Likelihoodbased approach

Before we go on to discuss our multivariate empirical Bayes methods, here we briefly comment on the likelihood-based approaches for longitudinal time course data. Given the very few replications in this context, the gene-specific sample variance-covariance matrix or its variant may be singular and the resulting analysis can be unstable. Guo et al. (38) proposed the gene-specific score based on the robust Wald statistic for one-sample longitudinal data, using an approach similar to Tusher et al. (22), adding a small positive number times the identity matrix in the denominator.

w(i) = [Lft(i)]T [LVS (i)LT + ^ Irxr ]-1[L/3(i)], (20.1)

where L is an r x p matrix of rank r, $ is the estimated p x 1 vector of unknown regression parameters $, VS is the estimated covariance matrix for ft, and Xw is a positive scalar. The way they estimated Xw is exploratory. Moreover, their approach is for a one-sample problem only, and the fact that the number of subjects is usually very small makes asymptotic theory inappropriate.

Storey et al. (39) also proposed a likelihood-ratio based approach, assuming gene expression values are composed of population mean and individual deviates. They constructed the P-statistics for both longitudinal and cross-sectional data in a standard way, and presented a careful treatment of the multiple testing issue. In contrast, Tai and Speed (34) suggested the moderated LR and Hotelling T2 statistics, with the smoothing of gene-

specific sample variance-covariance matrix making use of the replicate variability information across the whole gene set, and then they simply ranked genes according to one or the other statistic. These two statistics were shown in a simulation study to perform about as well as the MB-statistic described below.

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