# Chapter 2 Functional Enrichment Analysis Methods

## 2.1 Over Representation Analysis

Over Representation Analysis (ORA) (Boyle et al. 2004) is a widely used approach to determine whether known biological functions or processes are over-represented (= enriched) in an experimentally-derived gene list, e.g. a list of differentially expressed genes (DEGs).

The p-value can be calculated by hypergeometric distribution.

$$p = 1 - \displaystyle\sum_{i = 0}^{k-1}\frac{{M \choose i}{{N-M} \choose {n-i}}} {{N \choose n}}$$

In this equation, N is the total number of genes in the background distribution, M is the number of genes within that distribution that are annotated (either directly or indirectly) to the gene set of interest, n is the size of the list of genes of interest and k is the number of genes within that list which are annotated to the gene set. The background distribution by default is all the genes that have annotation. P-values should be adjusted for multiple comparison.

Example: Suppose we have 17,980 genes detected in a Microarray study and 57 genes were differentially expressed. Among the differential expressed genes, 28 are annotated to a gene set2.

d <- data.frame(gene.not.interest=c(2613, 15310), gene.in.interest=c(28, 29))
row.names(d) <- c("In_category", "not_in_category")
d
##                 gene.not.interest gene.in.interest
## In_category                  2613               28
## not_in_category             15310               29

whether the overlap of 25 genes are significantly over represented in the gene set can be assessed using hypergeometric distribution. This corresponding to a one-sided version of Fisher’s exact test.

fisher.test(d, alternative = "greater")
##
##  Fisher's Exact Test for Count Data
##
## data:  d
## p-value = 1
## alternative hypothesis: true odds ratio is greater than 1
## 95 percent confidence interval:
##  0.110242      Inf
## sample estimates:
## odds ratio
##  0.1767937

## 2.2 Gene Set Enrichment Analysis

A common approach in analyzing gene expression profiles was identifying differential expressed genes that are deemed interesting. The enrichment analysis we demonstrated in Disease enrichment analysis vignette were based on these differential expressed genes. This approach will find genes where the difference is large, but it will not detect a situation where the difference is small, but evidenced in coordinated way in a set of related genes. Gene Set Enrichment Analysis (GSEA)(Subramanian et al. 2005) directly addresses this limitation. All genes can be used in GSEA; GSEA aggregates the per gene statistics across genes within a gene set, therefore making it possible to detect situations where all genes in a predefined set change in a small but coordinated way. Since it is likely that many relevant phenotypic differences are manifested by small but consistent changes in a set of genes.

Genes are ranked based on their phenotypes. Given a priori defined set of gene S (e.g., genes shareing the same DO category), the goal of GSEA is to determine whether the members of S are randomly distributed throughout the ranked gene list (L) or primarily found at the top or bottom.

There are three key elements of the GSEA method:

• Calculation of an Enrichment Score.
• The enrichment score (ES) represent the degree to which a set S is over-represented at the top or bottom of the ranked list L. The score is calculated by walking down the list L, increasing a running-sum statistic when we encounter a gene in S and decreasing when it is not. The magnitude of the increment depends on the gene statistics (e.g., correlation of the gene with phenotype). The ES is the maximum deviation from zero encountered in the random walk; it corresponds to a weighted Kolmogorov-Smirnov-like statistic(Subramanian et al. 2005).
• Esimation of Significance Level of ES.
• The p-value of the ES is calculated using permutation test. Specifically, we permute the gene labels of the gene list L and recompute the ES of the gene set for the permutated data, which generate a null distribution for the ES. The p-value of the observed ES is then calculated relative to this null distribution.
• Adjustment for Multiple Hypothesis Testing.
• When the entire gene sets were evaluated, DOSE adjust the estimated significance level to account for multiple hypothesis testing and also q-values were calculated for FDR control.

We implemented GSEA algorithm proposed by Subramanian(Subramanian et al. 2005). Alexey Sergushichev implemented an algorithm for fast GSEA analysis in the fgsea(S., n.d.) package.

In DOSE(Yu et al. 2015), user can use GSEA algorithm implemented in DOSE or fgsea by specifying the parameter by="DOSE" or by="fgsea". By default, DOSE use fgsea since it is much more fast.

## 2.3 Leading edge analysis and core enriched genes

Leading edge analysis reports Tags to indicate the percentage of genes contributing to the enrichment score, List to indicate where in the list the enrichment score is attained and Signal for enrichment signal strength.

It would also be very interesting to get the core enriched genes that contribute to the enrichment.

DOSE supports leading edge analysis and report core enriched genes in GSEA analysis.

### References

Boyle, Elizabeth I, Shuai Weng, Jeremy Gollub, Heng Jin, David Botstein, J Michael Cherry, and Gavin Sherlock. 2004. “GO::TermFinder–open Source Software for Accessing Gene Ontology Information and Finding Significantly Enriched Gene Ontology Terms Associated with a List of Genes.” Bioinformatics (Oxford, England) 20 (18): 3710–5. https://doi.org/10.1093/bioinformatics/bth456.

S., Alexey. n.d. “An Algorithm for Fast Preranked Gene Set Enrichment Analysis Using Cumulative Statistic Calculation.” Biorxiv. https://doi.org/10.1101/060012.

Subramanian, Aravind, Pablo Tamayo, Vamsi K. Mootha, Sayan Mukherjee, Benjamin L. Ebert, Michael A. Gillette, Amanda Paulovich, et al. 2005. “Gene Set Enrichment Analysis: A Knowledge-Based Approach for Interpreting Genome-Wide Expression Profiles.” Proceedings of the National Academy of Sciences of the United States of America 102 (43): 15545–50. https://doi.org/10.1073/pnas.0506580102.

Yu, Guangchuang, Li-Gen Wang, Guang-Rong Yan, and Qing-Yu He. 2015. “DOSE: An R/Bioconductor Package for Disease Ontology Semantic and Enrichment Analysis.” Bioinformatics 31 (4): 608–9. https://doi.org/10.1093/bioinformatics/btu684.