Mixed Effects Designs: The Symmetry Assumption and Missing Data

Abstract

The classical F test for the hypothesis of no main fixed effects in the two-way crossed mixed effects design is derived under model assumptions that include normality, variance homogeneity, and symmetry. The symmetry assumption specifies that the random main and interaction effects are independent. While it is known that the F test is robust against violations of the normality assumption and/or the homoscedasticity assumption in the balanced case, the effects of the symmetry assumption have not been investigated. Our simulations demonstrate that the F test becomes very liberal under violations of the symmetry assumption. A new test procedure is developed under a more flexible model, which does not require the restrictive assumptions of the classical model. By considering the cell sample sizes to be random, the procedure applies equally well to unbalanced designs and to designs with empty cells, provided the missingness is completely at random. The asymptotic theory of the test statistic pertains to designs where the number of levels of both the fixed and random effects is large. The limiting distribution of the proposed test statistic is an infinite weighted sum of independent χ² ₁ random variables. An approximation to this limiting distribution is proposed. Extensive simulations indicate that the proposed test procedure achieves levels reasonably close to the nominal ones under violations of the symmetry and other classical assumptions, and under different patterns of missingness. An analysis of a dataset from the Mussel Watch Project is presented. The supplementary materials for this article are available online and contain the proofs pertaining to the asymptotic distribution of the proposed statistic, as well as the results of simulations with nonnormal errors and random effects.

KEY WORDS:

• Asymptotic theory
• Heteroscedasticity
• Hypothesis testing
• Main effects
• Spectrum approximation
• Unbalanced designs
• U-statistics

SUPPLEMENTARY MATERIALS

Appendix with Proofs and Results of Additional Simulations: Because the proofs used to establish the asymptotic distribution of the proposed statistic are quite lengthy, they are presented in this supplemental file. In addition, we include the results of simulations where the data are generated such that either the distribution of the random effects or the errors is nonnormal. (PDF type)

This research was supported in part by NSF grants SES-0318200 and DMS-0805598. This research was initiated during the second author's sabbatical visit at the Australian National University in the Spring of 2006. Helpful discussions with Peter Hall are gratefully acknowledged.