A class of statistics Ĵ(ν) is proposed for constructing tests and confidence intervals for a linear combination of means ν using an incomplete multivariate normal sample. The class includes statistics based on maximum likelihood estimates for all the data, and simpler statistics such as those based on a subset of complete observations. A simple measure of relative efficiency is proposed to decide when a simpler statistic is appropriate. For the case of two variables with extra observations on one variable, approximate t distributions are suggested for some members of Ĵ, and resulting test statistics are compared in a simulation study.