In order to improve the chi-squared approximation of the log likelihood ratio statistic for multivariate normal test problems concerning structured covariance matrices, the Bartlett adjustments are derived from the simple connection between the adjustment factor and the normalizing constant of the conditional density of the maximum likelihood estimator.
In particular, the Bartlett adjustments for 10 basic multivariate test problems are derived in detail. For three of these testing problems we discuss the accuracy of the chi-squared approximation of the log likelihood ratio statistic in its unadjusted form and adjusted with one of four asymptotically equivalent Bartlett adjustments. Finally, the Bartlett adjustments for multivariate normal test problems concerning a linear structure in the mean value are derived.