Parametric bootstrap tests for unbalanced nested designs under heteroscedasticity

Abstract

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.

Keywords:

• bootstrap re-sampling
• generalized p-values
• nested models
• unbalanced data

AMS Subject Classifications:

• 62F03
• 62F40
• 62J10

Acknowledgements

The authors are grateful to the referees and an Associate Editor for helpful comments and clarifying suggestions resulting in the present version. Xu's research was supported by grants from the National Natural Science Foundation of China (NSFC) (No. 11171002), Beijing Natural Science Foundation (The Theory of Mixed Effects Models of Multivariate Complex Data and Its Applications, No. 1112008).