Missing Data Mechanisms and Homogeneity of Means and Variances-Covariances

Unless data are missing completely at random (MCAR), proper methodology is crucial for the analysis of incomplete data. Consequently, methods for effectively testing the MCAR mechanism become important, and procedures were developed via testing the homogeneity of means and variances-covariances across the observed patterns (e.g., Kim & Bentler in Psychometrika 67:609-624, 2002; Little in J Am Stat Assoc 83:1198-1202, 1988). The current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data, and the normal-distribution-based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random, although the values may not equal those of the underlying population distribution. The results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the MCAR mechanism even when the population distribution is multivariate normal.