## 5.1 The difference between the estimates of fixed and random effects

In ordinary regression analysis and analysis of variance, the random-effect terms are always *nuisance variables*: residual variation or block effects. The effects of individual levels of such terms are not of interest. However, we have seen that in a mixed model, effects that are of intrinsic interest may be specified as random – for example, the effects of the individual breeding lines in the barley field trial discussed in Chapter 3. The decision to specify a term as random causes a fundamental change in the way in which the effect of each level of that term is estimated. We will illustrate this change and its consequences in the context of the barley breeding lines. However, the concepts introduced and the arguments presented apply equally to any situation in which replicated, quantitative evaluations are available for the comparison of members of some population – for example, new chemical entities to be evaluated as potential medicines by a pharmaceutical company, or candidates for admission to a university on the basis of their examination scores.

It is easy to illustrate the relationship between the estimates of fixed and random effects in data that are grouped by a single factor – for example, a fully randomized design leading to a one-way anova. The data from the barley field trial are classified by two factors, ...