Showing papers by "Francesco Bartolucci published in 2011"

Assessment of School Performance through a Multilevel Latent Markov Rasch Model.

Abstract: An extension of the latent Markov Rasch model is described for the analysis of binary longitudinal data with covariates when subjects are collected in clusters, such as students clustered in classes. For each subject, a latent process is used to represent the characteristic of interest (e.g., ability) conditional on the effect of the cluster to which he or she belongs. The latter effect is modeled by a discrete latent variable associated to each cluster. For the maximum likelihood estimation of the model parameters, an Expectation-Maximization algorithm is outlined. Through the analysis of a data set collected in the Lombardy Region (Italy), it is shown how the proposed model may be used for assessing the development of cognitive achievement. The data set is based on test scores in mathematics observed over 3 years on middle school students attending public and non-state schools.Manuscript received March 20, 2009 Revision received July 2, 2010 Accepted July 10, 2010

Modeling Partial Compliance Through Copulas in a Principal Stratification Framework

Abstract: Within the principal stratification framework for causal inference, modeling partial compliance is challenging because the continuous nature of the principal strata raises subtle specification issues. In this context, we propose an approach based on the assumption that the joint distribution of the degree of compliance to the treatment and the degree of compliance to the control follows a Plackett copula, so that their association is modeled in a flexible way through a single parameter. Moreover, given the two compliances, the distribution of the outcomes is parameterized in a flexible way through a regression model which may include interaction and quadratic terms and may also be heteroscedastic. In order to estimate the parameters of the resulting model, and then the causal effect of the treatment, we adopt a maximum likelihood approach via the EM algorithm. In applying this approach, the marginal distributions of the two compliances are estimated by their empirical distribution functions, so that no co...

Impact Evaluation of Job Training Programs by a Latent Variable Model

Abstract: We introduce a model for categorical panel data which is tailored to the dynamic evaluation of the impact of job training programs. The model may be seen as an extension of the dynamic logit model in which unobserved heterogeneity between subjects is taken into account by the introduction of a discrete latent variable. For the estimation of the model parameters we use an EM algorithm and we compute standard errors on the basis of the numerical derivative of the score vector of the complete data log-likelihood. The approach is illustrated through the analysis of a dataset containing the work histories of the employees of the private firms of the province of Milan between 2003 and 2005, some of whom attended job training programs supported by the European Social Fund.

Mixture latent autoregressive models for longitudinal data

Abstract: Many relevant statistical and econometric models for the analysis of longitudinal data include a latent process to account for the unobserved heterogeneity between subjects in a dynamic fashion. Such a process may be continuous (typically an AR(1)) or discrete (typically a Markov chain). In this paper, we propose a model for longitudinal data which is based on a mixture of AR(1) processes with different means and correlation coefficients, but with equal variances. This model belongs to the class of models based on a continuous latent process, and then it has a natural interpretation in many contexts of application, but it is more flexible than other models in this class, reaching a goodness-of-fit similar to that of a discrete latent process model, with a reduced number of parameters. We show how to perform maximum likelihood estimation of the proposed model by the joint use of an Expectation-Maximisation algorithm and a Newton-Raphson algorithm, implemented by means of recursions developed in the hidden Markov literature. We also introduce a simple method to obtain standard errors for the parameter estimates and a criterion to choose the number of mixture components. The proposed approach is illustrated by an application to a longitudinal dataset, coming from the Health and Retirement Study, about self-evaluation of the health status by a sample of subjects. In this application, the response variable is ordinal and time-constant and time-varying individual covariates are available.

Bayesian inference for a class of latent Markov models for categorical longitudinal data

Abstract: We propose a Bayesian inference approach for a class of latent Markov models. These models are widely used for the analysis of longitudinal categorical data, when the interest is in studying the evolution of an individual unobservable characteristic. We consider, in particular, the basic latent Markov, which does not account for individual covariates, and its version that includes such covariates in the measurement model. The proposed inferential approach is based on a system of priors formulated on a transformation of the initial and transition probabilities of the latent Markov chain. This system of priors is equivalent to one based on Dirichlet distributions. In order to draw samples from the joint posterior distribution of the parameters and the number of latent states, we implement a reversible jump algorithm which alternates moves of Metropolis-Hastings type with moves of split/combine and birth/death types. The proposed approach is illustrated through two applications based on longitudinal datasets.

An alternative to the Baum-Welch recursions for hidden Markov models

Abstract: We develop a recursion for hidden Markov model of any order h, which allows us to obtain the posterior distribution of the latent state at every occasion, given the previous h states and the observed data. With respect to the well-known Baum-Welch recursions, the proposed recursion has the advantage of being more direct to use and, in particular, of not requiring dummy renormalizations to avoid numerical problems. We also show how this recursion may be expressed in matrix notation, so as to allow for an efficient implementation, and how it may be used to obtain the manifest distribution of the observed data and for parameter estimation within the Expectation-Maximization algorithm. The approach is illustrated by an application to financial data which is focused on the study of the dynamics of the volatility level of log-returns.

An alternative to the Baum-Welch recursions for hidden Markov models

Abstract: We develop a recursion for hidden Markov model of any order h, which allows us to obtain the posterior distribution of the latent state at every occasion, given the previous h states and the observed data. With respect to the well-known Baum-Welch recursions, the proposed recursion has the advantage of being more direct to use and, in particular, of not requiring dummy renormalizations to avoid numerical problems. We also show how this recursion may be expressed in matrix notation, so as to allow for an efficient implementation, and how it may be used to obtain the manifest distribution of the observed data and for parameter estimation within the Expectation-Maximization algorithm. The approach is illustrated by an application to nancial data which is focused on the study of the dynamics of the volatility level of log-returns.