Split-Plot design: a robust analysis

This paper is devoted to the examination of the robustness of standard split-plot analysis for a two factors design with complete blocks under normality assumptions. For instance, in ceramics firing, the main factors are the oven temperature A and the clay mixture $B$. With a block factor it is possible to control different geometrical shapes of each piece. In a factorial design each piece would be treated separately and, in this case, at fixed temperature A for a particular clay mixture B. If the design is a split-plot, a batch (whole-plot) is defined with different clay mixture pieces (sub-plots) of the same shape (block) and is simultaneously treated at a fixed temperature in the oven. The simultaneous application of treatments to a group of experimental units violates the independence of error terms. 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. The standard 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 well-known procedure.