Accurate critical constants for the one-sided approximate likelihood ratio test of a normal mean vector when the covariance matrix is estimated.

Tang, Gnecco, and Geller (1989, Biometrika 76, 577-583) proposed an approximate likelihood ratio (ALR) test of the null hypothesis that a normal mean vector equals a null vector against the alternative that all of its components are nonnegative with at least one strictly positive. This test is useful for comparing a treatment group with a control group on multiple endpoints, and the data from the two groups are assumed to follow multivariate normal distributions with different mean vectors and a common covariance matrix (the homoscedastic case). Tang et al. derived the test statistic and its null distribution assuming a known covariance matrix. In practice, when the covariance matrix is estimated, the critical constants tabulated by Tang et al. result in a highly liberal test. To deal with this problem, we derive an accurate small-sample approximation to the null distribution of the ALR test statistic by using the moment matching method. The proposed approximation is then extended to the heteroscedastic case. The accuracy of both the approximations is verified by simulations. A real data example is given to illustrate the use of the approximations.