Statistical advances and challenges for analyzing correlated high dimensional SNP data in genomic study for complex diseases

Statisti cs Surve ys V ol. 2 (2008) 43–60 ISSN: 1935-7516 DOI: 10.1214/ 07-SS026 Statistic al adv ances and c h allenges for analyzing corr elated hi gh dimensional SNP data in genomic study for compl ex diseases ∗ Y ulan Liang Dep artment of Biostatistics University at Buffalo, The State University of Ne w Y ork, Buffalo, NY 14214, USA e-mail: yliang@b uffalo.e du Arpad Kelemen Dep artment of Neur olo gy, Buffalo Neur oimaging A nalysis Center, The Jac obs Neur olo g i c al Institute, Universit y at Buffalo, The State University of New Y ork, 100 High Str e et, Buffalo, NY 14203, USA e-mail: akelemen @buffalo .edu Abstract: Recen t adv ances of information tec hnology in biomedical sci- ences and other applied areas hav e created numerous l arge dive rse data sets with a high dimensional featu re space, whic h pro vide us a t remendous amoun t of inf ormation and new opportunities f or improving the quality of h uman l i fe. M ean while, great challenges are also created drive n by the con tin uous arr iv al of new data that requir es researche rs to conv ert these raw data in to scient ific knowledge in or der to benefit f rom it. Association studies of complex di seases using SNP data ha v e become more and more popular in biomedical r esearc h in r ecent years. In this pap er , we present a review of recen t st atistical adv ances and challenge s for analyzing correlated high dimensional SNP data in genomic asso ciation studies for complex dis- eases. The r eview includes b oth general feature r eduction approac hes for high dimensional correlated data and more sp ecific approache s for SNPs data, which i nclude unsupervised haplotype mapping, tag SNP selection, and supervised SNPs selection using stat istical testing/ scoring, statistical modeli ng and machine learning m ethods with an emphasis on ho w to iden- tify i n teracting lo ci . Keywords and phra ses: Complex disease, High dimensional dat a, Single Nucleotide Polymorphism, Statistical metho ds. Receiv ed June 2007. Con ten ts 1 Int ro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2 F ea ture selection metho ds for high dimensional problems . . . . . . . . 45 3 SNP selections in genome-wide asso ciation studies . . . . . . . . . . . 46 ∗ This pap er was accepted b y Mi c hael Kosorok, Ass o ciate Editor for the IMS. 43 Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 44 3.1 Statistical measures and testing for SNP-disease a sso ciation . . . 47 3.2 Super vised statistical mo dels and statistical learning algorithms . 48 3.3 Unsuper vised haplotype mapping approaches . . . . . . . . . . . 50 3.4 Computational int elligence approaches . . . . . . . . . . . . . . . 50 4 Other c hallenges in genetic asso ciation studies o f complex diseases . . 51 5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1. In tro duction Correla ting genetic v ariations in DNA sequences with phenotypic differences has bee n one of the grand challenges in biomedical resear ch. Substan tial efforts hav e bee n made to o btain all common g e netic v ariations in h umans, including single nu cleotide po lymorphisms (SNPs), dele tio ns and ins e r tions [ 13 ]. SNPs a re sin- gle base pair p ositio ns in genomic DNA at which different s equence alternatives (alleles) exist in normal individuals in some populatio n(s), wherein the least frequent allele ha s an abundance o f 1 % or gre ater [ 13 ]. In pra ctice, the ter m “SNP” is used more lo osely . Restricting the attention to co mmon SNPs with minor a llele frequency bigger than a certain cutoff, e.g. 1% will help to filter out so me “ r ecent” mutations. SNPs are b elieved to a lter the risk for develop- ing particular diseases. It is, how ev er, very unlikely that individual SNPs play an impo r tant role in the development of co mplex diseases. Instead, high-or der int eractions o f SNPs a re suppo sed to e x plain the difference s betw een low and high risk population gro ups. The HapMa p Pro ject has collected g enotypes of millions of SNPs from p opu- lations with ances tr y fr o m Africa, Asia a nd Europ e a nd makes this infor mation freely av ailable in the public domain [ 93 – 95 ]. T o find evidence of asso ciation in this huge data set is a gr and challenge no w. Therefore, there is a great need, con- ceptually as w ell as computationally , to develop adv a nced ro bust algo rithms and analytical metho ds for characterizing ge ne tic v ariations that ar e non-redunda nt and identif y the target SNPs that a r e most lik ely to affect the phenotypes and ultimately contribute to disease developmen t. Exploiting informa tion redundanc y due to a sso ciations between SNP markers po tent ially reduces the efforts in terms o f time and cost for gene tic a sso ciation studies [ 75 ]. How ev er, the efficacy of sear ching for an optimal set of SNPs ha s not b een as succes s ful as exp ected in theor y . One primary ca use is the high dimensionality with highly correla ted feature s /SNPs that ca n hinder the p ow er of the identification of small to mo derate genetic effects in co mplex dis eases. The need to incor p o rate cov ar iates of other en vironmental risk factors as effect mo difiers or confounders further worsens the “ curse o f dimensionality” problem in mapping genes for complex diseases [ 16 ]. Therefore , featur e se le c tion for mas- sive genomic data in high dimensions has be come one of the ma in tas ks to b e tackled with statistical and computational efforts in the past deca de . Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 45 2. F eature sel ection metho ds for hig h dimens ional problems The computational and s tatistical metho ds that address the “curs e of dimen- sionality” problem in ge nomic research can be group ed into three categor ies: filtering, wrapp er , a nd embedded metho ds. Filtering metho ds select feature s ub- sets indep endently from the lear ning clas s ifiers and do not incorp or ate learning. Therefore, filtering metho ds a re fast [ 10 ; 60 ; 6 9 ; 10 9 ]. A weakness of filtering metho ds is that they o nly consider the individual features in iso lation and ig - nore the p ossible interaction among them. Y et, the co m bination of these featur es may hav e a co m bined effect that do es not nece s sarily follow from the individual per formances of features in the group [ 73 ]. One of the consequences of filtering metho ds is that we may end up with many highly cor related features/ SNPs with highly redundant information that worsens the class ification and predic- tion per formance. If there is a limit o n the num b er of features to b e chosen, then w e ma y not b e able to include all the informative ones. T o a ddress this problem in filter ing metho ds, wrapp er metho ds wrap around a pa rticular learning algorithm that can asses s the selected feature subsets in terms of the estimated clas sification err o rs and then build the final classifier [ 44 ]. W r app er metho ds use a le arning machine to measure the quality of subsets of features. One of the well-kno wn wrapp er metho ds for feature selection is Sup- po rt V ector Machine Recursive F ea ture E limination, which r efines the optimum feature set by using a Supp or t V ector Machine, [ 33 ]. The idea of SVMRFE is that the or ientation of the sepa rating hyper - plane found by the SVM can b e used to se le ct informa tive features: if the plane is o rthogona l to a particular feature dimension, then that feature is infor mative, and vice versa. SVMRFE uses the weight s o f a SVM class ifier to pro duce a feature ranking, and then eliminates the feature with smallest w eigh t mag nitude recursively . W r app er metho ds ca n be used with arbitr ary classifiers and can nota bly re- duce the num ber of fea tur es and significantly improv e the classification accur acy [ 63 ; 79 ]. How ev er, w r app er methods hav e the dra wback that they do not incor- po rate kno wledge a b o ut the sp ecific structure of the cla ssification or re g ression function [ 52 ]. Moreover, they are mo re computatio nally exp ensive since they need to ev aluate a cross-v alidation scheme a t each itera tion. With muc h b etter computational efficiency and s imila r p e r formance to wrap- per metho ds, a r elatively new cla ss of approaches for featur e selection called “embedded metho ds” has be c o me av ailable in the literatur e. Lal et al. [ 52 ] pro - vide the detailed mathematical form ulations of embedded metho ds. Embedded metho ds pro ce s s feature s election simultaneously with the learning classifier and the fea ture selection can no t be separa ted from the learning. F or exa mple, W e- ston e t al. [ 107 ] mea sure the imp or tance of a feature using a b ound that is v a lid for Supp or t V e c tor Machines only , thus it is not po ssible to use this metho d with, for example, decision tr e es. Therefore the structure o f the class of functions under co nsideration pla ys a crucial r ole. F o r an embedded metho d, every subset of features is mo deled by a vector σ ∈ { 0 , 1 } n of indicator v ariables, σ i := 0 indicating that a feature is present in a subset and σ i := 1 indicating that a feature is absen t (i = 1, . . . ,n). Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 46 A parameterized family of classificatio n or regress ion functions are given as follows: f : Λ × ℜ n → ℜ , ( α, x ) ∝ f ( α, x ). The goal of an em bedded metho d is to find a vector of indicator v ariables σ ∗ ∈ { 0 , 1 } n and α ∗ ∈ Λ that minimize the exp ected risk R ( α, σ ) = R L [ f ( α, σ ∗ x ) , y ] dF ( x, y ), where ∗ denotes the po int wise pro duct, L is a loss function and P is a measure on the do main o f the training data ( X ; Y ). One may imp ose some additional cons tr aints for pena lt y or r egulariza tions to achiev e sparseness: s ( σ ) ≤ σ 0 , where s : [0 , 1] n → ℜ + measures the spar s ity of a given indicator vector σ . F or exa mple, s could b e defined as: s ( σ ) := l 0 ( σ ) ≤ σ 0 , that is to b ound the zero “ norm” l 0 ( σ ), which counts the num ber o f nonzero ent ries in σ . The L1-no rm, L2-norm, and L ∞ - norm or the elastic- net p enalty , a mixture of the L2-no rm and the L1-no rm p enalties [ 105 ] a r e also prop osed to achiev e automatic featur e s elections by s hr inking the fitted co efficients tow ard zero. These automatic fea ture selection metho ds a lso b enefit from the reduction in the fitted coefficients’ v ariance. One of the merits of an embedded metho d is that it int ends to find the feature subset of a certain size that leads to the b est po ssible genera lization or equiv alently to minimal r isk, which can b e seen fro m the ab ov e for mulation. Therefore, the function that measur es the quality of a scaling factor can be ev alua ted faster than a cro ss-v a lidation error e s timation pro cedur e. Moreover, they turn the m ultiple testing problems for feature selection in to an optimization problem in the no nparametric setting. Some recent studies [ 90 ; 10 5 ] hav e s hown that they are mor e co mputatio nal efficient and asymptotically optimal for high dimensional data. Embedded metho ds tend to hav e higher capacit y than filtering metho ds and are therefore mor e likely to ov erfit. W e th us exp ect filter ing metho ds to p erfor m better if o nly a small amount of training da ta is av aila ble. E mbedded meth- o ds even tually outp erform filtering metho ds as the num b er of training sam- ples increase. L ASSO prop os ed by Tibshir ani [ 97 ; 98 ], lo gic regression with the regular iz ed Laplacia n prior [ 51 ] a nd Bayesian regular ized ne ur al netw ork with automatic relev a nc e determination [ 56 ] are examples of em bedded metho ds. Note that the three featur e r eduction methods , filter, wra ppe r and embedded metho ds discus sed in this section may p erfo r m differently when a pplied to cate- gorica l SNP data instead of contin uous g ene ex pression data, in which there are only three genotypes, tw o ho mo zygous genotypes and o ne heterozygo us geno- t ype . Next we will fo c us on the rev iew of the r ecently develop ed categoric a l SNP data reduction methods in geno me wide ass o ciation studies. 3. SNP selectio ns in genome -wide asso ciatio n s tudies A ma jor aim of as so ciation studies is the identification of po lymorphisms, usu- ally sing le n ucleotide po lymorphisms (SNPs) asso ciated with a trait or disease status. There a re se veral ma jor computational and statistical tracks for SNP selections, which we will review next [ 3 ; 18 ; 25 ; 34 ; 35 ; 40 ; 46 ; 48 ; 59 ; 115 ]. Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 47 3.1. Statistic al me asur es and testing for SNP-dise ase asso ciation Spec ific a lly , in genome-wide disea s e asso cia tion studies, v arious s tatistical mea- sures and testing based approaches have b een prop osed for sele c ting a sub- set of SNPs [ 17 ; 30 ; 3 6 ; 5 7 ; 85 ; 89 ]. These include Link age Disequilibrium (LD) based SNP selection a nd supe rvised SNP selectio n. Link age Disequilibrium based methods for selecting a maxima lly informative set of SNPs for asso cia- tion a na lyses were develop ed first [ 24 ; 92 ; 10 1 ; 1 02 ; 1 08 ]. F or instance, Zhang and J in [ 11 4 ] ident ified tagSNPs fro m haplotype data in tw o steps; fir st, they ident ified ha plotype blo cks and then identified tagSNPs that b est distinguish the haplotypes w ithin a haplotype blo ck. This metho d is applicable for a ll types of asso ciatio n studies. Anderso n and No v em bre [ 1 ] and Mannila et al. [ 61 ] pro- po sed finding haplotype blo ck b ounda ries using minimum de s cription length. Entrop y-based meas ure for SNP selections were pr op osed by Hamp e, Schreiber , and Kr aw czak [ 36 ] and Zhao, Boe rwinkle, and Xiong [ 1 17 ]. Beckmann et al. [ 8 ] presented Mantel s tatistics for SNP selections a nd disease mapping purp oses by using haplotype sharing to corr elate temp oral and spatia l dis tr ibutions of cancer in a g e neralized regression mode l. A sliding window appro ach developed by Neale and Sham [ 68 ] combines p- v alues from m ultiple indep endent tests using χ 2 = − 2 P m i =1 l og ( p i ) ∼ χ 2 2 m . Here, p i is the p-v a lue o f as s o ciation b etw een S N P i and presence of disease, and m is the n um ber of SNPs in the sliding window. The test sta tistic χ 2 has a chi-square dis tr ibution with 2 m deg rees of freedom. The sliding window incorp ora tes the or dering of SNPs on the chromosome and merges res ults a cross adjacent windows to detect chromosome regions with significant ass o ciations [ 27 ; 84 ; 1 13 ]. How ev er, it do es not consider the distance b etw een them and the implicit assumption is that the SNPs ar e equally spa ced. The scan statistic [ 26 ; 3 9 ; 54 ; 91 ; 1 0 4 ] do es account for the spacing and order - ing of SNP s on the chromosome, but it do es not consider gene-g ene interactions. F o r instance, Sun, et al. [ 91 ] developed a ch romoso ma l scan statistic approa ch, which includes tw o par ts : (i) Identif ying SNP clusters; (ii) Identif ying SNP clus- ters with significa nt dis e a se ass o ciation. This sca n metho d assumes the po sition of ea ch SNP is ra ndomly determined by a Poisson pro cess . The lengths b etw een t wo a djacent SNPs ha v e an exp onential distribution and the sum-o f-lengths betw een SNPs has a Ga mma distribution. Under the ab ov e assumptions, the clusters of SNP s are fir st identified b y testing the hypothesis tha t whether the observed leng th b etw een a set of SNPs (co mbin ed interv al b etw een these SNP s ) is equal (null hyp o thesis) or less than (alternative hypothesis) the expected length. Rejection of the null hypothesis identifies this gr oup of SNPs a s clus- ter. T o further identify SNP clus ter s with sig nifica nt disease a sso ciation, diseas e outcomes ar e incorp o rated and Pearso n C hi- square p-v alues are co mputed fo r asso ciatio ns of significance. Other test statistic appro aches, such as the sc ore statistic [ 81 ; 82 ], a nd weigh ted- av erage s ta tistic [ 87 ] for disease mapping in case-control studies were als o pro- po sed fo r SNP selection in ge netic asso cia tio n studies. Cheng et al. [ 20 ] prop ose using the exp ectation maximization (EM) alg orithm to estimate haplotype fre- Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 48 quencies of multip le linked SNPs, and follow this by constructing a contingency table statistic S for LD analy sis based on the estimated haplotype frequencies. An empirica l p-v a lue is obtained base d on the null distribution o f the max im um of S (S*) from a large num b er (e.g., 1,000 or more) of r a ndomized p ermuta- tions. This metho d is develop ed fo r mapping functional sites or reg io ns from case-control data using haplotypes of m ultiple link ed SNP s. All these conv en tional test based filter approaches estimate the asso ciation betw een each SNP (or multiple SNPs) and a phenotype, and then use the cor - resp onding p- v alues to prioritize the res ults. O ne drawback is that one may end up with many highly correla ted SNPs or genes with high re dunda ncy informa- tion, which can b e h urdles for further classifica tions and predictions. Also the test based approa ches ca n no t incorp orate many environmental factors to ac- count for g ene-environment interactions. F urthermore, the no n-indep e ndence of SNPs in physical proximit y (Link age Disequilibrium) may cause pro blems for m ultiple testing sce na rios with corr elated tests [ 6 ; 7 ; 23 ; 70 ; 80 ; 112 ]. Simple co r- rections may lead to either co nserv ativ e p-v alues if Bonferro ni correction is used or b ecome computationally ex pens ive, if p ermutation is use d [ 84 ]. Nyholt [ 70 ] prop osed a method for efficient ly accounting for multiple tes ting of many SNPs in an asso ciatio n study that in v olves estimating an “effective num ber ” of inde- pendent tests, and then adjusting the smallest observed p-v a lue using Sida k’s formula based on this n um ber of tests. Salyakina et al. [ 80 ] further ev alua ted this method. Note that the “multiple testing problem” discussed here differs from the “curse of dimensio nality pro blem”, so it p oses different challenges. “Multiple testing problem” is ca used by the high dimensionality of the predictors (includ- ing features plus p o ssible interactions of features) and the complex c o rrelatio n structures of the predictors, while the “cur se o f dimensionality problem” ar is es when c o nsidering the interaction of many features, i.e., there are not enough observ ations in each combination o f those features. Last, but not least, thes e exis ting testing based appro a ches ignore s ome in- formation ab o ut the SNPs, suc h a s sub-s tructures of the underlying p opula tion (admixture proble m). This may lead to spurio us results as well as s uffer from low pow er. This may explain why repro ducibilit y has b ecome a ma jor issue in genomic a sso ciation studies for co mplex disea ses. The sa me data set can show a highly sig nificant a sso ciation with o ne method, wher e as a different metho d shows no o r only a marginal asso cia tio n. Also , g iven the low prior probabil- it y of causa lity for each SNP in the genome, r igoro us standar ds of statistica l significance ar e needed for genome-wide a sso ciation studies in order to avoid a flo o d of false-p ositive r esults. Multiple replications in lar g e s amples may pro- vide the most straightforw ard pa th in ident ifying r o bust and broadly r elev a nt asso ciatio ns. 3.2. Sup ervise d stati stic al mo dels and statistic al le arning algor ithms In order to incor p orate e nvironmental factors and o ther cov ariates /confounders int o the geno mic asso cia tion studies, v a rious mo del based approa ches hav e b een Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 49 developed. He a nd Zeliko vsky [ 37 ] pr op osed tagSNP s for unphased g enotypes based on mu ltiple linear regr essions. Durrant et al. [ 28 ] ado pted a log istic- regres s ion mo del applicable to whole-genome screens using sliding w indows; it controls for o ther (contin uous) confounders a nd gene-e n vironment interactions. Y et, they ha ve to make assumptions on the disease model, which is usually un- known in pra c tice. Moreover, the effects of violations of these assumptions are unpredictable in g eneral. Baker [ 5 ] applied a simple lo glinear mo del for haplo- t ype e ffects in a case-control study inv olving tw o unphased genotypes. The haplotype tr end regr ession, developed by Zaykin et al. [ 111 ], fits a mo del of additive effects of haplotypes that takes a s e r ies of marker genotypes, com- putes haplo t ype probabilities for e a ch o bserv ation us ing the co mpo site haplo type metho d (CHM), and for ms a linear regre s sion on the r esp onse using the ha p- lotype pro babilities a s the r egressio n matr ix. A nonpara metric metho d called Haplotype Pattern Mining (HPM) was pr op osed to ident ify disease asso ciated haplotype patterns from case-c o ntrol da ta. HPM has tw o steps: In step I, given the data-mar kers, haplotypes, and phenotypes , the g o al is to output all haplo- t ype patterns that are stro ngly asso ciated with the disease sta tus for a g iven v alue of the asso ciation threshold; In step I I, it is to find the “gene lo ca tion”, by counting fr equency that one marker app ears in the haplotype pa tterns identified in the first step. Since the HPM metho d utilizes the diseas e status (case/ control), it is a super vised mining appro a ch. T oivonen et al. [ 99 ] sho w ed that HPM do es not require any assumptions on the inheritance patterns a nd ha s go o d lo ca l- ization p ow er, even when the num ber of pheno co pies is la rge. Knorr- Held and Rue [ 49 ] develop ed Mar ko v ra ndom field mo de ls on blo ck updating for disease mapping. O ther mo del-based approaches that ca n take into a ccount the spatial correla tion betw een markers were also pro p o sed [ 14 ; 31 ; 32 ; 42 ; 9 6 ; 100 ; 103 ; 106 ]. Recently , Sch wender and Ickstadt [ 83 ] demonstra ted logic re gressio n based ident ification of SNP interactions for the dis ease status in cas e-control study and prop osed tw o measures for q ua ntif ying the imp ortance of featur e int eractions for classificatio n. In compa rison with some well-known classification metho ds such as CAR T [ 12 ], Rando m F orests [ 1 1 ], and other regress ion pro cedures [ 10 8 ], logic r egress ion ha s shown a go o d classifica tio n p erforma nce when applied to SNP data. When fitting with categor ical features/ v ar iables in the mo del based approaches, i.e. the genotype mea s urements with tw o homozygous g e notypes and one hetero zygous genotype, we often define a set of dummy v aria ble s that represent a single categorica l feature/v ar iable. In order to select the se t/ group of dummy v ariables that r epresent a single c ategorica l fea tur e/v a riable/SNP simult aneously , Y uan and Lin [ 110 ] pr op osed the gro up-Lars and the group- Lasso metho ds. P ark and Hastie [ 72 ] prop osed se veral regula rization path a lgo- rithms with g roup ed featur e /v ar iable selection for mo deling gene-g ene in terac- tions. Multifactor dimensio nality r eduction has b een prop o sed and implemen ted for SNPs data reduction by Coffey et a l. [ 22 ], Ritchie et al. [ 77 ] a nd Mo o re et al. [ 64 ]. Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 50 3.3. Unsup ervise d haplotyp e mapping appr o aches Haplotype density ba sed clustering algor ithms and clustering techniques ba sed on the deg ree of haplo type shar ing in affected individuals for haplo type map- ping were developed r e cently . These approaches have adv antages of robustness since they are nonpa rametric and require fewer as sumptions in mo deling. F u et al. [ 29 ] and Zhang et a l. [ 1 1 6 ] prop osed Bayesian mo dels for the ana ly sis of genetic structure when p opulations are corr elated. Liu et a l. [ 58 ] employ ed a Bay esian a pproach to mo del p o sitions of the histor ical recombinations and m utation even ts that pro duced the o bs erved haplotypes fro m an initial set of founders by acco un ting for all sources of uncertainties. They employ ed Monte Carlo Markov Chain (MCMC) metho d for parameter estimation a nd assig ned haplotypes to clus ters repr e senting allele heter ogeneity . Molitor et al. [ 62 ] mo d- eled haplotype ris ks using clusters obtained fr o m a proba bilit y mo del, but their metho d do e s not ta ke pheno co pies into consider a tion. Both metho ds were devel- op ed mainly for haplot ype fine mapping and do not scale up for whole-genome screens v ery w ell. Other alg orithms for SNPs a re hierarchical clustering and graph metho ds [ 2 ; 55 ]. Pr incipal Comp onent Analysis with multip le genotype frequencies was also applied to select a subset of cor related SNPs that capture multiple g enotype v aria bility in the r egion [ 9 ; 5 7 ]. Howev er, whether Principal Comp onent Analysis is a suitable to ol for ca tegorica l SNPs information is arg uable, since it is more appropria te fo r co nt in uous scale data . The related corr esp ondence ana lysis may be mor e suitable, but the interpretation of the results from corresp ondence analysis reveals many challenges. 3.4. Computational intel ligenc e appr o aches Computational in telligence systems [ 47 ; 7 4 ] hold a gr eat pr o mise for tackling the tasks and c hallenges posed by la rge, diverse, genomic data for complex diseases. Some of these challenges are the iden tification of gene-gene and gene- environmen t interactions [ 4 ; 4 3 ; 50 ; 66 ; 78 ], dealing with the notor ious “curse of dimensionality”, the uncer taint y , and unclear , fuzzy b oundarie s of phenotypes for complex diseases [ 76 ; 88 ]. T echniques include neura l netw orks [ 7 1 ], genetic algorithms [ 2 1 ], g enetic programming [ 65 ], evolutionary trees [ 53 ], evolutionary algorithms [ 41 ] and v arious h ybrid a pproaches. F or instance, Mo ore [ 64 ] de- veloped a hybrid genetic progra mming (GP) with a multif actor dimensiona lit y reduction method to pick SNPs for epistasis . Motsinger, et al. [ 67 ] applied a genetic progr amming neur al netw ork (GPNN) approach for detecting epistasis in case-co ntrol studies for SNPs data. They ev al- uated the p ow er of GPNN for iden tifying hig h- order gene-gene interactions a nd applied GPNN to a real data on Parkinso n’s disea se. They develop ed a Gram- matical Evolution Neural Netw ork (GENN), a machine-learning approa ch to detect gene - gene and gene- environmen t in teractions in high dimensiona l genetic epidemiological da ta. F urthermore, they pro p o sed an E nsemble Le a rning Ap- proach for Set asso ciation (ELAS) to detect a se t of interacting lo ci that predicts Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 51 the complex trait. An imp ortant adv antage of the hybrid approach is that a ny form of expe rt knowledge could b e used to guide the sto chastic sea rch a lgorithm to iden tify epistatic SNPs in the absenc e of marginal effects. 4. Other c hallenges in genetic asso ciation studie s of compl ex diseases An imp or tant challenge that faces molecular ass o ciation studies in the p ost ge- nomic era is to under stand the interconnections from a netw ork of genes and their pro ducts that a r e mo dified by a v ariety of en vironmental factors [ 15 ; 45 ]. The v ar iety of phenotype definitions leads to a m ultiplicit y of tests that inv olv e a larg e n um ber of compariso ns that often r esult in less p ow er. The need for a de- quate a lgorithms a nd mode ls for r educing biolog ical and statistical re dundancy from thousands of SNPs and finding an o ptimal s et of SNPs asso ciated with dis- eases ar e pressing for common complex disea ses. Dealing with man y depe ndent asso ciatio n tests is one of the emer ging issues on the statistical/ computational side. F o r SNP-disease data, in addition to b eing larg e, redundant, diverse and dis- tributed, three imp orta nt characteristics p ose c hallenges for data a nalysis and mo deling: (1) heterog eneity , (2) a c onstantly evolving biolog ical nature and (3 ) complexity . Firstly , there is the heter o geneity of SNP data, in the sense that i) the p opulatio n data inv o lves the po pulation substructur e or admixture problem and there is lo cus he ter ogeneity wher e a large fraction o f the prev a lence is due to pheno copies; and ii) there is a wide array of data types, including categor- ical, contin uo us, sequence data, as well as temp or a l, inco mplete and missing data. Such data sets are large with a lot of r edundancy in SNP and haplo- t ype databases. Sec o ndly , they a re very dynamic and contin uously evolving, which means that s pe c ia l knowledge is req uired when designing the mo deling techn iques. Lastly , but most imp or tantly , these SNP and haplotype da ta a re complex with intrinsic features a nd subtle patterns, in the sense that they ar e very rich in asso ciated co mplex phenot ype tra its. The difficulty in a SNPs asso ciation study is inc r eased by the nature of com- plex disea ses [ 38 ]. Typically , the contribution o f single genes as well as of single environmen tal risk facto r s is small to mo dera te. F urthermore, most complex diseases result from gene-gene and gene-environment in teractions [ 19 ]. By dis- regar ding interactions, rela tive r isks of individual gene tic v a riants are expected to be s mall. Disregarding gene-environmen t interaction also w eakens ex po sure- disease a nd gene-disea se asso ciations. In complex diseas es, it is likely that a combination of genes predisp os es for the disease and environmen tal factor s ag- grav ate the impact of these genes and therefor e a re jo intly resp onsible for disease developmen t in po pulations (known as epistatic effect). In addition, e nvironmen- tal factors, which s eem to have o nly a mo derate impact at the p opula tion lev el might have lar ger relative r isks in subp o pulations with certain genetic pr edis- po sitions. There a re ma jor metho dologica l challenges in the study o f gene-g ene and gene-environmen t interactions. Y. Liang and A. K e lemen/Statistic al advanc es for SNP data 52 Other op en questions and ch allenges for new computatio nal appr oaches in analyzing the asso ciations b etw e en genetic ma rkers, such as SNPs in complex diseases involv e several hierar chical levels. First lev el of complexity: How to analyze multiple SNPs in a s ingle gene? How to ana lyze in teractions a mong m ultiple SNPs in a single ge ne ? Second level of complexity: How to analyze m ultiple SNPs in multiple genes? How to analyze in teractions among multiple SNPs in multiple g e nes? Third le vel of c omplexity: How to analyze interactions among multiple SNPs in multip le g e nes and environmen tal factors? F ourth level of c omplexity: How to analyze a s so ciations b etw een SNPs in single or multi- ple genes and quantitativ e tr aits? How to identify and quantify the perce n tage of the ass o ciation b etw een genes and disea ses expla ined by the asso cia tion be- t ween the same gene and q uantit ative traits, taking into co nsideration single genes, multiple g enes and en vironmental factor s. Lastly , the ultimate g oal in ge- netic/genomic analysis is to build direct or indirec t ca usal asso ciation b etw e en genetic v ariants and phenotypes/dis e a se status, but the difficult y he r e is that we do not kno w if ther e is asso cia tion b etw een the SNPs and the disease. Ho w- ever, with the development o f computational/s tatistical approaches, we may b e able to iden tify these causal asso cia tions and co ns truct the path w ays related to complex diseases. 5. Discuss ion New a dv ances in human geno me resear ch hav e drawn tremendous a tten tion of resear chers fr om multiple fields, including b oth theor etical scientists a nd ap- plied researchers, esp ecia lly in the sta tis tica l field. Huge amounts of con tin- uously gr owing la rge-sc a le g enomic, pro teo mic a nd clinical data for complex diseases and phenotype tra its have p osed ever gr eater challenges for the compu- tational field. Multiple who le geno me wide ass o ciation studies hav e alre a dy b een completed and have resulted in nov el and promising genetic v ar iants for v ari- ous disea ses. In this pa pe r we presented a s urvey of recent adv a nces and some promises of designing, developing and implemen ting statistical/ computational metho ds for identifying SNP markers r esp onsible for common, co mplex, chronic diseases, suc h as diab etes, cancer, multiple s clerosis, and car diov ascular dis- ease and for tackling the challenges, such as gene- gene and gene-e nvironment int eractions a long with the notorio us “curse of dimensiona lit y” pr oblem. 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