A structural mixed model to shrink covariance matrices for time-course differential gene expression studies

Abstract

Time-course microarray studies require a particular modelling of covariance matrices when measures are repeated on the same individuals. Taking into account the within-subject correlation in the test statistics for differential gene expression, however, requires a large number of parameters when a gene-specific approach is used, which often results in a lack of power due to the small number of individuals usually considered in microarray experiments. Shrinkage approaches can improve this detection power in differential gene expression studies by reducing the number of parameters, while offering a good flexibility and a small rate of false positives. A natural extension of the shrinkage approach based on a structural mixed model to variance–covariance matrices is proposed. The structural model was used in three configurations to shrink (i) the eigenvalues in an eigenvalue/eigenvector decomposition, (ii) the innovation variances in a Cholesky decomposition, (iii) both the variances and correlation parameters of a gene-by-gene covariance matrix using a Fisher transformation. The proposed methods were applied both to a publicly available data set and to simulated data. They were found to perform well, compared to previously proposed empirical Bayesian approaches, and outperformed the gene-specific or common-covariance methods in many cases.