in the latent state 1 have a fifty percent probability of success in the DCCS tasks but low probabilities of success in the day/night task. The interpretation of the ability on the DCCS tasks for such participants must be made with care, since the experimenter has reported that she believes that some participants adopted a strategy, i.e. alternating the card placement regardless of the game’s rules, to perform the card tasks that led to a medium size number of successes in the 6-trial sequences. Importantly, our analysis strongly supports the hypothesis that there are two separable abilities, namely inhibitory control and attentional flexibility, underlying performance at the items administered to participants. We next address developmental questions regarding these abilities by exploring the transition probabilities of the assumed latent process.

We examine the effect of age on performance by modelling the initial probabilities of the latent process in a logistic regression with age as explanatory variable. The estimates of the regression coefficients of age, corresponding to the logistic curves of the initial probabilities for the second and third latent states, are 0.111 and 0.361 respectively. That is, the model classifies older participants into states of higher performance. Averaged over all the subjects in the sample, the three latent states have initial probabilities equal to 0.261, 0.360 and 0.379. The second and third state have comparable dimension, whereas the first is smaller and contains around one fourth of the sample.

Table 7 reports the estimated transition probabilities matrices. Recall that earlier in this section we identified four transition matrices to represent the change patterns on participants’ performance. The matrix denoted by “within IC items” in Table 7 encompasses changes in performance within the inhibitory control items and transitions to the attention flexibility items. It corresponds to . The “within AF items” represents performance changes within the attention flexibility items only and corresponds to . The remaining two transition matrices represent transition of states after a week and six-month periods respectively, that is and .

The within IC items transition matrix show evidence of a tiring effect as reflected by some tendency of participants to move from higher to lower states within the sequences of inhibitory control trials. In contrast, the within AF items transition matrix primarily indicates the presence of learning effects as participants tend to move from the first to the higher states. Similarly, there is a learning effect during the transition after a week’s time. Finally, the probability matrix concerning transitions over a six-month period shows strong developmental effects for participants in states 1 and 2. There are also some evidence of participants moving back from the second and third states to lower states.

A final point concerns the computation effort involved in the analysis based on the LM model described in this paper. We found that, although estimation is based on the EM algorithm which may require a large number of iterations, fitting the model on the data described above is rather fast even with a large number of latent states. This is because we use an efficient implementation based on recursions taken from the hidden-Markov literature. Implementing these recursions is rather easy in mathematical packages such as Matlab. On the other hand, we found that for the data considered here, most of the fitted models have a multimodal likelihood. It is then crucial to try different starting values for the EM algorithm as described at the end of Section 4.1. We make our Matlab implementation of the EM algorithm, and the related routine to choose the starting values, available to the readers upon request.

## 6 Discussion

We have developed an approach based on a multidimensional latent Markov model suitable for the analysis of long binary response sequences that measure two or more latent traits.

For each subject in the sample, our approach relies on Markov chains with transition probabilities that are of substantive interest, in order to measure dynamic changes in the responses during repeated trials and over fixed periods of time. Hence the applicability of the method lies on studies where the aim is at studying developmental or detrimental changes over time in an intuitive way. Importantly, we show how to test the dimensionality of the problem by developing valid LR tests.

The modelling approach has been illustrated through the analysis of data from a developmental study. Throughout the statistical analysis we emphasized the importance of taking into consideration experimental aspects of the study, for instance to account for potential ordering effects. Our main finding was evidence that supports that the tasks measure two distinct psychological constructs, namely inhibitory control and attentional flexibility. An obvious extension of the applicability of our method would be to investigate hypotheses on the relationship between other executive skills such as working memory and planning.

In addition, we were able to demonstrate developmental aspects of the two abilities. In particular that these abilities develop at an early age, as the covariate age played an important role at performance. The dynamics within sequences were different among the two types of tasks considered. The presence of tiring effects was clear within the inhibitory control sequences. In contrast, participants where either able or unable to perform the attentional flexibility task. It is plausible that a third group showed a spurious medium probability of success at this task. Further investigation on the nature of this task is warranted.

A limitation of our approach is the assumption of a first order Markov chain for the latent process. However, this assumption seems to be suitable for the data analysed in this article. Indeed, a quick inspection of time dependence through the assumption of a continuous latent process favours a short-range dependence. Note also, that we do not require to make any distributional assumptions on the latent process as opposed to the conventional Gaussian assumption made for continuous processes.

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