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In generalized linear models (GLMs), measures of lack of fit are typically defined as the deviance between two nested models, and a deviance-based R2 is commonly used to evaluate the fit. In this paper, we extend deviance measures to mixtures of GLMs, whose parameters are estimated by maximum likelihood (ML) via the EM algorithm. Such measures are defined both locally, i.e., at cluster-level, and globally, i.e., with reference to the whole sample. At the cluster-level, we propose a normalized two-term decomposition of the local deviance into explained, and unexplained local deviances. At the sample-level, we introduce an additive normalized decomposition of the total deviance into three terms, where each evaluates a different aspect of the fitted model: (1) the cluster separation on the dependent variable, (2) the proportion of the total deviance explained by the fitted model, and (3) the proportion of the total deviance which remains unexplained. We use both local and global decompositions to define, respectively, local and overall deviance R2 measures for mixtures of GLMs, which we illustrate—for Gaussian, Poisson and binomial responses—by means of a simulation study. The proposed fit measures are then used to assess, and interpret clusters of COVID-19 spread in Italy in two time points. [ABSTRACT FROM AUTHOR]

Surveys are used to infer the level of social integration of immigrants. Item response theory helps to describe the relationship among responses to test items and latent traits of interest. However, in the presence of nonignorable missing data, which are omitted responses depending on the latent traits to be measured, estimates of the model parameters are biased. To account for nonignorable missing data, the quantity and quality of contacts between immigrants and natives (so called “social integration”) are taken into account through a linear function of the response propensity. Higher education, no intention to migrate again, young age, Albanian nationality, and declaring a non-Muslim religion or none, comparatively favor social integration. [ABSTRACT FROM AUTHOR]