Bayesian Statistics: An Introduction, 4th Edition

6.6 The two way layout

6.6.1 Notation

Sometimes we come across observations which can be classified in two ways. We shall consider a situation in which each of a number of treatments is applied to a number of plots in each of a number of blocks. However, the terminology need not be taken literally; the terms treatments and blocks are purely conventional and could be interchanged in what follows. A possible application is to the yields on plots on which different varieties of wheat have been sown and to which different fertilizers are applied, and in this case either the varieties or the fertilizers could be termed treatments or blocks. Another is to an analysis of rainfall per hour, in which case the months of the year might be treated as the blocks and the hours of the day as the treatments (or vice versa).

We consider the simplest situation in which we have N=IJK observations

to which I treatments have been applied in J blocks, the observations having a common variance . For simplicity, we assume independent reference priors uniform in the and , that is,

The likelihood is

where