## Abstract

Survey data typically contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. The most widely used statistic for evaluating the adequacy of a SEM model is T_{ML}, a slight modification to the likelihood ratio statistic. Under normality assumption, T_{ML}approximately follows a chi-square distribution when the number of observations (N) is large and the number of items or variables (p) is small. However, in practice, p can be rather large while N is always limited due to not having enough participants. Even with a relatively large N, empirical results show that T_{ML}rejects the correct model too often when p is not too small. Various corrections to T_{ML}have been proposed, but they are mostly heuristic. Following the principle of the Bartlett correction, this paper proposes an empirical approach to correct T_{ML}so that the mean of the resulting statistic approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics follow the nominal chi-square distribution much more closely than previously proposed corrections to T_{ML}, and they control type I errors reasonably well whenever N≥max(50,2p). The formulations of the empirically corrected statistics are further used to predict type I errors of T_{ML}as reported in the literature, and they perform well.

Original language | English |
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Pages (from-to) | 379-405 |

Number of pages | 27 |

Journal | Psychometrika |

Volume | 80 |

Issue number | 2 |

DOIs | |

Publication status | Published - 9 Jun 2015 |

## Keywords

- Bartlett correction
- Bayesian information criterion
- maximum likelihood
- type I errors