With the development of modern human and medical genetics , it has been widely accepted that genetic variation plays an important role in the pathogenesis of genetic inherited diseases 1 ., The identification of causative genetic variants therefore becomes the primary step towards the understanding of genetic principles underlying these diseases ., For Mendelian diseases in which an individual genetic variant in a single gene is both sufficient and necessary to cause a disease , classical statistical approaches such as linkage analysis 2–5 and association studies 6 , 7 have shown remarkable successes in the identification of causative genetic variants ., Nevertheless most common diseases are complex ones that are supposed to be caused by multiple genetic variants , their interactive effects , and/or their interaction with environment factors 7 , 8 ., The detection of such interactive effects therefore plays a key role in the understanding of these diseases ., The interactive effects of multiple genetic variants underlying complex diseases are often referred to as epistasis or epistatic interactions ., Recent advances in biomedical studies have been confirming the contribution of epistasis to complex diseases ., For example , Tiret et al reported synergistic effects of polymorphisms in the angiotensin-converting enzyme and the angiotensin-II type 1 receptor gene on the risk of myocardial infarction 9 ., Ritchie et al identified the association of a high-order interaction among four polymorphisms in three estrogen-metabolism genes with breast cancer 10 ., Williams et al reported the influence of a two-locus interaction between polymorphisms in the angiotensin converting enzyme and the G protein-coupled receptor kinase on hypertension susceptibility 11 ., Tsai et al identified the association of a three-locus interaction among polymorphisms in renin-angiotensin system genes with atrial fibrillation 12 ., Cho et al reported the association of a two-locus interaction between polymorphisms in the uncoupling protein 2 gene and the peroxisome proliferator-activated receptor gamma gene with Type 2 diabetes mellitus 13 ., Martin et al reported the influence of a two-locus interaction between polymorphisms in KIR3DL1 and HLA-B on both AIDS progression and plasma HIV RNA 14 ., With these examples , epistasis between multiple genetic variants is now widely believed to be the causative pattern of human complex diseases ., In order to detect epistasis , a number of multi-locus approaches have been developed ., For example , Hoh et al proposed a trimming , weighting , and grouping approach that used the summation of statistics on the basis of single-locus marginal effects and the Hardy-Weinberg equilibrium ( HWE ) for hypothesis testing 15 ., Nelson et al proposed a combinatorial partitioning method ( CPM ) that exhaustively searched for a combinatory genotype group that had the most significant difference in the mean of the responding continuous phenotype 16 ., Culverhouse et al proposed a restricted partitioning method ( RPM ) which modified CPM by ignoring partitions that combined individual genotypes with very different mean trait values 17 ., Millstein et al proposed a focused interaction testing framework ( FITF ) in which a prescreening strategy was developed to reduce the number of tests 18 ., Chatterjee et al used Turkeys 1-degree-of-freedom model to detect interacting loci from different regions 19 ., Ritchie et al proposed a multifactor-dimensionality reduction ( MDR ) method in which exhaustive search was performed to detect combinations of loci with the highest classification capability 10 ., Although these methods have shown their successes in association studies for small scale candidate genes 10 , 15–19 , their effectiveness for large scale case-control data has not yet been validated ., Besides , most of the methods rely strongly on exhaustive search for combinations of multiple loci ., This search strategy , though feasible when the number of candidate genetic variants is small , can hardly be computationally practical for large scale or whole-genome association studies in which the number of candidate genetic variants is typically very huge ., For example , a study on Age-related Macular Degeneration ( AMD ) has genotyped more than 100 thousand single nucleotide polymorphism ( SNP ) markers for 96 patients and 50 unaffected people 20 , and a recent genome-wide association study on Parkinsons disease has genotyped more than 400 thousand SNP markers for 270 patients and 271 unaffected people 21 , 22 ., With such dense SNPs being genotyped , methods based on exhaustive search are computationally impractical due to the vast number of possible combinations of the SNP markers ., The main challenge for genome-wide association studies is therefore to design computational approaches that are capable of avoiding the “combinatorial explosion” curse to identify epistatic interactions ., A recent breakthrough in genome-wide epistasis mapping is the introduction of the Bayesian epistasis association mapping ( BEAM ) method 23 that integrates a Bayesian model with the Metropolis-Hastings algorithm to infer the probability that each locus is associated with the susceptibility of a specified disease ., BEAM classifies SNP markers into three types: SNPs unassociated with the disease , SNPs contributing to the disease susceptibility independently , and SNPs influencing the disease risk jointly with each other ., However , the genetic models for complex diseases could be far more complicated than that proposed by BEAM ., For example , the disease-associated SNPs that jointly influence the disease risk may be further divided into subgroups , in which a SNP interacts with other SNPs in the same subgroup , but not with those in the other subgroups ., This situation could be very common in real data , making BEAM ineffective in the exploration of true interactive effects of multiple loci ., To overcome this limitation , in this paper , we give an explicit presentation of “epistasis” and define “epistatic modules” as basic units of disease susceptibility loci ., On the basis of this notion , we put forward a Bayesian marker partition model to explain the observed case-control data and further generalize this model to account for the existence of linkage disequilibrium ( LD ) between genetic variants ., To facilitate the identification of epistatic modules , we develop a Gibbs sampling strategy with a reversible jump Markov chain Monte Carlo ( RJ-MCMC ) procedure to simulate the posterior distribution that genetic variants belong to the epistatic modules and further resort to hypothesis testing to screen out statistically significant modules ., In contrast to most of the existing methods that entirely or partially rely on exhaustive search for combinations of loci , the proposed approach , named epiMODE ( epistatic MOdule DEtection ) , natively identifies interactive loci ( epistatic modules ) without enumerating their combinations , thereby being capable of detecting interactive effects of multiple loci from a vast number of genotyped genetic variants ., We systematically compare the proposed approach with three existing methods on seven simulated disease models ., The results show the superior performance of our approach over the other methods ., We further apply the proposed approach to a genome-wide case-control data set for Age-related Macular Degeneration ( AMD ) that contains more than 100 thousand SNPs genotyped for 96 cases and 50 controls 20 and successfully identify two SNPs that are known to be associated with the disease ., Besides , the results also suggest that two other SNPs ( rs1394608 and rs3743175 ) may have interactive effects on the susceptibility of the disease ., We also apply the proposed approach to a genome-wide case-control data set for Parkinsons disease ( 400 thousand SNPs genotyped for 270 cases and 271 controls ) 21 , 22 and identify seven SNP markers that may be associated with the disease ., The concept of epistasis implies that the phenotypic effect of one locus is dependent on one or more other loci ., Nonetheless the definitions of epistasis in biology and statistics are not exactly consistent ., Even from the statistical perspective only , researchers have different understandings of epistasis 24 , 25 ., Considering these inconsistencies , it is necessary to first give a clear definition of epistasis , for the purpose of developing a computational method for identifying multiple loci that contribute to the disease susceptibility ., In this paper , a locus stands for a SNP ., A genotype stands for a set of two alleles ( one inherited from father and the other from mother ) at a locus and has three possible values: homozygosity of common alleles , homozygosity of minor alleles , and heterozygosity ., A combinatory genotype represents the genotype of a combination of multiple loci ., For a combination of t loci , the number of all possible combinatory genotypes is 3t ., The penetrance of a combinatory genotype is the probability/risk that an individual with this combinatory genotype is affected , given the combinatory genotype of the multiple loci ., We first assume that all loci are in linkage equilibrium , also known as independent , and then we generalize the definitions to the situation with linkage disequilibrium between multiple loci ., Let be the set of all L loci under investigation , and be the set of all s disease susceptibility loci that determine the disease risk ., For any two subsets , S1 and S2 , of S ( , , and ) , their penetrance given the combinatory genotypes and respectively , can be described aswhere represents the penetrance of a given combinatory genotype , a combinatory genotype of the multiple loci , and the function denoting how combinatory genotypes determine the disease penetrance ., For any given combinatory genotypes of S1 and S2 , ifis always true , the relationship between the two subsets of loci S1 and S2 is defined as “independently contributing” to the disease ., Otherwise , the relationship between S1 and S2 is defined as “epistasis . ”, Particularly , the relationship between a set of loci and a null set is defined as epistasis ., A set of loci is an “epistatic module” if and only if the relationship between S1 and its complement , , is “independently contributing , ” that is , for any given genotype and , and the relationship between any subset of S1 , , and its complement is epistasis ., Obviously , the set of disease susceptibility loci S consists of one or more epistatic modules ., We further verify that there is no overlap between any two epistatic modules , and epistatic modules are independent in both case and control populations ( Text S1 ) ., In genome-wide association studies where the SNPs are quite dense , it is common that a SNP may be in LD with other SNPs ., To account for this situation , we define a group of SNPs that are in LD with each other as an “LD set” and extend the above definition of epistatic modules by replacing individual loci with LD sets ., Note that with this extension , all properties of epistatic modules remain unchanged , as long as we treat an LD set as an individual locus in the previous derivation ., The mechanism how a number of susceptibility SNPs contribute to the disease risk through epistatic modules is shown in Figure 1 ., The disease risk is determined by a number of epistatic modules , each of which contributes to the disease independent of the others ., An epistatic module is composed of one or more susceptibility SNPs , each of which may be in LD with some other SNPs , forming an LD set ., A disease susceptibility SNP , together with the SNPs that are in LD with it , relies on other disease susceptibility SNPs or LD sets in the same epistatic module to affect the disease susceptibility ., An epistatic module cannot be further divided into smaller epistatic modules; hence epistatic modules are the smallest genetic units that independently influence the disease risk ., Suppose that in a population-based case-control study , cases and controls are genotyped at a number of L SNP markers ., The genotypes for cases and controls are represented as and , respectively , where and denote the genotypes of the i-th patient and the j-th unaffected individual at the L markers , respectively ., With the understanding of epistatic modules , the L markers can be partitioned into modules M0 , M1 , … , MS , with M0 containing markers unlinked to the disease and M1 to MS being epistatic modules ., Let ( ) be an indicator of the assignment of the i-th marker into one of the modules , and be a vector representing the assignments for all of the L markers ., Obviously , has possible values ., Let lm be the number of markers falling into the m-th module ( ) ., We have that ., Let Dm and Um be the genotypes of the sets of markers that belong to the m-th module in the case and the control populations , respectively ., Obviously , we have that when and for the case population , and when and for the control population ., With these concepts , the problem of finding markers that have epistatic interactions on the disease risk is equivalent to a problem of assigning the markers to epistatic modules ., Particularly , the assignment for a marker can be done by first calculating the probability of the observed data given a certain marker partition pattern and then obtaining the posterior probability that the marker belongs to each module using some sampling strategy ., For a clear presentation , we first derive a Bayesian model that assumes independence between SNPs and then generalize the model to account for the existence of LD sets ., The module M0 consists of markers that are unlinked to the disease ., Therefore , markers in D0 ( the case population ) should follow the same distribution as those in U0 ( the control population ) ., Let , , be the probabilities of occurrence of the three possible genotypes for the i-th marker in M0 , and be the vector that is composed of all probabilities of genotypes of the l0 markers belonging to M0 ., Let and be the number of individuals that have the k-th genotype at the i-th marker in the case and the control populations , respectively ., The joint distribution of the observed genotypes D0 and U0 , given the partition I and the parameters can then be written as ( 1 ) Following the Bayesian approach , we assume that every ( ) follows a Dirichlet distribution with the hyper-parameter , that is , ., Integrating out in Equation ( 1 ) , we obtain ( 2 ) where is the Gamma function ., For an epistatic module Mm ( ) containing lm SNPs , there are a total of combinatory genotypes ., Let and be the probabilities of occurrences of all combinatory genotypes in the case and the control populations , respectively ., Let and be the numbers of occurrences of the k-th combinatory genotype in the case and the control populations , respectively ., The distributions of Dm and Um , given the parameters and , can be written asrespectively ., Assuming that and follow Dirichlet prior distributions with hyper-parameters and , respectively , we integrate out and and obtain ( 3 ) and ( 4 ) As the distributions of Dm and Um are independent , we havePutting the above likelihood functions together , we have the posterior distribution of I , given the observed genotypes , as The prior distribution need to be determined in advance ., For simplicity , we assume that the partition of the loci are independent , and for each locus , without prior knowledge , the probability that it belongs to the m-th module is ( and ) ., With these two assumptions , we have ., Note that when prior knowledge that can be used to infer the relationship between a locus and the disease risk is available , the corresponding could be updated accordingly ., We assume that all Dirichlet hyper-parameters are equal to 0 . 5 unless otherwise specified ., We use a first-order Markov model to account for the situation in which a set of SNPs are in LD with a disease susceptibility SNP in an epistatic module , say , an LD set ., For a clear presentation , we refer to the disease susceptibility SNP as the core SNP and SNPs in LD with it as peripheral SNPs ., Given a core SNP , the likelihood of the genotypes of a peripheral SNP in the case population iswhere is the probability that the peripheral SNP has the k-th genotype conditional on that the core SNP has the j-th genotype , and is the number of cases for which the core and peripheral SNPs have the j-th and k-th genotypes , respectively ., Assuming Dirichlet priors with hyper-parameters for , we integrate out and obtain the posterior distribution of the genotypes of the peripheral SNP in the case population conditional on the core SNP as Suppose that in a module with SNPs , there are core SNPs and peripheral SNPs ( ) ., Let be the set of peripheral SNPs that are in LD with the c-th core SNP ( ) ., We have that the intersection of any two of these sets is empty , while the union of all these sets contains all peripheral SNPs ., The posterior distribution of the genotypes of the set of peripheral SNPs in the case population conditional on the c-th core SNP is given by ( 5 ) where ( , , ) are Dirichlet hyper-parameters , and is the number of cases for which the c-th core SNP has the j-th genotype , and the i-th peripheral SNP has the k-th genotypes ., Putting Equations ( 3 ) and ( 5 ) together , the likelihood of the genotypes in the case population iswhere pcore is given by Equation ( 3 ) as Similarly , by replacing the case population with the control population , the likelihood of the genotypes in the control population can be obtained as where is given by Equation ( 4 ) asand is given by ( 6 ) where ( , , ) are Dirichlet hyper-parameters , and is the number of controls for which the c-th core SNP has the j-th genotype , and the i-th peripheral SNP has the k-th genotypes ., Finally , the likelihood of observing both the case and the control populations is given by ( 7 ) We also assume that all Dirichlet hyper-parameters are equal to 0 . 5 unless otherwise specified ., The Bayesian marker partition model described above assumes independence between SNPs that are unlinked to the disease ., Nevertheless the existence of LD may make distributions of genotypes of these SNPs dependent ., In the model discussed above , there is no specific module for these linked disease-unassociated SNPs ., As a result , these SNPs could be partitioned into some epistatic modules and negatively affect the correct partition of these modules ., We therefore propose the use of LD modules to account for the existence of LD between disease-unassociated SNPs ., Although the distributions of genotypes for markers in LD are dependent in both the case and the control populations , as those for markers in epistatic modules , the underlying principle between LD markers and epistatic modules are quite different ., For LD markers , the distributions of genotypes are almost the same for the case and the control populations , while for epistatic modules the distributions of genotypes are different between the case and the control populations ., In order to incorporate this understanding into the Bayesian partition model , we assume that other than the S epistatic modules , there further exist T LD modules , labeled by {} , in each of which loci are in strong LD with each other ., We also use a first-order Markov model to account for LD between the SNPs in an LD module ., For an LD module ( ) , we assume that there exists a core SNP c , and the distributions of genotypes of all other ( peripheral ) SNPs in this LD module depend on the genotype of this core SNP ., Let be the set of the peripheral SNPs that are in LD with the core SNP ., Using similar reasoning as for the epistatic modules , we obtain that is derived with a similar way as Equation ( 2 ) and is given bywhere and are the numbers of individuals that have the k-th genotype at the core SNP in the case and the control populations , respectively ., is derived with a similar way as Equation ( 6 ) and is given bywhere are Dirichlet hyper-parameters , and and are the numbers of individuals for which the core SNP has the j-th genotype , and the i-th peripheral SNP has the k-th genotypes in the case and the control populations , respectively ., We also assume that all hyper-parameters are equal to 0 . 5 unless otherwise specified ., With LD modules being incorporated , the posterior distribution for the generalized indicator vector under the generalized Bayesian model is then The posterior distribution of the partition I given by the above Bayesian partition model suggests the following Gibbs sampler ( 8 ) where and ., In order to calculate this sampler in an efficient way , we computefor , and then obtainWith this sampler , a Gibbs sampling algorithm can be performed as follow ., In order to calculate the Gibbs sampler , i . e . , Equation ( 8 ) , we need to partition SNPs in epistatic and LD modules into core SNPs and peripheral SNPs , say , to obtain structures of the modules ., Besides , the numbers of modules ( S and T ) are also unknown ., We will address these two questions in the following two sections ., Given a set of SNPs in an epistatic module , we need to partition the SNPs into non-overlap LD sets ., For each LD set , we need to assign a core SNP ., The partition of LD sets , together with the assignment of a core SNP for each LD set , is referred to as the structure of an epistatic module ., A naïve method for obtaining the structure of a module is to exhaustively search for all possible structures of the module and then select the one with the maximum likelihood ., Specifically , for an epistatic module Mm ( ) containing SNPs , there are ways for selecting the core SNPs , corresponding to the different ways of selecting non-empty subsets from the SNPs ., Furthermore , in the case that the number of core SNPs is , the number of ways for associating the rest peripheral SNPs to the core SNPs is , since each peripheral SNP can be assigned to one of the core SNPs , and the assignments are mutually independent ., Obviously , the number of all possible structures of an epistatic module grows rapidly , making the exhaustive search strategy practical only when the module contains a small number of SNPs ., We therefore propose the following sampling approach to search for a reasonable module structure when the exhaustive search strategy is hard to apply ., For an epistatic module with SNPs , in which are core SNPs , and the rest are peripheral ones , we index the core SNPs by numbers from 1 to , and we index the peripheral SNPs by numbers from to ., We further introduce an indicator vector , representing the status of all SNPs in the module ., In this vector , ( ) means that the i-th SNP is a core SNP , and means that the i-th SNP is a peripheral SNP of the k-th core SNP ., Consider a peripheral SNP indexed by i ( ) ., The posterior distribution of the indicator , given the rest of the indicators and the observation and , can be written aswhere the likelihood function can be calculated in a similar way as Equation ( 7 ) ., Assuming equal prior probabilities for all possible structures of the module , the above posterior distribution suggests the following Gibbs sampler for the peripheral SNP , ( 9 ) Consider a core SNP indexed by i ( ) ., There are two situations: ( 1 ) the core SNP has some peripheral SNPs , and ( 2 ) the core SNP has no peripheral SNPs ., In the former case , we need to fix the indicator ., In the latter case , a Gibbs sampler can be obtained as ( 10 ) where we exclude the situation in which the core SNP becomes its own peripheral SNP ., The above Gibbs samplers suggest the following sampling strategy: To further reduce the computational burden , we propose the following forward and backward strategies that are very economy in terms of computation time ., In the forward strategy , we consider three situations of adding a SNP into an existing epistatic module ., First , the SNP is itself a core SNP , and there are no other SNPs in LD with it ., Second , the SNP is in LD with an existing core SNP , and this core SNP remains unchanged ., Third , the SNP is in LD with an existing core SNP , but this core SNP needs to be updated as the added SNP ., To deal with the first case , we try to create a new LD set to include the new SNP as the core SNP in constant time complexity ., To deal with the second case , we try to add the new SNP as a peripheral SNP to every existing LD set in linear time complexity , proportional to the number of existing LD sets ., To deal with the third case , we try to add the new SNP as the core SNP and downgrade the previous core SNP to a peripheral SNP for every existing LD set in linear time complexity , also proportional to the number of existing LD sets ., Finally , we compare likelihood values of resulting structures of the above efforts and select the structure with the highest likelihood as the new module structure ., In the backward strategy , we also consider three situation of removing a SNP from an existing epistatic module ., First , the SNP is in LD with a core SNP ., Second , the SNP is itself a core SNP with no other SNPs in LD with it ., Third , the SNP is a core SNP with some other SNPs in LD with it ., The first and second cases can be dealt with in constant time complexity ., The third case can be exhaustively searched for the new core SNP in linear time complexity , proportional to the number of SNPs in LD with the removed SNP ., By comparing the likelihood values of these three cases , we can obtain a new structure for the module ., The exhaustive search strategy can provide optimal module structures , but its computation time is acceptable only when a module contains a small number of SNPs ., The sampling strategy takes uncertainty in the partitioning process into consideration and can alleviate the computational burden when a module contains a large number of SNPs ., The forward and the backward strategies can greatly reduce the computational burden and offer sub-optimal module structures ., To achieve a reasonable trade-off between the computational burden and the optimality of module structures , we also propose a hybrid strategy in which we mainly perform the forward and the backward strategies and periodically apply the exhaustive search or the sampling methods ., According to our experience , the hybrid strategy is much faster than the exhaustive search and the sampling methods and can yield similar results as the other two methods in most cases ., Therefore , we suggest the use of the hybrid strategy ., Similar to epistatic modules , we need to also assign a core SNP for each LD module ., However , the situation is quite simple for obtaining structures for LD modules , because an LD module has only one core SNP , and thus the number of possible structures for an LD module is equal to the number of SNPs in the module ., In the exhaustive search strategy , we can search for the core SNP in linear time complexity , proportional to the number of SNPs in the module ., In the forward strategy , we consider the situation of adding a SNP into an LD module , and determine the structure by comparing the likelihood values of two cases: ( 1 ) the added SNP is a peripheral SNP , and ( 2 ) the added SNP is the core SNP ., This can be done in constant time complexity ., In the backward strategy , we consider the situation of removing a SNP from the module ., If the removed SNP is not the core SNP , we simply remove it ., In the case that the deleted SNP is the core SNP , we select a new core SNP from the previous peripheral SNPs by exhaustive search , which can be done in linear time complexity , proportional to the number of SNPs remaining in the module ., Since the exhaustive search strategy is straightforward and already computationally economy ( linear complexity ) , we simply apply the exhaustive search strategy to obtain structures for LD modules ., With the module structures being obtained , we are now able to calculate the Gibbs sampler defined by Equation ( 8 ) ., We assume that the numbers of epistatic modules ( S ) and LD modules ( T ) are already known in the Gibbs sampling strategy for marker partitioning ., Nevertheless the values of S and T are usually unknown in real applications ., To address the uncertainty of S and T , we adopt a reversible jump Markov chain Monte Carlo ( RJ-MCMC ) procedure 26 as follows ., With a sufficient number of the above RJ-MCMC sampling procedure being repeated , the Markov chains for S and T could achieve their stable distributions ., In our studies , we use , , and ., The RJ-MCMC procedure samples the posterior distributions of the numbers of epistatic and LD modules , while the Gibbs sampling algorithm gives us the posterior probability that a locus belongs to a module and enables us to sample the indicators with the use of their conditional distributions in a sequential way ., Starting from an initial ( random ) assignment of the indicators , the Gibbs sampling procedure simulates a Markov chain whose stationary distribution follows the distribution of the indicator vector ., When the Markov chain reaches its stationary distribution after a number of burn-in iterations , we record candidate epistatic modules and their posterior probabilities ., The posterior probability of an epistatic module represents the strength that the module is associated with the disease and thus can be directly used to make statistical inference ., For example , biologists can select epistatic modules with top posterior probabilities for further functional analysis or biological experiments ., Nevertheless , the statistical significance of epistatic modules might be more desired by geneticists ., We therefore provide in the following parts a permutation test method and a “selection-testing-correction” approach for assessing the statistical significance of candidate epistatic modules ., The URL for the software presented herein is as follows: http://bioinfo . au . tsinghua . edu . cn/epiMODE, In order to verify the capability of the proposed approach in the detection of epistatic interactions in real genome-wide association studies , we apply epiMODE to an Age-related Macular Degeneration ( AMD ) data set 20 , which contains 103 , 611 SNPs genotyped with 96 cases and 50 controls ., The authors of the original paper reported that two SNPs , rs380390 and rs1329428 , were believed to be significantly associated with AMD ., Our method successfully indentifies both of the two SNPs through the identification of an epistatic module that included these two SNPs ( two more SNPs are also indentified in the same epistatic module , and the posterior probability of the module is above 0 . 9 , see Figures 6 and 7 ) ., The nominal p-values for rs380390 and rs1329428 are 1 . 75×10−6 and 3 . 64×10−6 , respectively , according to the Chi-squared test with two degrees of freedom ., Our method also indentifies two novel SNPs , rs1394608 and rs3743175 , by detecting an epistatic module that includes both loci ( two more SNPs in LD with them are also indentified in the same epistatic module , and the posterior probability of the module is greater than 0 . 9 , see Figures 6 and 7 ) ., The nominal p-values for these two SNPs are 8 . 81×10−5 and 1 . 76×10−3 , respectively , according to the Chi-squared test with two degrees of freedom ., Note that the p-value for the combination of rs1394608 and rs3743175 is 7 . 39×10−7 , while the p-value for the combination of rs380390 and rs1329428 is only 1 . 84×10−5 , according to the Chi-squared test with eight degrees of freedom ., The distributions of the combination of rs1394608 and rs3743175 in cases and controls are shown in Figures 8A and 8B , respectively ., According to Chi-squared tests with four degrees of freedom , these two SNPs are independent in controls ( p-value\u200a=\u200a7 . 50×10−1 ) and dependent in cases ( p-value\u200a=\u200a3 . 27×10−3 ) ., We also infer the genotype frequencies of the combination of these two SNPs according to their distributions in contro
The detection of epistatic interactive effects of multiple genetic variants on the susceptibility of human complex diseases is a great challenge in genome-wide association studies ( GWAS ) ., Although methods have been proposed to identify such interactions , the lack of an explicit definition of epistatic effects , together with computational difficulties , makes the development of new methods indispensable ., In this paper , we introduce epistatic modules to describe epistatic interactive effects of multiple loci on diseases ., On the basis of this notion , we put forward a Bayesian marker partition model to explain observed case-control data , and we develop a Gibbs sampling strategy to facilitate the detection of epistatic modules ., Comparisons of the proposed approach with three existing methods on seven simulated disease models demonstrate the superior performance of our approach ., When applied to a genome-wide case-control data set for Age-related Macular Degeneration ( AMD ) , the proposed approach successfully identifies two known susceptible loci and suggests that a combination of two other loci—one in the gene SGCD and the other in SCAPER—is associated with the disease ., Further functional analysis supports the speculation that the interaction of these two genetic variants may be responsible for the susceptibility of AMD ., When applied to a genome-wide case-control data set for Parkinsons disease , the proposed method identifies seven suspicious loci that may contribute independently to the disease .
Although genome-wide association studies ( GWAS ) have been quite popular due to recent advances in low-cost genotyping techniques , most of the reported studies only analyze single-locus effects because traditional multi-locus methods are not computationally practical in the detection of epistatic interactive effects of multiple loci ., Here , on the basis of a rigorous definition of epistatic modules that describe interactive effects of multiple loci , we take advantage of a Bayesian model with a properly designed Gibbs sampling strategy to facilitate the detection of such modules ., We confirm via extensive simulation studies that the proposed method , named epiMODE , is not only feasible in detecting multi-locus effects but also more powerful than three representative methods on seven disease models ., We apply the proposed method to an Age-related Macular Degeneration ( AMD ) data and discover that a combination of two loci—one in the gene SGCD and the other in SCAPER—might be associated with AMD ., Considering its advantages , we suggest that the proposed method be applied to more GWAS data for the detection of multi-locus interactive effects .
computational biology/population genetics, genetics and genomics/complex traits, genetics and genomics/disease models, genetics and genomics/genetics of disease, mathematics/statistics, genetics and genomics/population genetics