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166 Volume LXIII nn. 3-4 – Luglio-Dicembre 2009 contextual covariates. We define the discrete latent random variable X ij that represents the individual condition of social exclusion. Given their response patterns to the selected indicators, individuals will be classified in a probabilistic way in one of the t = 1,..., T latent classes of X ij . Moreover, we assume the existence of a discrete latent random variable W j at regional level, with m= 1,..., M classes, conditionally on which the individual responses are assumed to be mutually independent. The second level latent variable has the role of a random effect in the model for X ij , and it aims to identify latent types of regions for which parameters in the specified model differ. This multilevel specification of the latent class probability structure is built by introducing a finite mixture model at each level of nesting (Vermunt 2003), i.e. at the individual and regional level: M n ( ) j T K ⎡ g ⎡ ⎤⎤ P Yj Z j = ∑⎢PW ( = mZ ) ( , ) ( , j j ⎢∏∑P X = tW Z PY s X W ij j ij ∏ = ijk k ij j ) ⎥⎥ m= 1⎣ ⎣ i= n t= 1 k= 1 ⎦⎦ The three components of the right-hand side of the previous equation are specified using multinomial logit models, and represent: a) the probability that region j belongs to a particular level of the latent g variable W , given the three selected regional covariates Z ; j j b) the probability that respondent i belongs to a particular class of the latent variable at the first level X ij , given regional latent class membership and the two individual covariates Z ; ij c) the joint probability that the i -th respondent follows the pattern s i given individual and regional latent class membership. In order to account for variability of item response among different regions, we hypothesized some direct effects of the group-level latent variable. 4. Results The model discussed here identifies 6 different respondent types as regard their deprivation status in all the relevant domains, i.e. T = 6 latent classes at individual level, and of 4 classes at regional level, i.e. M = 4 (subsequently indicated as “clusters”) which enable to differentiate rather well among regions. The model has been selected by means of the BIC statistics. The characteristics of each class, in terms of their similarities and differences, can be discussed referring to the class-specific marginal probabilities associated
Rivista Italiana di Economia Demografia e Statistica 167 with each indicator PY ˆ ( ijk sk X t) = = , which show how the latent classes are related to the 12 indicator variables used in the analysis. In this sense, we can characterize each class of the latent variable in term of response probability to each level of the indicators, and thus describe the different typologies that emerge. At the individual level, the model identifies a class of people excluded from all the dimensions. Individuals in class 6 (11.2% of the population) have high risk to be in the lowest income quartiles, to perceive to be poor, to feel excluded from the society, to have low personal relationships, and to have a negative perception of the institutional system. More than one-third of the population (the 37.9%) is in the opposite situation: for individuals in class 1 the probability to give an answer that corresponds to a positive situation is the highest for almost all the indicators. In this sense, class 1 represent the “not excluded class”. The other classes present intermediate situations. An interesting characteristic of class 2 (size equal to 18.3%) is the disagreeing between the objective measure of the income and the perception to get by with that income. This class has low probability to include people who feel unhelpful or excluded, or people who are unsatisfied with the social and security system. A low level of income does not represent per se an element that affect the capability of these individuals to integrate in the mainstream society. The profile of latent class 3 identifies individuals (11.9%) who perceive the risk of social exclusion and the difficulty to have a useful role in the society in a measure higher than the overall mean. The critical aspect of this class is represented by the social relationships: people in this class are in a situation in which the risk of marginalization and the feeling of social exclusion is not linked to a lack of economic stability, but rather to a lack of a stable and positive social network. Class 4, for which membership probability is 0.14, is characterized by a high probability to be in the lowest income quartiles and to perceive the income as not sufficient. Individuals in this class feel a sense of inferiority with regards the others due to their income or their job situation, and feel to be left out of the society and to not have a useful role in the society. Class 4 thus identifies a kind of exclusion that is mainly economic. Finally, class 5 identifies a situation of exclusion referred to the “institutional dimension”: the probability to be dissatisfied with the social assistance and health care system is the highest, together with the assessment about the strong presence of violence and theft in the area. Moreover, people in this class not participate in association activities. This class is the smallest one (6.6%). Let us now move to the second level by computing the region classification based on these cluster membership (prior) probabilities conditional on group-level covariates ˆ g PW ( = mZ j j ). We consider the classification into 4 clusters of regions. In cluster number 1 are classified almost all UK regions, Ireland, two German regions (Bremen and Baden-Wurttemberg), the North-East of Italy, the East and
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