Brassica napus . A population of 131 doubled haploid lines derived from a cross between European and Australian parents was evaluated for days to flowering, thermal time to flowering (measured in degree-days) and the number of leaf nodes at flowering in a compact and efficient glasshouse-based experiment with replicated short and long day treatments. All three traits were under strong genetic control with heritability estimates ranging from 0.85–0.93. There was a very strong photoperiod effect with flowering in the population accelerated by 765 degree-days in the long day versus short day treatments. However, there was a strong genetic correlation of line effects (0.91) between the long and short day treatments and relatively low genotype x treatment interaction indicating that photoperiod had a similar effect across the population. Bivariate analysis of thermal time to flowering in short and long days revealed three main effect quantitative trait loci (QTLs) that accounted for 57.7% of the variation in the population and no significant interaction QTLs. These results provided insight into the contrasting adaptations of Australian and European varieties. Both parents responded to photoperiod and their alleles shifted the population to earlier flowering under long days. In addition, segregation of QTLs in the population caused wide transgressive segregation in thermal time to flowering. Potential candidate flowering time homologues located near QTLs were identified with the aid of the Brassica rapa reference genome sequence. We discuss how these results will help to guide the breeding of summer annual types of B. napus adapted to new and changing environments.Time of flowering is a key adaptive trait in plants and is conditioned by the interaction of genes and environmental cues including length of photoperiod, ambient temperature and vernalisation. Here we investigated the photoperiod responsiveness of summer annual-types of  Timing of life history events (phenology) such as flowering time and maturity is crucial for the successful adaptation of a flowering plant to its environment Arabidopsis thaliana  is a close relative of Arabidopsis. Like Arabidopsis, rapeseed has both winter and summer annual types. Summer annual-type rapeseed has little or no requirement for vernalisation in order to flower, in contrast to winter annual-type rapeseed Rapeseed  analysis. Most studies have focused on populations with contrasting vernalisation responsiveness  and Northern Europe (grown over spring and summer), and the reproductive isolation of these breeding pools, we expected that the Australian and European parents would contribute contrasting alleles for photoperiod sensitivity in flowering to the population. Using the published B. napus population (n = 131) derived from reciprocal F1 hybrids created from two genetically-distinct parents  used in this study was a doubled haploid (DH) http://www.timeanddate.com/worldclock/sunrise.html). Day time light intensity  was recorded every 15 minutes and is summarised in ±0.53 µmol.m–2.s–1. Light pollution from the LD treatment benches, which may have affected neighbouring SD treatment benches, was prevented by drawing black curtains around the LD treatment benches at dusk every night. During the day, the curtains were drawn open to allow full sunlight on all LD and SD treatment plants. Supplementary lights in the LD treatment were withdrawn once all plants had flowered or were bolting with visible floral buds . Ambient temperature in degrees Celsius was recorded at two points within the glasshouse every 15 minutes throughout the experiment. The change in season between winter (June to September) and spring (October to December) was reflected in the sharp temperature rise in the monthly mean temperature between September and October (Seeds were sown on 10 June 2009 (approximately two weeks before the winter solstice) into potting mix in 50 mm×50 mm tall pots and placed on benches in a glasshouse at The University of Western Australia . Plants were watered daily throughout the experiment. There were four benches, each with 200 pots, arranged as 10 columns by 20 rows. The two day-length treatments were allocated to benches with two replicates (benches) per treatment. The entries were randomised to pots within benches using a partially replicated design  October . Days toB. napus as reported by Raman et al. B. napus genome. The linkage mapping strategy used to generate the linkage map The genetic map developed using this DH population was based on 135 SSR markers developed by Aslam The approach used in this study was a hybrid of those presented in Verbyla et al. After fitting the baseline model, denoted as M0, individual markers were scanned to establish a final multi-QTL model. The steps involved in this process were as follows:all marker and marker x treatment interaction effects are included simultaneously, and that effects for the ith marker are fitted as fixed effects, whereas the remainder are fitted as random effects. This differs from Pastina et al. i+mi×trt where mi represents the (fixed) main effect of the marker and mi×trt the (fixed) marker by treatment interaction effect. For each model a P-value is obtained for the Wald statistic for both of these fixed effects. In order to respect marginality, the interaction effects were first examined. All marker by treatment interaction effects with P-values of less than 0.05 were chosen. The associated set of markers was then thinned to exclude markers with a high degree of collinearity using a threshold of 20 cM . Suppose that the resultant set of markers is labelled a1…aa. The main effects and interactions for all markers in this set were then added to M0 as fixed effects to produce a working multi-QTL model, M1. Schematically this could be written as M1 = M0+a1+…+aa+a1×trt+…+aa×trt.In a manner similar to Pastina et al. i+mi×trt was fitted and significant marker by treatment interaction effects identified. Suppose that the resultant set of markers is labelled b1…bb. The sets of markers identified in (1) and (2) were combined and thinned for collinearity and the resultant set of main effects and interactions were fitted as fixed effects in working multi-QTL model M2. For ease of presentation of a schematic model, the case of no such collinearity is considered here, so that the model can be written as M2 = M0+a1+…+aa+b1+…+bb+a1×trt+…+aa×trt+b1×trt+…+bb×trt. This second scan was needed to ensure that the working multi-QTL model captured a substantial amount of marker by treatment interaction variance.Step (1) was repeated but with M1 as the base model. Thus for each marker not already included with fixed effects in M1, the model M1+m1…cc identified in the first scan and d1…dd in the second, and assuming no collinearity between them, the model is then given by M3 = M0+a1+…+aa+b1+…+bb+c1+…+cc+d1+… dd+a1×trt+…+aa×trt+b1×trt+…+bb×trt.Steps (1) and (2) were repeated but in the context of examining marker main effects and commencing with M2 as the base model. This resulted in the identification of additional main effects to be added to the fixed effects to produce working multi-QTL model M3. Thus for markers c1…aa, b1…bb identified in this way, will be labelled as e1…ee. This produced working multi-QTL model M4 = M0+e1+…+ee+c1+…+cc+d1+…dd+e1×trt+…+ee×trt.Backward elimination of fixed marker effects was then performed to obtain a parsimonious model. Once again, marker by treatment interaction effects were considered first, with backward elimination performed on M3. Effects were eliminated until the percentage of marker by treatment interaction variance fell to a nominated threshold. The resultant set of markers, that is, the subset of a1…cc, d1…dd identified in this way, will be labelled as f1…ff. This produced the final multi-QTL model Mf = M0+e1+…+ee+f1+…+ff+e1×trt+…+ee×trt.Step (4) was repeated on M4 for marker main effects. The resultant set of markers, that is, the subset of cP = 0.00001 level were deemed important and only these were reported. All other fixed marker effects in the final multi-QTL model were retained as ‘co-factors’ in the sense of composite mapping.The final multi-QTL model was fitted to obtain Wald statistics for all marker by treatment interactions and main effects in the fixed part of the model. Effects that were significant at the These analyses were conducted using ASReml B. napus genetic map to the A-genome of B. rapa−20) of sequenced marker clones Brassica A genome were determined by BLASTn analysis using coding sequences from 27 Arabidopsis thaliana genes listed in −30).Alignment of the P<0.0001) for all traits; line means for all traits in LD and SD treatments are presented in Seeds germinated successfully for all parents, F1 and control lines, and for 128 out of 131 DH lines, which were scored for thermal time to flowering (THERM), days to flowering (DTF) and leaf nodes at flowering (LNF) traits in long day (LD) and short day (SD) treatments . A so-caThe extent of genetic control of traits was then investigated by calculating line mean heritability for each treatment as the mean of the squared accuracy of the predicted DH line effects see . All traThe residual maximum likelihood (REML) estimates of the genetic variance parameters  from the phenotypic mixed model for each trait are given in The proportion of total variance that could be attributed to individual marker loci (i.e. QTLs) was then investigated for each trait. The probability of marker by treatment interactions (i.e. QTL×E) was first considered. The main effect was then calculated for those QTLs showing no interaction effect. Three main effect QTLs controlling THERM were detected which together accounted for 57.7% of variance in both LD and SD treatments . Both paThree main effect QTLs controlling DTF were detected which together accounted for 52.7% of variance in both LD and SD treatments . All thrFour QTLs controlling LNF were detected which together accounted for 63.3% of the variance in the LD treatment and 90.4% in the SD treatment . Three LB. napus LMDH genetic map were aligned to the B. rapa A-genome sequence via BLASTn analysis of sequenced genetic markers  QTLs in summer annual-types of rapeseed. Despite the very compact nature of the experiment, spatial analysis improved accuracy of the estimation of line effects, and heritability estimates of THERM were extremely high: 0.91 in long days (LD) and 0.92 in short days (SD) . The strPhotoperiod had a profound effect on flowering, which was accelerated by an average of 765 degree-days in LD compared with SD treatments . HoweverGiven the genetic distinctiveness of Australian and European summer annual type rapeseed varieties It is unclear at this stage which floral initiation pathway accounted for the non-photoperiod related (i.e. main effect) QTLs. While both Australian and European rapeseed parents used in this experiment were summer annual types, residual vernalisation requirement could conceivably underlie some of these QTLs. The minimum temperature in this glasshouse-based experiment was 11.3°C and there were just 9 hours below 12°C in the whole growing period, which would be insufficient to fulfil vernalisation requirements. However, the variety Monty  is known to have a very low vernalisation requirement but is highly responsive to changes in ambient temperature In this study, we included two other measures of flowering time: days to flowering (DTF), which is the normal measure of flowering time in previous rapeseed flowering time studies; and the number of leaves produced by each plant prior to flowering , which is more commonly used in the Arabidopsis research community Brassica A-genome allowed the identification of flowering time homologues for all 6 QTLs located on A-genome chromosomes, and an FLC-based molecular marker co-localised with three QTLs in the C-genome  in the LMDH population. Plants were grown under long day (LD) and short day (SD) conditions in a glasshouse-based experiment.(TIF)Click here for additional data file.Figure S2Histogram of residual errors for days to flowering (DTF) in the LMDH population. Plants were grown under long day (LD) and short day (SD) conditions in a glasshouse-based experiment.(TIF)Click here for additional data file.Figure S3Histogram of residual errors for the number of leaf nodes at flowering (LNF). Plants from the LMDH population were grown under long day (LD) and short day (SD) conditions in a glasshouse-based experiment.(TIF)Click here for additional data file.Figure S4Frequency distribution for days to flowering (DTF) in the LMDH population. Plants were grown under long day (LD) and short day (SD) conditions in a glasshouse-based experiment. The mean DTF for Monty-028DH was 90.5 days (LD) and 104.4 days (SD). The mean DTF for Lynx-037DH was 110.6 days (LD) and 141.0 days (SD).(TIF)Click here for additional data file.Figure S5Frequency distribution for number of leaf nodes at flowering (LNF) in the LMDH population. Plants were grown under long day (LD) and short day (SD) conditions in a glasshouse-based experiment. The mean LNF for Monty-028DH was 15.3 (LD) and 18.8 (SD). The mean LNF for Lynx-037DH was 18.4 (LD) and 23.9 (SD).(TIF)Click here for additional data file.Table S1Graphical genotype information for 128 lines in the Brassica napus LMDH population. Data are presented for 329 non-redundant, high quality framework markers on 19 chromosomes (A01–A10 and C1–C9).(XLSX)Click here for additional data file.Table S2Arabidopsis thaliana flowering time gene homologues and BLASTn matches in the Brassica rapa genome.(XLSX)Click here for additional data file.Table S3Raw data for flowering time-related traits in the LMDH population and controls. Flowering time traits comprised thermal time to flowering , days to flowering (DTF) and number of leaf nodes at flowering (LNF).(XLSX)Click here for additional data file.Table S4Brassica napus LMDH linkage groups A01–A10 aligned to the Brassica rapa genome.(XLSX)Click here for additional data file.Text S1Base-line marker model, M0, in asreml syntax.(DOCX)Click here for additional data file.


Understanding the genetic basis of traits involved in adaptation is a majorchallenge in evolutionary biology but remains poorly understood. Here, we usegenome-wide association mapping using a custom 50 k single nucleotidepolymorphism (SNP) array in a natural population of collared flycatchers toexamine the genetic basis of clutch size, an important life-history trait inmany animal species. We found evidence for an association on chromosome 18 whereone SNP significant at the genome-wide level explained 3.9% of thephenotypic variance. We also detected two suggestive quantitative trait loci(QTLs) on chromosomes 9 and 26. Fitness differences among genotypes weregenerally weak and not significant, although there was some indication of asex-by-genotype interaction for lifetime reproductive success at the suggestiveQTL on chromosome 26. This implies that sexual antagonism may play a role inmaintaining genetic variation at this QTL. Our findings provide candidateregions for a classic avian life-history trait that will be useful for futurestudies examining the molecular and cellular function of, as well asevolutionary mechanisms operating at, these loci. A claA long-standing paradox that needs resolution is the observation that life-historytraits, such as clutch size, often seem to harbour abundant genetic variationdespite being under directional or stabilizing selection . There aFicedula albicollis), a small migratorypasserine that breeds in eastern and central Europe, and is an important ecologicalmodel organism [While previous studies on birds have demonstrated significant heritability of clutchsize e.g. ,18,19), ,19, 18,1organism ,25. Specorganism  and a cuorganism ,28 to ma2.(a)F. hypoleuca) and collared flycatchers [Individuals included in this study were breeding on the island of Ölandduring the years 2003–2010 and were monitored as part of a long-termstudy on pied (catchers . A range(b)Eight hundred and sixty-four collared flycatchers were genotyped with acustom-made 50 k Ilumina iSelect BeadChip with 45 138 SNPs successfully includedon the chip . For fur(c)p-value for rejectionof Hardy–Weinberg equilibrium (HWE) of less than 0.001, we had 37 309SNPs available for downstream analysis. Markers deviating from expected HWE wereremoved to safeguard against potential genotyping errors [After removing markers with a call rate of less than 95%, minor allelefrequency (MAF) of less than 0.01 and a g errors . Ten indX is the fixed-effect design matrixfor non-genetic fixed effects (age and year) and β isthe corresponding fixed effects, XSNP is theSNP-covariate  and βSNP is theSNP fixed effect. The model includes a random genetic effect gand a permanent environmental effect p on each individual thatare linked to observations by incidence matrices Z andW, respectively. For further details regarding the fittedmodel, see the electronic supplementary material. The estimated kinship matrixis the proportion of shared alleles identical by state across all markersweighted by the allele frequencies [p-values are from the above modelbased on Wald tests, and are corrected for relatedness among individuals and therepeated observations on the same individual in the sample. Genome-widesignificance threshold was estimated by dividing the significance value (chosenhere as 0.05) by the number of markers (37 309), resulting in a significancethreshold of p = 1.34 × 10−6,which is conservative because it assumes all markers are independent. Similarly,a suggestive threshold was estimated allowing for one false positive, resultingin a threshold of p = 2.68 ×10−5.Current statistical methods for GWAS do not allow including repeated measures onthe same individual , andwe quencies . ReporteVSNP =2pq(a +d(q −p))2, where a is half thedifference between the two homozygotes , d isthe dominance deviation (which in our case is zero because an additive model wasfitted), p is the MAF and q the major allelefrequency [The additive effect of a marker was calculated asrequency .r2) wascalculated between markers within the candidate regions including all genotypedindividuals using PLINK [Linkage disequilibrium We examined the association between the genotype at the three candidate loci andLRS in males and females using a generalized linear model with Poisson errorstructure using R. Lifetime reproductive success was fitted as a responsevariable and the genotype at the candidate locus was fitted as a three-levelfactor with inferences about differences in fitness between genotypes evaluatedusing an ANOVA table. Information on LRS from genotyped individuals wasavailable from the same 313 females as for the analysis of clutch size above andfrom 301 males that were also genotyped on the SNP array. Similarly,associations between genotype and annual reproductive success was examined usinga generalized linear mixed model (GLMM) with Poisson error structure fittingindividual identity as a random effect to account for repeated observationsacross years on the same individuals. For these analyses, we had informationfrom 815 records from 301 males and 656 records from 313 females.3.h2 = 0.14 (±0.03)  see . Table 1λ = 1.008; see electronic supplementary material,figure S1). After adjustment for multiple testing using strict Bonferronicorrection, we found one SNP on chromosome 18 that was genome-wide significant to detectloci where marker genotype was significantly correlated with observed clutch size,while controlling both for the realized relatedness between individuals as well asthe repeated records on individuals  identified the same QTL region on chromosome 18 andthe two suggestive QTLs as the most significant association signals .p = 0.53; chr 9:p =0.63) or females . Interestingly, however, for the QTL at chromosome 26, males homozygous forthe G allele had significantly higher LRS compared with the other genotypes and therefore reduced LRS (p = 0.13) or lifespan . To learn more about the evolutionary processes operating at the detected QTLs, weexamined selection acting among the genotypes in both sexes using LRS as a compositeestimate of fitness. As we study a wild population, we can investigate the selectionpressures operating on these QTLs within the natural habitat experienced by thebirds. For the QTLs at chromosomes 18 and 9, there were no differences in LRSbetween genotypes in males  means effects are clearly overestimated [Parus major), and founda strong positive correlation between chromosome size and the proportion of geneticvariance in clutch size attributed to that chromosome [A general expectation for traits closely related to fitness, such as clutch size, isthat genetic variance should be low compared with traits that are less closelyassociated with fitness . Howquiosophila , QTL maposophila . While tosophila  and for osophila .Before stimated ,43 and tstimated . Apolygromosome . In thatRAB11FIP4 (RAB11 Family Interacting Protein 4), whichis needed for the completion of cytokinesis [FIP4 is part of the class IIFIPs (FIP4 and FIP3) that localize to the cleavagefurrow/midbody during cytokinesis and are probably required for abscission, thefinal separation of the two cells [Urotensin (UTS2B/D)gene, which is important for vasoconstriction in regulation of blood pressure [We detected one genome-wide significant and two suggestive QTLs that were associatedwith variation in clutch size, and examined whether the detected QTL regionscontained any potential candidate genes. The QTL region on chromosome 18 overlapswith the gene okinesis –47. FIP4wo cells . The sugpressure . Future In general, there were few fitness differences among genotypes at the identified QTLswhen examining a more complete measure of fitness than clutch size, LRS. While wedid discover an interesting sex-by-genotype interaction for LRS at the QTL onchromosome 26, it is important to keep in mind this is a suggestive QTL.Nevertheless, the preliminary analyses indicate that intra-locus conflict may playsome role in maintaining genetic variation at this locus because females homozygousfor the G allele at this QTL had significantly lower LRS compared with GG males. That thPrevious studies that have detected fitness differences among genotypes at adaptiveloci, such as horn morphology in Soay sheep  or armouQTL studies on complex traits in natural populations are rare –58 and,


Arabidopsis thaliana, we find that environmentally predicted traits lead to higher recovery of functional loci in multivariate GWAS and are more strongly correlated to allele frequencies at adaptive loci than individual environmental variables. Our results provide an example of the use of environmental data to obtain independent and meaningful information on adaptive genetic variation.Current methods for studying the genetic basis of adaptation evaluate genetic associations with ecologically relevant traits or single environmental variables, under the implicit assumption that natural selection imposes correlations between phenotypes, environments and genotypes. In practice, observed trait and environmental data are manifestations of unknown selective forces and are only indirectly associated with adaptive genetic variation. In theory, improved estimation of these forces could enable more powerful detection of loci under selection. Here we present an approach in which we approximate adaptive variation by modeling phenotypes as a function of the environment and using the predicted trait in multivariate and univariate genome-wide association analysis (GWAS). Based on computer simulations and published flowering time data from the model plant  Arabidopsis thaliana, we show that this approach may find adaptive genes more effectively compared to existing methods. We also demonstrate that predicted traits can be used to identify relevant loci in individuals for which no phenotypic data is available.Finding genes involved in adaptation to the environment has long been of interest to evolutionary biologists and ecologists. Most commonly, researchers look for loci whose differences in allelic state correlate with differences in a particular trait or environmental variable such as temperature. The implicit assumption behind such methods is that natural selection by the environment will shape variation in adaptive traits through associated changes in allele frequencies. This means that both environmental and phenotypic variation are relevant for detecting adaptive genes, although we have incomplete knowledge of how the two types of variation relate to adaptation. Here we present a method that aims to identify adaptive genes by combining phenotypic and environmental data. We first predict trait variation from a set of environmental variables as a way to extract the most biologically relevant information from the environment and then look for genes associated with both the predicted and observed trait. Using simulations and published data from the model plant  Arabidopsis thaliana  3]  [3] 2] [  [3] 2] thaliana . AlthougAn alternative approach, which we explore here, is to extract information from ecological data by modeling traits as a function of multiple environmental variables  13] and and13] aArabidopsis thaliana . We. We15]. thaliana  and showthaliana , making thaliana  19]. We. WeArabithaliana , an apprETM first models the observed phenotype as a function of environmental data, producing a combination of the environmental variables which we call the predicted phenotype. The trait prediction model is fit on the set of accessions for which both phenotypic and environmental data are available, but the resulting prediction can be extended to the accessions for which there are only environmental data. In case of non-constant prediction, bivariate ETM then performs multitrait association mapping on the observed and predicted phenotype, using all available accessions. In univariate ETM we perform single trait association mapping for the accessions with missing phenotypic data.As proof of concept, we simulated a simple scenario in which an adaptive trait is modeled as a linear function of a random subset of ten out of 30 simulated environmental variables . The frequency of the causative SNP was set to be a monotone function of the true adaptive trait. The observed trait was then defined as the sum of a SNP effect and polygenic and residual noise.Four trait prediction methods were implemented: linear model (LM) prediction with backward variable selection, elastic nets (EN) , random 10(p) threshold of 5, versus 0.64 for univariate mapping). Other methods perform similarly with power ranging between 0.68–0.73 at the same threshold, and achieving larger gains over univariate mapping at higher −log10(p) thresholds.We first simulated a scenario where the heritability is 0.5 and the causative SNP explains 5% of the phenotypic variance; correlations between true and observed environmental variables was set to 0.8. For both types of tests, bivariate ETM using predicted traits shows a clear gain in power over univariate mapping . Bivaria10(p) thresholds.There is a clear relationship across simulated traits between the significance of ETM and correlation between the predicted trait and the simulated true adaptive trait : ETM is Similar differences between methods are observed in 8 additional scenarios with heritabilities 0.2, 0.5 and 0.8 and the causative SNP explaining 2%, 5% and 10% of the phenotypic variance . As expeNext, we modified the scenario of Given the similar performance of the two tests we chose to present all subsequent results for the common marker effect only. We consider this test to be conceptually more appropriate for the detection of loci associated with both the observed trait and its selective environment, which are expected to be positively correlated.Arabidopsis thaliana accessions , fl, flArabi = 0.98) –S11 Figs−3 to 4.9 ⋅ 10−6 and was achieved after evaluating varying numbers of genes . Bivariate ETM, by contrast, recovers 9–13 flowering genes within the first 400 candidates (p = 5.6 ⋅ 10−3 − 2.6 ⋅ 10−5), with all prediction methods providing higher enrichment than summer day length . The four types of bivariate ETM all recover the genes SVP, GA1, DFL2, LDL1, SPA2, FPF1, DOG1, within the first 400 candidates  identifies only 5 unique genes compared to univariate association mapping  genes is shared between prediction methods , compare mapping .Environmental prediction of trait values offers the possibility of association mapping when phenotypic data is incomplete. Traits of interest can be predicted across geographic space using geographic information and association mapping may then be performed on any set of georeferenced individuals for which genotypic data are available. SVP, CRP, SPA2, DOG1, PIE1 and FRI are recovered by more than one method  has a higher correlation with SNP frequency than either summer day length or latitude. In fact, our data provides no evidence for a latitudinal trend for either FRI or FLC, while the correlation with predicted flowering time is weak but significant (p < 1 ⋅ 10−9).Recovery of known flowering genes is somewhat lower compared to bivariate ETM . Althouge method  and for  thaliana 25], wh, whSVP, ing time  17]  [31 [3114] .Although selective forces determining trait variation may sometimes shape allele frequencies at non causal loci (e.g. those affecting an unmeasured adaptive trait), independent estimates of these selective forces can at the same time help to find true associations, particularly when combined with the trait itself. Bivariate ETM is designed to detect genes whose frequencies correlate with selective forces that have shaped a trait of interest. These are likely to affect the target trait directly, although they may also be genes affecting correlated adaptive traits. In our case an average of 87% of the top 2000 SNPs for bivariate ETM had p-values below 0.05 for flowering time itself. Since our primary aim is to find genes related to adaptation however, any gene that is affected by the same selective environment is of interest, regardless of its causal relation to the trait.The success of this approach does require that traits and the environment provide complementary estimates of underlying selective forces, something that may not always be the case. The result that enrichment for known flowering genes is higher for bivariate ETM than for univariate mapping on the trait itself, and that this is not observed for randomly simulated variables with the same correlation to the observed trait, suggests that predicted and observed traits indeed complement each other. One thing to observe, is that our definition of recovery as the closest gene to a detected SNP, deviates from Atwell et al.’s decision to consider SNPs within 20kb of their candidate genes . Our criAnother application of environmental trait prediction is the mapping of adaptive genes in individuals with missing phenotypic information. It offers potential for mining the growing genomic data available for many species without the need for complete phenotypic data, and exploiting the wealth of publicly available geographic and environmental data. Our results on mapping flowering genes in unphenotyped individuals are encouraging in the sense that more genes are found than expected at random. On the other hand, the improvement achieved over single environmental variables such as latitude is rather modest. This probably reflects the fact that environmental variables related to latitude are the dominant selective agents affecting flowering time, making it hard to improve over the use of well chosen single environmental variables. Nonetheless, at several genes with known association with latitude, estimated gene frequencies are more strongly correlated with predicted flowering time than with latitude. This observation provides evidence that mapping on predicted traits has the potential of producing more relevant association results than single environmental variables chosen a priori.genetic rather than phenotypic correlation. This however implies an optimization problem for which no algorithms currently seem to be available. Alternatively, bivariate MTMM could be replaced by multivariate MTMM, including all environmental variables individually (as well as the observed trait), but state-of-the art approaches  or  or 36] oIn terms of application, it will be interesting to test the added value of our approach for traits that are more weakly correlated with known environmental factors, such as is the case for disease or drought resistance. We hope that the present work may serve as a first step in moving adaptation mapping beyond the traditional univariate analysis of traits and environmental variables and towards a more integrated use of all available data.A. thaliana [We used two datasets from two highly cited examples of trait association and environmental association in thaliana  7]. The. TheA. thttps://www.mpipz.mpg.de/14637/Arabidopsis_flowering_genes). SNP positions with the highest frequency differentiation at functional variants of the flowering genes FLC and FRI were identified based on 85 accessions for which functional haplogroups were available  be the n × p predictor matrix, and let i = 1, …, n) denote its rows.We made environmental trait predictions using standard linear regression with backward selection (LM), elastic nets (EN), random forests (RF) and canonical correlation analysis (CCA), which we describe below. Suppose that for accessions xjxj′, 1 ≤ j < j′ ≤ p, giving in total 2p + p(p − 1)/2 predictors. Since this number exceeds the sample size, we did not include interactions for LM and CCA. For ease of notation, we will now assume that p is the total number of predictors  or the total number of variables plus the number of interactions (EN and RF)).For most of our examples, between 10 and 50 predictors are available. This number is smaller than the number of accessions, but to avoid over-fitting and allow for interactions, some form of variable selection is desirable. LM, elastic nets and random forests select, in different ways, the environmental variables most relevant for prediction. CCA does not perform variable selection. In the case of elastic nets and random forests we also include the second moments y = XS)(βS), using λ > 0 is the amount of penalization and α ∈  determines the weight of the L1- and L2-penalty. The term βj to be exactly zero. We fixed α = 0.5 and chose λ by 10-fold cross-validation. We used the implementation in the R-package glmnet 15], whics. As in , we haveollowing , the folα = β = 0).for any genetic effect (‘full’): the full model against the null-model (α = 0 against the null-model (α = β = 0).for a common genetic effect (‘common’): the model with α = 0), which will not be considered here. In Eqs   α = 0), For all methods  SNPs were ordered by their significance and the 2000 SNPs with lowest p-values were considered as candidate SNPs. We assigned each of these SNPs to the gene(s) overlapping with its position or to the closest gene in the case of non-genic SNPs. This criterion differs from that used by Atwell et al. 2010) [0 [1], whWe counted how many out of the 240 known flowering genes were recovered as a function of the number of unique genes considered when going down the ordered list of candidate genes. At each point, enrichment was calculated as the hypergeometric probability of finding (at least) the number of unique flowering genes, given the number of genes evaluated so far, the total of flowering genes (240) and the total of 29,477 genes assigned to any of the SNPs.n = 300 accessions taken from the regmap, of which we randomly selected 100 Swedish, 100 French, 50 German and 50 Czech accessions. Each simulation consists of k = 30 simulated environmental variables and 1 simulated trait.We simulate traits and environmental variables for a fixed set of n × k matrix XT, containing the true (unobserved) environmental variables at the locations of origin of the accessions. XT specifies what we will call the true environment. First we randomly draw a subset S ⊂ {1, …, k}, containing s = 10 environmental variables, which will later form the environmental gradient. We will use the notation XT(S) for the submatrix of XT with columns defined by S.Each simulation starts by drawing a Gaussian XT contain polygenic components, such that their heritabilities are 0.5. Specifically, XT is the sum of Genv and Eenv, which are drawn from zero mean matrix variate normal distributions (see e.g. . For simplicity we assume that yT is the (unobserved) adaptive phenotype, although more complex relations between environmental gradients and phenotypes can be expected in nature.We then define the environmental gradient as f of causal allele frequencies at each simulated location, is defined as λ = 3, and hence has a correlation of 1 with yT. A corresponding genotypic vector g is formed by sampling a single allele for each location from a Bernoulli distribution with probability f. Finally, we simulate the vector of observed phenotypes yO = βsnpg + Gtrait + Etrait, where βsnp represents the SNP-effect on the trait, Gtrait is the polygenic effect defined above, and Etrait is residual noise.The vector βsnp and the variance of Etrait are chosen such that the SNP explains 5% of the phenotypic variance, while Gtrait and Etrait explain respectively 45% and 50%, i.e. the heritability of the observed trait is 0.5. The correlations between Gtrait and Genv(S) are set to 0.The main set , where βIn In Gtrait and Genv(S) being 0.5.In Gtrait and Genv(S) being 0.5. Additionally, the SNP effect (βsnp) was set to 0, and Gtrait explained 50% of the variance.In g of SNP scores randomly from independent Bernoulli(0.5) distributions, i.e. independent of any environmental variable.In We performed the following sets of 2000 simulations:In all cases, ETM p-values from simulations yielding constant trait predictions were set to their corresponding univariate GWAS p-values.S1 FigThe regression line is shown in red and the line marking equal performance of the two methods is marked in blue. Only results for non-constant predictions are shown.(PDF)Click here for additional data file.S2 FigBivariate ETM was performed by testing for a common marker effect (top) and by testing whether there is any effect on environment or trait (bottom).h2 = 0.2. The causal SNP explained 2% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 18% and 80%. Correlations between true and observed environmental variables were 0.8.(a) h2 = 0.2. The causal SNP explained 5% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 15% and 80%. Correlations between true and observed environmental variables were 0.8.(b) h2 = 0.2. The causal SNP explained 10% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 10% and 80%. Correlations between true and observed environmental variables were 0.8.(c) h2 = 0.5. The causal SNP explained 2% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 48% and 50%. Correlations between true and observed environmental variables were 0.8.(d) h2 = 0.5. The causal SNP explained 5% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 45% and 50%. Correlations between true and observed environmental variables were 0.8.(e) h2 = 0.5. The causal SNP explained 10% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 40% and 50%. Correlations between true and observed environmental variables were 0.8.(f) h2 = 0.8. The causal SNP explained 2% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 78% and 20%. Correlations between true and observed environmental variables were 0.8.(g) h2 = 0.8. The causal SNP explained 5% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 75% and 20%. Correlations between true and observed environmental variables were 0.8.(h) h2 = 0.8. The causal SNP explained 10% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 70% and 20%. Correlations between true and observed environmental variables were 0.8.(i) (PDF)Click here for additional data file.S3 FigBivariate ETM was performed by testing for a common marker effect (top) and by testing whether there is any effect on environment or trait (bottom). The causal SNP explained 45% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 45% and 50%.(PDF)Click here for additional data file.S4 FigBivariate ETM was performed by testing for a common marker effect (top) and by testing whether there is any effect on environment or trait (bottom). The causal SNP explained 5% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 45% and 50%.(PDF)Click here for additional data file.S5 FigThe SNP scores were independently drawn from the Bernoulli(0.5) distribution. Polygenic background (adaptive) and residual variance each explained 50% of the phenotypic variance. The genetic correlation between the observed trait and each of the 10 environmental variables defining the environmental gradient was 0.5. (a) Bivariate ETM performed by testing for a common marker effect. (b) Bivariate ETM was performed by testing whether there is any effect on environment or trait.(PDF)Click here for additional data file.S6 FigBivariate ETM was performed by testing for a common marker effect (top) and by testing whether there is any effect on environment or trait (bottom). The causal SNP explained 5% of the variance of the simulated trait, while polygenic background and residual variance explained respectively 45% and 50%. Correlations between true and observed environmental variables were 0.8.(PDF)Click here for additional data file.S7 FigNumbers in red indicate squared Pearson correlations.(PDF)Click here for additional data file.S8 FigNumbers in red indicate squared Pearson correlations.(PDF)Click here for additional data file.S9 FigNumbers in red indicate squared Pearson correlations.(PDF)Click here for additional data file.S10 FigNumbers in red indicate squared Pearson correlations.(PDF)Click here for additional data file.S11 FigNumbers in red indicate squared Pearson correlations.(PDF)Click here for additional data file.S12 Fig(PDF)Click here for additional data file.S13 Fig(PDF)Click here for additional data file.S14 Fig(PDF)Click here for additional data file.S15 FigThe numbers of co-factors selected using the extended BIC criterion were 1,2,0,1,3 and 2, for respectively the observed traits (first row) and for predicted traits (LM and EN (middle row); RF and CCA (bottom row)).(PDF)Click here for additional data file.S16 FigThe bottom right panel shows results for univariate mapping of the observed trait when calling a maximum of one gene per SNP (i.e. correction for “double hits”).(PDF)Click here for additional data file.S1 File(XLS)Click here for additional data file.S2 File(CSV)Click here for additional data file.


Unlike case-control studies, family-based tests for association are protected against population stratification. Complex genetic traits are often governed by quantitative precursors and it has been argued that it may be a more powerful strategy to analyze these quantitative precursors instead of the clinical end point trait. Although methods have been developed for family-based association tests for single quantitative traits, it is of interest to develop such methods for multivariate phenotypes. We propose a novel transmission-based approach based on a trio design using a simple logistic regression to test for association with a multivariate phenotype. We use our proposed method to analyze data on systolic and diastolic blood pressure levels provided in Genetic Analysis Workshop 18. However, we find that the bivariate analysis of the two phenotypes did not provide more promising results compared to univariate analyses, suggesting a possibility of a different set of major genetic variants modulating the two phenotypes. The family-based design  for deteFor our analyses, we use pedigree data on systolic blood pressure (SBP) levels and diastolic blood pressure (DBP) levels at four different time points for 453 individuals along with their genotypes at all of the available 456,752 variant sites distributed over 11 autosomal chromosomes. In addition to age, we used smoking status and medication indicator (both defined as binary variables) at each time point of examination as covariates, as these factors could be potential confounders in the association analyses. Both the SBPand the DBP levels were adjusted for these covariates for each time point and the tests for transmission disequilibrium were performed on the adjusted phenotypes.p-values are very close to the threshold of 0.05. We thus perform a logarithmic transformation on each of the phenotypes to induce normality. We use an unrelated set of 142 individuals from the pedigrees for whom data on all the variables are available to estimate the missing log-transformed phenotype values using data on the available phenotype values. Suppose the vector of log-transformed values of any of the two phenotypes at the four time points is represented as X = . If Y denotes the vector comprising those components of X that are missing and Zis the components that are available for an individual, Y is estimated via an EM algorithm as the expectation of Y conditioned on Z and is given by μY ΣYZ ΣZZ−1(Z-μZ), where, μY and μZ are the mean vectors of Y and Z, respectively; ΣYZ is the matrix of covariance between Y and Z, while ΣZZ is the dispersion matrix of Z. We perform a linear regression of the log-transformed values of each of the two phenotypes (available as well as imputed) at each time point on age, smoking status, and medication indicator. We plug-in the parameter estimates of the mean vector and variance-covariance matrix of the log-transformed phenotypes obtained via the EM algorithm to estimate the missing log-transformed values of each phenotype conditioned on the available log-transformed values of that phenotype at every time point for the remaining individuals in the pedigree. We then use the regression equation at each time point to obtain the residuals for all individuals in the pedigree for whom data are available on all the covariates.Data on the two phenotypes and the covariates are not available for all individuals at every time point. The assumption of multivariate normality provides a computationally elegant framework for the expectation maximization (EM) algorithm  to estimX =  denotes a vector of k phenotypes and Wis anindicator random variable (1 or 0) denoting whether a heterozygous parent at a SNP transmits the minor allele or not. We model the conditional distribution of W given X using a logistic link function given by:The phenotypes for our association analyses are the adjusted SBP and DBP levels at each time point obtained using the algorithm described in the preceding section. We use a novel binary logistic regression framework to test for association of a SNP with a multivariate phenotype. For each SNP, we consider all trios in the pedigree with at least 1 heterozygous parent at that SNP, selecting one sib at random from each sibship. Suppose iis the mean of i Xin the population thatis estimated by the sample mean and the parameters where, μ0H: 1β=2 β= ... =kβ= 0 versus H1: not H0 and the log-likelihood ratio test statistic is distributed as chi-squares with k degrees of freedom under the null hypothesis. We compare the relative performances of 3phenotype vectors in detecting association: (a) T1: the adjusted SBP levels summarized by the first two principal components across the four time points; (b) T2: the adjusted DBP levels summarized by the first two principal components across the four time points; and (c) T3: a bivariate phenotype comprising the adjusted SBPand the adjusted DBP levels summarized by the first two principal components corresponding to each of the phenotypes across the four time points. The above choice of principal components is motivated by the fact that 75% of the variation in each of the two phenotypes is explained by the corresponding first two principal components. To correct for multiple testing, we use the false discovery rate procedure [We note that even though this model is in similar lines as Waldman , it captThe pedigree is made up of 95 distinct pairs of parents. Thus, our transmission disequilibrium analyses are based on 95 independent trios. Given that most parents have multiple offspring, there exists a large number of possible sets of trios if 1 sib is selected at random from each sibship made up of two or more sibs. We consider 1000 such possible sets of trios at random. Because transmissions only from heterozygous parents are relevant for the proposed test for transmission disequilibrium, we analyze only those SNPs that are made up of at least 25 informative trios for efficient estimation of parameters in the logistic regression. We also exclude those SNPs that show significant deviation from the Hardy-Weinberg equilibrium based on the unrelated set of 139 individuals for whom genotype data are available, and use Bonferroni correction for multiple testing.T3  would be more powerful in detecting association, T1  provides the most promising association finding. The SNPs rs4754220 and rs12419678 on chromosome 11 attains genome-wide significance  with T1 in 37 and 35 of the 1000 sets of trios, respectively. On the other hand, the SNP rs13301156 on chromosome 9 exhibits significant evidence of transmission disequilibrium with T2  in 24 sets of trios. These three SNPs also rank among the top five SNPs significantly associated with T3, although in less than 10 sets of trios.The tests for association based on the proposed logistic regression are carried out on 426,193 SNPs. Among the phenotype vectors considered, contrary to our expectation that We have developed a simple binary logistic regression model that incorporates multiple phenotypes for transmission-based association analyses of the multivariate phenotype vector. The method does not involve any modeling of the correlation structure within the components of the multivariate phenotype as required in likelihood-based approaches and, consequently, is more robust with respect to distributional assumptions. On the other hand, the method does not reduce the multivariate phenotype vector to principal components, thus circumventing the problem of biological interpretations of derived phenotypes.rs4754220 and rs12419678 that exhibited the most significant evidence of linkage disequilibrium with SBP values are located in the intronic region of the gene CWF19L2  on 11q22.3. Studies show that RNA expression of this gene is upregulated in humans for inflammatory cardiomyopathy [rs13301156 that yields significant evidence of association with DBP levels is located in the intergenic region between the genes RPS6P13  and GAS1 (growth arrest-specific 1) on 9q21.3. The RNA expression of RPS6P13 has been reported to bedownregulated in humans for coronary collateralization [GAS1 has been reported to be upregulated for arrhythmogenic right ventricular cardiomyopathy in humans [The SNPs n humans .It is expected that if a genetic variant modulates multiple phenotypes, a multivariate analysis will be more powerful than separate univariate analyses in detecting association with the genetic variant. However, we find that the association test for the bivariate phenotype is less powerful than the tests for SBP levels and DBP levels separately. Moreover, the most significant association findings obtained for the bivariate phenotype form a disjoint union of those obtained for the two phenotypes separately. Consequently, it is possible that althoughthere may be common genes modulating both SBP and DBP levels, the major genetic variants for the two phenotypes may be different and the bivariate phenotype contains minimal additional information on the variants compared to any of the two phenotypes.The proposed transmission-based association test can incorporate multiple sibs within a sibship by considering the transmission to each sib separately. However, such a test is strictly a valid test only for linkage. Althoughthe presence of association increases the power to detect transmission disequilibrium, the rejection of the null hypothesis does not necessarily imply the presence of linkage disequilibrium. When we perform our proposed test with all sibs within each sibship, we obtain large clusters of significant SNPs since linkage exists over much larger distances on the genome compared to linkage disequilibrium. However, we find that the clusters on chromosomes 9 and 11 include the three SNPs that provided the most significant evidence of association. We are currently exploring the theoretical properties of various methods to integrate the test statistics (such as the mean or the maximum order statistic) for the different sets of trios (considering 1sib at random from each sibship) into a combined test statistic.The authors declare that they have no competing interests.SG and TH developed the proposed method. AMand TH wrote the computer codes and performed the data analyses. TH participated in the compilation and interpretation of the results. SG drafted the manuscript. All authors read and approved the final manuscript.


In similar fashion, we developed the extended T2 tests for longitudinal whole genome sequencing data for family-based association studies. The new methods simultaneously incorporate three correlation sources: from linkage disequilibrium, from pedigree structure, and from the repeated measures of covariates. We assess and compare these methods using the simulated data from Genetic Analysis Workshop 18. We show that, in general, the extended T2 tests incorporating longitudinal repeated measures have higher power than the single-time-point T2 tests in detecting hypertension-associated genome segments.Family data and rare variants are two key features of whole genome sequencing analysis for hunting the missing heritability of common human diseases. Recently, Zhu and Xiong proposed the generalized T Compared with traditional genome-wide association studies (GWAS), whole genome sequencing (WGS) is a more efficient way of finding the missing heritability of diseases . While GMost of the recent discoveries from sequencing studies were based on the Mendelian trait model . Genetic2 tests for family-based WGS data [2 tests to address these limits. By applying these methods to an analysis of the Genetic Analysis Workshop 18 (GAW18) simulation data, we showed that the asymptotic null distributions of Zhu and Xiong [In a recent celebrated work, Zhu and Xiong proposed a set of generalized TWGS data . These mWGS data ,10 are a2 test [2 test that incorporates these covariance matrices, which are estimated separately by using the same data. Depending on various strategies of collapsing of rare variants, here we consider three generalized T2 tests of Zhu and Xiong.Zhu and Xiong  showed t2 test ,12, the 2T: The genotypes of rare variants between adjacent common variants are summed up, and one covariance matrix is estimated for both common and collapsed rare variants.CMC.ZXpaper (CMC test): The rare variants are collapsed in the same way as above, but the covariance matrices are estimated separately for common and rare variants (assuming they are uncorrelated).CMC.ZXcode: Rare variants are collapsed by the maximum of their genotypes, and one covariance matrix is estimated for both common and collapsed rare variants. This strategy follows the R function pedCMC of Zhu and Xiong (https://sph.uth.edu/hgc/faculty/xiong/software-D.html).2 tests to account for the longitudinal repeated covariates. Figure 2 tests compare the blocks of common variants, rare variants, and covariates with repeated measures in cases and in controls, while simultaneously accounting for the correlations among genetic factors, among pedigree individuals, and among longitudinal repeated measures. The response is the occurrence of the event at any of the measurement points.Building on the idea of Zhu and Xiong, we further extend the generalized Ttth common variant is T common variants is represented by sth rare variant is S rare variants. Considering the covariates with longitudinal repeated measures, the column vector of the cth covariate at the jth repeated measurement point is C covariates, each measured for J times. Similarly, the row vectors are denoted as follows. For Following the notations in Figure 2 test is simply to compare the difference between the row average of the case blocks and the row average of the control blocks. Let η. That isThe idea of the extended Tn and k, then the matrixwhere if we define 2 test, the extended T2 test is Following the idea of the generalized TZ and V are independent, we consider two cases. In the first case, assume A is also independent with Z and V. Then The key problem is to estimate Λ. Following the assumption of Zhu and Xiong  that Z aΦ is the kinship matrix, and ⊗ denotes the Kronecker product. For the covariate block, we consider 2 test for estimating the covariance matrix (which is different from equation (6) in Ref. [To better account for the heterogeneity of the data in cases and in controls, we applied the method in Hotelling's T in Ref. ). Then e2 tests by Zhu and Xiong [2 tests are denoted T2.longi, CMC.ZXpaper.longi, and CMC.ZXcode.longi, respectively.We consider two simplification assumptions: (a) nd Xiong , the corp values as shown below. Thus, we applied a permutation test for the type I error rate being well controlled. Specifically, let gth genome window from the original data and from the lth permutation, respectively. The empirical p value for the gth window is L =1000. Because the target is to find the associations with genetic variants, not with the covariates, the permutations are applied only to the genotype data to retain the relationship between response and the covariates.Zhu and Xiong derived asymptotic chi-square distribution for the null. In their paper , the degFor evaluating the above methods, we used the "dose" genotype data of 1,215,399 single-nucleotide variants (SNVs) on chromosome 3 and the 200 simulation replicates of hypertension outcomes and covariates . As an arbitrary, yet simple, way to group variants, we split chr3 into 19,080 windows, each 10 kilobase pairs (kbp) long. In each window, rare variants (MAF <0.05) between adjacent common variants were collapsed, leaving 654,415 genetic factors  to be analyzed. The average number of genetic factors contained in the windows is 34.3, the median is 32, the minimum is 1, and the maximum is 330. For the simulated phenotypes, the number of individuals is 849 in 20 families, where the family sizes are from 21 to 74, with the mean 42.45 and the median 36.5. There are 188 simulated true SNVs contained in 129 true windows  on chr3. The knowledge of these true SNVs was used only for evaluating the power of these association tests, not for designing data analysis strategy.p values of these tests for all false windows in chr3. The Q-Q plot of Figure p values with large genomic inflation factors λ[p value cutoff of 0.05, the actual  error rate is approximately 0.1. At the same time, the following results show that permutation test controls the type I error rate well. Thus, the inflated type I error rate is likely caused by the inappropriate asymptotic null distributions, not by possible stratification.To assess the asymptotic null distributions of the tests provided in Zhu and Xiong , we obtafactors λ. For exap value cutoffs. In general, the power is low at small or moderate p values. This phenomenon indicates that the sample size is still relatively too small for detecting many weak genetic effects simulated in the data. At the same time, it is clear that the 3 extended T2 tests that incorporate longitudinal information are significantly better than the generalized T2 tests that only consider the measures at the first time point. Because the two setups: We studied the power of these tests in detecting true windows over chr3. Based on the phenotype data in the simulation replicate 1, the right panel of Figure 2 tests more powerful than generalized T2 tests . For all windows, the type I error rates of all methods were well controlled (results are available upon request).To compare the overall capabilities of these tests, we studied their power  in detecting each of all windows over 200 simulation replicates. As illustrated in Figure In the simulation data of GAW18, true SNVs are always allocated on genes. Using genes as windows to group SNVs may concentrate the true SNVs and has the potential to improve the detection power. However, the idea of WGS, instead of exome sequencing, is that the disease-related genetic factors might allocate outside of genes. So we did not use the knowledge that true SNVs are in genes; instead, we evaluated the methods based on fixed-genome segment windows.2 test. This is a maximal likelihood estimate if observations are independent. Unfortunately, independency is not true for family data in the first place. Besides the correlation issue, for a high-dimensional matrix with a potentially sparse structure, there are better estimates of the covariance matrix and its inverse [p value calculation. Third, necessary modifications of these tests are needed to handle missing data and unequal numbers of repeated measures, which are common problems.There are several limitations and future research topics based on the current work. First, matrix estimation is a key issue in this methodology development. Good estimation of matrices and their inverses can better incorporate correlation structures' potential to improve the performance. Here we simply adopted the same variance matrix estimate in Hotelling's T inverse . Second,2 tests to incorporate the covariates with longitudinal repeated measures. These methods account for 3 sources of correlation structures among genetic variants, family members, and time series observations. Compared with the T2 test methods for snapshot observations, the new methods have higher power to detect hypertension-related genome segments according to the GAW18 simulation data.We have extended Zhu and Xiong's  generaliThe authors declare that they have no competing interests.ZW designed the overall study. ZW, YL, and JX conducted statistical analyses and drafted the manuscript. All authors read and approved the final manuscript.


With the recent development of high-throughput DNA microarray and next-generation sequencing techniques for detecting various genomic variants , genome-wide association studies (GWASs) have become a popular strategy to discover genetic factors affecting common complex diseases. Many GWASs have successfully identified genetic risk factors associated with common diseases and have achieved substantial success in unveiling genomic regions responsible for the various aspects of phenotypes.However, identifying the underlying mechanism of disease susceptible loci has proven to be difficult due to the complex genetic architecture of common diseases. The previously associated variants through GWASs only explain a small portion of the genetic factors in complex diseases. This rather limited finding is partly ascribed to the lack of intensive analysis on undiscovered genetic determinants such as rare variants and gene-gene interactions. Unfortunately, standard methods used to test for association with single common genetic variants are underpowered for detection of rare variants and genetic interactions.This special issue is dedicated to presenting state-of-the-art statistical and computational methods for finding missing heritability underlying complex traits with massive genetic data including GWAS, next-generation sequencing, and DNA microarray data. The main focus of this special issue is on data mining and machine learning for advanced GWAS analysis. The advanced GWAS analysis includes multi-SNP analysis, gene-gene and gene-environment interaction analysis, estimation of missing heritability, and analysis of population heterogeneity. This special issue provides a platform to the researchers with expertise in data mining to discuss recent advancements in analytic approach of post-GWAS association analysis in field of statistics and bioinformatics.The paper by W. Lee et al. proposes an approach to identifying clinically interesting subgroups in a heterogeneous population. The identification step uses a clustering algorithm and proposes an improved false discovery rate- (FDR-) based measure to remedy the overestimation of the ordinary FDR-based approach. The paper by Y. Kim et al. performs heritability estimation by using population- and family-based samples. The main idea lies in utilizing genetic relationship matrix to parameterize the variance of a polygenic effect for population-based samples.Three other papers consider gene-gene and gene-environment analysis. First, J. Yee et al. proposed interaction analysis for quantitative traits using entropy. Although there have been several methods proposed for gene-gene interaction using entropy, this is a robust entropy-based gene-gene interaction analysis that does not necessarily require an assumption on the distribution of trait for quantitative traits. Second, S. Y. Lee et al. focused on identifying multi-SNP effects or gene-gene interactions for survival phenotypes. In the framework of the multifactor dimensionality reduction (MDR) method, several extensions for the survival phenotype are considered and compared to the earlier MDR method through comprehensive simulation studies. Third, the paper by H. Xu et al. proposes a new GWAS strategy for detecting gene-gene and gene-environment analysis by combining the generalized multifactor dimensionality reduction-graphics processing unit (GMDR-GPU) algorithm with mixed linear model approach. It was further employed to investigate the genetic architecture of important quality traits in rice. The reliability and efficiency of the model and analytical methods were verified through Monte Carlo simulations.The next two papers discuss multi-SNP analysis. Y. J. Yoo et al. propose a new multi-bin linear combination (MLC) test for multiple SNP analysis. It first performs clustering analysis to find cliques, complete subnetworks of SNPs with all pairwise correlations above a threshold, and then performs MLC test. Through simulation studies, the clique-based algorithm was shown to produce smaller clusters with stronger positive correlation than other MLC tests. The paper by S. Won et al. focuses on comparing penalized and nonpenalized methods for disease prediction with large-scale genetic data. It was shown that penalized regressions are usually robust and provide better accuracy than nonpenalized methods for disease prediction.p values of the discovery and replication studies were demonstrated.Next, the work of J. Joo et al. considers robust genetic association tests for GWAS. How these robust tests can be applied to the replication study of GWAS and how the overall statistical significance can be evaluated using the combined test formed by Finally, the paper by L. Li and M. Xiong proposes a dynamic model for RNA-seq data analysis. To extract biologically useful transcription process from the RNA-seq data, the ordinary differential equation (ODE) model was proposed for modeling the RNA-seq data. Differential principal analysis was developed for estimation of location-varying coefficients of the ODE.This special issue discusses the most challenging issues in multiple SNPs approaches including gene-gene interaction and introduces statistical and computational methods for data mining and machine learning for revealing hidden association network of genotype-phenotype relationship. The nine papers in this special issue provide scientists with an overview on the recent advancements in multiple SNP analysis for GWASs. We hope the papers can encourage researchers towards a more extensive use of statistical genetics and bioinformatics techniques for research in biology and medical sciences.