The authors have declared that no competing interests exist.
Genomewide association study (GWAS) aims to discover genetic factors underlying phenotypic traits. The large number of genetic factors poses both computational and statistical challenges. Various computational approaches have been developed for large scale GWAS. In this chapter, we will discuss several widely used computational approaches in GWAS. The following topics will be covered: (1) An introduction to the background of GWAS. (2) The existing computational approaches that are widely used in GWAS. This will cover singlelocus, epistasis detection, and machine learning methods that have been recently developed in biology, statistic, and computer science communities. This part will be the main focus of this chapter. (3) The limitations of current approaches and future directions.
The background of Genomewide association study (GWAS).
The existing computational approaches that are widely used in GWAS. This will cover singlelocus, epistasis detection, and machine learning methods.
The limitations of current approaches and future directions.
With the advancement of genotyping technology, genomewide highdensity single nucleotide polymorphisms (SNPs) of human and other organisms are now available
Genomewide association study is an interdiscipline problem of biology, statistics and computer science
A human genome contains over 3 billion DNA base pairs. There are four possible nucleotides at each base in the DNA: adenine (A), guanine (G), thymine (T), and cytosine (C). In some locations in the genome, a genetic variation may be found which involves two or more nucleotides across different individuals. These genetic variations are known as
More formally, let
In the remainder of the chapter, we will first discuss the singlelocus methods. We will then study epistasis detection (multilocus) approaches which are designed for association studies of complex traits. For epistasis detection, we will mainly focus on exact twolocus association mapping methods.
As the rapid development of highthroughput genotyping technology, millions of SNPs are now available for genomewide association studies. Singlelocus association test is a traditional way for association studies. Specifically, for each SNP, a statistical test is performed to evaluate the association between the SNP and the phenotype. A variety of tests can be applied depending on the data types. The phenotype involved in a study can be casecontrol (binary), quantitative (continuous), or categorical. We categorize the statistical tests based on what kind of phenotypes they can be applied on.
Let
In a casecontrol study, the phenotype can be represented as a binary variable with 0 representing controls and 1 representing cases.
A contingency table records the frequencies of different events.



Totals  















Many tests can be used to assess the significance of the association between a single SNP and a binary phenotype. The test statistics are usually based on the contingency table. The null hypothesis is that there is no association between the rows and columns of the contingency table.
Pearson's
The value of the test statistic is
Gtest is an approximation of the loglikelihood ratio. The test statistic is
The null hypothesis is that the observed frequencies result from random sampling from a distribution with the given expected frequencies. The distribution of G is approximately that of
When the sample size is small, the Fisher exact test is useful to determine the significance of the association. The pvalue of the test is the probability of the contingency table given the fixed margins. The probability of obtaining such values in
For complex traits, contributions to disease risk from SNPs are widely considered to be roughly additive. In other words, the heterozygous alleles will have an intermediate risk between two homozygous alleles. CochranArmitage test can be used in this case
There is no overall winner of the introduced tests. CochranArmitage test may not be the best if the risks are deviated from the additive model. Meanwhile,
In addition to casecontrol phenotypes, many complex traits are quantitative. This type of study is also often referred to as the quantitative trait locus (QTL) analysis. The standard tools for testing the association between a single marker and a continuous outcome are analysis of variance (ANOVA) and linear regression.
The Ftest in oneway analysis of variance is used to assess whether the expected values of a quantitative variable within several predefined groups differ from each other.
For each SNP
The total sum of squares (SST) can be divided into two parts, the betweengroup sum of squares (SSB) and the withingroup sum of squares (SSW):
In the linear regression model, a leastsquares regression line is fit between the phenotype values and the genotype values
We have the sums of squares as follows:
To achieve least squares, the estimator of
In a typical GWAS, the test needs to be performed many times. We should pay attention to a statistical issue known as the multiple testing problem. In the remainder of this section, we will discuss the multiple testing problem and how to effectively control error rate in GWAS.
Type 1 error rate, is the possibility that a null hypothesis is rejected when it is actually true. In other words, it is the chance of observing a positive (significant) result even if it is not. If a test is performed multiple times, the overall Type 1 Error rate will increase. This is called the multiple testing problem.
Let
Because of the multiple testing problem, the test result may not be that significant even if its pvalue is less than a significant level
For the singlelocus test, we denote the pvalue for a association test of a SNP
Many methods can be used to control FWER. Bonferroni correction is a commonly used method, in which pvalues need to be enlarged to account for the number of comparisons being performed. Permutation test
In Bonferroni correction, the pvalue of a test is multiplied by the number of tests in the multiple comparison.
In the permutation test, data are reshuffled. For each permutation, pvalues for all the tests are recalculated, and the minimal pvalue is retained. After
Let
False discovery rate (FDR) controls the expected proportion of type 1 error among all significant hypotheses. It is less conservative than the familywise error rate. For example, if 100 observed results are claimed to be significant, and the FDR is 0.1, then 10 of results are expected to be false discoveries.
One way to control the FDR is as follows
The vast number of SNPs has posed great computational challenge to genomewide association study. In order to understand the underlying biological mechanisms of complex phenotype, one needs to consider the joint effect of multiple SNPs simultaneously. Although the idea of studying the association between phenotype and multiple SNPs is straightforward, the implementation is nontrivial. For a study with total
In this section, we will focus on the recently developed exact method for twolocus epistasis detection. Different from the singlelocus approach, the goal of twolocus epistasis detection is to identify interacting SNPpairs that have strong association with the phenotype. FastANOVA
FastANOVA utilizes an upper bound of the twolocus ANOVA test to prune the search space. The upper bound is expressed as the sum of two terms. The first term is based on the singleSNP ANOVA test. The second term is based on the genotype of the SNPpair and is independent of permutations. This property allows to index SNPpairs in a 2D array based on the genotype relationship between SNPs. Since the number of entries in the 2D array is bound by the number of individuals in the study, many SNPpairs share a common entry. Moreover, it can be shown that all SNPpairs indexed by the same entry have exactly the same upper bound. Therefore, we can compute the upper bound for a group of SNPpairs together. Another important property is that the indexing structure only needs to be built once and can be reused for all permutated data. Utilizing the upper bound and the indexing structure, FastANOVA only needs to perform the ANOVA test on a small number of candidate SNPpairs without the risk of missing any significant pair. We discuss the algorithm in further detail in the following.
Let
For any SNP
The basic idea of ANOVA test is to partition the total sum of squared deviations


group 
group 




group 
group 

group 
group 
Let
Let
Let
The notations in the bound can be found in
Symbols  Formulas 












We now discuss how to apply the upper bound in Theorem 1 in detail. The set of all SNPpairs is partitioned into nonoverlapping groups such that the upper bound can be readily applied to each group. For every
Note that
If there are
To efficiently retrieve the candidates, the SNPpairs
Suppose that there are 32 individuals, and the genotype of
For any SNP
For multiple tests, permutation procedure is often used in genetic analysis for controlling familywise error rate. For genomewide association study, permutation is less commonly used because it often entails prohibitively long computation times. Our FastANOVA algorithm makes permutation procedure feasible in genomewide association study.
Let
As our initial attempt to develop scalable algorithms for genomewide association study, FastANOVA is specifically designed for the ANOVA test on quantitative phenotypes. Another category of phenotypes is generated in casecontrol study, where the phenotypes are binary variables representing disease/nondisease individuals. Chisquare test is one of the most commonly used statistics in binary phenotype association study. We can extend the principles in FastANOVA for efficient twolocus chisquare test. The general idea of FastChi is similar to that of FastANOVA, i.e., reformulating the chisquare test statistic to establish an upper bound of twolocus chisquare test, and indexing the SNPpairs according to their genotypes in order to effectively prune the search space and reuse redundant computations. Here we briefly introduce the FastChi algorithm.
For SNP
Symbols  Formulas 












For given phenotype
Suppose that there are 32 individuals,
Similar to FastANOVA, in FastChi, we can index the SNPpairs in
Both FastANOVA and FastChi rework the formula of ANOVA test and Chisquare test to estimate an upper bound of the test value for SNP pairs. These upper bounds are used to identify candidate SNP pairs that may have strong epistatic effect. Repetitive computation in a permutation test is also identified and performed once those results are stored for use by all permutations. These two strategies lead to substantial speedup, especially for large permutation test, without compromising the accuracy of the test. These approaches guarantee to find the optimal solutions. However, a common drawback of these methods is that they are designed for specific tests, i.e., chisquare test and ANOVA test. The upper bounds used in these methods do not work for other statistical tests, which are also routinely used by researchers. In addition, new statistics for epistasis detection are continually emerging in the literature. Therefore, it is desirable to develop a general model that supports a variety of statistical tests.
The COE algorithm takes the advantage of convex optimization. It can be shown that a wide range of statistical tests, such as chisquare test, likelihood ratio test (also known as Gtest), and entropybased tests are all convex functions of observed frequencies in contingency tables. Since the maximum value of a convex function is attained at the vertices of its convex domain, by constraining on the observed frequencies in the contingency tables, we can determine the domain of the convex function and get its maximum value. This maximum value is used as the upper bound on the test statistics to filter out insignificant SNPpairs. COE is applicable to all tests that are convex.
The methods we have discussed so far provide promising alternatives for GWAS. However, there are two major drawbacks that limit their applicability. First, they are designed for relatively small sample size and only consider homozygous markers (i.e., each SNP can be represented as a
To address these limitations, TEAM is proposed for efficient epistasis detection in human GWAS. TEAM has several advantages over previous methods. It supports to both homozygous and heterozygous data. By exhaustively computing all twolocus test values in permutation test, it enables both FWER and FDR controlling. It is applicable to all statistics based on the contingency table. Previous methods are either designed for specific tests or require the test statistics satisfy certain property. Experimental results demonstrate that TEAM is more efficient than existing methods for large sample studies.
TEAM incorporates the permutation test for proper error controlling. The key idea is to incrementally update the contingency tables of twolocus tests. We show that only four of the eighteen observed frequencies in the contingency table need to be updated to compute the test value. In the algorithm, we build a minimum spanning tree
As a summary of the exact twolocus algorithms, FastANOVA and FastChi are designed for specific tests and binary genotype data. The COE algorithm is a more general method that can be applied to all convex tests. The TEAM algorithm is more suitable for large sample human GWAS.
Multifactor dimensionality reduction (MDR)
Divide the set of factors into 10 equal subsets.
Select a set of
Create a contingency table for these
Compute the casecontrol ratio in each combination. Label them as “highrisk if it is greater than a certain threshold, and otherwise, it is marked as “lowrisk”.
Use the labels to classify individuals. Compute the misclassification rate.
Repeat previous steps for all combinations of
Choose the model whose average misclassification rate is minimized and crossvalidation consistency is maximized as the “best” model.
MDR designs a constructive induction method that combines two or more SNPs before testing for association. The power of the MDR approach is that it can be combined with other methodologies including the ones described in this chapter.
Logistic regression is a statistical method for predicting binary and categorical outcome. It is widely used in GWAS
Let
The potential of genomewide association study for the identification of genetic variants that underlying phenotypic variations is well recognized. The availability of large SNP data generated by highthroughput genotyping methods poses great computational and statistical challenges. In this chapter, we have discussed serval computational approaches to detect associations between genetic markers and the phenotypes. For further readings, the readers are encouraged to refer to
Answers to the Exercises can be found in
Cantor RM, Lange K, Sinsheimer JS (2008) Prioritizing GWAS results: a review of statistical methods and recommendations for their application. Nat Rev Genet 9(11): 855–867.
Cordell HJ (2009) Detecting genegene interactions that underlie human diseases. Nat Rev Genet 10(6): 392–404.
Manolio TA, Collins FS, Cox NJ, Goldstein DB, Hindorff LA, et al. (2009) Finding the missing heritability of complex diseases. Nature 461(7265): 747–753.
Moore JH, Williams SM (2009) Epistasis and its implications for personal genetics. Am J Hum Genet 85(3): 309–320.
Phillips PC (2010) Epistasis  the essential role of gene interactions in the structure and evolution of genetic systems. Am J Hum Genet 86(1): 6–22.
Wang K, Li M, Hakonarson H (2010) Analysing biological pathways in genomewide association studies. Nat Rev Genet 11: 843–854.
Answers to Exercises
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