On the likelihood ratio test in structural equation modeling when parameters are subject to boundary constraints.

Abstract

The authors show how the use of inequality constraints on parameters in structural equation models may affect the distribution of the likelihood ratio test. Inequality constraints are implicitly used in the testing of commonly applied structural equation models, such as the common factor model, the autoregressive model, and the latent growth curve model, although this is not commonly acknowledged. Such constraints are the result of the null hypothesis in which the parameter value or values are placed on the boundary of the parameter space. For instance, this occurs in testing whether the variance of a growth parameter is significantly different from 0. It is shown that in these cases, the asymptotic distribution of the chi-square difference cannot be treated as that of a central chi-square-distributed random variable with degrees of freedom equal to the number of constraints. The correct distribution for testing 1 or a few parameters at a time is inferred for the 3 structural equation models mentioned above. Subsequently, the authors describe and illustrate the steps that one should take to obtain this distribution. An important message is that using the correct distribution may lead to appreciably greater statistical power.

Testing Inequality Constrained Hypotheses in SEM Models

“…An appealing property for any p value, and consequently for our plug-in p value, is, considered as a random variable, to be asymptotically uniform [0,1] under the null hypothesis: P.p < 'jH 0 / D '. However, in some situations exact uniformity of p values cannot be attained (Andrews, 2000;Bayarri & Berger, 2000;Galindo-Garre & Vermunt, 2004Stoel et al, 2006). Andrews (2000), for example, showed that the results of the bootstrap procedure are not coherent when inequality constraints are imposed on the model parameters.…”

“…Bootstrapping is an approach for statistical inference falling within a broader class of resampling methods (Efron & Tibshirani, 1993). Various authors have suggested using the parametric bootstrap when the parameter space is restricted (Galindo-Garre & Vermunt, 2004Ritov & Gilula, 1993;Stoel et al, 2006;Tsonaka & Moustaki, 2007).…”

“…However, testing order constraints has received relatively little attention in the structural equation modeling (SEM) literature (Gonzalez & Griffin, 2001; Stoel, Galindo-Garre, Dolan, & Van den Wittenboer, 2006). SEM is often used and its attractiveness is largely due to its flexibility in specifying and testing hypotheses among both observed and latent variables in multiple groups.…”

The Effects of Educational Diversity in a National Sample of Law Students: Fitting Multilevel Latent Variable Models in Data With Categorical Indicators

“…In this case, regularity conditions for a likelihood ratio test are violated (see Stoel, Garre, Dolan, & van den Wittenboer, 2006). Because the between-level residual variances are likely to be zero or very close to zero, this violation cannot be ignored, and so the nested models cannot be compared with this method.…”

A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling.

“…Thus, we should reject the null hypothesis when the observed p value is larger than .10 for ␣ ϭ .05. Readers may refer to Viechtbauer (2007b) and Stoel, Galindo-Garre, Dolan, and van den Wittenboer (2006) for a discussion on this issue in the context of meta-analysis and SEM. The estimates of 2 can sometimes be negative because of the sampling fluctuation.…”