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

Chapter 3Estimation of the variances of random-effect terms

3.1 The need to estimate variance components

In Chapters 1 and 2, we recognized random-effect terms and took them into account when fitting models to data, in order to interpret the fixed-effect terms more reliably. In Chapter 1, the emphasis was on the effect of latitude (a fixed-effect term) on house prices: the additional variation among towns (a random-effect term) was treated as a nuisance variable that reduced the precision with which the effect of latitude was estimated, and that must be taken into account in order to assess the statistical significance of this effect realistically. In Chapter 2, the emphasis was on the effects of brand and assessor (fixed-effect terms) on the perceived saltiness of ravioli, and the additional variation among days, presentations and servings (random-effect terms) was treated as a set of nuisance variables. However, a random-effect term may be of interest in its own right. Sometimes the means of individual levels of a random-effect factor are of interest, but even if this is not the case, it may still be useful to estimate the variance of the effects among the population of levels from which the sample studied was drawn. This chapter examines the methods for estimating the variance due to each random-effect term, and the interpretation of the results.

3.2 A hierarchical random-effects model for a three-stage assay process

In an assay process, several sources of random variation ...