Split-Plot Design: Extension and Robustness Aspects

Within a general linear model framework, optimality of OLS estimators may be obtained if and only if the covariance structure of the stochastic error has a special pattern. Rao (1967) and Zyskind (1967) gave a complete characterization of this pattern. This paper is devoted to the examination of the robustness of standard split-plot analysis for a two-factor design with complete blocks under normality assumptions. Real-world applications considerably modify the randomized nature of treatments within blocks. A second factor B is nested with common levels of the first one, A, so that the independence assumption of error terms is violated. The usual equicorrelation assumption, characterizing split-plot designs, may no longer be assumed when an unknown spatial dependence is acting on the error term. Under this general assumption it is proved that the usual tests for block and main effects of A are exact and equivalent to the standard case. Canonical tests for the main effect of B and interaction A x B are not F-distributed. Nevertheless, with an approximation it is possible to recover robustly the usual statistical procedure.