ABSTRACT
A distinction was made in §1.7.5 between fixed, random, and mixed effects models based on the number of random variables and fixed effects involved in the statistical model. A mixed effects model — or mixed model for short — contains fixed effects as well as at least two random variables (one of which is the obligatory model error). Mixed models arise quite frequently in designed experiments. In a completely randomized design with subsampling, for example, treatments are assigned at random to experimental units. A random subsample of observations is then drawn from every experimental unit. This design is practical if an experimental unit is too large to be measured in its entirety, for example, a field plot contains twelve rows of a particular crop but only three rows per plot can be measured and analyzed. Soil samples are often randomly divided into subsamples prior to laboratory analysis. The statistical model for such a design can be written as
, [7.1]
where are zero mean random variables representing the experi- mental and observational errors, respectively. If the treatment effects are fixed, i.e., the levels of the treatment factor were predetermined and not chosen at random, this is a mixed model.
