Fit indices and fit measures commonly used to determine the accuracy and desirability of structural equation models are expected to be insensitive to nonlinearity in the data. This includes measures as ubiquitous as the CFI, TLI, RMSEA, SRMR, AIC, and BIC. Despite this, some software will report these measures when certain models are used. Consequently, some researchers may be led to use these fit measures without realizing the impropriety of the act. Alternative fit measures have been proposed, but these measures require further testing. As part of this thesis, a large simulation study was carried out to investigate alternative fit measures and to confirm whether the traditional measures are practically blind to nonlinearity in the data. The results of the simulation provide conclusive evidence that fit statistics and fit indices based on the chi-square distribution or the residual covariance matrix are entirely insensitive to nonlinearity. The posterior predictive p-value was also insensitive to nonlinearity. Only fit measures based on the structural residuals (i.e., HFI and R-squared) showed any sensitivity to nonlinearity. Of these, the R-squared was the only reliable measure of nonlinear model misspecification. This thesis shows that an effective strategy for determining whether a nonlinear model is preferable to a linear one involves using the R-squared to compare models that have been fit to the same data. An R-squared that is much larger for the nonlinear model than the linear model suggests that the linear model may be less desirable than the nonlinear model. The proposed method is intended to be supplementary to substantive theory. It is argued that any dependence on fit indices or fit statistics that places these measures on a higher pedestal than substantive theory will invariably lead to blindness on the part of the researcher. In other words, unwavering adherence to goodness-of-fit measures limits the researchers vision to what the measures themselves can detect.



College and Department

David O. McKay School of Education; Instructional Psychology and Technology

Date Submitted


Document Type





structural equation modeling, goodness-of-fit, nonlinear statistical models, Bayesian analysis