Variances are not always nuisance parameters

In classical problems, e.g., comparing two populations, fitting a regression surface, etc., variability is a nuisance parameter. The term nuisance parameter is meant here in both the technical and the practical sense. However, there are many instances where understanding the structure of variability is just as central as understanding the mean structure. The purpose of this article is to review a few of these problems. I focus in particular on two issues: (a) the determination of the validity of an assay; and (b) the issue of the power for detecting health effects from nutrient intakes when the latter are measured by food frequency questionnaires. I will also briefly mention the problems of variance structure in generalized linear mixed models, robust parameter design in quality technology, and the signal in microarrays. In these and other problems, treating variance structure as a nuisance instead of a central part of the modeling effort not only leads to inefficient estimation of means, but also to misleading conclusions.