Let be a set of SNPs for individuals and . We use 0, 1, 2 to represent the homozygous major allele, heterozygous allele, and homozygous minor allele respectively. Thus we have that (). Let be the phenotype. Note that the values that can take depend on its type.

In a case-control 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. Table 1 is an example contingency table. For a SNP and a phenotype , and we use to denote the number of individuals whose equals and equals . Also, we have and . The total number of individuals .

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.

In addition to case-control 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.

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 be the type 1 error rate for a statistical test. If the test is performed times, the experimental-wise error rate is given byFor example, if and , then . In this case, the chance of getting at least one false positive is .

Because of the multiple testing problem, the test result may not be that significant even if its p-value is less than a significant level . To solve this problem, the nominal p-value need to be corrected/adjusted.

For the single-locus test, we denote the p-value for a association test of a SNP and a phenotype to be , and the corrected p-value to be . Family-wise error rate (FWER), or the experiment-wise error rate, is the probability of at least one false association. We use to denote family-wise error rate, and it is given bywhere is the total number of tests and is the hypothesis that all the are true.

Many methods can be used to control FWER. Bonferroni correction is a commonly used method, in which p-values need to be enlarged to account for the number of comparisons being performed. Permutation test [13] is also widely used to correct for multiple testing in GWAS.

False discovery rate (FDR) controls the expected proportion of type 1 error among all significant hypotheses. It is less conservative than the family-wise 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 [14]. The p-values of SNPs and the phenotype are ranked from smallest to largest. We denote the ordered p-values to be . Starting from the largest p-value to the smallest, the original p-value is multiplied by the total number of SNPs and divided by its rank. For the p-value , its corrected p-value is given byIn this section, we have discussed commonly used methods in single-locus study, the multiple testing problem and how to control error rate in GWAS. In the next section, we will introduce methods used for two-locus association studies. We will focus on one class work that finds exact solution when searching for SNP-SNP interactions in GWAS.

FastANOVA utilizes an upper bound of the two-locus 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 single-SNP ANOVA test. The second term is based on the genotype of the SNP-pair and is independent of permutations. This property allows to index SNP-pairs 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 SNP-pairs share a common entry. Moreover, it can be shown that all SNP-pairs indexed by the same entry have exactly the same upper bound. Therefore, we can compute the upper bound for a group of SNP-pairs 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 SNP-pairs without the risk of missing any significant pair. We discuss the algorithm in further detail in the following.

Let be the set of SNPs of individuals () and be the quantitative phenotype of interest, where () is the phenotype value of individual .

For any SNP (), we represent the F-statistic from the ANOVA test of and as . For any SNP-pair , we represent the F-statistic from the ANOVA test of and as .

The basic idea of ANOVA test is to partition the total sum of squared deviations into between-group sum of squared deviations and within-group sum of squared deviations :In our application of the two-locus association study, Table 2 and Table 3 show the possible groupings of phenotype values by the genotypes of and respectively.

Let , , , , , represent the groups as indicated in Table 2 and Table 3. We use and to distinct the one locus (i.e., single-SNP) and two locus (i.e., SNP-pair) analyses. Specifically, we haveThe F-statistics for ANOVA tests on and are:(1.1)(1.2)where in Equation (1.2) is the number of groups that the genotype of partitions the individuals into. Possible values of are 3 or 4, assuming all SNPs are distinct: If none of groups , , , , , is empty, then . If one of them is empty, then .

Let be the sum of all phenotype values. The total sum of squared deviations does not depend on the groupings of individuals:Let be the sum of phenotype values in a specific group, and be the number of individuals in that group. and can be calculated as follows:Note that for any group of , , , , , , if , then is defined to be 0.

Let be the phenotype values in group . Without loss of generality, assume that these phenotype values are arranged in ascending order, i.e.,Let be the phenotype values in group . Without loss of generality, assume that these phenotype values are arranged in ascending order, i.e.,We have the overall upper bound on :

Theorem 1 (Upper bound of )

The notations in the bound can be found in Table 4. The upper bound in Theorem 1 is tight. The tightness of the bound is obvious from the derivation of the upper bound, since there exists some genotype of SNP-pair that makes the equality hold.

We now discuss how to apply the upper bound in Theorem 1 in detail. The set of all SNP-pairs is partitioned into non-overlapping groups such that the upper bound can be readily applied to each group. For every , let be the set of SNP-pairsFor all SNP-pairs in , , , , and are constants. Moreover, , are determined by , and , are determined by . Therefore, in the upper bound, and are the only variables that depend on and may vary for different SNP-pairs in .

Note that is the number of 1's in when takes value 1, and is the number of 1's in when takes value 0. It is easy to prove that switching and does not change the F-statistic value and the correctness of the upper bound. This is also true if we switch and . Therefore, without loss of generality, we can always assume that is the smaller one between the number of 1's and number of 0's in when takes value 1, and is the smaller one between the number of 1's and number of 0's in when takes value 0.

If there are 1's and 0's in , then for any , the possible values that can take are . The possible values that can take are .

To efficiently retrieve the candidates, the SNP-pairs in are grouped by their values and indexed in a 2D array, referred to as .

Suppose that there are 32 individuals, and the genotype of consists of half 0's and half 1's. Thus for the SNP-pairs in , the possible values of and are . Figure 1 shows the array, , whose entries represent the possible values of for the SNP-pairs . The entries in the same column have the same value. The entries in the same row have the same value. The value of each column is noted beneath each column. The value of each row is noted left to each row. Each entry of the array is a pointer to the SNP-pairs having the corresponding values.

For any SNP , the maximum number of the entries in is . The proof of this property is straightforward and omitted here. In order to find candidate SNP-pairs, we scan all entries in to calculate their upper bounds. Since the SNP-pairs indexed by the same entry share the same value, they have the same upper bound. In this way, we can calculate the upper bound for a group of SNP-pairs together. Note that for typical genome-wide association studies, the number of individuals is much smaller than the number of SNPs . Therefore, the additional cost for accessing is minimal compared to performing ANOVA tests for all pairs .

For multiple tests, permutation procedure is often used in genetic analysis for controlling family-wise error rate. For genome-wide association study, permutation is less commonly used because it often entails prohibitively long computation times. Our FastANOVA algorithm makes permutation procedure feasible in genome-wide association study.

Let be the permutations of the phenotype . Following the idea discussed above, the upper bound in Theorem 1 can be easily incorporated in the algorithm to handle the permutations. For every SNP , the indexing structure is independent of the permuted phenotypes in . The correctness of this property relies on the fact that, for any , and only depend on the genotype of the SNP-pair and thus remain constant for different phenotype permutations. Therefore, for each , once we build , it can be reused in all permutations.

As our initial attempt to develop scalable algorithms for genome-wide association study, FastANOVA is specifically designed for the ANOVA test on quantitative phenotypes. Another category of phenotypes is generated in case-control study, where the phenotypes are binary variables representing disease/non-disease individuals. Chi-square test is one of the most commonly used statistics in binary phenotype association study. We can extend the principles in FastANOVA for efficient two-locus chi-square test. The general idea of FastChi is similar to that of FastANOVA, i.e., re-formulating the chi-square test statistic to establish an upper bound of two-locus chi-square test, and indexing the SNP-pairs 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 , we represent the chi-square test value of and the binary phenotype as . For any SNP-pair and , we use to represent the chi-square test value for the combined effect of with . Let represent the following events respectively: ; ; ; . Let denote the observed value of an event. , , , , , and represent the formulas shown in Table 5. We have the upper bound of stated in Theorem 2.

Theorem 2 (Upper bound of )

For given phenotype and SNP, , , , , and are constants. and are the only variables that depend on and may vary for different SNP-pairs . (Recall that .) Thus for a given , we can treat equation as a straight line in the 2-D space of and . The ones whose values fall below the line can be pruned without any further test.

Suppose that there are 32 individuals, contains half 0's, and half 1's. For the SNP-pairs in , the possible values of (and ) are . Figure 2 shows the 2-D space of and . The blue stars represent the values that can take. The line is plotted in the figure. Only the SNP-pairs whose values are in the shaded region are subject to two-locus Chi-square test.

Similar to FastANOVA, in FastChi, we can index the SNP-pairs in according to their genotype relationships, i.e., by the values of . Experimental results demonstrate that FastChi is an order of magnitude faster than the brute force alternative.

Both FastANOVA and FastChi rework the formula of ANOVA test and Chi-square 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., chi-square 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 chi-square test, likelihood ratio test (also known as G-test), and entropy-based 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 SNP-pairs. 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 binary variable). In human study, the sample size is usually large and most SNPs contain heterozygous genotypes and are coded using . These make previous methods intractable. Second, although the family-wise error rate (FWER) and the false discovery rate (FDR) are both widely used for error controlling, previous methods are designed only to control the FWER. From a computational point of view, the difference in the FWER and the FDR controlling is that, to estimate FWER, for each permutation, only the maximum two-locus test value is needed. To estimate the FDR, on the other hand, for each permutation, all two-locus test values must be computed.

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 two-locus 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 two-locus 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 [19] on the SNPs. The nodes of the tree are SNPs. Each edge represents the genotype difference between the two connected SNPs. This tree structure can be utilized to speed up the updating process for the contingency tables. A majority of the individuals are pruned and only a small portion are scanned to update the contingency tables. This is advantageous in human study, which usually involves thousands of individuals. Extensive experimental results demonstrate the efficiency of the TEAM algorithm.

As a summary of the exact two-locus 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.