Differential Expression Analysis for RNA-Seq: An Overview of Statistical Methods and Computational Software <s> statistical testing <s> SUMMARY The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Therefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferronitype procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples. <s> BIB001 </s> Differential Expression Analysis for RNA-Seq: An Overview of Statistical Methods and Computational Software <s> statistical testing <s> Multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously. One multiple hypothesis testing error measure is the false discovery rate (FDR), which is loosely defined to be the expected proportion of false positives among all significant hypotheses. The FDR is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests. In this work, we introduce a modified version of the FDR called the “positive false discovery rate” (pFDR). We discuss the advantages and disadvantages of the pFDR and investigate its statistical properties. When assuming the test statistics follow a mixture distribution, we show that the pFDR can be written as a Bayesian posterior probability and can be connected to classification theory. These properties remain asymptotically true under fairly general conditions, even under certain forms of dependence. Also, a new quantity called the “q-value” is introduced and investigated, which is a natural “Bayesian posterior p-value,” or rather the pFDR analogue of the p-value. 1. Introduction. When testing a single hypothesis, one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists. In statistical terms, we maximize the power conditional on the Type I error rate being at or below some level. The field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously. One must define an appropriate compound error measure according to the rate of false positives one is willing to encounter. Then a procedure is developed that allows one to control the error rate at a desired level, while maintaining the power of each test as much as possible. The most commonly controlled quantity when testing multiple hypotheses is the family wise error rate (FWER), which is the probability of yielding one or more false positives out of all hypotheses tested. The most familiar example of this is the Bonferroni method. If there are m hypothesis tests, each test is controlled so that the probability of a false positive is less than or equal to α/m for some chosen value of α. It then follows that the overall FWER is less than or equal to α .M any <s> BIB002
After performing parameter estimation for a statistical model, significance of differential expression can be assessed compar ing the expression of gene g among K groups. Assume that λ gk(i) is the expression level of gene g in sample i belonging to sample group k. φ g is the dispersion parameter. DE tests are proposed below for the null hypothesis (H 0 ): In parametric regime, one can employ classic log likelihood ratio test. In absence of overdispersion, where l g denotes the loglikelihood function for the gth λ 0 denote the MLE of biologi cal and experimental effects under the full model and null model, respectively. An exact test for negative binomial, analogous to the Fisher's exact test, is used by methods, such as edgeR and DESeq. By conditioning on the total sum, one can calculate the probability of observing counts as extreme or more extreme than what is really obtained, resulting in an exact Pvalue. Note that a sum of gene counts from all replicates in each group that is either too large or too small indicates a differential expres sion, so a twosided test is used. A score statistic is used by PoissonSeq, which tests for the significance of the association of gene g with expression of groups. In the context of gene count with unknown dispersion parameters, a score test is as follows: where M denotes a model. Posterior probability of DE to non DE ratio is often used. The choice of a testing strategy is a decision that often depends on the chosen method and other factors such as sam ple size. With a small sample size, the largesample approxi mations based on the Wald test, score test, and likelihood ratio test are questionable and an exact test is usually preferred.  We summarize testing strategies that are plausible for each method in Table 2 . Finally, almost all the methods we mentioned in this paper use standard approaches for multiple hypothesis correc tion to control false discoveries. BIB001 BIB002 PoissonSeq is an exception that builds its own estimation of false discovery rate (FDR) from a permutation test. Permutation test calculates a score test per gene, S g , for H 0g vs H ag , each time when the outcome is permuted. For B permutations, the same procedure is applied to calculate null statistics Note that I{π 0g , t} = I{π 1g $ t} for small t of interest.