Structural Equation Models: Fit and generalisation (22)

Chair: Roger Millsap, Tuesday 21st July, 15.25 - 16.45, Uppercroft, School of Pytahgoras. 

Johan Lyhagen, Department of Information Science, Divison of Statistics, Uppsala University, Sweden. Non-normal structural equation modeling: Regression augmented maximum likelihood. (185)

Daniel Bauer, Ruth Mathiowetz, Nisha Gottfredson, Jolynn Pek and Diane Losardo, L.L. Thurstone Psychometric Laboratory, Department of Psychology, University of North Carolina at Chapel Hill, USA. Modeling non-linear relationships among latent variables via mixtures of linear structural equations. (054)

Ab Mooijaart, Leiden University, The Netherlands, and Albert Satorra, Universitat Pompeu Fabra and Barcelona GSE, Spain. Testing for non-linear relationship in Structural Equation Modeling. (151)

Roger Millsap and Soyoung Lee, Department of Psychology, Arizona State University, USA. Approximate fit in SEM without a priori cutpoints. (087)

ABSTRACTS

Non-normal structural equation modeling: Regression augmented maximum likelihood. (185)
Johan Lyhagen
In applied work data are usually not normal, hence, using maximum like-lihood assuming normality yields in the best case an e¢ ciency loss. In line with Im and Schmidt (2008, Journal of Econometrics) we propose augmenting the maximum likelihood estimator in structural equation modeling with suitable functions of the observational residual. We prove the consistency of the estimator. A Monte Carlo simulation analyzing the small sample properties shows very promising results outperforming the pseudo/quasi maximum likelihood estimator.

Modeling non-linear relationships among latent variables via mixtures of linear structural equations. (054)
Daniel Bauer, Ruth Mathiowetz, Nisha Gottfredson, Jolynn Pek and Diane Losardo
A key assumption of the standard structural equations model is that the latent variables are linearly related. In the past two decades, many approaches have been proposed to relax this assumption and allow for the incorporation of nonlinear effects, typically through the inclusion of low-order powers or products of latent variables. Often, researchers are cautioned to use these approaches only when there is strong theory regarding the presence and form of the putative nonlinear effects. In practice, however, theory is seldom specific enough to imply a quadratic curve or bilinear interaction or even the existence of a nonlinear effect, so a more exploratory, diagnostic approach may be required. Accordingly, we propose a semiparametric approach that allows for the estimation of nonlinear effects of unknown functional form among latent variables. A mixture of linear structural equations is first fit to the data. The nonlinear regression surface is then approximated by aggregating over the mixing components. We show that this approach can effectively recover nonlinear relationships of various forms and that it compares favorably to ad hoc diagnostics based on factor score estimates. We also propose methods for generating pointwise confidence intervales for the nonlinear regression surface.

Testing for non-linear relationship in Structural Equation Modeling. (151)
Ab Mooijaart and Albert Satorra
In this paper we first show that for some structural equation models (SEM), the classical chi-square goodness-of-fit test is unable to detect the presence of non-linear terms in the model. Not only the model test has zero power against that type of misspecifications, but even the theoretical (chi-square) distribution of the test is not distorted when severe interaction term misspecification is present in the postulated model. This may lead to the wrong conclusion that a proposed linear model fits the data well according to the chi-square goodness-of-fit test, while the underlying model may in fact be severely non-linear (see Mooijaart & Satorra, Psychometrika (2009), for an explanation of these phenomena). Secondly, we show that introducing in the analysis higher-order cross-product terms (beyond the usual means and covariances) we are able to test for the presence of non-linear terms in the model. A relevant issue, however, is which set of specific higher order-order moments induces more power in the analysis of a specific model with interaction terms. In relation to different test statistics, we assess the power for interaction terms in the model for different added higher-order moments. We investigate the use of the power as the criterion for selection of the third-order moments to be included in the first- and second-order moments. The statistical theory and a small Monte Carlo study will be presented.

Approximate fit in SEM without a priori cutpoints. (087)
Roger Millsap and Soyoung Lee
Recent research has questioned the generality of some of the cutpoints in standard use for approximate fit indices in SEM. A general problem is that over different model structures and data conditions, it is difficult to determine what types of model misspecifications are consistent with a given level of approximate fit. As a consequence, it is difficult to specify cutpoints for common indices of fit. Here we present a simulation-based method for determining whether the fit of a given model is consistent with particular forms of misspecification. The method does not require the investigator to specify a priori cutpoints for the approximate fit index of interest. The method is described and an example of an application is given.