A flexible family of mixture models known as cluster-weighted models (CWMs) arise when the joint distribution of a response variable and a set of covariates can be modelled by a weighted combination of several component distributions. We introduce an extension to CWMs where changepoints are present. Similar to the finite mixture of regressions (FMR) with changepoints, CWMs with changepoints are more flexible than standard CWMs if we believe that changepoints are present within the data. We consider changepoints within the linear Gaussian CWM, where both the marginal and conditional densities are assumed to be Gaussian. Furthermore, we consider changepoints within the Poisson and Binomial CWM. Model parameter estimation and performance of some information criteria are investigated through simulation studies and two real-world datasets. / Thesis / Master of Science (MSc)

Governance of the healthcare systems is one of the most important challenges forWestern countries. Within this, an accurate assessment of the quality is key to policy makers and public managers, in order to guarantee equity, effectiveness and efficiency. In this thesis, we investigate aspects and methods related to healthcare evaluation by focussing on the healthcare system in Lombardy (Italy), where public and private providers compete with each other, patients are free to choose where to be hospitalized, and a pay-for-performance program was recently implemented. The general aim of this thesis is to highlight the role of statistics within a quality evaluation framework, in the form of advancing the statistical methods used to measure quality, of evaluating the effectiveness of implemented policies, and of testing the effect that mechanisms of competition and cooperation can have on the quality of a healthcare system. We firstly advance a new methodological approach for measuring hospital quality, providing a new tool for managers involved in performance evaluations. Multilevel models are typically used in healthcare, in order to account for the hierarchical structure of the data. These models however do not account for unobserved heterogeneity. We therefore propose an extension of the cluster-weighted models to the multilevel framework and focus in particular on the case of a binary dependent variable, which is common in healthcare. The resulting multilevel logistic cluster-weighted model is shown to perform well in a healthcare evaluation context. Secondly, we evaluate the effectiveness of a pay-for-performance program. Differently from the existent literature, in this thesis we evaluate this program on the basis of five health outcomes and across a wide range of medical conditions. Availability of data pre and post-policy in Lombardy allows us to use a difference-in-differences approach. The statistical model includes multiple dependent outcomes, that allow quantifying the joint effect of the program, and random effects, that account for the heterogeneity of the data at the ward and hospital level. The results show that the policy has overall a positive effect on the hospitals' performance. Thirdly, we study the effect of pro-competition reforms on the hospital quality. In Lombardy, competition between hospitals has been mostly driven by the adoption of a quasi-market system. Our results show that no association exists between hospital quality and competition. We speculate that this may be the result of asymmetric information, i.e. the lack of transparent information provided to citizens about the quality of hospitals. This is bound to reduce the impact of pro-competition reforms on quality and can in part explain the conflicting results found in the literature on this subject. Our results should motivate a public disclosure of quality evaluations. Regardless of the specifics of a system, hospitals are altruistic economic agents and they cooperate in order to improve their quality. In this work, we analyse the effect of cooperation on quality, taking the network of patients' transfers between hospitals as a proxy of their level of cooperation. Using the latest network models, we find that cooperation does lead to an increase in quality and should therefore be encouraged by policy makers.

The poverty rate among veterans in US has increased over the past decade, according to the U.S. Department of Veterans Affairs (2015). Thus, it is crucial to veterans who live below the poverty level to get sufficient benefit grants. A study on prudently managing health benefit grants for veterans may be helpful for government and policy-makers making appropriate decisions and investments. The purpose of this research is to find an underlying group structure for the veterans' benefit grants dataset and then estimate veterans' benefit grants sought using incomplete data. The generalized linear mixed cluster-weighted model based on mixture models is carried out by grouping similar observations to the same cluster. Finally, the estimates of veterans' benefit grants sought will provide reference for future public policies. / Thesis / Master of Science (MSc)

Mixture model-based clustering has become an increasingly popular data analysis technique since its introduction almost fifty years ago. Families of mixture models are said to arise when the component parameters, usually the component covariance matrices, are decomposed and a number of constraints are imposed. Within the family setting, it is necessary to choose the member of the family --- i.e., the appropriate covariance structure --- in addition to the number of mixture components. To date, the Bayesian information criterion (BIC) has proved most effective for this model selection process, and the expectation-maximization (EM) algorithm has been predominantly used for parameter estimation. We deviate from the EM-BIC rubric, using variational Bayes approximations for parameter estimation and the deviance information criterion (DIC) for model selection. The variational Bayes approach alleviates some of the computational complexities associated with the EM algorithm. We use this approach on the most famous family of Gaussian mixture models known as Gaussian parsimonious clustering models (GPCM). These models have an eigen-decomposed covariance structure. Cluster-weighted modelling (CWM) is another flexible statistical framework for modelling local relationships in heterogeneous populations on the basis of weighted combinations of local models. In particular, we extend cluster-weighted models to include an underlying latent factor structure of the independent variable, resulting in a novel family of models known as parsimonious cluster-weighted factor analyzers. The EM-BIC rubric is utilized for parameter estimation and model selection. Some work on a mixture of multivariate t-distributions is also presented, with a linear model for the mean and a modified Cholesky-decomposed covariance structure leading to a novel family of mixture models. In addition to model-based clustering, these models are also used for model-based classification, i.e., semi-supervised clustering. Parameters are estimated using the EM algorithm and another approach to model selection other than the BIC is also considered. / NSERC PGS-D

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