Some Aspects of the Statistical Analysis of 'Split Plot' Experiments in Completely Randomized Layouts
The statistical analysis of completely randomized 'split-plot' experiments is discussed from the point of view of the underlying multivariate model. In doing this, first certain well-known aspects of the parametric case are reviewed; however, attention is primarily directed toward the development of appropriate non-parametric procedures. The general structure of 'split-plot' experiments involves $N$ randomly chosen subjects to whom treatments have been assigned according to a completely randomized design and from each of whom is obtained an observation vector, the components of which represent the responses of the subject to each one of several conditions. Hence, the data matrix has the appearance of a set of mixed models, each one of which corresponds to a particular treatment. The different conditions correspond to the 'split plot' treatments in agricultural experiments while the different treatments correspond to the 'whole plot' treatments. Alternatively, such designs may be interpreted simply as multivariate one-way layouts in which the components of the observation vector have been measured in the same units (or on the same scale) and hence are comparable. In such experiments, a number of hypotheses are of interest-the hypothesis of no treatment effects, the hypothesis of no condition effects, the hypothesis of no interaction between treatments and conditions. Various formulations of these hypotheses are discussed under several different combinations of assumptions concerning the joint distribution of the components of the observation vector. In each case considered, appropriate parametric or non-parametric test procedures are indicated. Some of the methods considered in the paper are illustrated in a numerical example. The example is representative of a situation in which some of the standard assumptions regarding normality and variance homogeneity may not hold. In this part of the paper, certain aspects of the efficient computation of the various test criteria are indicated. Corresponding algorithms are presented in Koch and Sen  and Koch . Finally, computer programs based on these algorithms have been written and can be made available to interested persons
Koch, GG. (1969). Some Aspects of the Statistical Analysis of 'Split Plot' Experiments in Completely Randomized Layouts. Journal of the American Statistical Association, 64(326), 485-505.