Correcting for selection bias via cross-validation in the classification of microarray data

There is increasing interest in the use of diagnostic rules based on microarray data. These rules are formed by considering the expression levels of thousands of genes in tissue samples taken on patients of known classification with respect to a number of classes, representing, say, disease status or treatment strategy. As the final versions of these rules are usually based on a small subset of the available genes, there is a selection bias that has to be corrected for in the estimation of the associated error rates. We consider the problem using cross-validation. In particular, we present explicit formulae that are useful in explaining the layers of validation that have to be performed in order to avoid improperly cross-validated estimates.