In the context of a normal model, testing problems with missing data are considered. Tests on means are treated when independent extra data on the first $p_1$ variates of $p$ variates is available in addition to complete data. For testing that the mean of the first $p_1$ variates is zero, the LRT is UMP invariant, but for testing that the whole mean is zero, no LMPI (locally most powerful invariant) test exists. Second, tests for independence are treated in similar situations, and an LMPI test for each situation is derived. In some situations it is found that the LRT for independence ignores the extra data.
"Multivariate Tests with Incomplete Data." Ann. Statist. 11 (2) 654 - 665, June, 1983. https://doi.org/10.1214/aos/1176346170