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The cluster-weighted model (CWM) is a member of the family of mixture of regression models and is also known as mixtures of regressions with random covariates. CWMs refer to the framework of model-based clustering and have their natural application when the research interest requires modeling the relationship between a response variable and a set of covariates using a regression-based approach such as a generalized linear model and the sample is suspected to be composed by heterogeneous latent classes. Software for estimating these models is not yet available in Stata. The aim of this article is to introduce the Stata package \code{cwmglm}, which allows fitting CWMs based on the most common generalized linear models with random covariates. Moreover, \code{cwmglm} allows the estimation of parsimonious models of Gaussian distributions, with the parametrization of the variance-covariance matrix based on the eigenvalue decomposition. These features are completely new for Stata users. The \code{cwmglm} package features goodness-of-fit, bootstrapping and model selection tools. We illustrate the use of \code{cwmglm} with real and simulated datasets

The COVID-19 pandemic caused an unprecedented excess mortality. Since 2020, many studies have focussed on the characteristics of COVID-19 patients who did not survive. From the statistical point of view, what seems to dominate is the large heterogeneity of the populations affected by COVID-19 and the extreme difficulty in identifying subpopulations who died affected by a plurality of contemporary characteristics. In this paper, we propose an extremely flexible approach based on a cluster-weighted model, which allows us to identify latent groups of patients sharing similar characteristics at the moment of hospitalisation as well as a similar mortality. We focus on one of the hardest hit areas in Italy and study the heterogeneity in the population of patients affected by COVID-19 using administrative data on hospitalisations in the first wave of the pandemic. Results highlighted that a model-based clustering approach is essential to understand the complexity of the COVID-19 patients treated by hospitals and who die during hospitalisation.

In observational studies, one of the main difficulties consists in the comparison of treat- ment effects. In fact, receiving a treatment is not a “purely random” event, and there could be relevant differences between treatment groups. Propensity score is a popular tool to account for this source of bias. However, its use requires a careful modelization of the dependence relationships of the treatment on the covariates. In this work, we consider a general setting with multiple treatments and discrete multi-valued outcome. We propose to estimate the propensity score by using Bayesian Networks and, based on this, we develop an inferential methodology to evaluate the treatments effect. The performance of the pro- posed approach have been studied through a simulation study with very promising results.

In the model-based clustering literature, we find several methodologies to study latent structures underlying the data, among which mixtures of factor analyzers. However, none of them can detect hierarchical relationships among latent variables. The Ultrametric Gaussian Mixture Model (UGMM) is intended to reach this goal by identifying a hierarchy of variables, starting by partitioning the variables into a reduced number of groups per mixture component. Nonetheless, up to now, it requires complete observations, which is often not the case in real data collections. In this paper, we propose the extension of UGMM in the missing data framework. The proposal is applied to a real data set for inspecting the relationships among features of songs of different genres.

Road accidents represent a concern for modern societies, especially in poor and developing countries. In this paper, we develop a road safety model assuming that the car crashes recorded in Milan (Italy) during 2019 can be appropriately modelled as a realisation of a spatio-temporal point process on a linear network. We adopt a separable first-order intensity function with spatial and temporal components. The temporal dimension is estimated semi-parametrically using an additive Poisson regression model. The spatial dimension is estimated semi-parametrically considering a b-spline transformation of two potentially relevant space-varying covariates, namely the traffic flows and the distance to the closest road sign. This approach permits us to analyse traffic accidents at a very granular spatial scale, hence avoiding potential biases due to data aggregation.

Despite the existence of a normative space as the European Union (EU), the rates of recognition for asylum seekers’ protection statuses remarkably vary across different countries (Van Wolleghem, Sicakkan, 2022). Several studies have attempted to examine the potential explanations of these different between-countries rates of recognition, considering both economics and political reasons, without finding any clear driver (Toshkov, 2014), up to the point that it has been commented that «asylum decisions in Western Europe are highly arbitrary» (Bronkhorst, 1991, p.151). At the same time, it has been argued that changes in asylum policies within a given country may be associated with changes in recognition rates on asylum application flows (Thielemann, 2003), modifying the level of deterrence characterizing a determined socio-economic context (Holzer et al., 2000). On December 3rd 2018, the Italian government approved the so-called “Immigration and security” decree. Among its intentions, regarding a wide range of matters (contrast to mafia and terrorism, urban security), the decree remarkably changed the regulation of asylum, immigration and citizenship (De Petris, 2019). This contribution aims to evaluate the association between the “Immigration and security” decree and recognition rates of asylum applicants in Italy and if and how this association has been conditional on the applicant’s socio-demographic characteristics. This goal will be accomplished by performing some statistical analyses on a dataset created from the annual Eurostat database on the first instances decisions on applications.

During recent years, the concept of sustainability has been spread like one of the most cited and interesting trend topics giving a lot of definitions and good practices to perceive it. One of the aspects less frequently faced is the perception of sustainability for the population and the companies. In this paper, this issue has been analysed using results from a survey in which a sample of 1,000 respondents answered to the perception of sustainability for Italian companies. Since items in the questionnaire also asked for the behaviour of the respondent regards to the company declaring to be sustainable in terms of preferences and availability to spend, it is possible to identify different behaviours from the respondents. From a methodological point of view, these behaviours have been achieved creating some decision trees where each node represents a different group of respondents.

Hierarchical relationships among manifest variables can be detected by analyzing their correlation matrix. To pinpoint the hierarchy underlying a multidimensional phenomenon, the Ultrametric Correlation Model (UCM) has been proposed with the aim of reconstructing a nonnegative correlation matrix via an ultrametric one. In this paper, we illustrate the mathematical advantages that a simple structure induced by the ultrametric property entails for the estimation of the UCM parameters in a maximum likelihood framework.

Given a sample of unlabeled observations, the goal of a novelty detection method is to identify which units substantially deviate from the observed labeled patterns. Therefore, in a model-based framework, it is firstly of paramount importance to learn the components that correspond to the manifest groups in the training set. Secondly, one needs to take into account the lack of knowledge regarding the statistical novelties. Thirdly, contaminated elements in the known classes could greatly jeopardize the identification of new groups. Motivated by these challenges, we propose a two-stage Bayesian non-parametric novelty detector. At stage one, robust estimates are extracted from the training set and, subsequently, such information is employed to elicit informative priors within a flexible semiparametric mixture. This general paradigm can be easily adapted to complex modeling frameworks: we provide here an application to functional data from a food authenticity study.

This paper presents a review about the usage of eigenvalues restrictions for constrained parameter estimation in mixtures of elliptical distributions according to the likelihood approach. The restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) local maximizers, related to complex likelihood surfaces. The paper shows how the constraints may play a key role in the theory of Euclidean data clustering. The aim here is to provide a reasoned survey of the constraints and their applications, considering the contributions of many authors and spanning the literature of the last 30 years.

Cluster-weighted models (CWMs) are a flexible family of mixture models for fitting the joint distribution of a random vector composed of a response variable and a set of covariates. CWMs act as a convex combination of the products of the marginal distribution of the covariates and the conditional distribution of the response given the covariates. In this paper, we introduce a broad family of CWMs in which the component conditional distributions are assumed to belong to the exponential family and the covariates are allowed to be of mixed-type. Under the assumption of Gaussian covariates, sufficient conditions for model identifiability are provided. Moreover, maximum likelihood parameter estimates are derived using the EM algorithm. Parameter recovery, classification assessment, and performance of some information criteria are investigated through a broad simulation design. An application to real data is finally presented, with the proposed model outperforming other well-established mixture-based approaches.

This paper presents a review about the usage of eigenvalues restrictions for constrained parameter estimation in mixtures of elliptical distributions according to the likelihood approach. The restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non- interesting (spurious) local maximizers, related to complex likelihood surfaces. The likelihood function may present local spurious maxima when a fitted component has a very small variance or generalized variance (i.e., the determinant of the covariance matrix), compared to the others (Day, 1969). Such a component usually corresponds to a cluster containing few data points either relatively close together or almost lying in a lower-dimensional subspace, in the case of multivariate data. These solutions are of little practical use or real world interpretation (McLachlan and Peel, 2000). To get the best from the EM algorithm widely used for model estimation, a constrained estimation of the covariance matrices can help in driving the maximum likelihood approach toward sensible solutions and far from the ”wild”, non-useful ones. We begin presenting the strongest constraints, i.e. equality among spherical covariance matrices, and the well-known homoscedastic assumption. Afterwards, some authors reinterpreted the safe, above mentioned methods, looking for milder conditions, like the lightest assumption of equality of covariance determinants (McLachlan and Peel, 2000, Section 3.9.1). Gaussian parsimonious mixture models have been proposed by Banfield and Raftery (1993), to get intermediate component covariance matrices lying between homoscedasticity and heteroscedasticity. A broad family of contributions in the literature arise from Hathaway’s seminal paper and deals with setting constraints on the ratio between the maximum and the minimum eigenvalue (among many others, Ingrassia, 2004 and García-Escudero et al. 2008). The same approach, simultaneously with impartial trim

A robust estimator for a wide family of mixtures of linear regression is presented. Robustness is based on the joint adoption of the cluster weighted model and of an estimator based on trimming and restrictions. The selected model provides the conditional distribution of the response for each group, as in mixtures of regression, and further supplies local distributions for the explanatory variables. A novel version of the restrictions has been devised, under this model, for separately controlling the two sources of variability identified in it. This proposal avoids singularities in the log-likelihood, caused by approximate local collinearity in the explanatory variables or local exact fits in regressions, and reduces the occurrence of spurious local maximizers. In a natural way, due to the interaction between the model and the estimator, the procedure is able to resist the harmful influence of bad leverage points along the estimation of the mixture of regressions, which is still an open issue in the literature. The given methodology defines a well-posed statistical problem, whose estimator exists and is consistent to the corresponding solution of the population optimum, under widely general conditions. A feasible EM algorithm has also been provided to obtain the corresponding estimation. Many simulated examples and two real datasets have been chosen to show the ability of the procedure, on the one hand, to detect anomalous data, and, on the other hand, to identify the real cluster regressions without the influence of contamination.

Mixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering-at the same time-dimension reduction and model-based clustering. The high prevalence of spurious solutions and the disturbing effects of outlying observations in maximum likelihood estimation may cause biased or misleading inferences. Restrictions for the component covariances are considered in order to avoid spurious solutions, and trimming is also adopted, to provide robustness against violations of normality assumptions of the underlying latent factors. A detailed AECM algorithm for this new approach is presented. Simulation results and an application to the AIS dataset show the aim and effectiveness of the proposed methodology.

In recent years, increasing attention has been directed toward problems inherent to quality control in healthcare services. In particular, it is necessary to measure effectiveness with respect to improving healthcare outcomes of diagnostic procedures or specific treatment episodes. The performance of hospitals is usually evaluated by multilevel models and other methods for risk adjustment. However, these approaches are not suitable for data with large unobserved heterogeneity. A potentially large source of unobserved heterogeneity comes from the variation of the regression coefficients between groups of individuals sharing similar but unobserved characteristics. To overcome such drawbacks, we propose the multilevel cluster-weighted model, a new mixture model approach for handling hierarchical data.

Mixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering at the same time dimension reduction and model-based clustering. Unfortunately, the high prevalence of spurious solutions and the disturbing effects of outlying observations, along maximum likelihood estimation, open serious issues. We consider restrictions for the component covariances, to avoid spurious solutions, and trimming, to provide robustness against violations of normality assumptions of the underlying latent factors. A detailed AECM algorithm for this new approach is presented. Simulation results and an application to the AIS dataset show the aim and effectiveness of the proposed methodology

Mixtures of factor analyzers are a flexible way for modelling multivariate data with local dependences. It is well known that trimming and restrictions are useful tools for getting robust estimators when estimating mixture models. It will be introduced a new robust estimator for estimating a mixture of factor analyzers based in the joint application of these mentioned tools. The asymptotic and robustness properties of the proposed methodology will be shown. A feasible AECM algorithm for this estimator will be provided. There will be shown evidences about the effectiveness of this methodology when it is applied to real and to artificial data sets.

Cluster-weighted models (CWMs) are a flexible family of mixture models for fitting the joint distribution of a random vector composed of a response variable and a set of covariates. CWMs act as a convex combination of the products of the marginal distribution of the covariates and the conditional distribution of the response given the covariates. In this paper, we introduce a broad family of CWMs in which the component conditional distributions are assumed to belong to the exponential family and the covariates are allowed to be of mixed-type. Under the assumption of Gaussian covariates, sufficient conditions for model identifiability are provided. Moreover, maximum likelihood parameter estimates are derived using the EM algorithm. Parameter recovery, classification assessment, and performance of some information criteria are investigated through a broad simulation design. An application to real data is finally presented, with the proposed model outperforming other well-established mixture-based approaches.

Mixtures of factor analyzers are becoming more and more popular in the area of model based clustering of multivariate data. According to the likelihood approach in data modeling, it is well known that the unconstrained likelihood function may present spurious maxima and singularities. To reduce such drawbacks, in this paper we introduce a procedure for parameter estimation of mixtures of factor analyzers, which maximizes the likelihood function under the mild requirement that the eigenvalues of the covariance matrices lie into some interval [a,b]. Moreover, we give a recipe on how to select appropriate bounds for the constrained EM algorithm, directly from the handled data. We then analyze and measure its performance, compared with the usual non-constrained approach, and also with other constrained models in the literature. Results show that the data-driven constraints improve the estimation and the subsequent classification, at the same time.