ANOVA and the Fstatistic

An intuitive way to select differentially expressed genes from time course data is to use the classical ANOVA (cross-sectional) or mixed-effect ANOVA (longitudinal) model (14, 18). In the one-sample case, one includes time as a factor and possibly replicate as another factor, and calculates the F-statis-tic corresponding to time. Similarly, for two- and D > two-sample cases, one includes the factors time and biological condition, and their interaction term in the model, and possibly replicate as another factor, and computes the F-statistic for the interaction term. Wang and Kim (30) has a one-sample example using classical one-way ANOVA. In order to obtain approximately valid p-values, care needs to be taken to deal with multiple testing (26 and references therein).

Park et al. (31) proposed a modified ANOVA approach and some variants without the normality assumption. Their basic model is like the usual two-way ANOVA with time, biological condition, and their interaction as effects. Genes which are not significant (after p-value adjustments) in the time x condition interaction term will be fitted with the second model, removing the interaction term. Then genes with significant time effect after p-value adjustments are selected. However, their method ignores the potential correlations among times from longitudinal time course data. Even if there are no biological correlations, the fact that there are usually only very few replicates makes the estimation of gene-specific variances unstable. Romagnolo et al. (32) and Wang and Kim (30) used mixed-effect ANOVA to identify genes with different temporal profiles between the heat-shock-treated let-60 and wildtype, and dauer exit and L1 starvation C. elegans, respectively. Himanen et al. (9) also used mixed-effect ANOVA for their one-sample problem. Chapter 6 of Diggle et al. (14) discusses some standard ANOVA methods for longitudinal data, and we refer the reader there.

A number of questions are not adequately addressed by classical ANOVA methods or the variants of Park et al. (31). First, to obtain an ^-distribution for the F-statistic, we require that the samples at different time points are independent, an assumption easily violated by longitudinal time course data, and we also require normality of the observations. The robustness of the F-distribution to deviations from normality is well-studied (see e.g. 33), but for gene expression measurements on the log scale this may not be a great concern. Secondly, as a result of the very large number of genes, and relatively small number of replicates, the F-statistic may lead to more false positives and false negatives than would normally be the case, because of poorly estimated variances in the denominator. See Tai and Speed (34) for the results of a simulation study. This issue can be addressed using the notion of moderation (see below).

Despite these reservations concerning the F-statistic, it should be understood that it has been and will continue to be effective for identifying genes of interest to researchers. However, we believe that we can do this job better with alternative statistics.

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