For decomposable covariance selection models, stochastic inequalities which relate the null distribution of the log-likelihood ratio statistic to its asymptotic $\chi^2$ distribution are obtained. The implications are twofold: First, the null distribution of the log-likelihood ratio statistic is seen to be stochastically larger than its asymptotic $\chi^2$ distribution. Extremely large samples apart, for the $\chi^2$ approximation to be valid, a deflation of the log-likelihood ratio statistic is then necessary. Second, a simple adjustment to the log-likelihood ratio statistic, similar in spirit to the Bartlett adjustment, yields a conservative test.
"Stochastic Inequalities Relating a Class of Log-Likelihood Ratio Statistics to their Asymptotic $\chi^2$ Distribution." Ann. Statist. 17 (4) 1723 - 1734, December, 1989. https://doi.org/10.1214/aos/1176347390