Statistica Sinica 24 (2014), 43-61

TWO-FOLD NESTED DESIGNS:

THEIR ANALYSIS AND CONNECTION WITH

NONPARAMETRIC ANCOVA

Shu-Min Liao and Michael G. Akritas

Amherst College and The Pennsylvania State University

Abstract: In the context of a nonparametric model for the unbalanced heteroscedastic two-fold nested design, we consider the problem of testing for the sub-class effect. The asymptotic theory pertains to cases with a large number of sub-classes, and small number of classes. It is shown that the classical $F$-statistic is very sensitive to departures from homoscedasticity, even in balanced designs. We propose new test statistics when heteroscedasticity is of the between-classes type, as well as for the general heteroscedastic design. Their asymptotic distributions, both under the null and local alternative hypotheses, are established. The ramifications of these results to the hypothesis of no covariate effect in the nonparametric analysis of covariance are discussed. Simulation studies compare the finite sample performance of the proposed statistics with those of the classical $F$-test and the GEE test. Two data sets are analyzed.

Key words and phrases: Asymptotic distributions, heteroscedasticity, nonparametric testing, non-normality, sub-class effect, unbalanced designs.