The Noncentral Chi-square Distribution in Misspecified Structural Equation Models: Finite Sample Results from a Monte Carlo Simulation

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Abstract
The noncentral chi-square distribution plays a key role in structural equation modeling (SEM). The likelihood ratio test statistic that accompanies virtually all SEMs asymptotically follows a noncentral chi-square under certain assumptions relating to misspecification and multivariate distribution. Many scholars use the noncentral chi-square distribution in the construction of fit indices, such as Steiger and Lind's (1980) Root Mean Square Error of Approximation (RMSEA) or the family of baseline fit indices (e.g., RNI, CFI), and for the computation of statistical power for model hypothesis testing. Despite this wide use, surprisingly little is known about the extent to which the test statistic follows a noncentral chi-square in applied research. Our study examines several hypotheses about the suitability of the noncentral chi-square distribution for the usual SEM test statistic under conditions commonly encountered in practice. We designed Monte Carlo computer simulation experiments to empirically test th...