Estimation of Generalized Linear Latent Variable Models

Abstract

Generalized Linear Latent Variable Models (GLLVM), as de ned in Bartholomew and Knott (1999) enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and lead to biased estimators. In this paper, we propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as a M -estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent nite sample properties, in particular when compared with a well established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimentional inequality is analysed to highlight the importance of the methodology.

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“…Some methods also exist in the context of latent variable modeling when the outcomes are of mixed types; see, for example, Catalano & Ryan [8], Fitzmaurice & Laird [9], Sammel et al [10], Regan & Catalano [11], Moustaki & Knott [12], and Huber et al [13]. However, a limitation of these models for mixed outcomes is that the relationship between the measurable outcomes and the Gaussian latent variable must be known a priori.…”

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“…In that approach, polychoric correlations (Muthén 1984, Poon andLee 1987) between the manifest variables are first estimated and then used as sufficient statistics in the normal model (3) with (4) and (5). Huber, Ronchetti, and Victoria-Feser (2004) show that in the case of mixtures between normal and binary manifest variables, this procedure leads to biased estimators and incorrect inference.…”

“…Note that Huber, Ronchetti, and Victoria-Feser (2004) pointed out that Proposition 2 (Consistency) Let θ ∈Θ. If Θ is compact,…”

“…given by (9), we use a Laplace approximation (see De Bruijn 1981) to approximate the likelihood function of the sample as it has been done in Huber, Ronchetti, and Victoria-Feser (2004) for other types of variables.…”