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La libreria LMest permette di specificare e stimare i modelli Hidden o Latent Markov (LM) per l'analisi di dati longitudinali sia continui che categoriali. Le covariate sono presenti nei modelli in base ad adeguate parametrizzazioni. Varie tipologie di modelli possono essere stimati utilizzando la struttura dei dati sia in formato long che wide. La stima di massima verosimiglianza è ottenuta con l'algoritmo Expectation-Maximization implementato attraverso routines Fortran. La libreria permette di trattare risposte mancanti, drop-out e valori mancanti secondo una struttura non-monotona. Gli errori standard per i parametri stimati vengono calcolati attraverso la matrice di informazione ottenuta in modo esatto oppure approssimato. La libreria include alcuni esempi e dei dati sia reali che simulati., The package LMest is a framework for specifying and fitting Latent (or Hidden) Markov (LM) models, which are tailored for the analysis of longitudinal continuous and categorical data. Covariates are also included in the model specification through suitable parameterizations. Different LM models are estimated through specific functions requiring a data frame in long format. Maximum likelihood estimation of model parameters is performed through the Expectation-Maximization algorithm, which is implemented by relying on Fortran routines. The package allows us to deal with missing responses, including drop-out and non-monotonic missingness, under the missing-at-random assumption. Standard errors for the parameter estimates are obtained by exact computation of the information matrix or through reliable numerical approximations of this matrix. The package also provides some examples and real and simulated data sets.

Many statistical methods currently employed to evaluate the effect of a marketing campaign in dealing with observational data advocate strong parametric assumptions to correct for endogeneity among the participants. In addition, the assumptions compromise the estimated values when applied to data in which the research expects endogeneity but this is not realized. Based on the recent advances in the literature of causal models dealing with data collected across time, we propose a dynamic version of the inverse-probability-of-treatment weighting within the latent Markov model. The proposal, which is based on a weighted maximum likelihood approach, accounts for endogeneity without imposing strong restrictions. The likelihood function is maximized through the Expectation-Maximization algorithm which is suitably modified to account for the inverse probability weights. Standard errors for the parameters estimates are obtained by a nonparametric bootstrap method. We show the effects of multiple mail campaigns conducted by a large European bank with the purpose to influence their customers to the acquisitions of the addressed financial products

We consider a latent class model especially tailored for an ordinal response derived by comparing two continuous variables. We propose a general method to estimate the model parameters with survey data when there are missing responses and survey weights. First, we estimate the model with the missing responses without covariates with a weighted likelihood function maximised through the Expectation-Maximization algorithm. In order to determine the suitable number of latent classes we rely on the Akaike Information Criterion. Second, by fixing the parameters of the measurement model we estimate the remaining parameters by adding the full set of covariates. We make predictions on the basis of the maximum a posteriori probability. In the application, we consider Japanese survey data collected at four waves covering 40 years with the aim to study changes on couples' breadwinning patterns.

We examine how couples'breadwinning patterns are classified and how they have changed over the past four decades, during which we have seen increase in women's labor force participation. We explore how the latent variable of spousal breadwinning types is associated with the socioeconomic statuses. To this end, we consider a latent class model especially tailored for an underlying ordinal response derived by comparing two continuous variables. We develop method to estimate the model parameters accounting for the informative sampling design and missing responses. We estimate the measurement model parameters by means of a weighted likelihood function maximised through the Expectation-Maximization algorithm. In order to determine the suitable number of latent classes we rely on the Akaike Information Criterion. Then, fixing the obtained parameters we estimate the latent model parameters by adding the full set of covariates. We make predictions on the basis of the maximum a-posteriori probability. Using data from the Japanese surveys covering the period from 1985 to 2015 breadwinning patterns are examined. Our model discloses two latent classes, each of which represents distinct breadwinning types, namely ``traditional couples" and ``new couples". Interestingly, the two-class pattern persists across the four waves covering the past 40 years.

We aim to monitor financial asset price series by a generalized version of the moving average convergence/divergence (MACD) trend indicator which is currently employed as technical indicator in trading systems. We use the proposed indicator to test the semi-strong form of market efficiency hypothesis stating that asset prices follows a martingale model when, on the basis of the information available up to a given time, the expected returns are equal to zero. By assuming a martingale model with drift for prices, we propose an MACD-based test statistic for the local drift and derive its main theoretical properties. The semi-strong market efficiency hypothesis is assessed through a nonparametric bootstrap test, where under the null hypothesis the prices have no drift. A simulation study shows that the empirical size and power of the bootstrap test rapidly converges to the chosen nominal levels. Finally, we use our test to derive a long/short equity trading strategy where no active positions are taken or maintained under the null hypothesis (no drift). We apply this strategy to monitor 1,469 daily quotes of the crude oil prices over the six year period 2010-2016 finding that our methodology is often capable to detect market inefficiencies by accruing positive returns.

We use survey data collected over 40 years to understand changes related to traditional inequalities in couples' breadwinning patterns. Implicitly, we investigate the wife's role in the Japanese society by accounting for factors influencing the active role of the women in the labour force. While accounting for informative sampling and missing responses we advance research and theory on the field by adopting a model based on an underlying ordinal latent variable to explore the connections with demographic and social-related factors.

Statistical methods currently employed for marketing are based on strong parametric assumptions to correct for endogeneity. These assumptions may com- promise the results when applied to data in which the research expects endogeneity, but this is not the case. By considering the recent advances in the literature of causal models for panel data we propose to employ a latent Markov model which is tailored to estimate causal effects of a marketing campaign and to deal dynamically with endogeneity. Our proposal does not require the strong assumption of strictly absence of unmeasured confounders. The endogeneity correction imposes no restrictions on the data and it does not bias results when the endogeneity is not in. We assess the effects of multiple mail campaigns conducted by a large European bank with the purpose of predicting customer acquisitions of financial products.

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 six years period.

Latent Markov (LM) models represent an important class of models for the analysis of longitudinal data, especially when response variables are categorical. These models have a great potential of application in many fields, such as economics and medicine. We illustrate the R package LMest that is tailored to deal with the basic LM model and some extended formulations accounting for individual covariates and for the presence of unobserved clusters of units having the same initial and transition probabilities (mixed LM model). The main functions of the package are tailored to parameter estimation through the expectation-maximization algorithm, which is based on suitable forwardbackward recursions. The package also permits local and global decoding and to obtain standard errors for the parameter estimates. We illustrate the use of the package and its main features through some empirical examples in the fields of labour market, health, and criminology.

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

Objective: We review two alternative ways of modeling stability and change of longitudinal data by using time-fixed and time-varying covariates for the observed individuals. Both the methods build on the foundation of finite mixture models, and are commonly applied in many fields but they look at the data from different perspectives. Our attempt is to make comparisons when the ordinal nature of the response variable is of interest. Methods: The latent Markov model is based on time-varying latent variables to explain the observable behavior of the individuals. It is proposed in a semiparametric formulation as the latent process has a discrete distribution and 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 Gaussian random variables to account for the variability in the growth factors. We refer to a real data example on self-reported health status to illustrate their peculiarities and differences.

We illustrate how to detect causal effects of multiple treatments on subpopulations of individuals from a dynamic perspective. The proposed Latent Markov (LM) model is based on the propensity score weights corresponding to the individual pretreatment covariates. The weights contribute to the likelihood function of the LM model and allow us to balance the groups receiving different treatments. The likelihood function is maximized through a modified version of the Expectation-Maximization algorithm, while standard errors for the parameter estimates are obtained by a nonparametric bootstrap method. The asymptotic properties of the causal effect estimator are illustrated in details as well as its finite sample properties through a series of simulations showing that the estimator has the expected behaviour. The empirical illustration is devoted to assess the relative effectiveness of certain degree programs on three ordinal response variables: contract type, skill and gross income of the graduates

Latent Markov (LM) models represent an important tool of analysis of longitudinal data. We illustrate the main functions of the R package LMest that is tailored to fit the basic LM model, and some of its extended formulations, on longitudinal categorical data. The illustration is based on empirical analyses of datasets from a socio-economic perspective.

We show the multilevel logistic cluster-weigthted model and we make a comparison with the standard multilevel model when they are employed to provide an hospitals’ ranking in order to evaluate the performance related to the 30-day mortality

Individuals’ life courses may be represented as ordered collections of the activities experienced during a given period. We consider data on the school-work transition for a set of young individuals from Northern Ireland, and show how some interesting features of their careers can be described by a dynamic latent variable model accounting for some unobserved individuals’ characteristics, such as attitudes

The latent Markov model and the growth mixture model for longitudinal data are 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 behaviour of the individuals. It is proposed in a semiparametric formulation as the latent process has a discrete distribution and 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 Gaussian random variables to account for the variability in the growth factors. We refer to a real data example on self-reported health status to illustrate their peculiarities and differences

In many contexts, sample units are clustered in groups according to a certain criterion, for instance employees in firms, students in classes, or patients in hospitals. These data are analyzed by multilevel models (Goldstein, 2011) and have important applications in the evaluation of public services, particularly in education and health. For instance, it may be of interest to make comparisons between schools or classes at national and international level on the basis of the students’acquired knowledge. Accountability systems in education have been promoted in the statistical literature mainly since the 90’s by Goldstein and Spiegelhalter (1996), who supported the idea that the performance monitoring approach may improve efficiency. In this work, we focus on models in which the multilevel structure is accounted for by a hierarchical set of discrete latent variables, even in the presence of multivariate responses; these latent variables are used to represent the unobserved heterogeneity between clusters (i.e., groups) of units and between units in each cluster, extending the Latent Class (LC) approach (Lazarsfeld and Henry, 1968) to the multilevel setting. In particular, two cases are of interest. The first is when the observed outcomes are polytomous, as they correspond to item responses, and data are collected at the same time occasion. This approach has been applied by many authors in the educational context, see among others Vermunt (2008) and Gnaldi et al. (2016). The second case of interest is when the data have a longitudinal dimension and heterogeneity between units is represented in a dynamic fashion by a Latent Markov (LM) chain, as proposed in Bartolucci et al. (2011); see also Bartolucci et al. (2013). While maximum likelihood estimation through the Expectation-Maximization algorithm (Dempster et al. 1977) of the models mentioned above is already well established, an issue that still deserves attention is that of predicting the latent variables at cluster

We test the hypothesis that divorce may be a positive experience for children with parents in high-distress unions, while the dissolution of low-distress unions may have negative effects. We use the first three waves of the Millennium Cohort Study (MCS), a longitudinal and representative British survey, to explore parental divorce and parent relationship quality on several children's outcomes at age five. By using the augmented inverse propensity weighted estimator we show that the dissolution of high-quality parental unions has the most harmful effects on children, especially on conduct problems. Among children whose parents have the highest-quality relationships, those that experience parental separation/divorce have higher conduct problems than children with a stable family. Our findings indicate that early childhood programs and interventions should particularly target children whose parents report high quality relationship before parental separation since they are especially at risk

We illustrate a multivariate multilevel analysis in a complex setting of large-scale assessment surveys, dealing with plausible values and accounting for the survey design. We analyse the Italian sample of the TIMSS and PIRLS 2011 Combined International Database on fourth grade students. We jointly considers educational achievement in Reading, Mathematics and Science, thus we test for differential associations of the covariates with the three response variables, and we estimate the residual correlations among pairs of responses within and between classes. 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 data 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

We show how to handle the information which is acquired in a dynamic framework when there are multiple items in a survey collected on the same individuals at different time occasions. The response variables are commonly measured on an ordinal scale and the data may show a non-monotone missing pattern. The underlying phenomenon which is related to the interest of the survey may be modelled by a latent stochastic process. The latter having dependences according to a Markov structure is able to capture the heterogeneity of the response behaviour and to account for the measurement errors that naturally arise in the survey. The maximum likelihood estimation of the model parameters allows us to handle the missing data and to take advantage of the information provided by those individuals not showing complete responses. In a similar way, it is possible to consider a counterfactual framework in which the outcomes of interest are not directly observable even for the selected treatment. Such a flexible modelling approach is illustrated with two examples based on real data.

In the following we test the hypothesis that divorce may be a positive experience for children with parents in high-distress unions, while the dissolution of low-distress unions may have negative effects. Despite the importance of this subject, only nine studies have examined this hypothesis, and all suffer limitations and they are based on statistical method not properly tailored to enhance causal relationships. In this paper, we use the first three waves of the Millennium Cohort Study (MCS), a longitudinal and representative British survey, to explore parental divorce and parent relationship quality on several children’s outcomes at age 5: three measures of cognitive abilities and five psychological dimensions. By using the augmented inverse propensity weighted estimator we show that the dissolution of high-quality parental unions has the most harmful effects on children, especially on conduct problems. Among children whose parents have the highest-quality relationships, those that experience parental separation/divorce have higher conduct problems than children with an intact family. Our findings indicate that early childhood programs and interventions should particularly target children who did experience good relationship quality before parental separation since they are especially at risk.

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.

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.

We propose the use of the latent Markov model in a context of the estimation of multiple causal effects when dealing with observational studies and there are unobserved baseline differences between individuals. The proposed model, tailored for longitudinal data analysis in its basic formulation, has been first introduced by Wiggins (1951) and then formalized in his Ph.D thesis, Wiggins (1955). In Bartolucci et al. (2013) several extensions of the first basic formulation are given and new models have been proposed. The fact that the assumptions encoded by the model may be represented with the help of a path diagram contributes to make such class of models a powerful tool for the analysis of statistical data. In fact, as stated in Pennoni (2014), such models may be seen as built on the foundation of graphical causal models first proposed by Wright (1921) in genetics. Many statistical models tailored for the estimation of the causal effects have been proposed from that period. The potential outcome framework resulted to be one of the most useful tool. However, in a longitudinal setting the latter it is not still well developed as well as for some powerful models developed in the econometric context, see also Romeo (2014). Building on the foundation of the above models and on a recent proposal of Lanza et al. (2013), we introduce a new use of the propensity score weighting (Rosenbaum and Rubin, 1983) when dealing with a multivariate responses observed at multiple time occasions. We show some assumptions which have to be sustainable for the use of the proposed approach in the context of study. The use of the latent Markov model helps to get a reliable estimate of the average causal effect. An interesting feature of the proposed approach is its flexibility given by the adopted parameterization which allows us to deal with any kind of response variable. The model is fitted by a maximum likelihood estimation procedure based on first estimating a multinomial logit model for

We propose a multivariate multilevel model to analyse the Italian sample of the TIMSS&PIRLS 2011 Combined International Database on 4th grade students (Foy, 2013). The multivariate model jointly considers educational achievement on Reading, Mathematics and Science, thus allowing us to test for differential effects of the covariates on the three scores. This approach represents an advance with respect to official reports, where the three scores are analysed separately. Moreover, the multilevel approach allows us to disentangle student and contextual factors affecting achievement, also considering territorial differences in wealth.

We propose a novel model for longitudinal studies based on random effects to capture unobserved heterogeneity. We propose an extension of the latent Markov Rasch model, which is specially tailored to deal with confounders and missing data on the primary response. The model is based on both time-fixed and time-varying latent variables having a discrete distribution. In particular, time-varying latent variables are assumed to follow a Markov chain. The model estimation is performed by the maximum likelihood procedure through the EM algorithm. This estimation based on a set of weights associated to each subject to balance the composition of different subsamples corresponding to different treatments/exposures in a perspective of causal inference. These weights are computed by the propensity score method. The model is applied to the analysis of epidemiological and molecular data from the Normative Aging Study (NAS), a longitudinal cohort of older individuals to identify key epigenetic pathways in humans that reflect air pollution exposure and predict worse cognitive decline. The participants are assigned estimates of black carbon exposure, a measure of diesel particles, since 2010; have epigenome-wide Illumina Infinium 450K Methylation BeadChip data for methylation at ~486,000 DNA sites measured at two different time points; and are administered cognitive testing assessing multiple functional domains every 3-5 years. We will consider DNA methylation as a possible intermediate variable mediating the effects of air pollution on cognitive aging. Epigenetic profiles may represent cumulative biomarkers of air pollution exposures and aid in the early diagnosis and prevention of air pollution-related diseases.

We focus on the problem of the estimation of the credit migration matrices which are a widely used instrument in the context of risk management to account for the credit portfolio. We propose to apply a latent Markov model in which the possible movement directions are modeled as a continuous or discrete-valued stochastic process following a non-homogenous Markov chain of first or second order. It is a novel approach in this context, which outperform the existing models mainly based on an observed Markov process. The latter do not satisfactory represent the nature of the credit ratings which is mainly governed by underlying forces. We illustrate the traditional techniques employed in this field and their limitations in comparison to our proposal by applying them to simulated data which reproduce the credit ratings histories of Standard and Poor rated firms and sovereigns around the world from 1982 to 2014. We also show how the model may be of interest for prediction of default

We compare two alternative approaches of modeling longitudinal data in order to find patterns of change according to the available covariates, accounting for the unobserved heterogeneity in the population of interest. We refer to the Latent Markov (LM) model and to the Growth Mixture Model (GMM). We illustrate the main results by using real data on the self-perceived health status.

One of the main topic of the sociological literature of divorce is to gain insight on the children’s well-being during the phases of the parents’s relationship crisis and after the divorce. Some studies aimed to make comparisons between the children of intact and divorced families. They are inspecting the family characteristics such as parental conflict before separation (see among others Sigle Rushton and McLanahan, 2004; Amato, 2010). It is quite well stated that one of the main interesting feature is the parents’ previous relationship quality, as well as the divorce (see Jekielek, 1998; Hanson, 1999). One way to investigate this topic is to control for many characteristics before the marriage dissolution and also to include some measures related to the children’s wealth before and after the event of divorce. In fact, in such a context, to assess experimental results is methodologically difficult or even impossible. As suggested in Amato (2010) the findings of studies that focus on the average effects of parental divorce may be contradictory and open to multiple interpretations, since researchers can view marital discord either as a cause of divorce or as part of the divorce processes. Another method of investigation is that to study the effects of divorce on children. According to the results of many researches the children’s wealth of those living in high-conflict marriages is better after the divorce. On the other hand, the dissolution of low-conflict marriages can damage the children’s lives (see Amato et al., 1995; Booth and Amato, 2001). By relaying on European data of the Millennium Cohort Study we attempt to understand the features illustrated above taking into account as suggested by Hanson (1999) the well- being and cognitive ability of young children. To our knowledge, only the study of Fomby and Osborne (2010) has focused on very young children. We also consider the children’s externalizing problems which are related to: conduct and hyperactivity-inatt

We evaluate the use of two different model formulations by proposing a modeling framework which extends the stochastic volatility models and the stochastic frontier models by considering an hidden Markov model formulation or a model made by a mixture of latent auto-regressive stochastic processes both of first order. Those models are suitable statistical tools to be fitted to many available panel data in various applicative cases. The proposed model formulation is especially tailored for ordinal data when they are derived as a grouping of a different scale. We show some features of the models estimation which is carried out by means of the maximum likelihood. In the illustrative example we recall the available function of the library LMest on the R environment which is tailored to carry out the estimation of the models. Further, we provide some results of a case study to evaluate efficiency of a public organization by showing how the results can help policy makers.

We develop a suitable reweighting approach to deal with outliers when maximum-likelihood estimation is used to estimate latent class models. In such a context, the EM algorithm is used and the presence of singularities and spurious local maxima is common. The proposed method is motivated by an application aimed at finding clusters of offending behaviours.

The natural family planner methods are becoming widespread to consider the fertile or infertile period of a woman. The post ovulation phase of the woman’s menstrual cycle is considered as an important period of time due to the fact that the ovum survive maximum two days after the ovulation. However to determine such period of time is a difficult as the ovulation is not really linked to the overall length of the cycle. By relying on historical data we show some evidences by comparing the calendar method and the Ogino-Knaus method. We propose a suitable statistical model which uses the basal body temperature and the characteristics of vulval secretions in order to capture unobserved heterogeneity in the population of interest.

Latent Markov (LM) models represent an important class of models for the analysis of longitudinal data (Bartolucci et. al., 2013), especially when response variables are categorical. These models have a great potential of application for the analysis of social, medical, and behavioral data as well as in other disciplines. We propose the R package LMest, which is tailored to deal with these types of model. In particular, we consider a general framework for extended LM models by including individual covariates and by formulating a mixed approach to take into account additional dependence structures in the data. Such extensions lead to a very flexible class of models, which allows us to fit different types of longitudinal data. Model parameters are estimated through the expectation-maximization algorithm, based on the forward-backward recursions, which is implemented in the main functions of the package. The package also allows us to perform local and global decoding and to obtain standard errors for the parameter estimates. We illustrate its use and the most important features on the basis of examples involving applications in health and criminology.

We extend to the longitudinal setting a latent class approach that has beed recently introduced by Lanza et al. (2013) to estimate the causal effect of a treatment. The proposed approach permits the evaluation of the effect of multiple treatments 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 pre-treatment 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 non-parametric 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 when the work path of a graduate is considered as the manifestation of his/her human capital level across time

This book is made of different reseach problems which have in common the presence of latent variables. In the first part undirected and directed graphical models are considered and in the case of Gaussian continuous variables the author shows that the specification of a complex multivariate distribution through univariate regressions induced by a Directed Acyclic Graph (DAG) can be regarded as a simplification. The Expectation-Maximization algorithm is considered for the maximum likelihood estimation of the model parameters and the author illustrates a method for obtaining an explicit formula of the observed information matrix using the missing information principle. An essential background on the latent class model is given as well its extension to study latent changes over time. The Hidden Markov model is presented consisting of hidden states and observed variables both varying over time. The latent class cluster model is extended by proposing a latent model that also incorporates the longitudinal structure of the data by using a local likelihood approach. Some examples illustrate the use of the models in criminology and education. A detailed bibliography is provided

This book is made of different reseach problems which have in common the presence of latent variables. In the first part undirected and directed graphical models are considered and in the case of Gaussian continuous variables the author shows that the specification of a complex multivariate distribution through univariate regressions induced by a Directed Acyclic Graph (DAG) can be regarded as a simplification. The Expectation-Maximization algorithm is considered for the maximum likelihood estimation of the model parameters and the author illustrates a method for obtaining an explicit formula of the observed information matrix using the missing information principle. An essential background on the latent class model is given as well its extension to study latent changes over time. The Hidden Markov model is presented consisting of hidden states and observed variables both varying over time. The latent class cluster model is extended by proposing a latent model that also incorporates the longitudinal structure of the data by using a local likelihood approach. Some examples illustrate the use of the models in criminology and education. A detailed bibliography is provided

A brief overview of the generalized latent Markov models will be illustrated by showing some of the many available modelling options specially tailored for categorial and ordinal data. A list of the analyzed problems in real applications will be presented.

We propose a multivariate multilevel model for the analysis of the Italian sample of the TIMSS&PIRLS 2011 Combined International Database on 4th grade students. The multivariate model jointly considers educational achievement on Reading, Mathematics and Science, thus allowing us to test for differential effects of the covariates on the three scores. This approach represents an advance with respect to official reports, where the three scores are analysed separately. Moreover, the multilevel approach allows us to disentangle student and contextual factors affecting achievement, also considering territorial differences in wealth. The proportion of variability at class level is relevant even after controlling for the observed factors. The residual analysis allows us to locate classes with extremely high or low effectiveness.

Outlier detection in mixture models where there may be multiple outliers contaminating the main data is considered. Methods in the literature for continuous data have grown and now include fitting mixtures of t-distributions rather then normal or robustifying through modifying the likelihood in various ways. The focus in contrast is on discrete data and considers the approach of obtaining better estimates for the main underlying structure in latent class models in large data sets by detecting them and fitting them as separate mixture and latent class components. We propose a method that uses latent class outliers as seeds for new groups, and we propose a new algorithm for fitting latent class models where some of the latent classes are small. We show that often such latent class solutions are better than those obtained from random start points.

We propose a comparison between two alternative approaches of modeling longitudinal data, mainly focusing on the case of univariate ordinal responses when time fixed and time-varying covariates are available: latent Markov model and growth mixture model. Both approaches try to model patterns of changes through time and are built on the finite mixture model. They are commonly applied in many fields and explain the dynamic change from a different perspective. In the latent Markov model the latent process fully explains the observable behavior of the subjects. This process is made by time-varying latent variables having a discrete distribution which are assumed to follow a first-order Markov chain. The available covariates may affect the conditional distribution of the response variables given the latent process (measurement model) or the distribution of the latent process (latent model). The model estimation is performed by the maximum likelihood procedure through the EM algorithm. The growth mixture model is tailored to study the evolution of a latent characteristic of the subjects with the idea of estimating trajectories across all time points and it is designed to account for individual variability about a mean population trend. It aims at estimating class specific variance components (intercept variance, slope variance). A latent categorical variable is used to account for the heterogeneity in observed development trajectories and the population variability in growth is modeled through a mixture of different subpopulations. The comparison we propose is focused on the basic assumptions, the inferential properties, the computational aspects and the predictive features of each methodology. Strengths and weaknesses of the latent Markov and of the growth mixture model are illustrated having in mind a specific research question related to an applicative example concerning the perception of the health status of elderly people.

This talk takes as a starting point outlier detection in mixture models where there may be multiple outliers. Methods in the literature for continuous data have grown and now include fitting mixtures of t-distributions rather than normal distributions or robustifying through modifying the likelihood in various ways. The focus of this work in contrast is on discrete data and considers latent class models with a large number of binary or multinomial indicators. In large data sets and with a large number of indicators, poor identification of the underlying structure can easily occur. Moreover, outliers may not be detectable through classic latent class analysis, and classes will be contaminated. We propose a method that uses latent class outliers as seeds for new groups, and we propose a new algorithm for fitting latent class models where some of the latent classes are small. We show that often such latent class solutions are better than those obtained from random start points. We take the view that outliers are interesting and informative, and need to be used rather than ignored or removed. We illustrate this problem with a latent class analysis of the criminal typologies of female offenders in England and Wales. This extends statistical work on criminal typologies. Methods of detecting outliers in such problems will be presented and discussed, and the benefit of developing more comprehensive approaches to latent class analysis will be emphasised.

The paper analyses the Italian sample of the TIMSS&PIRLS 2011 Combined International Database, containing the results of the assessment on Reading, Mathematics and Science for a sample of 4th grade pupils, alongside with several variables about the pupils, their teachers and their schools. This is the first time that TIMSS and PIRLS surveys are conducted on the same sample, thus allowing to jointly analyse the achievement for the considered subjects. We propose a multivariate multilevel model, with pupils nested within classes, to explore the determinants of achievement in the three subjects.

The paper analyses the Italian subsample of the TIMMS&PIRLS 2011 Combined International Database, containing the results of the assessment on Reading, Mathematics and Science for a sample of 4th grade pupils, alongside with several variables about the pupils, their teachers and their schools. This is the first time that TIMMS and PIRLS survey are conducted on the same sample, thus allowing to jointly analyse the achievement in three subjects. Indeed, the Official Reports analyse the three results separately, without exploiting relationships among them. Even research papers usually analyse the results subject by subject. We propose a multivariate multilevel model to explore the determinants of the achievements in the three subjects, focusing on the characteristics of the teachers. The model considers the three scores on reading, math and science as a joint outcome, measured at pupil level, with pupils nested within classes. The classes are nested in 202 schools, with only 37 schools having two classes. Thus, the school is not considered as a further hierarchical level with its own random effect. However, the school characteristics are included in the model as covariates. This multivariate analysis allows us to decompose the residual variances and covariances among the three scores into a pupil-level component and a class-level component. Moreover, the multivariate model allows us to test whether the covariates have different effects on the three responses. The five plausible values for each subject convey the uncertainty inherent in test scores. We account for this source of variability by combining the results through multiple imputation procedures. Preliminary results show that the three scores are highly correlated, in particular at class level. Moreover, the proportion of variability of scores at class level is relevant (from 0.20 for reading to 0.30 for science), thus calling for an analysis of contextual factors and teacher effects.