Cross-validation for linear model with unequal variances in genomic analysis

In recent years, genomic studies are usually conducted to identify genes that may have an impact on clinical outcomes. The identified genes are then used to establish a predictive model for identifying subjects who are most likely to respond to the test treatment in clinical trials. This information is useful in early and later phases of clinical development. The United States Food and Drug Administration (FDA) requires that such a predictive model be validated before it can be used in clinical development. Shao [Shao, J. (1993). Linear model selection by cross-validation. J. Amer. Statist. Assoc. 88(422):486-494] proposed a cross-validation method for linear model with equal variances, which is found useful in genomic studies. In practice, however, genomic data may be obtained from different sources with unequal variances. As a result, Shao's method may not be applied directly. In this paper, we extend Shao's method for cross-validation of a linear model with unequal variances. Along this line, two re-sampling methods were proposed to account for the heterogeneity in variance. Several simulations were performed to evaluate the finite samples performances of the proposed methods. An example concerning a breast cancer research is present to illustrate the use of the proposed methods.