Computational Statistics & Data Analysis 51 (2007) 4164 – 4177 www.elsevier.com/locate/csda Parameter constraints in generalized linear latent variable models R. Tsonaka a , I. Moustaki b, ∗ a Biostatistical Centre, Catholic University of Leuven, Belgium b Department of Statistics, Athens University of Economics and Business, 76 Patission street, 104 34 Athens, Greece Received 25 March 2005; received in revised form 8 April 2006; accepted 10 April 2006 Available online 11 May 2006 Abstract Parameter constraints in generalized linear latent variable models are discussed. Both linear equality and inequality constraints are considered. Maximum likelihood estimators for the parameters of the constrained model and corrected standard errors are derived. A significant reduction in the dimension of the optimization problem is achieved with the proposed methodology for fitting models subject to linear equality constraints. © 2006 Elsevier B.V.All rights reserved. Keywords: Generalized linear latent variable models; Linear equality and inequality constraints; Lagrange multipliers; Adaptive barrier method 1. Introduction Latent variable models are widely used in social science research where variables of major interest such as ability, attitudes, behavior, cannot be directly measured. For example, in educational testing the performance of students on a number of tests is used as an indicator of ability or in economics variables such as income, expenditures, ownership of car, summer house, etc., are used to measure wealth. In some cases, a construct can be represented by a single latent variable but often it is multidimensional and therefore more than one latent variable is needed. In latent variable modelling there is often the need to impose parameter constraints. More specifically, model parameters such as intercepts and factor loadings are constrained to be equal to or greater/less than a fixed value or other parameters according to some linear or non-linear function. Constraints are mainly required in confirmatory analysis where factor loadings follow a pre-specified pattern. In addition, fixed value constraints provide one way of identifying the scale of a latent variable (see Bollen, 1989, p. 183). Equality constraints among parameters in different groups are used for testing measurement invariance. Finally, constraints need to be imposed for making a model identified. Lack of identification implies that the model contains insufficient information for the purpose of attaining a determinate solution. Fitting models under parameter constraints has been studied in structural equation modelling where limited informa- tion estimation methods are mainly used (Jöreskog, 1971; Bentler and Weeks, 1980; McDonald, 1980; Lee, 1980; Lee and Tsui, 1982; Bentler and Lee, 1983; Rindskopf, 1983, 1984; Jamshidian and Bentler, 1993). Constraints have also ∗ Corresponding author. E-mail addresses: spyridoula.tsonaka@med.kuleuven.be (R. Tsonaka), moustaki@aueb.gr (I. Moustaki). 0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.04.023