Maximum likelihood and Bayesian estimation for ….pdf

Maximum likelihood and Bayesian estimation for nonlinear structural equation models Melanie Wall Email: ******@, phone: (612)625-2138, fax: (612)626-0660 Division of Biostatistics School of Public Health University of Minnesota A460 Mayo Building, MMC 303 Minneapolis, MN 55455-0378, USA December 31, 2007 1 Introduction Structural equation modeling (SEM) began at its roots as a method for modeling linear rela- tionships among latent variables. The well-known software for SEMname LISREL (J¨ oreskog and S¨ orbom, 1996) stands for “ Linear Structural Relations”. But, in many cases, the re- striction to linearity is not adequate or ?exib le enough to explain the phenomena of interest. For example, if the slope between two continuous latent variables is directly a?ected or “mod- erated” by a third continuous latent variable, this relationship which can be modeled via a cross-product term between the two latent variables, cannot be estimated via the traditional SEM methods. The di?culty is that traditional estimation methods appropriate for ?tting linear structural models are focused on minimization of a discrepancy function between the observed and modeled covariance matrix and this cannot be extended in a straightforward way to handle nonlinear structural models. Tha t is, estimation of parameters in a nonlinear structural model cannot be plished usi ng only the sample covariance matrix of the observed data. Kenny and Judd (1984) introduced the ?rst statistical method aimed at producing es- timates of parameters in a nonlinear structur al equation model (speci?cally a quadratic or cross-product structural model with a linear m easurement model). The basic idea of Kenny and Judd (1984) was to create new “observed variables” by taking products of existing vari- ables and then using them as additional indicators of the nonlinear terms in the model. The method as described by Kenny and Judd (1984) resulted in many tedious constraints on the model covariance