Partial Mixed Effects Split-Plot Design under Unknown Spatial Dependence

It is well-known that sequential nested treatments give rise to unknown dependent error effects. Factorial designs assume independence of error term so that each experiment must be performed one at a time. Grouping experimental units is a common practice, therefore, statistical analysis must be conveniently modified. For instance, in determining the effect of annealing temperature, factor A, in breaking strength of experimental metal alloys, the laboratory ovens are arranged with each different metal samples, factor B, (one for each alloy). The temperature levels assigned to each oven are different. The experiment may be replicated (blocking factor) during three different tours (shifts). This is not a factorial design with blocking factors, because annealing is performed simultaneously (whole-plots) on different metal samples (sub-plots). Standard split-plot design is based upon fixed effects and a special additive version of independent error term components (equicorrelation) under normality. In many situations the effects of factors A and B may be no longer assumed fixed and, moreover, the spatial arrangement of sub-plots may not agree with the simple symmetric equicorrelation structure of residuals. The paper is devoted to the combined statistical analysis under partial mixed effects and unknown spatial dependence. Under these extended assumptions, it is proved that the usual standard significancy tests for block and factor $A$ effects are exact in probability. The corresponding standard tests for main effects of B and interaction A x B are not F-distributed, nevertheless, a simple robust procedure may be applied.