Analyzing categorical data from split-plot and other multi-stratum experiments

Many factorial experiments yield categorical response data. Moreover, the experiments are often run under a restricted randomization for logistical reasons and/or because of time and cost constraints. The combination of categorical data and restricted randomization necessitates the use of generalized linear mixed models. In this paper, we demonstrate the use of Hasse diagrams for laying out the randomization structure of a complex factorial design involving seven two-level factors, four three-level factors and a five-level factor, and three repeated observations for each experimental unit. The Hasse diagrams form the basis of the mixed model analysis of the ordered categorical data produced by the experiment. We also discuss the added value of categorical data over binary data and difficulties with the estimation of variance components and, consequently, with the statistical inference. Finally, we show how to deal with repeats in the presence of categorical data, and describe a general strategy for building a suitable generalized linear mixed model.