Recovery of interblock, intergradient, and intervariety information in incomplete block and lattice rectangle designed experiments

Spatial analysis and blocking analysis of experimental results are treated separately in the literature. Here we combine these analyses into a single analysis. The information arising from the random nature of different gradients within incomplete blocks is used to adjust treatment means. We extend Cox's (1958, Journal of the Royal Statistical Society, Series B 20, 193-204) idea of differential gradients within columns of a Latin square to within blocks for incomplete block and row-column designed experiments and, in addition, treat them as random effects. With this analysis, the restrictions on randomization due to blocking are taken into consideration whereas they are often ignored in spatial analysis literature. Some comments on designing experiments and analyzing experimental results to control heterogeneity are presented. A numerical example illustrates the computational procedure and indicates effect of alternative analyses. The class of augmented experiment designs has been found useful for experiments involving comparisons of standard check treatments with a set of new and untried treatments, usually with one replicate. Interreplicate, interblock, interrow, and/or intercolumn information is available to use in obtaining solutions for new treatment effects. Since the new treatment effects are often considered to be random effects, their distributional properties may be used to increase the efficiency of the experiment. We demonstrate the statistical procedures for recovering this information in block and row-column designs using mixed model procedures.