Uncorrelated residuals and an exact test for two variance components in experimental design

The error contrasts from an experimental design can be constructed from uncorrelated residuals normally associated with the linear model. In this paper uncorrelated residuals are defined for the linear model that has a design matrix which is less than full rank, typical of many experimental design representations. It transpires in this setting, that for certain choices of uncorrelated residuals, corresponding to recursive type residuals, there is a natural partition of information when two variance components are known to be present. Under an assumtion of normality of errors this leads to construction of appropriate F-tests for testing heteroscedasticity. The test, which can be optimal, is applied to two well known data sets to illustrate its usefullness.