Consistent estimators of the variance-covariance matrix of the gmanova model with missing data

Tests for gaussian repeated measures with missing data in small samples

For small samples of Gaussian repeated measures with missing data, Barton and Cramer recommended using the EM algorithm for estimation and reducing the degrees of freedom for an analogue of Rao's F approximation to Wilks' test. Computer simulations led to the conclusion that the modified test was slightly conservative for total sample size of N=40. Here we consider additional methods and smaller sample sizes, Nin¿12,24¿. We describe analogues of the Pillai-Bartlett trace, Hotelling-Lawley trace and Geisser-Greenhouse corrected univariate tests which allow for missing data. Eleven sample size adjustments were examined which replace N by some function of the numbers of non-missing pairs of responses in computing error degrees of freedom. Overall, simulation results allowed concluding that an adjusted test can always control test size at or below the nominal rate, even with as few as 12 observations and up to 10 per cent missing data. The choice of method varies with the test statistic. Replacing N by the mean number of non-missing responses per variable works best for the Geisser-Greenhouse test. The Pillai-Bartlett test requires the stronger adjustment of replacing N by the harmonic mean number of non-missing pairs of responses. For Wilks' and Hotelling-Lawley, an even more aggressive adjustment based on the minimum number of non-missing pairs must be used.