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One-way ANOVA and Post-hoc Tests

T-tests are appropriate for two-condition comparisons. However, with more than two conditions, for example, given three types of tissues, a one-way Analysis of Variance (ANOVA) is the proper way to handle the analysis.

In an ANOVA, the null-hypothesis is that all conditions come from the same distribution. Therefore, if for a given gene, we decide to reject the null-hypothesis (i.e., the p-value is below some significance threshold), then we can conclude that the gene is different across conditions. However, we cannot conclude anything about differences between any pair of conditions. For example, say we have tissue samples from the liver, the heart, and the lung. If for a given gene, the ANOVA p-value is less-than 0.5, we can conclude that the gene is differentially expressed between the three tissues. However, we cannot conclude that the gene is differentially expressed between the heart and the lung. To make this determination, we must use pairwise post-hoc tests. In essence, post-hoc tests examine differences across all pairs of conditions. Tukey's Honestly Significant Difference (TukeyHSD) and Scheffe's Method are available as post-hoc test options.

Note: the pairwise post-hoc p-values are not corrected for multiple testing. One should only examine pairwise post-hoc p-values for those measurements that are significant (after multiple testing correction) at the ANOVA level. Following this step-wise procedure should provide sufficient protection against Type I errors due to multiplicity. Low pairwise post-hoc p-values on their own are not indicitave of significant differential behavior.