Introduction to Mixed Modelling: Beyond Regression and Analysis of Variance, 2nd Edition

Chapter 9The use of mixed models for the analysis of unbalanced experimental designs

9.1 A balanced incomplete block design

In Chapter 2, we saw that the analysis of the variance of a standard, balanced experimental design, the split plot design, could be viewed as the fitting of a mixed model, the treatment terms being the fixed-effect terms and the block terms being the random-effect terms. One of the major uses of mixed modelling is the analysis of experiments that cannot be tackled by the analysis of variance, because it has not been possible to achieve exact balance during the design phase.

A balanced incomplete block design presented by Cox (1958, Sections 11.1 and 11.2, pp. 219–230) provides the starting point for a simple illustration of this problem. The data are displayed in a spreadsheet in Table 9.1. (Data reproduced by kind permission of Wiley and Sons, Inc.) This experiment comprises five treatments, T1 to T5. It is expected that there will be random variation in the response from day to day, and ‘day’ is therefore to be included in the analysis as a block term. However, each treatment does not occur on every day: though there are five treatments, there are only three observations per day, so this is clearly not an ordinary randomized complete block design. Each block is incomplete. Such a compromise is often necessary: nature does not always provide groups of experimental units (areas of land, litters of animals) that are the right size to permit the application ...