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Goodness-of-fit for generalized linear latent variables models

Conne, David
Defense Thèse de doctorat : Univ. Genève, 2008 - SES 681 - 2008/10/09
Abstract Generalized Linear Latent Variables Models (GLLVM) enable the modelling of relationships between manifest and latent variables. These models are widely used in the social sciences. In a latent variable framework, one works with several unobservable quantities (latent scores, parameters) and it is herefore essential to choose a model as close as possible to the data. To test the appropriateness of a particular model, ne needs to define a Goodness-of-fit test statistic (GFI). Available GFI can be separated in two groups: first, GFI based on a comparison between the sample covariance and the model covariance of the manifest variables, which implies reducing the information that is contained in the data to their covariance structure, and secondly emph{Pearson}-type statistic when manifest variables are binary. In this work, we propose an alternative Goodness-of-fit statistic based on some distance comparison between the latent scores and the original data. This GFI takes into account the nature of each manifest variable and can in principle be applied in various situations and in particular with models with both discrete and continuous manifest variables. We propose two procedures to compute the p-values of our GFI. The first one is based on the asymptotic distribution of a U-statistic and appears to be quite difficult to implement numerically. The second one is based on resampling techniques and requires a consistent estimator of the loadings, the scores, and a corresponding asymptotic covariance matrix. A simulation study based on the second procedure shows that our GFI has good performance in terms of empirical level and empirical power, especially compared to the one proposed by citeN{bentsat}. Finally, a real dataset is analyzed to highlight the application of the methodology. In most health surveys the state of health of individuals is measured through several qualitative, discrete quantitative or dichotomic variables. From these variables, one aims at building univariate indicators of health that summarize the information. To do so, we propose to use a GLLVM, in which the latent variables are the health indicators. We consider the data from the 1997 Swiss Health Survey and we define a new model with two health indicators. The first one describes the health status induced merely by the age of the subject, while the second one captures another dimension of the health status. This latter model is not rejected by our GFI and gives another insight into the understanding of the health status of the population.
URN: urn:nbn:ch:unige-67331
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CONNE, David. Goodness-of-fit for generalized linear latent variables models. Université de Genève. Thèse, 2008. https://archive-ouverte.unige.ch/unige:6733

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Deposited on : 2010-06-01

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