Fulvia Pennoni

Assessment of School Performance Through a Multilevel Latent Markov Rasch Model

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

A comparison of some criteria for states selection in the latent Markov model for longitudinal data

We compare different selection criteria to choose the number of latent states of a multivariate latent Markov model for longitudinal data. This model is based on an underlying Markov chain to represent the evolution of a latent characteristic of a group of individuals over time. Then, the response variables observed at different occasions are assumed to be conditionally independent given this chain. Maximum likelihood estimation of the model is carried out through an Expectation–Maximization algorithm based on forward–backward recursions which are well known in the hidden Markov literature for time series. The selection criteria we consider are based on penalized versions of the maximum log-likelihood or on the posterior probabilities of belonging to each latent state, that is, the conditional probability of the latent state given the observed data. Among the latter criteria, we propose an appropriate entropy measure tailored for the latent Markov models. We show the results of a Monte Carlo simulation study aimed at comparing the performance of the above states selection criteria on the basis of a wide set of model specifications.

Exploiting TIMSS and PIRLS combined data: multivariate multilevel modelling of student achievement

We exploit a multivariate multilevel model for the analysis of the Italian sample of the TIMSS\&PIRLS 2011 Combined International Database on fourth grade students. The multivariate approach jointly considers educational achievement on Reading, Mathematics and Science, thus allowing us to test for differential associations of the covariates with the three outcomes, and to estimate the residual correlations between pairs of outcomes at student and class levels. Multilevel modelling allows us to disentangle student and contextual factors affecting achievement. We also account for territorial differences in wealth by means of an index from an external source. The model residuals point out classes with high or low performance. As educational achievement is measured by plausible values, the estimates are obtained through multiple imputation formulas. The results, while confirming the role of traditional student and contextual factors, reveal interesting patterns of achievement in Italian primary schools.

Latent Markov and growth mixture models for ordinal individual responses with covariates: A comparison

We propose a short review between two alternative ways of modeling stability and change of longitudinal data when time-fixed and time-varying covariates referred to the observed individuals are available. They both build on the foundation of the finite mixture models and are commonly applied in many fields. They look at the data by a different perspective and in the literature they have not been compared when the ordinal nature of the response variable is of interest. The latent Markov model is based on time-varying latent variables to explain the observable behavior of the individuals. The model is proposed in a semi-parametric formulation as the latent Markov process has a discrete distribution and it is characterized by a Markov structure. The growth mixture model is based on a latent categorical variable that accounts for the unobserved heterogeneity in the observed trajectories and on a mixture of normally distributed random variable to account for the variability of growth rates. To illustrate the main differences among them we refer to a real data example on the self reported health status.

Causal Latent Markov Model for the Comparison of Multiple Treatments in Observational Longitudinal Studies

We extend to the longitudinal setting a latent class approach that was recently introduced by Lanza, Coffman, and Xu to estimate the causal effect of a treatment. The proposed approach enables an evaluation of multiple treatment effects on subpopulations of individuals from a dynamic perspective, as it relies on a latent Markov (LM) model that is estimated taking into account propensity score weights based on individual pretreatment covariates. These weights are involved in the expression of the likelihood function of the LM model and allow us to balance the groups receiving different treatments. This likelihood function is maximized through a modified version of the traditional expectation–maximization algorithm, while standard errors for the parameter estimates are obtained by a nonparametric bootstrap method. We study in detail the asymptotic properties of the causal effect estimator based on the maximization of this likelihood function, and we illustrate its finite sample properties through a series of simulations showing that the estimator has the expected behavior. As an illustration, we consider an application aimed at assessing the relative effectiveness of certain degree programs on the basis of three ordinal response variables in which the work path of a graduate is considered as the manifestation of his or her human capital-level across time.

Impact Evaluation of Job Training Programs by a Latent Variable Model

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.

A Generalized Moving Average Convergence/Divergence for Testing Semi-strong Market Efficiency

We propose a generalized version of the moving average converge divergence (MACD) indicator widely employed in the technical analysis and trading of financial markets. By assuming a martingale model with drift for prices, as well as for their transformed values, we propose a test statistic for the local drift and derive its main theoretical properties. The semi-strong market efficiency hypothesis is assessed through a bootstrap test. We conclude by applying the indicator to monitor the crude oil prices over a 6 years period.

A Latent Markov Model with Covariates to Study Unobserved Heterogeneity among Fertility Patterns of Couples Employing Natural Family Planning Methods

Purpose We use the historical data from the European Study of Daily Fecundability and we develop an algorithm to determine the fertile window in a woman’s cycle according to the rules of the C.A.Me.N. symptothermal method proposed by the Centro Ambrosiano Metodi Naturali. Our aim is to identify variables acting on the probability of conception by considering the fertile window and factors that cannot be explained by employing the observed covariates of individuals and couples. Methods We adopt the latent Markov model with covariates tailored for data collected at times when a latent process detects the dependence across fertile periods of each woman’s cycle. We consider measurement errors, transitions between conception and non-conception, and the prediction of conception rate over the fertile windows. Conclusion We find that the conception pattern is mainly related to sexual intercourse behavior during the fertile window and to previous pregnancies. For the cohort under study, we predict a steep decline in the average conception rate across fertile windows.