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We propose a hidden Markov model for multivariate continuous longitudinal responses with covariates that accounts for three different types of missing pattern: (I) partially missing outcomes at a given time occasion, (II) completely missing outcomes at a given time occasion (intermittent pattern), and (III) dropout before the end of the period of observation (monotone pattern). The missing-at-random (MAR) assumption is formulated to deal with the first two types of missingness, while to account for the informative dropout, we rely on an extra absorbing state. Estimation of the model parameters is based on the maximum likelihood method that is implemented by an expectation-maximization (EM) algorithm relying on suitable recursions. The proposal is illustrated by a Monte Carlo simulation study and an application based on historical data on primary biliary cholangitis.

Background Previous Enhanced Recovery After Surgery (ERAS®) studies have not always taken into account that ERAS interventions depend on baseline covariates and that several confounding variables affect the composite outcomes. Method A causal latent variable model is proposed to analyze data obtained prospectively concerning 1261 patients undergoing elective colorectal surgery within the ERAS protocol. Primary outcomes (composite of any complication, surgical site infection, medical complications, early ready for discharge (TRD), early actual discharge) and secondary outcomes (composite of late bowel function recovery, IV fluid resumption, nasogastric tube replacement, postoperative nausea and vomiting, re-intervention, re-admission, death) are considered along with their multiple dimensions. Results Concerning the primary outcomes, our results evidence three subpopulations of patients: one with probable good outcome, one with possibly prolonged TRD and discharge without complications, and the other one with probable complications and prolonged TRD and discharge. Epidural anesthesia, waiving surgical drainage, and early ambulation, IV fluid stop and urinary catheter removal act favorably, while preoperative hospital stay and blood transfusion act negatively. Concerning the secondary outcomes our results evidence two subpopulations of patients: one with high probability of good outcome and one with high probability of complications. Epidural anesthesia, waiving surgical drainage, early ambulation and IV fluid stop act favorably, while blood transfusion acts negatively also with respect to these secondary outcomes. Conclusion The multivariate causal latent class two-parameter logistic model, a modern statistical method overcoming drawbacks of traditional models to estimate the average causal effects on the treated, allows us to disentangle subpopulations of patients and to evaluate ERAS interventions.

We review the discrete latent variable approach, which is very popular in statistics and related fields. It allows us to formulate interpretable and flexible models that can be used to analyze complex datasets in the presence of articulated dependence structures among variables. Specific models including discrete latent variables are illustrated, such as finite mixture, latent class, hidden Markov, and stochastic block models. Algorithms for maximum likelihood and Bayesian estimation of these models are reviewed, focusing, in particular, on the expectation–maximization algorithm and the Markov chain Monte Carlo method with data augmentation. Model selection, particularly concerning the number of support points of the latent distribution, is discussed. The approach is illustrated by summarizing applications available in the literature; a brief review of the main software packages to handle discrete latent variable models is also provided. Finally, some possible developments in this literature are suggested.

The Expectation-Maximization (EM) algorithm is undoubtedly one of the most widely used techniques to estimate a discrete latent variable (DLV) model. However, while it is possible to prove that this algorithm converges to a local maximum of the log-likelihood function, there is no guarantee of convergence to the global maximum of this function. We propose two modifications to the EM algorithm to tackle this serious problem. The first one incorporates a tempering scheme into the EM algorithm: the log-likelihood is initially flattened to escape local maxima and then warped back to its original shape in a gradual way. The second uses evolutionary computation to encourage more accurate parameter space exploration. The performance of the resulting tempered EM (T-EM) and evolutionary EM (E-EM) algorithms is assessed for latent class and hidden Markov models in terms of both ability to reach the global maximum and computational time; a comparison with the standard EM algorithm is carried out through an extensive Monte Carlo simulation study. We show that the proposed algorithms outperform the standard EM, significantly increasing the chance of reaching the global maximum in almost all the examined cases. This improvement remains considerable, even accounting for the inflated overall computing time.

Maximum likelihood estimation of discrete latent variable (DLV) models is usually performed by the expectation-maximization (EM) algorithm. A well-known drawback is related to the multimodality of the log-likelihood function so that the estimation algorithm can converge to a local maximum, not corresponding to the global one. We propose a tempered EM algorithm to explore the parameter space adequately for two main classes of DLV models, namely latent class and hidden Markov. We compare the proposal with the standard EM algorithm by an extensive Monte Carlo simulation study, evaluating both the ability to reach the global maximum and the computational time. We show the results of the analysis of discrete and continuous cross-sectional and longitudinal data referring to some applications of interest. All the results provide supporting evidence that the proposal outperforms the standard EM algorithm, and it significantly improves the chance to reach the global maximum. The advantage is relevant even considering the overall computing time.

In order to evaluate the effect of a policy or treatment with pre- and post-treatment outcomes, we propose an approach based on a transition model, which may be applied with multivariate outcomes and accounts for unobserved heterogeneity. This model is based on potential versions of discrete latent variables representing the individual characteristic of interest and may be cast in the hidden (latent) Markov literature for panel data. Therefore, it can be estimated by maximum likelihood in a relatively simple way. The approach extends the difference-in-difference method as it is possible to deal with multivariate outcomes. Moreover, causal effects may be expressed with respect to transition probabilities. The proposal is validated through a simulation study, and it is applied to evaluate educational programs administered to pupils in the sixth and seventh grades during their middle school period. These programs are carried out in an Italian region to improve non-cognitive skills (CSs). We study if they impact also on students’ CSs in Italian and Mathematics in the eighth grade, exploiting the pretreatment test scores available in the fifth grade. The main conclusion is that the educational programs aimed to develop noncognitive abilities help the best students to maintain their higher cognitive abilities over time.

A novel variable and model selection method is proposed to analyze multiple time-series and panel data based on a Hidden Markov (HM) model for multivariate continuous responses. We consider an approach for inference, under the missing-at-random assumption to account for missing data, focusing on the maximum likelihood estimation of the model parameters through a modified Expectation-Maximization (EM) algorithm. We develop a greedy forward-backward algorithm based on the Bayesian Information Criterion (BIC) seen as an approximation of the Bayes factor. In this way, we achieve a dimensionality reduction of the complete set of response variables to a smaller subset and thus we select the most useful variables for clustering purposes. The BIC is also used to choose the optimal number of latent states during the steps of the greedy search algorithm. In applying the selection method, the estimation of multivariate linear regression models is required. In the presence of missing values in the set of independent variables, we adopt a sort of multiple imputations based on the posterior expected values obtained at the convergence of the EM algorithm of the estimated HM model. To illustrate the proposal we use a collection of macroeconomic indicators provided by the World Bank related to 217 countries followed over a long period of time. The chosen HM model allows us to dynamically characterize countries’ transitions between hidden states representing different levels of development.

The Italian market of sparkling wines increases as volume and assortment (such as brands, appellations, typologies) mainly because of sparkling Prosecco consumption. We investigate the repeated purchase behavior of sparkling wines in two years within the supermarket channel through scanner data collected from a consumer panel. We propose a Hidden Markov Model to analyze these data, assuming an unobservable process to capture consumers’ preferences and allowing us to consider purchases sparsity over time. We consider multivariate responses defining types of purchases, namely price, appellation, and sugar content. Customers’ covariates influence the initial and transition probabilities of the latent process. We identify five market segments, and we track their evolution over time. One segment includes Prosecco-oriented consumers, and we show that loyalty to Prosecco changes strongly over time according to the region of residence, income, and family type. The findings improve the understanding of the market and may provide evidence to design successful marketing strategies. (JEL Classifications: C33, C51, D12, L66)

We propose a hidden Markov model for longitudinal multivariate continuous responses, accounting for missing data under the missing at random assumption. Maximum likelihood estimation of this model is carried out through the Expectation-Maximization algorithm. To address the problem of dimensionality reduction, we develop a greedy search algorithm based on the Bayesian Information Criterion. We illustrate the proposal through a dataset collected by the World Bank and UNESCO Institute for Statistics on the basis of which we dynamically cluster countries according to the selected variables observed during the period 2000-2017.

We investigate public trust among the society by a statistical model suitable for panel data. At this aim, using trust’s levels measured from individual items recorded through a long-term survey we dispose of key variables with appropriate meaning. We account for the repeated and missing item responses by a hidden Markov model using longitudinal sampling weights. Since trust may be conceived as a psychological unobservable process of each person that fluctuates over time we consider observed time-varying and time-fixed individual covariates. We estimate the model parameters by a weighted log-likelihood through the Expectation–Maximization algorithm by using data collected in an East-Central European country like Poland. The latter is a country where the level of support to the national and international institutions is one of the lowest among the European member states. We apply a suitable algorithm based on the posterior probabilities to predict the best allocation to each latent typology. The proposed model is validated by generating out-of-sample responses and we find reasonable predictive values. We disentangle four hidden groups of Poles: discouraged, with no opinion, with selective trust and with fully public trust. We reveal an increasing number of people that are going to trust only some selected institutions over time.

A latent variable model included in the class of finite mixture item response theory models is employed to disentangle some recent sociological aspects of the Japanese society. Data collected via the Stratification and Social Psychology survey project in 2018 are analyzed to examine how people are classified in terms of fourteen different life domains: satisfaction, patriotism, social exclusion, anxiety, fatalism, relationships, system justification, social dominance, opinion on inequality, authoritarianism, gender ideology, political opinion, and religious attitudes. We elucidate how the multidimensional latent variables concerning social and psychological traits that differ with respect to the pattern of response to each attitudinal question measured in an ordinal scale are associated with people’s objective socio-economic and socio-demographic characteristics such as gender, age, income, education, and Japanese geographical block. On the basis of the model, we disclose three latent classes, each of which represents distinct attitudinal traits and ideological dimensions.

The Italian market of sparkling wines has undergone a strong expansion driven by what can be defined as the “Prosecco phenomenon.” It has extended consumption reaching new and more complex segments with a wide offer of appellations, brands, and prices. We aim to evaluate the Italian market of sparkling wines to figure out the competitive associations among the major brands. We propose two different analyses to disentangle distinctive groups of brands. First, using the information on scanner purchases of sparkling wines recorded by a consumer panel over a 2‐year period, and appropriate specifications of the latent class model, we cluster homogeneous groups of winery brands for the product attributes they propose to the market. Then, we analyze consumers' brand preferences from a dynamic perspective by employing a hidden Markov model to identify segments of brands perceived as similar. These results shed light on loyalty behavior and its evolution over time in the market.

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.

Summary Motivated by an application to a longitudinal data set coming from the Health and Retirement Study about self-reported health status, we propose a model for longitudinal data which is based on a latent process to account for the unobserved heterogeneity between sample units in a dynamic fashion. The latent process is modelled by a mixture of auto-regressive AR(1) processes with different means and correlation coefficients, but with equal variances. We show how to perform maximum likelihood estimation of the proposed model by the joint use of an expectation–maximization algorithm and a Newton–Raphson algorithm, implemented by means of recursions developed in the hidden Markov model literature. We also introduce a simple method to obtain standard errors for the parameter estimates and suggest a strategy to choose the number of mixture components. In the application the response variable is ordinal; however, the approach may also be applied in other settings. Moreover, the application to the self-reported health status data set allows us to show that the model proposed is more flexible than other models for longitudinal data based on a continuous latent process. The model also achieves a goodness of fit that is similar to that of models based on a discrete latent process following a Markov chain, while retaining a reduced number of parameters. The effect of different formulations of the latent structure of the model is evaluated in terms of estimates of the regression parameters for the covariates.

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., 25 pages, 5 figures, Presented at the VI European Congress of Methodology, 23-25 July 2014, Utrecht, Netherlands; Presented at the 47th Scientific meeting of the Italian Statistical Society, 11-13 June 2014, Cagliari, Italy

Following Cox & Wermuth (1994, 2002), we show that the distribution of a set of binary observable variables, induced by a certain discrete latent variable model, may be approximated by a quadratic exponential distribution. This discrete latent variable model is equivalent to the latent-class version of the two-parameter logistic model of Birnbaum (1968), which may be seen as a generalized version of the Rasch model (Rasch, 1960, 196). On the basis of this result, we develop an approximate maximum likelihood estimator of the item parameters of the two-parameter logistic model which is very simply implemented. The proposed approach is illustrated through an example based on a dataset on educational assessment. Copyright 2007, Oxford University Press.

We propose an extension of the latent class model for the analysis of capture-recapture data which allows us to take into account the effect of a capture on the behavior of a subject with respect to future captures. The approach is based on the assumption that the variable indexing the latent class of a subject follows a Markov chain with transition probabilities depending on the previous capture history. Several constraints are allowed on these transition probabilities and on the parameters of the conditional distribution of the capture configuration given the latent process. We also allow for the presence of discrete explanatory variables, which may affect the parameters of the latent process. To estimate the resulting models, we rely on the conditional maximum likelihood approach and for this aim we outline an EM algorithm. We also give some simple rules for point and interval estimation of the population size. The approach is illustrated by applying it to two data sets concerning small mammal populations. © 2006, The International Biometric Society.

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 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.

Drawing on the authors’ extensive research in the analysis of categorical longitudinal data, Latent Markov Models for Longitudinal Data focuses on the formulation of latent Markov models and the practical use of these models. Numerous examples illustrate how latent Markov models are used in economics, education, sociology, and other fields. The R and MATLAB® routines used for the examples are available on the authors’ website. The book provides you with the essential background on latent variable models, particularly the latent class model. It discusses how the Markov chain model and the latent class model represent a useful paradigm for latent Markov models. The authors illustrate the assumptions of the basic version of the latent Markov model and introduce maximum likelihood estimation through the Expectation-Maximization algorithm. They also cover constrained versions of the basic latent Markov model, describe the inclusion of the individual covariates, and address the random effects and multilevel extensions of the model. After covering advanced topics, the book concludes with a discussion on Bayesian inference as an alternative to maximum likelihood inference. As longitudinal data become increasingly relevant in many fields, researchers must rely on specific statistical and econometric models tailored to their application. Through a complete overview of latent Markov models, this book demonstrates how to use the models in three types of analysis: transition analysis with measurement errors, analyses that consider unobserved heterogeneity, and finding clusters of units and studying the transition between the clusters. Features: -Presents the theory and applications of latent Markov modeling in a common conceptual framework - Incorporates the authors’ recent work in analyzing longitudinal data - Shows how the models have been successfully applied in several field of social, behavioral and health science

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.

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

Background Alcohol consumption causes an estimated 4% of the global disease burden, prompting goverments to impose regulations to mitigate the adverse effects of alcohol. To assist public health leaders and policymakers, the authors developed a composite indicator—the Alcohol Policy Index—to gauge the strength of a country's alcohol control policies. Methods and Findings The Index generates a score based on policies from five regulatory domains—physical availability of alcohol, drinking context, alcohol prices, alcohol advertising, and operation of motor vehicles. The Index was applied to the 30 countries that compose the Organization for Economic Cooperation and Development and regression analysis was used to examine the relationship between policy score and per capita alcohol consumption. Countries attained a median score of 42.4 of a possible 100 points, ranging from 14.5 (Luxembourg) to 67.3 (Norway). The analysis revealed a strong negative correlation between score and consumption (r = −0.57; p = 0.001): a 10-point increase in the score was associated with a one-liter decrease in absolute alcohol consumption per person per year (95% confidence interval, 0.4–1.5 l). A sensitivity analysis demonstrated the robustness of the Index by showing that countries' scores and ranks remained relatively stable in response to variations in methodological assumptions. Conclusions The strength of alcohol control policies, as estimated by the Alcohol Policy Index, varied widely among 30 countries located in Europe, Asia, North America, and Australia. The study revealed a clear inverse relationship between policy strength and alcohol consumption. The Index provides a straightforward tool for facilitating international comparisons. In addition, it can help policymakers review and strengthen existing regulations aimed at minimizing alcohol-related harm and estimate the likely impact of policy changes., Using an index that gauges the strength of national alcohol policies, a clear inverse relationship was found between policy strength and alcohol consumption., Editors' Summary Background. Alcohol drinking is now recognized as one of the most important risks to human health. Previous research studies (see the research article by Rodgers et al., linked below) have predicted that around 4% of the burden of disease worldwide comes about as a result of drinking alcohol, which can be a factor in a wide range of health problems. These include chronic diseases such as cirrhosis of the liver and certain cancers, as well as poor health resulting from trauma, violence, and accidental injuries. For these reasons, most governments try to control the consumption of alcohol through laws, although very few countries ban alcohol entirely. Why Was This Study Done? Although bodies such as the World Health Assembly have recommended that its member countries develop national control policies to prevent excessive alcohol use, there is a huge variation between national policies. It is also very unclear whether there is any link between the strictness of legislation regarding alcohol control in any given country and how much people in that country actually drink. What Did the Researchers Do and Find? The researchers carrying out this study had two broad goals. First, they wanted to develop an index (or scoring system) that would allow them and others to rate the strength of any given country's alcohol control policy. Second, they wanted to see whether there is any link between the strength of control policies on this index and the amount of alcohol that is drunk by people on average in each country. In order to develop the alcohol control index, the researchers chose five main areas relating to alcohol control. These five areas related to the availability of alcohol, the “drinking context,” pricing, advertising, and vehicles. Within each policy area, specific policy topics relating to prevention of alcohol consumption and harm were identified. Then, each of 30 countries within the OECD (Organization for Economic Cooperation and Development) were rated on this index using recent data from public reports and databases. The researchers also collected data on alcohol consumption within each country from the World Health Organization and used this to estimate the average amount drunk per person in a year. When the researchers plotted scores on their index against the average amount drunk per person per year, they saw a negative correlation. That is, the stronger the alcohol control policy in any given country, the less people seemed to drink. This worked out at around roughly a 10-point increase on the index equating to a one-liter drop in alcohol consumption per person per year. However, some countries did not seem to fit these predictions very well. What Do These Findings Mean? The finding that there is a link between the strength of alcohol control policies and amount of alcohol drinking does not necessarily mean that greater government control causes lower drinking rates. The relationship might just mean that some other variable (e.g., some cultural factor) plays a role in determining the amount that people drink as well as affecting national policies for alcohol control. However, the index developed here is a useful method for researchers and policy makers to measure changes in alcohol controls and therefore understand more clearly the factors that affect drinking rates. This study looked only at the connection between control measures and extent of alcohol consumption, and did not examine alcohol-related harm. Future research might focus on the links between controls and the harms caused by alcohol. Additional Information. Please access these Web sites via the online version of this summary at http://dx.doi.org/10.1371/journal.pmed.0040151. A Perspective in PLoS Medicine by Alison Ritter accompanies this article: “Comparing alcohol policies between countries: Science or silliness?” Facts and figures on alcohol are available from the World Health Organization, including information about the burden of disease worldwide as a result of alcohol Information from the US Centers for Disease Control and Prevention is available about alcohol and public health A 2004 PLoS Medicine research article includes discussion of the health burdens of alcohol: Rodgers A, Ezzati M, Vander Hoorn S, Lopez AD, Lin RB, et al. (2004) Distribution of major health risks: Findings from the global burden of disease study. PLoS Medicine 1(1): e27. doi:10.1371/journal.pmed.0010027 Current information about research on alcohol and alcoholism is available from the National Institute on Alcohol Abuse and Alcoholism