Sphericity Test on Variance-Covariance Matrix with Monotone Missing Data

This study considers the sphericity test, a specific test of variance-covariance matrix under monotone missing data for a one-sample problem. We provide the likelihood ratio (LR) and derive an asymptotic expansion of the likelihood ratio test (LRT) statistic and modified LRT statistic for the null distribution. We also derive the upper percentiles of the LRT statistic and modified LRT statistic when the null hypothesis holds, and provide approximate upper percentiles. Furthermore, we prove that the LR under monotone missing data is affine invariant under the null hypothesis. For complete data, we provide an asymptotic expansion of the LRT statistic and modified LRT statistic for the null distribution. Furthermore, we numerically evaluate the actual type I error rates for the approximate upper percentiles using Monte Carlo simulation and provide examples of the LRT statistic and modified LRT statistic and approximate upper percentiles under monotone missing data.