Global Model Fit for Mixed-Effect Models

• In Event: Past, Present, and Future of Model Fit: Modern Solutions to Historical Challenges

Abstract

It has long been suggested that modeling repeated measures via a mixed effects framework and factor analytic framework are highly similar with subtle differences (Curran, 2003; McNeish & Matta, 2019) . One key difference was the factor analytic framework enjoyed a formal test of global model fit as well as a set of global model fit indices. The mixed effect framework only had tests of relative model fit and pseudo-R-squared metrics. This paper develops the test of global model fit for the mixed effect modeling framework.

The paper begins by presenting two competing concepts of the “saturated model” for repeated measures data. The first conception, which is dominant in the regression literature, is parameter saturation relative to the data. Such a saturated model “explains all the variation by the systematic component of the model” (Agresti, 2013, p. 136) and leads to the test of relative model fit—the deviance test (Pawitan, 2001). The second conception, which is dominant in the SEM literature, reflects parameter saturation of the population distribution function. “The fit of a considered structural equation model is evaluated via comparison with a particular model that reproduces perfectly the analyzed [covariance matrix and means]” (Raykov, Marcoulides, & Patelis, 2013, p. 163). It is this conception of the saturated model that leads to the so-called test of global model fit.

Next, the paper describes how the likelihood for the saturated population distribution can be formed and calculated under the mixed effect parameterization, as well as the degrees of freedom between the saturated population distribution and the hypothesized model. With the likelihood for the saturated model, likelihood for the hypotheses model, and degrees of freedom, we show how the difference between this saturated likelihood and the model likelihood has a chi-square distribution, leading to the formal test of global model fit used in the SEM literature. The paper concludes by showing how this information can be used to derive many of the global model fit indices used throughout the SEM literature.

Author

• Tyler Matta, NWEA