ZCN8 gene, which is the closest maize homologue to Arabidopsis major flowering time (FT) gene, had the strongest effect. This polymorphism is in the vicinity of the causal factor of Vgt2 QTL. Diversity was lower, whereas differentiation and LD were higher for associated loci compared to the rest of the genome, which is consistent with selection acting on flowering time during maize migration. Selection tests also revealed supplementary loci that were highly differentiated among groups and not associated with flowering time in our panel, whereas they were in other linkage-based studies. This suggests that allele fixation led to a lack of statistical power when structure and relatedness were taken into account in a linear mixed model. Complementary designs and analysis methods are necessary to unravel the architecture of complex traits. Based on linkage disequilibrium (LD) estimates corrected for population structure, we concluded that the number of SNPs genotyped should be at least doubled to capture all QTLs contributing to the genetic architecture of polygenic traits in this panel. These results show that maize flowering time is controlled by numerous QTLs of small additive effect and that strong polygenic selection occurred under cool climatic conditions. They should contribute to more efficient genomic predictions of flowering time and facilitate the dissemination of diverse maize genetic resources under a wide range of environments.The migration of maize from tropical to temperate climates was accompanied by a dramatic evolution in flowering time. To gain insight into the genetic architecture of this adaptive trait, we conducted a 50K SNP-based genome-wide association and diversity investigation on a panel of tropical and temperate American and European representatives. Eighteen genomic regions were associated with flowering time. The number of early alleles cumulated along these regions was highly correlated with flowering time. Polymorphism in the vicinity of the  Maize was domesticated in tropical conditions in the lowlands of southwest Mexico and later adapted to the broadest range of climatic conditions of all crops, from 40°S in Chile to 50°N in Canada and Russia, from sea level in the West Indies to elevations above 3400 m in the Andes Despite this limited loss of diversity, phenotypes have been dramatically modified by domestication, large-scale migration/adaptation and selection cycles that gave rise to modern hybrids. In particular, flowering time evolved to adapt to short growing seasons, long days and low temperatures under temperate climates. Among the germplasm available, for maize landraces, the time from planting to the mature grain stage ranges from 2 to 11 months Flowering time plays a key role in the acclimation of plants to different environments by integrating diverse environmental and endogenous signals that control the optimal moment for the transition from the vegetative to the reproductive phase. It allows the plant to avoid drought and thus kernel abortion, or to optimize the light interception period and yield for instance. The synchrony of male and female flowering is also an important adaptive trait in maize as high asynchrony can result in yield losses, especially in modern uniform varieties ID1) gene is an important regulator of maize autonomous flowering that acts in leaves to mediate the expression of mobile signals that are hypothetical flowering hormones called florigens ID1 and to express a florigen in leaves FLOWERING LOCUS T (FT), a kinase regulator ZCN8 have been published Zea CENTRORADIALIS (ZCN) genes. Expression analysis demonstrated that some of them are involved in developmental processes. A second gene that has been shown to have a major downstream effect is Dfl1 (Delayed flowering1), a transcription factor that expresses in the shoot apical region ID1. Another major factor in flowering time variation, Vgt1, was detected in both linkage-based QTL analyses and association genetics studies ZmRap2.7Compared to some other species like Arabidopsis et al.Arabidopsis. Steinhoff et al.Further investigations to identify the main factors controlling maize flowering time in a panel representative of diverse migration routes would be beneficial: (i) to gain insight into the adaptation mechanisms under changing environments, (ii) to identify alleles for introgression into existing varieties in order to adapt them to different environmental conditions, and (iii) to better predict flowering time to the benefit of global crop management and local breeding programs, since flowering time is often considered as a major covariate in yield estimation. This latter objective can presently be approached through genomic selection models. The choice of suitable models should consider the complexity of trait architectures in terms of relative individual effects of loci and possible existence of non-additive effects due to interaction with the genetic background (GxQ epistasis) or gene-environment interactions (GxE) High density genotyping tools available today are expected to help in the discovery, fine mapping and allele diversity characterization of regions involved in flowering time. However, the choice of panel is very important as the level of polymorphism in each genetic group will determine the power of the analysis. In domesticated species like maize, loci that are critical to both local adaptation and yield performance, such as flowering time loci, are often targets of both natural and artificial selection, leading to complex forms of allele sharing and admixture among diverse genetic groups. Differentiation of flowering time between maize genetic groups is actually clear at the QTL level This study was thus designed to assess the potential of currently available mid-density arrays A sample of 375 maize lines representing the worldwide diversity was considered The panel of maize lines was described using two different sets of SSR markers. The first one was described in Vgt1 on chromosome 8 D8 on chromosome 1 As a reference, we considered 535 SNPs discovered on regions that were fully sequenced in the same panel around We computed two different similarity matrices between lines (kinship), one considering the identity by state (IBS) with r-Emma 6 iterations. Allele frequencies in each of the K clusters (from 2 to 15) were estimated, and the percentage of genome derived from each cluster was estimated for each accession.Population structure was investigated in order to define suitable covariates for association genetics models and investigate genetic diversity trends among genetic groups. We thus used two different sets of SSR markers (55 SSRs and 94 SSRs) and STRUCTURE The structure matrix built with 29911 Panzea SNP markers was estimated by ADMIXTURE, which computes maximum likelihood estimations of individual ancestries from multilocus biallelic genotype datasets using the same statistical model as STRUCTURE with a very fast numerical optimization algorithm OH), expected (EH) heterozygosity, differentiation (STF) corresponding to the ratio of inter-group diversity over the total genetic diversity according to Nei STFDiversity parameters were estimated for the total panel and within each of the five main genetic groups determined from 55 SSRs k at locus l in group q, i to group q, k at locus l for line I, coded as 0 and 1, respectively.Finally, we considered the classification based on 55 SSRs as a reference to estimate SNP allelic frequencies in each genetic group, as,They were calculated for all lines using only lines with assignment to one group above 0.8.STBF) is decomposed into a population-specific component (beta) shared by all loci and a locus-specific component  shared by all populations using a logistic regression. Departure from neutrality at a given locus is assumed when the locus-specific component is necessary to explain the observed diversity pattern . A positive alpha value suggests diversifying selection, whereas negative values suggest balancing or purifying selection.When taking the structure obtained with 55 SSRs as the reference, we identified loci under selection according to differences in allele frequencies between genetic groups using BayeScan 1M and 2M being the models without and with selection and N the number of loci tested. It copes with unequal population sizes and posterior probabilities allow control of the false discovery rate (FDR), i.e. the expected proportion of false positives among outlier markers. We used Jeffrey's scale of evidence 10(PO)>1. These values were highly correlated to Q-values, which are the minimum FDR at which the loci may be deemed significant.BayeScan uses posterior odds (PO) instead of Bayes factors to make decisions about the chance that each locus is under selection. The ratio of posterior probabilities is 2r using Plink 2sr measure developed by Linkage disequilibrium (LD) was first calculated as the squared correlation between allelic doses at two loci (2r and 2sr) according to the increase in physical distance was modeled with two non-linear regression models The LD curve of both statistics . The two latest groups of lines were also evaluated at France_Mauguio . French locations were evaluated over 3 years (2002–2004) and the Einbeck location over 1 year (2005). Note that only 2002 data were considered in a priori information. Each individual plot consisted of a row of 15 plants planted at a density of approximately six plants per square meter.Lines were repeated twice at each location using a complete block design. In order to limit competition effects, each block was organized into four sub-blocks corresponding to earliness groups based on To = 30°C) that maximized correlations between sites .Days to anthesis for male flowering (MFLW) and days to silking for female flowering (FFLW) and anthesis to silking interval (ASI) were measured in thermal time (GDD: growing degree-days) according to A global ANOVA of the data was performed to test the genotype, location and genotype-by-location interaction significance. For association analyses, considering that the genotype-by-location interaction was low compared to the genotype effect, we estimated the adjusted mean of each genotype in the total trial network.ijkly was the phenotype of the l individual of the i inbred line, in the j field trial, in the k subplot. μ indicates the mean. Inbred lines (ix) were considered as fixed effects. Field trial (jz) and subplot (jkv) were considered as random effects. ijkε was the residual error. For each trait, LSMEANS of each inbred line i was calculated with SAS PROC GLM as We fitted the mixed modelTo investigate the effect of population structure, the proportion of genetic variance of each trait explained by the first four columns of the five group Q matrix obtained with 55 SSRs 55SSRs), or ADMIXTURE 30KSNPs). We compared the results obtained with four different kinship matrices obtained with 94 SSRs, using (1) identity by state (IBS) similarity obtained with the Rogers index IBS(94SSRs)) and (2) the normalized IBS index IBD(94SSRs)). We also tested the kinship (4) obtained with 29911 SNPs and IBS ).As the structure of a population and/or the relatedness within a sample can increase the long-range LD, and consequently the rate of false positives, we corrected tests using a linear mixed model using the approach described by y is the vector of phenotypes, μ is the mean, S the vector of individual genotypes, α the SNP fixed effect, Q the matrix of assignment of each line to each genetic group, v the vector of genetic groups fixed effects, Z the matrix of line occurrences, u the vector of line genetic background effects and e the vector of residuals. Var(u) = 2KVg, where Vg is the genetic variance and K is a matrix of similarity between lines.We considered at each SNP the association model:55SSRs+KIBS(94SSRs) model as the reference to avoid having to use SNP as candidate loci and for population structure and kinship estimations. We compared P-values as obtained with the naive model (only the marker was included in the model) and those obtained with Q55SSRs, KIBS(94SSRs), Q55SSRs+KIBS(94SSRs), Q30KSNPs+KIBS(94SSRs), Q55SSRs+KIBD(94SSRs), Q55SSRs+KIBS.NORM(94SSRs), KIBS(30KSNPs), Q55SSRs+KIBS(30KSNPs) models.We took the QWe then considered associations obtained within the three main groups  obtained with STRUCTURE and 55 SSRs, using the corresponding K obtained with 94 SSRs in the linear mixed model.Vgt1 was shown to be a major QTL involved in flowering time variation 55SSRs+KIBS(94SSRs) model to further test the significance of SNPs that may be involved in flowering. Second, a forward variable selection was applied to the 96 markers with P-value<10−5, using the same mixed linear model used above (Q55SSRs+KIBS(94SSRs)). In order not to eliminate an increasing number of lines during the procedure, we used genotypes imputed with FastPHASE. At each step, a marker i chosen among 96-i+1 markers was added in the model. The relative quality of the statistical models was estimated for all data sets with the Akaike Information Criterion (AIC) that penalizes the likelihood by the increasing number of parameters to estimate . The marker i added at each step was chosen to minimize this criterion. We stopped the procedure when the AIC criterion stabilized and the last marker added was not significant conditionally to the n–i markers included before in the model. All models were analyzed using r-Asreml In addition to these single locus models we used two complementary multi-locus models. First, since Genes located in the vicinity of QTLs or regions presenting non-neutral patterns were identified according to maize annotation version 2 (maizegenome.org).Centromeric regions were consensually defined from maize GDB (maizegdb.org) and The SNP genotyping reproducibility was assessed with 20 DNA replicates and was above 0.999. The mean interval between successive markers was 50 kb . The observed heterozygosity was low (0.036), as expected for inbred lines. The average genetic diversity of the panel was 0.36. A lower genetic diversity value was obtained in NF (0.28) and EF (0.30) genetic groups relative to CBD (0.34) and Trop (0.33). The Wilcoxon Signed-Rank Test that does not assume normal distribution −15).Among the 43224 polymorphic SNP markers or the 29911 Panzea SNPs see . In ordeAmong the 987 markers which were monomorphic in tropicals, 921 (93%) presented a different allele in dents, 610 (62%) in flints, with 589 (57%) being common to dents and flints. Considering the 3346 alleles that were rare (<0.05) in tropicals, 30% were lost in NF and EF, 57% in SS and 6% in CBD. Looking at the frequency of these alleles that were rare in tropicals , 94 SSRs (2R = 0.77) and 30K SNPs (2R = 0.81). IBS similarities calculated with 30K SNPs were more correlated with IBS calculated with 94 SSRs (2R = 0.32) than with IBS calculated with 55 SSRs (2R = 0.29) , 0.70 for 29911 Panzea SNPs .Group assignments were consistent between 55 SSRs and 29911 Panzea SNPs for structure levels ranging from 2 to 5  based on non-admixed lines (assignment to groups >0.8) ranged from 0 to 0.95. According to Hierfstat, 86% of FST values were significant  and 5% were above 0.42 log above 0, including 34 substantially significant (ranging from 0.5 to 1) and 18 highly significant (ranging from 1 to 3.1). All of these markers had a positive alpha value corresponding to diversifying selection. Their STF values ranged from 0.36 to 0.72 when calculated with Hierfstat and from 0.38 to 0.51 when calculated with BayeScan. Therefore, only STF found within the range 0.4–0.6 when computed with BayesScan (STBF) were detected as outliers. Figure S14 in STF and STBF are constrained in a different way by MAF, with only loci with MAF above 0.3 being detected as outliers (10(PO)log>1) according to BayeScan.For SNPs, Hierfstat 2r = 0.1 after 200 kb according to the Hill and Weir model 2r  of 10−5. Some long distance LD was nevertheless observed between centromeric regions .Based on LD analysis, we plotted all diversity and LD statistics using a sliding window of 1 Mb by steps of 500 kb. Centromeric regions showed specific patterns  and S12,When assigning each line to 20 estimated clusters for each SNP position using FastPHASE algorithm STF) . For other chromosomes, centromeres did not display a specific differentiation pattern.Total diversity was higher in centromeric regions , ASI8 (sd = 26 GDD or 2.2 days). This variation appeared to be closely related to the population structure . For FFLW8, the group adjusted means were 762 (±41), 763 (±35), 943 (±73), 889 (±23) and 1181 (±33) for NF, EF, SS, CBD and Trop, respectively  as reference to avoid using SNPs as candidate loci and for population structure and kinship estimations. As illustrated by Figure S18 in P-values obtained with model Q55SSRs+KIBS(30KSNPs) were globally higher than with Q55SSRs+KIBS(94SSRs). However, most associations significant with P-values<10−5 obtained with model Q55SSRs+KIBS(94SSRs) remained highly significant  with model Q55SSRs+KIBS(30KSNPs). P-values of the two models were globally strongly related, the most notable exception being the centromeric region of chromosome 8, for which markers with a P-value of 10−7 Q55SSRs+KIBS(94SSRs) were never below 10−3 with Q55SSRs+KIBS(30KSNPs). This probably relates to the large number of markers in high LD in this region, which consequently have an important contribution to the KIBS(30KSNPs) kinship estimation .The analysis of the present design included eight additional field experiments compared to our previous studies which included only two experiments P-value<10−3. Considering this threshold, 673 markers were associated with FFLW8, 843 with MFLW8 and 145 with ASI8. Among associations for FFLW8, 96 (corresponding to 18 regions) had a P-value below 10−5, 50 below 10−6 (corresponding to Bonferroni correction) and 7 below 10−7 , which has a strong effect on flowering time (7 days) according to −5 ranged from 44 (3.5 days) to 123.5 GDD (9.9 days), with a majority of small effects and a few strong effects . The first marker included was ZCN8 (#23) as expected. The nine first markers added in the model were significant with  conditionally to the previous model. The last marker entering the model with  was the 29th. AIC then continued to decrease until 71 markers were included in the model  compared to dents (75 GDD) and flints (53 GDD). Among them, 70 were significant in tropicals, 34 in dents, 6 in flints . All significant effects had the same sign in the three genetic groups. In addition, 110 markers that were not significant in the entire panel were highly significant  in at least one group, 9 in flints, 63 in dents and 39 in tropicals, with a mean effect of 80, 120 and 230 GDD, respectively.Finally, to avoid possible confusion between population structure and marker effects, we estimated marker effects with an ANOVA in three different genetic groups . For the 96 markers with 55SSRs+KIBS(94SSRs) and Q55SSRs+KIBS(30KSNPs), as well as 10 additional loci strongly under selection. Regarding the first aspect, the marker that explained the highest genetic variance  with the highest P-value  was located on chromosome 8 . There was no other marker in the 50K Illumina array closer to this gene. Its effect was 108 GDD (9.7 days). It was 1 Mb apart from PZB02155.1 (124657056 bp), which was found to be associated with flowering time in the NAM population Mite (131984851 bp), a major flowering time QTL that has been cloned Mite presented an association P-value<10−14 with SAS GLM −9 with lmm in this study. LD (2r) between ZCN8 (marker #25) and Vgt1 Mite was 0.3. This marker remained significant when adding Vgt1 Mite in the linear mixed model. ZCN8 was located between two flowering time meta QTL defined in the meta analysis of Vgt1 in Vgt1 QTL.ZCN8 #23). Note that some other markers with slightly lower P-values  were close to NAM associations ZCN4, 400 kb from NAM PHM3457.6, in meta QTL 2_4; #17, 200 kb from NAM PHM4748.16, 1 Mb from ZCN7, in meta QTL 6_3; #23 5000 bp from ZCN8, 1 Mb from NAM PZB02155.1; #16, in meta QTL 6_1, 1 Mb from ZCN15, 5 Mb from NAM PZA00355.2; #5: centromere of chromosome 2, 5 Mb from ZCN21, in meta QTL 2_4). Note that #15 may be involved in the sugar pathway as observed in Arabidopsis Other markers with a main contribution to FFLW8 were located on chromosomes 1 (#2), 2 (#5), 3 , 4 (#11), 5 (#14), 8 (#24), 9 , with a genetic variance ranging from 10 to 23% according to lmm and 3 to 6% according to lm. Ten additional loci displayed intermediary effects. According to the projection of flowering time meta QTL STF were higher at markers associated with FFLW8  as compared to other parts of the genome . Within group diversity (SH) was also lower in these regions than for the whole genome in NF (0.17) and EF (0.22) and in contrast higher in tropicals (0.4) . ZCN8 (Vgt2 locus) appeared to be strongly under diversifying selection (10(PO)log = 3.1). Allelic frequencies at Vgt2 were slightly more differentiated than those observed for Vgt1 . As shown in Vgt2 and Vgt1 was characterized by low diversity in flints and a low recombination rate. Differentiation was high only around Vgt2.Diversity and 2r = 0.2 compared to 0.14 for the entire genome) and lower recombination (0.02 compared to 0.04). The major haplotype in these regions was shared by 65% of lines (compared to 50% for the whole genome)  .P-value 10−5; #3 1 Mb from Phyc1; #18, 200 kb from NAM PZA02722.1. This suggests that the association mapping power at these loci may have been hindered by fixation in some of the groups.In addition, some markers were found to be under selection although not related to flowering time in this panel. They nevertheless were related to flowering time according to other studies ×Mo17 (CBD) SNPs lead to overestimated genetic distances for dent (CBD and SS) lines  at the SNP discovery step. Other groups passed comparable sampling procedures, so the differences that were observed suggest the involvement of different evolutionary processes.Fifteen percent of loci were monomorphic in flints, compared to 2% only in CBD and tropicals . When loAs stated above, the level of polymorphism observed in CBD presented only limited loss compared to tropical materials. Alleles that were observed in flints while being rare or absent in tropicals were also generally noted in dents (CBD), whereas the reverse was not true. This is consistent with CBD originating in part from flints (NF) and in part from a group relatively close to tropicals, approx. 400 ago When comparing European flint diversity to that of Northern flints, we found that 93% of alleles were common to both groups, which was higher than in previous studies with multiallelic markers (75%) The genotyping method we used did not allow us to estimate LD at the gene level as obtained by −5 between different chromosomes) with, however, notable exceptions between centromeric regions of chromosomes 1, 5, 8 and a few additional regions, like the extremity of chromosome 10 (2sr) clearly removed long distance LD . Differentiation at some centromeric regions was also observed by Visualization of LD as a sliding window (1 Mb) approach revealed high variation along chromosomes . It confSu12 of 0.10 is considered to be the minimum LD value needed to detect associations with complex traits. This reasoning is based on a large effect QTL that explains 10% of phenotypic variation (2qh = 0.1), which is a rather high expectation according to present knowledge, as also discussed by Van Inghelandt et al.2 = 0.1) and thus 1%  were difficult to interpret biologically and may have resulted from confusion between estimations of fixed effects and variance in random effects, quencies . IndividP-value<10−5  conditionally to the previous model and the last one to enter the model with  was the 29th. The augmentation in P-values relative to those observed in individual tests suggests that some colinearity exists between flowering time QTLs beyond that expected when considering relatedness and population structure only. This is possibly due to differential selection. However, the main QTLs detected in the single marker analysis remained significant in the multi-locus analysis, which confirmed their effect. The fact that AIC continued to decrease until 71 markers were included in the model confirms that numerous regions are involved. Interestingly, the multi-locus analysis also highlighted several cases where linked markers entered the model. This suggests that several SNP markers are necessary in these regions to tag allelic series with a gradient of effects >0.5) and 18 markers or 9 regions that were highly significant (10(PO)log>1). Most markers associated with flowering time (34/35) showed high differentiation among groups , which was clearly associated with flowering time in a NAM design From an evolutionary standpoint, geographical distance along with flowering time divergence ultimately results in reproductive barriers between lines. Even if admixture is still possible between compatible lines, genomic regions involved in adaptive traits will tolerate only restricted gene flow, resulting in lower efficient recombination, lower diversity and higher LD compared to the rest of the genome. This pattern can be dramatic when it occurs in centromeric regions (see chromosome 8 in this study) that have mechanical barriers against crossover. Recent empirical and theoretical studies suggest that restricted recombination regions play an important role in the formation of new genetic groups and species. They could contain clusters of genes involved in reproduction isolation and centromeres may be privileged regions from this standpoint Vgt1-Vgt2/ZCN8, centromere of chromosome 8), and the cumulative effects of many loci with small to intermediate effects. In view of selection, this suggests that predictive methods offering flexibility in effect magnitude such as Bayes B a priori be more adapted than RA-blups Our data generally show that major differences in flowering time among inbred maize lines are caused by a few genes with relatively marked effects Click here for additional data file.Figure S11Supporting Figure including one page per chromosome (10).(PDF)Click here for additional data file.Figure S12Supporting Figure including one page per chromosome (10).(PDF)Click here for additional data file.Table S8Supporting Table including information about markers.(XLS)Click here for additional data file.


Estimation of genetic covariance matrices for multivariate problems comprising more than a few traits is inherently problematic, since sampling variation increases dramatically with the number of traits. This paper investigates the efficacy of regularized estimation of covariance components in a maximum likelihood framework, imposing a penalty on the likelihood designed to reduce sampling variation. In particular, penalties that "borrow strength" from the phenotypic covariance matrix are considered.An extensive simulation study was carried out to investigate the reduction in average 'loss', i.e. the deviation in estimated matrices from the population values, and the accompanying bias for a range of parameter values and sample sizes. A number of penalties are examined, penalizing either the canonical eigenvalues or the genetic covariance or correlation matrices. In addition, several strategies to determine the amount of penalization to be applied, i.e. to estimate the appropriate tuning factor, are explored.It is shown that substantial reductions in loss for estimates of genetic covariance can be achieved for small to moderate sample sizes. While no penalty performed best overall, penalizing the variance among the estimated canonical eigenvalues on the logarithmic scale or shrinking the genetic towards the phenotypic correlation matrix appeared most advantageous. Estimating the tuning factor using cross-validation resulted in a loss reduction 10 to 15% less than that obtained if population values were known. Applying a mild penalty, chosen so that the deviation in likelihood from the maximum was non-significant, performed as well if not better than cross-validation and can be recommended as a pragmatic strategy.Penalized maximum likelihood estimation provides the means to 'make the most' of limited and precious data and facilitates more stable estimation for multi-dimensional analyses. It should become part of our everyday toolkit for multivariate estimation in quantitative genetics. Estimation of genetic parameters, i.e. the partitioning of phenotypic variation into (co)variances due to genetic effects and other sources, is one of the basic tasks in quantitative genetics. Increasingly, livestock improvement schemes consider a multitude of traits. In turn, this requires complex, multivariate analyses that consider more than just a few traits simultaneously. Advances in modelling, improvements in computational algorithms and of corresponding software, paired with the capabilities of modern computer hardware have brought us to a point where large-scale analyses comprising numerous traits and records on tens of thousands of individuals are feasible. For example, Tyrisevä et al.  recentlyIt has long been known that the eigenvalues of estimated covariance matrices are over-dispersed, i.e. that the largest sample eigenvalues are systematically biased upwards and the smallest values are biased downwards, while their mean is expected to be unbiased . MoreoveHence, any avenue to 'improve' estimates, i.e. to obtain estimates which are on average closer to the population values, should be given serious consideration. To begin with, we have accumulated a substantial body of knowledge about genetic parameters for various traits. However, typically this is not used. While the Bayesian paradigm directly provides the means to incorporate such prior information, applications in estimating covariance components often assume flat or uninformative priors , i.e. doHowever, there has been little interest in regularized estimation in estimating genetic parameters. An early proposal, due to Hayes and Hill , was to Recently, Meyer and Kirkpatrick  proposedloss. A well known quantity is the mean square error which is a quadratic loss, comprised of the sampling variance and the square of the bias in the estimator. We talk about improving an estimator when we are able to modify it in some way so that it has reduced loss, i.e. is closer to the true value. Usually this involves a trade-off between a reduction in sampling variance and additional bias. For covariance matrices, commonly employed measures of divergence are the entropy (L1) and quadratic (L2) loss  (with x1 and x2 the lower and upper limits for x) can be taken into account by fitting a four parameter beta function . As we expect standard estimates of eigenvalues to be over-dispersed, a suitable, if somewhat inflated, estimate of the range may be given by the estimates of the extreme values from an unpenalized analysis (i.e. ψ = 0), denoted henceforth by a superscript of 0. Assuming eigenvalues are numbered in descending order of magnitude, this gives with Γ(·) denoting the gamma function, and mean function  or by reα = β = 2, 3,..., i.e. a symmetric distribution for A suitable choice for the shape parameters might be α and β can be estimated from estimates α/(α + β) and αβ(α + β)-2(α + β + 1)-1, results in the method of moment estimators α and β less than unity, implying probability distributions that are U- or J-shaped with a high mass at the extremes. To ensure a uni-modal beta distribution, we add a constant z (z ≥ 0). This gives penaltyAlternatively, f the λi0. This mai-th order statistic of a q-variate sample is the i-th smallest value. Assuming a uniform distribution, the order statistics on the unit interval have marginal beta distributions with scale parameters i and q - i + 1. Treating values Penalties considered so far implied that estimated eigenvalues were samples from a distribution with a common mean ad apart ,15,19. Hz in (Eq. 9). For the distribution of order statistics, z = 0. A value of z > 0 causes individual distributions to be 'squashed' together, i.e. yields a compromise between the assumption of a common mean for the Again, we allow for a modifying constant Motivated by the historical emphasis on the role of sample eigenvalues of covariance matrices, we have concentrated on penalties on these characteristics so far. A conceptually simpler alternative is to consider the covariance matrix as a whole and its prior distribution.Ω denoting the scale parameter and v the degree of belief we assign to the prior. Omitting terms not depending on Σ or Ω and taking logarithms gives (v + q + 1) log |Σ| + v tr(Σ-1 Ω). Corresponding to the penalties that 'borrow strength' from the phenotypic covariance matrix considered above, a penalty which regularizes the estimate of ΣG by shrinking it towards Σp can be obtained by using Σp as a scale matrix. Adopting an empirical Bayes approach, we substitute the estimate from an unpenalized REML analysis, ΣP  e.g. , with Ω ce of ΣP . FurtherC = (ψ + q + 1)/ψ. If C is approximated with unity, ΣG and L1(·) with Σ and ΣG and ΣE simultaneously usingwith xchanged . The relG C= EC) or separate tuning factors.weighted by either a joint (G r) and phenotypic (Pr) correlations are often similar, Cheverud .REML estimates of kpatrick , restraiz = 1 and z = 1, and four penalties on matrices, A total of 10 penalties were examined, six penalties on the canonical eigenvalues, ψ) for each analysis, estimates of ΣG and ΣE, denoted as ψ. A total of 311 values were used, comprising values of 0 to 2 in steps of 0.1, 2.2 to 5 in steps of 0.2, 5.5 to 10 in steps of 0.5, 11 to 100 in steps of 1, 102 to 250 in steps of 2, 255 to 500 in steps of 5 and 510 to 1000 in steps of 10. The 'best' value was then chosen using three different approaches.To determine the tuning factor  and within (MW) sires, as from an analysis of variance. For strategy V1, MB and MW were obtained by sampling one additional data set from the same distribution as the data used in the analysis. For strategy V∞, MB and MW were constructed from the population parameters. This is equivalent to sampling an infinite number of additional data sets, hence the notation V∞. For both strategies, the value of ψ which maximised with K-fold cross-validation (CV) was used to estimate ψ using only the data available. For this, data were split into K folds of approximately equal size by sequentially assigning complete sire families to subsets. For i = 1, K, the i-th subset was set aside for validation, while all the remaining K-1 subsets were used to obtain estimates ψ considered. Corresponding values for the unpenalized likelihood, ψ was chosen as the value for which the average likelihood, K = 3 and 5 were considered, with the corresponding strategies denoted as CV3 and CV5 in the following.Secondly, ψ as the largest value of ψ for which the reduction in the unpenalized likelihood due to penalization from the maximum at γ, 1.92 for γ = 0.05. This will be referred to as strategy L5%.The third approach used simply involved choosing As suggested by Lin and Perlman , the effΣX and X = G, E and P, and i θwere calculated as where s = 100. Using known population values (strategy V∞), achieved reductions in average loss in estimates of ΣG were substantial, ranging from about 60% to more than 72%. The main exception was ΣE were substantially higher than for ΣG. On average PRIAL values were somewhat smaller for q = 9 than q = 5 traits because cases for q = 9 comprised more unfavourable scenarios, i.e. population values with a large and uneven spread of the canonical eigenvalues.Mean PRIAL values across all cases for individual covariance matrices and all penalties considered are summarized in Table ΣG and ΣE, all penalties on i λyielded marked improvements in estimates of ΣE as well as ΣG. Considering log(1 - iλ) in addition to log(iλ)  would yield a ψ that was too high, as it pertained to the sample size of the subset, while a larger number of folds (high K) might off-set potential inabilities to ascertain optimal values for ψ due to the limited size of the validation set. However, results for CV5 were consistently worse than for CV3. Additional analyses for K = 10 (not shown) yielded even lower PRIAL than CV5. Inspection of the mean tuning factors K. For penalties ψ determined using V∞ tended to be very high for cases with little spread in the population iλ, while corresponding values using CV tended to be substantially lower, so that the average, ψ. For strategy CV3, ΣG for 35% of the individual cases (q = 5 and s = 100), compared to 2% for strategy V∞.Examining regularized estimation of a single covariance matrix, Rothman et al.  reportedse Table . Somewhaχ2 distribution corresponding to the number of parameters tested and an error probability γ. The smallest number of parameters which can be tested is p = 1. Hence, choosing ψ as the largest value for which the resulting change in log Difficulties in deriving the optimal 'bending' factor theoretically, led Hayes and Hill  to suggeψ in this way yielded substantially improved estimates of ΣG, with PRIAL consistently higher than for CV. Values for the average tuning factor I, H-I, I-I, M-I and R-I). As illustrated in Figure ψ using this strategy also blurred differences between penalties. In a number of cases, in particular for q = 9 traits, PRIAL for ΣG from strategy L5% were higher than those from V∞.As shown in Table q = 5. Clearly, penalization was most advantageous for small samples, with mean PRIAL for ΣG decreasing substantially as the number of sire families increased. There were marked differences between penalties and strategies to determine ψ, especially in the rate of decline of PRIAL with increasing s. This rate was least for penalty ΣG and ΣE for all sample sizes, when using the change in log ψ.The effect of sample size on the efficacy of regularized estimation is illustrated in Figure ΣG when using CV to determine the tuning factor were substantially less than for the other strategies. This difference tended to increase with sample size. Whilst consistently performing worst for strategy V∞, the penalty on i λderived from the distribution of order statistics ΣG for strategy CV3. It is not clear what this comparatively larger robustness against noise in estimates of ψ can be attributed to. The decline in PRIAL with sample size was clearly a function of the number of traits considered, with reductions for q = 9 markedly smaller. For instance, for ΣG declined from 69.4% for s = 100 to 64.1% for s = 400 and 60.2% for s = 1000. Respective values for As noted above, PRIAL in s = 100 sires and strategy V∞. Figure i λand their true values for selected penalties and strategy V∞, with the solid line showing a one-to-one correspondence (unbiased estimates) and the dashed line representing the linear regression of estimates on population values. Patterns obtained when selecting the tuning factor using L5% or CV were similar. As expected, without penalization, estimates of the largest values were biased upwards and those of the smallest values biased downwards. Whilst the mean was expected to be estimated without bias, a small upwards bias in the average eigenvalue, i λat zero were evident, reflecting the effects of constraints on the parameter space. A penalty on canonical eigenvalues tended to result in over-shrinkage, i.e. causing a downward bias of the largest and an upward bias of the smallest values. This was the more pronounced the further the population i λwere spread apart. Similar results for a single matrix were reported by Daniels and Kass [iλ, absolute changes tended to be small and penalization clustered estimates closer to the one-to-one line.As emphasized earlier, regularized estimation entails a trade-off between sampling variance and bias. Table and Kass . While tΣG were comparable. For λ1 was of a similar magnitude and individual estimates exhibited the same pattern . Without penalty, a slight bias in estimates that corresponded to the highest and lowest population values was evident, arising from constrained estimation. Penalized estimation biased estimates of h2, with the pattern of biases and differences between penalties similar to those observed for iλ. For instance, for h2 were substantially biased upwards, while estimates for the largest values were similar to those from unpenalized analyses. Penalties on the canonical eigenvalues resulted in marked underestimates of the highest h2. Taking the average of absolute deviations across traits yielded the lowest values for However, it should be stressed that bias in estimates of eigenvalues does not directly translate into bias in the corresponding covariance components or genetic parameters. Eigenvalues of sample covariance matrices are systematically over-dispersed and biased, but the sample covariance matrix is an unbiased estimator e.g. . REML esgh Table .VI  and s = 100. Shown is a box-and-whisker plot of individual estimates across replicates, with correlations in ascending order of their population values, depicted by horizontal bars. Not surprisingly for such small samples, unpenalized estimates were subject to substantial sampling variation and were most variable for pairs of traits with lowest h2. Again, unpenalized estimates were clearly biased due to constraints on the parameter space, with mean deviations from the population values ranging from -0.504 (8-9) to 0.035 (3-8) and a mean, absolute bias across replicates of 0.064. Penalization dramatically reduced the spread of estimates, but increased bias to a range of -0.734 (8-9) to 0.103 (4-8), with a mean absolute value of 0.142. In all cases, genetic correlations were shrunk towards the corresponding phenotypic correlations . In spite of the increase in bias, penalized estimation reduced the loss in the estimate of RG by 77.3%. The corresponding value for ΣG was less, 58.1% for V∞, i.e. this was a scenario for which penalization was less effective  difference between true and estimated covariance matrices, can be achieved through penalized estimation. As expected, this reduction in loss is at the cost of increasing bias, over and above that introduced by constraining estimates to the parameter space in standard REML analyses. The magnitude and direction of the additional bias depend on the population parameters and penalty applied but in general, penalization leads to reduced estimates of the highest heritabilities and increases estimates of the smallest heritabilities while estimates of genetic correlations are reduced in absolute value. With comparable (or better) reductions in loss to other penalties, those which shrink the genetic towards the phenotypic correlation matrix (Λ and T [Penalized REML estimation for penalties on canonical eigenvalues is best implemented by parameterising to the elements of the canonical decomposition, Λ and T . In contΛ and T . Moreoveψ limiting achieved PRIAL and increasing the proportion of replicates for which penalization was detrimental. These errors appeared especially important for larger samples, i.e. in small samples any degree of penalization is likely to have a substantial effect, while over-penalization becomes more harmful as sample size increases. An added problem with CV for data with a genetic family structure is that of representative sampling of data subsets. In our setting, assigning whole sire families to individual folds was a natural choice and yielded higher PRIAL values than splitting families evenly across folds. In practical data sets with arbitrary relationships and fixed effects, choices are less obvious and guidelines to good sampling strategies in a mixed model setting are scarce.CV is a widely used technique to estimate the tuning factor in regularization problems from the data at hand. For our application, however, it was found to be only moderately successful, with errors in estimating Moreover, CV is laborious and increases the number of analyses required by orders of magnitude. A sequential search for the optimal tuning factor was used in our simulation study. A more efficient strategy would be to use one of the many structured optimization methods available, e.g. a quadratic approximation of the average likelihood from the validation sets. However, this relies on the 'validation' curves to be smooth, increasing monotonically to a maximum and then decreasing again. This was not always the case in the simulations presented - some jagged curves were encountered, in particular for the smallest sample sizes. Presumably this was due to likelihood surfaces being very flat around their maxima, resulting in inaccurate location of these points. Use of such techniques was thus disregarded here.ψ based on the decrease in the unpenalized likelihood from its maximum at ψ = 0 can result in penalized estimates that are closely related to those which would be obtained if population values were known. As demonstrated, such strategies yielded average loss reductions for estimates of the genetic covariance matrix that were substantially higher than loss reductions obtained when estimating ψ by CV, and loss reductions comparable to those achieved when using knowledge of the population parameters for some penalties. Choosing the limit to the change in likelihood so that it was just not statistically significant appeared to be a sensible choice to select a mild degree of penalization. Although this choice did not perform quite as well for individual cases where all population canonical eigenvalues were very similar, this is a scenario which is unlikely to be of practical relevance in quantitative genetic applications.Fortunately, choice of Work so far has considered a balanced scenario, in which all traits in a multivariate analysis were measured for all individuals. However, we often have a substantial discrepancy between the number of observations available for different traits. For instance, we may have a number of traits recorded on a substantial number of individuals whilst records for difficult to measure traits are available for a small subset only. It is then necessary to penalize parts of the genetic covariance matrix corresponding to such groups of traits differently. An extension of the penalties on the divergence between genetic and phenotypic matrices allowing this can be derived assuming a generalized inverse Wishart prior distribution, and will be considered in future work.Even with today's computational resources, there are problems for which analyses that consider all traits of interest are not feasible, so that elements of the complete covariance matrix have to be obtained through a series of analyses of subsets of traits. This yields multiple estimates of variance and some covariance components, which need to be pooled whilst ensuring the resulting matrix is positive definite. Typically, this is done by considering one matrix at a time, using methods such as 'iterative summation of expanded part matrices' , or by cPenalized maximum likelihood estimation provides the means to 'make the most' of limited and precious data and facilitates more stable estimation for multi-dimensional analyses, even when samples sizes are larger. We anticipate that penalized maximum likelihood estimation will become part of our everyday toolkit, as truly multivariate estimation for quantitative genetic problems becomes routine. At the present state of knowledge, a mild penalty on the divergence of the genetic from the phenotypic correlation matrix, chosen on the basis of the change in likelihood from an unpenalized analysis, appears the most suitable option for practical applications.The author declares that she has no competing interests.KM carried out all the tasks associated with this paper. All authors read and approved the final manuscript.


In animal breeding, the genetic potential of an animal is summarized as its estimated breeding value, which is derived from its own performance as well as the performance of related individuals. Here, we illustrate why estimated breeding values are not suitable as a phenotype for genome-wide association studies. We simulated human-type and pig-type pedigrees with a range of quantitative trait loci (QTL) effects (0.5–3% of phenotypic variance) and heritabilities (0.3−0.8). We analyzed 1000 replicates of each scenario with four models: (a) a full mixed model including a polygenic effect, (b) a regression analysis using the residual of a mixed model as a trait score , (c) a regression analysis using the estimated breeding value as a trait score, and (d) a regression analysis that uses the raw phenotype as a trait score. We show that using breeding values as a trait score gives very high false-positive rates (up 14% in human pedigrees and >60% in pig pedigrees). Simulations based on a real pedigree show that additional generations of pedigree increase the type I error. Including the family relationship as a random effect provides the greatest power to detect QTL while controlling for type I error at the desired level and providing the most accurate estimates of the QTL effect. Both the use of residuals and the use of breeding values result in deflated estimates of the QTL effect. We derive the contributions of QTL effects to the breeding value and residual and show how this affects the estimates. Genome-wide association studies (GWAS) are now commonplace in humans, livestock, plants, and model organisms. A commonality among these studies is that genetic links exist between genotyped subjects and these must be accounted for in statistical analyses. Several approaches have been proposed to take account of these genetic structures in GWAS  for the given trait based on its own performance and/or that of genetically related individuals. In the genome-wide rapid association using mixed model and regression (GRAMMAR) approach, the observed phenotype is analyzed under a mixed model resulting in an EBV and a residual, with the latter being used as the trait value for GWAS  and a residual. The EBV is an estimated measure of the additive genetic merit of an individual  can give much greater false-positive rates than ignoring family relationships altogether.We will show here with a straightforward simulation study that the “information from all measured relatives” is a prime source of false-positive results in GWAS. We note that this insight is neither profound nor novel, but our aim is to provide a clear and concise illustration that using EBV comprising familial information , which accounted for familial relatedness of individuals in the pedigree using the additive genetic relationships among individuals. The SNP effect and polygenic effect were estimated together using this model:y is the vector of trait values, μ is the overall mean, a is the additive QTL effect, u and e are vectors of additive polygenic effects (random), and random residuals, respectively; u ~ N, where A is the additive genetic relationship matrix based on pedigree information and e ~ N, where I is an identity matrix; σ2a and σ2e are the additive genetic and residual error variance, respectively. w is a vector of marker genotypes  and Z is an incidence matrix related to polygenic effects.where Step 1: For the simulated trait score, we fitted the following mixed model [with the same variable definitions as (1)] without the marker effect:y*), the marker genotype effect of each SNP on the trait was tested by linear regression:Step 2: Using the estimated residuals from Step 1 as the new quantitative trait (GRAMMAR: The GRAMMAR approach consists of two steps (y) on the SNP data and does not take account of family relationships.Ignoring family structure (IF): The IF analysis is comparable with the second step of the GRAMMAR analysis. It uses a direct regression of the phenotypic observation (y*) for the association study (3).EBV: Similar to GRAMMAR but in this analysis the EBV from the polygenic model  is used as a trait score . The statistical power to detect the causal SNP was estimated using either the tabulated F-threshold of 3.84 or an empirical threshold based on a 5% error rate for the unlinked SNP. In order to facilitate the computational load of 1000 replicates of 46 scenarios, the simulations and analyses were run on the Edinburgh Compute and Data Facility.All analyses were performed in ASReml  for the four methods are summarized in Table S1), albeit slightly lower than for the simulated pedigree. This was due to the smaller family sizes in the sample from the real pedigree compared to the simulated pedigree. The use of five generations of pedigree in the mixed model (2) gave a greater FPR when we used EBV as the trait score compared to using only two generations (Table S1). The use of GRAMMAR was more conservative when applying five generations of pedigree compared with two. There was no clear trend in type I error when using the measured genotype approach (1) or ignoring family structure: in some scenarios using five generation of pedigree gave a more conservative type I error, whereas in other scenarios it was more liberal than using two generations of pedigree information (Table S1).The simulations based on the commercial pig pedigree also showed a large type I error when using EBV  usually, but not always, had the greatest power, irrespective of whether empirical or tabulated thresholds were used (Table S2). Using EBV or ignoring family relationships had greater power when using tabulated  thresholds (Table S2), but at the cost of high FPRs (Table S2), in agreement with . For the simulations based on the commercial pig pedigree, the measured genotype approach had the highest power regardless of whether the thresholds were tabulated or based on empirical results (Table S1). The two-generation pedigree generally gave slightly greater power to detect QTL than the five-generation pedigree.For scenarios in which the heritability and QTL effect sizes were low, The estimates of the QTL effect and the empirical standard deviation more than 1000 replicates are summarized in A clear trend was apparent for the effect of the heritability on the estimates when using GRAMMAR or EBVs. With increasing heritability, the GRAMMAR estimates were increasingly biased downward while those from the EBV approach became less biased . FurtherTable S1 and Table S2) as many sibs may have received alternative QTL alleles. This dilutes the SNP effect, and it will have greater impact in situations in which family information makes a greater contribution to the EBV. Second, there is a greater risk of FPRs, as any SNP that differs in frequency between families risks being correlated by chance with the family mean polygenic value for the trait and hence shows a significant association with the EBV. This finding implies that in analyses of real human pedigrees the type I errors are expected to be even more serious that those shown here in the simulated pedigrees as a result of minor ethnic variations from one pedigree to the next.Our simulations have shown clearly that the use of EBV in association studies, incorporating information from relatives from the same or previous generations, can result in several problems, most notably huge increases in the type I error. This finding is attributed to the fact that when an individual’s EBV is estimated with familial information, the estimate is a linear combination of the individual’s phenotype, expressed as a deviation from the family mean, and the family mean itself. Although the individual phenotype captures the within-family segregation of the QTL, the family mean contains information on the QTL allele expressed by other family members. This “contamination” of the EBV by family information can affect both power and the FPR as follows. First, power can be reduced  effects. With EBVs, it is apparent that the weighting given to the family mean is always greater than that given to the Mendelian sampling term, hence the risks of FPRs and reduced power described previously, with the converse true for residuals. The relative weightings, and the expected value of the regression of phenotype on marker, are slightly complex and depend on the trait heritability, the accuracy with which the family mean is estimated, and the QTL frequency. Derivations are shown in the Appendix for family information estimated solely from sib means; however, the same principles apply to information obtained from other sources. This is seen in e.g., different numbers of daughter records per sire. However, de-regressed EBVs do not remove the component of the EBV coming from information on other relatives. To remove effects from other relatives from the (deregressed) EBV, In species in which the bulk of the information is derived from progeny testing, such as dairy cattle breeding, the contribution to the EBV coming from relatives other than the direct offspring becomes smaller and using EBV will be less detrimental than in the cases considered here. This is because offspring information directly estimates the Mendelian sampling term of the animal being evaluated, and hence assists in estimation of the QTL effect. We acknowledge that for many species, most commonly in dairy cattle where EBV are derived for bulls using a large number of daughter records, EBV for a wide range of traits are routinely available and are a convenient source of information. In these cases, the use of deregressed EBV has been proposed for GWAS and genomic prediction , we recommend the use of a measured genotype approach (1) or an approximation using GRAMMAR. Although GRAMMAR was once again shown to be conservative and give an underestimate of the QTL effect, recent developments of the GenABEL software have accounted for this in the GRAMMAR-Lambda module that provides an adjusted test static and a correction for the estimated QTL effect (


This was further corroborated by QTL analysis for body weight, which narrowed this region to 4.43 cM with flanking markers D5Rat256 & D5Wox37. Interval mapping of body weight QTL shows that the LOD score peak maps upstream of leptin receptor and shows an additive effect suggesting this as a novel mutation and signifying the model as a valuable resource for studies on obesity and metabolic syndrome. WNIN/Obese (WNIN/Ob) rat a new mutant model of metabolic syndrome was identified in 1996 from an inbred Wistar rat strain, WNIN. So far several papers are published on this model highlighting its physical, biochemical and metabolic traits. WNIN/Ob is leptin resistant with unaltered leptin or its receptor coding sequences - the two well-known candidate genes for obesity. Genotyping analysis of F Obesity, referred to as overweight due to excess fat accumulation has assumed epidemiological proportion in the last decade ,2 along National Centre for Laboratory Animal Science (NCLAS) at National Institute of Nutrition (NIN), India is maintaining one of the oldest Wistar rat stocks (since 1920), designated as WNIN. During 1990’s, a ‘Fat’ rat with about 47% of its body weight representing fat was identified from this stock, and later by pedigree and back cross analysis, a mutant strain was established as a uni-locus model for obesity trait, designated as WNIN/Ob. The pattern of inheritance was found to be autosomal incomplete dominance and the strain exhibited three phenotypes viz., lean, carrier (heterozygote) and obese, inherited in a Mendelian fashion, identifiable by morphology, body weight as well as by a kinky tail association (only in carriers and homozygous obese). Incomplete dominance and co-segregation of kinky tail are unique characteristics shown by this model in comparison to the existing rodent models of obesity . It is tAnimal experiments in the study were conducted after obtaining consent from Institutional Animal Ethical committee (IEAC), NCLAS, NIN.0 generation were crossed to raise F1 progeny. F1 progeny rats  were inter-crossed to raise F2 progeny.Rat strains employed for the study were bred and maintained at NCLAS, NIN. Food restricted male obese rats (competent for reproduction) from WNIN/Ob (n= 5) and female rats from Fischer-344(n=13) representing FRats were housed in clean polypropylene cages with sterilized paddy husk as bedding material. Room temperature was maintained at 22±2°C with 14-16 air changes per hour and 55±5% relative humidity. Rats were maintained with 12 hour light-dark cycles and the animals had free access to food and water.2) test was conducted in F2 progeny to test for the expected 1:2:1 ratio of lean, carrier and obese phenotypes respectively. Observed ratios were considered significant for χ2 value < 5.99 . Phenotypes were identified based on morphological characteristics, body weights and kinky tail association.Chi-square (χ2) and Wilcoxon rank sum tests (W) were performed to infer body weight differences. Test statistics with p-value ≤ 0.05 were considered significant. Analyses were carried out on ‘R’ software through ‘R Commander’ [Body weights of the rats were recorded at 3 months age using standard electronic balance with a precision of 1gm . Descriptive statistics like range, inter-quartile range and Median values were used to describe body weights of the observed phenotypes. Kruskal-Wallis (χmmander’ .0 and F2 rats and genomic DNA was isolated from them using Qiagen genomic DNA isolation kit.Tails were clipped from FFive microsatellite DNA markers polymorphic between WNIN/Ob and Fischer-344(F-344) spanning genetic distance of 31cM  and fla2 progeny (66 obese+2carriers) were genotyped. Genotypes were constructed by scoring for alleles of WNIN/Ob and F-344. Rats showing alleles of WNIN/Ob were coded as ‘A’, Rats showing alleles of F-344 were coded as ‘B’ and Rats showing alleles of both WNIN/Ob and F-344 were coded as ‘H’.Genotyping was performed on ABI-377 machine at Rockefeller University. A total of 68 F2 obese rats at each marker comprised about 63-66 (126 to 132 meioses). Recombination fractions (θ) were calculated as: number of meiotic recombinations detected / number of meioses scored. χ2 tests were conducted for each marker to determine deviation of the observed number of meiotic recombinations  from the expected 50% . χ2 value > 10.83  were considered significant. Logarithm of Odds (LOD) scores were calculated to determine linkage using the formula, Z = (((1-θ)^NR)*( θ^R))/(0.5^NR+R), where NR is the number of non-recombinants and R is the number of recombinants [2 obese rats showing ‘A’ genotype at a marker were scored as non-recombinants and ‘H’ or ‘B’ genotype as recombinants. LOD scores ≥ 3 were considered significant. As LOD scores cannot be calculated for ‘θ’ = 0, the value was set to 0.001 for the respective marker assuming less than 0.01% recombination fraction. Estimates of linkage disequilibrium (Dꞌ) and its correlation coefficient (r2) were obtained through ‘Gentics’ package for ‘R’ software [2 = 1 suggest for perfect linkage disequilibrium. The genotyped Fsoftware . As calc2 inter-cross data was prepared as per format and piped for the following estimations. Recombination fractions were estimated between all pairs of markers using two point tests with threshold LOD score of 4 and maximum recombination fraction of 3.5. Markers were assigned to linkage groups using the same above thresholds. Kosambi mapping function was used to display the genetic map. Markers of the linkage group were ordered by two-point based algorithm (Seriation). A framework of ordered markers was derived by comparing all possible orders using ‘Compare’ function. ‘Ripple’ function was used to permute sequentially with 4 markers per subset to check for any plausible alternative orders.ONEMAP package for ‘R’software  was used2 progeny rats with ‘A’ genotype versus ‘H’ genotype. Microsoft Excel-2007 was used for the above analysis. Association was considered significant for p< 0.001. To determine association between the marker and bodyweight phenotype, single factor analysis of variance was conducted on body weights of F2) for the linked markers was determined using QTL cartographer [R/QTL package was used for the analysis . Markersographer . (+/+), carrier(+/-) and obese(-/-) [2 progeny in the present study represented a total of 305 rats among which lean were 77, carriers were 151 and obese were 77. Test for Mendelian inheritance of obese trait suggested that F2 progeny follows 1:2:1 ratio (χ2=0.052). The probability of occurrence of Mendelian ratio calculated from the present study is more than 95% . The mutation causing obesity in WNIN/Ob was reported as incomplete dominant where, rats of carrier phenotype were comparatively heavier than their lean counterparts [2 progeny in the present analysis showed a similar trend. Body weights (gms) of F2 progeny ranged from 156 - 432 in lean rats, 150 - 480 in carrier rats, and 330 - 650 in obese rats. The likely range of variation i.e., Inter Quartile Range was 214 - 314 in lean, 232 - 361 in carriers and 440 - 546 in obese rats. Median values were 238, 289, and 501 for lean, carrier and obese rats respectively  between the phenotypes. Pairwise comparison by Wilcoxon Rank Sum test showed significant differences for lean versus obese , lean versus carrier  and carrier versus obese . These results re-confirm or previous findings on Mendelian inheritance and incomplete dominant nature of obese mutation identified in WNIN/Ob. The mutation reported in WNIN/Ob is uni-locus with three differentiating phenotypes, i.e., leanese(-/-) . The F2 ectively . Kruskal2 progeny, identified a genetic marker (D5Rat256) associated with obese phenotype. With the objective to narrow down towards mutation causing obesity, five additional markers were genotyped in the vicinity in F2 progeny. All the six markers, D5Rat98, D5Got131, D5Rat256, D5Wox37, D5Rat235, D5Rat69 showed less than 50% recombination with obese locus as indicated by their recombination fractions (θ) suggesting for an association. Chi-square test conducted for the above, showed a significant association for five markers D5Rat98, D5Got131, D5Rat256, D5Wox37, and D5Rat235. Linkage analysis revealed a significant linkage of the markers, D5Got131, D5Rat256, D5Wox37, and D5Rat235 with obese locus as indicated by their LOD scores. Highest LOD score was observed for D5Wox37 suggesting for a tight linkage. All the markers showed linkage disequilibrium with obese locus where as only one marker, D5Wox37 showed perfect disequilibrium as explained by the r2 value  explained by the two markers was 10.35% and 27.48% respectively. Rat genome database shows 42 QTLs [To further analyze the genomic region that is associated with obese trait, quantitative trait locus (QTL) mapping of the incomplete dominant phenotype (body weight) was carried out. As discussed earlier, rats with ‘Obese’ phenotype showed significantly higher body weights compared to their ‘Carrier’ or ‘Lean’ counter parts and ‘Carriers’ were comparatively heavier than their ‘Lean’ counterparts. To determine whether a quantitative trait locus for the excess body weight exist in the currently analyzed genomic region, association tests were conducted between the genotyped markers and the phenotype (Body weight). Two markers D5Rat256 and D5Wox37 showed a significant association with excess bodyweight as shown by the ‘F’ ratios  suggesti 42 QTLs  for vari 42 QTLs -36. FurtTable S12 progeny genotype data and corresponding animal body weights (g).FA= WNIN/Ob alleles; H= WNIN/Ob alleles & Fisher -344 alleles; - missing data.(DOCX)Click here for additional data file.Table S2Gene annotation - physical versus genetic distance.Physical position of the genes was mined from Rat Genome Database. Physical position is indicated in Million bases (Mb) and represent the centre position of gene ((start position + stop position) /2). Genetic distance (cM) was obtained by fitting linear regression for Physical vs Genetic distance of five markers D5Rat98, D5Got131, D5Rat256, D5Wox37, D5Rat235. Equation obtained from trend line, y = 0.3356(x)-49.378. R2=0.9899.(DOCX)Click here for additional data file.


Genome-wide Complex Trait Analysis (GCTA) that estimates the population variance explained by additive effects of DNA differences among unrelated individuals. For 1118 unrelated individuals in our sample with genotyping information on 1.7 million DNA markers, GCTA estimated zero heritability for number sense, unlike other cognitive abilities in the same twin study where the GCTA heritability estimates were about 25%. The low heritability of number sense, observed in this study, is consistent with the directional selection explanation whereby additive genetic variance for evolutionary important traits is reduced.Basic intellectual abilities of quantity and numerosity estimation have been detected across animal species. Such abilities are referred to as ‘number sense’. For human species, individual differences in number sense are detectable early in life, persist in later development, and relate to general intelligence. The origins of these individual differences are unknown. To address this question, we conducted the first large-scale genetically sensitive investigation of number sense, assessing numerosity discrimination abilities in 837 pairs of monozygotic and 1422 pairs of dizygotic 16-year-old twin pairs. Univariate genetic analysis of the twin data revealed that number sense is modestly heritable (32%), with individual differences being largely explained by non-shared environmental influences (68%) and no contribution from shared environmental factors. Sex-Limitation model fitting revealed no differences between males and females in the etiology of individual differences in number sense abilities. We also carried out  •Modest heritability of numerosity discrimination skills at 16 years•GCTA analysis supports directional selection for numerosity discrimination skills.•No sex differences in the etiology of individual variation in numerosity at age 16 Numerals are an efficient way to keep track of discrete quantities and numerosities. This is particularly useful if the numerosities to be represented are relatively large. An alternative way to represent quantities and numerosities is to evaluate them in terms of ‘more’ or ‘less’; this approach does not require the use of symbols or any learned system and is based on approximation. The mechanism supporting such approximations, the approximate number system, is also often referred to as ‘number sense’ see . This paNumber sense has attracted considerable attention as individual differences in this ability have been found to be associated with mathematical ability e.g. .One of the theories underlying mathematical learning is that numeracy skills partially originate from non-symbolic numerosity ability interfacing with the taught symbolic numerical system e.g. . It has Some studies, however, challenge the view of a significant relationship between symbolic and non-symbolic representation of numbers. In one study, mathematical achievement in 6-to 7-year-old children correlated with Numerical Distance Effect  in symbolic, but not in non-symbolic comparisons . Similar1.1The approximate number system is not unique to humans. Many animal species can approximate numerosities and can remember discrete number of objects and events. Basic numerical competences have been reported in social and non-social animals ants: ; mosquitAnimal evidence suggests that basic numerical competences are independent from language and are present at birth. Studies of human infants also show that this ability is preverbal. Using habituation paradigms it has been shown that babies as old as 6 months are able to distinguish between arrays of items or sequences of sounds of 4 from 8, and 8 from 16 (ratio 1:2) . Older bSuch evidence from animal and infant studies suggests that basic estimation skills involved in number sense are evolutionarily conserved. However, this does not imply that individual differences in number sense are genetic in origin. Behavioural genetic studies have shown that in almost every aspect of human behaviour and cognition, individual variation is a product of both, environmental and genetic influences . GeneticEvidence from both animals and humans suggests that for morphological traits , individual differences are under stronger genetic influence than for fitness-related traits  . In othe1.2One fundamental parameter in estimation skills, used to assess an individual's number sense acuity, is the ratio of the items in the arrays that are being compared. Discrimination of numerosities in animals, infant humans and adult humans follows the Weber Law . The WebIndividual differences in numerosity discrimination emerge early in life. Although at 6 months, infants can, on average, distinguish between ratios not finer than 1:2, one study showed that at this age babies already exhibit stable individual differences in numerosity discrimination. Further, the study found that individual differences in the ability to detect changes in numerosity at 6 months predicted this ability at 9 months independently of short-term memory skills . IndividIt is possible that individual differences in numerosity estimation in children are driven by differences in the processing of perceptual characteristics of the stimuli rather than numerical information per se. Pre-school children have difficulties in ignoring continuous, non-numerical irrelevant information  in non-symbolic numerical comparisons . For exaAdults also seem to automatically process irrelevant non-numerical information (the area occupied by the dots for example) in numerosity discrimination, e.g. . NeverthWhether individual differences in the processing of numerosity stem from perceptual processing of continuous or discrete information, accuracy in a simple task of judging which of two arrays has more items has been associated with mathematical abilities e.g. . These sResearch also suggests that numerosity discrimination is associated with measures of general cognitive ability. For example, the numerosity discrimination measure at age 16 in this study correlated with contemporaneous measures of mathematics (.33), speed of processing (.25), visuo-spatial working memory (.22), verbal and non-verbal ability (.19 and .27 respectively), language (.21), reading fluency and comprehension (.16). These relationships were explored using factor analysis. The measures are clustered into 3 factors: a verbal, a non-verbal and a perceptual dimension. Number sense loaded on the non-verbal factor, together with non-verbal ability, memory and mathematics and on the perceptual factor together with speed of processing — suggesting that variation in number sense is not only limited to variation in perceptual discrimination, but is also related to variance in other cognitive abilities. Further, after controlling for mathematics and a range of other cognitive abilities, numerosity discrimination significantly correlated with speed of processing (results are available from the authors). In the same sample, correlations of number sense (measured at age 16) with non-verbal ability, measured at different ages, remained significant (~ .15), even after controlling for longitudinal measures of mathematics. An association between numerosity discrimination and memory was found in a study of 4 year-old children . Further1.3The present study is the first large scale genetic investigation into the etiology of individual differences in number sense. We assessed number sense in 16-year-old twins and conducted genetic analyses in order to estimate the relative contribution of genetic and environmental factors to variation in number sense. The large and representative sample, which included both same-sex and opposite-sex twin pairs, also allowed the investigation of any sex differences in the etiology of the variation in number sense.Given the association of number sense with mathematical and other cognitive abilities, for which moderate to high heritabilities have been shown by previous research e.g., , it coulIn addition to estimating heritability of number sense using the twin method, we used Genome-wide Complex Trait Analysis (GCTA), to estimate heritability directly from DNA using 1.7 million DNA markers available for 1118 unrelated individuals in our sample .22.1Twins Early Development Study (TEDS) is a large longitudinal study of twins born in England and Wales in 1994, 1995 and 1996 . The samThe analyses for this investigation were carried out on the data collected from 3799 twin pairs of the 1994 and 1995 birth cohorts of TEDS when the twins were 16 years old. For the purpose of this study, twins with specific medical problems and whose English was not the first language were excluded from the analyses. The final sample consisted of 4518 twins (2259 pairs): 836 monozygotic (MZ), 733 dizygotic same-sex (DZss), and 689 dizygotic opposite-sex (DZos) pairs. The mean age for the sample was 16.6 years (SD = .28).2.2Since the wave of testing at age 10, TEDS' assessments have been mainly conducted via the Internet as it provides a cost-effective, quick and reliable method to collect data in such a large and widespread sample. The advantages and disadvantages of Internet testing have been reviewed in Number Sense Task, that assessed the ability to discriminate numerosities, was embedded in the web-based battery of the TEDS assessment at age 16. The battery could be accessed online at the TEDS website (www.teds.ac.uk) using a unique anonymized log-in. The task was an online implementation of the test described in The 2.3Standard quantitative genetic analyses were used to estimate genetic and environmental influences on individual differences in number sense . SimilarAnalyses of twin data are carried out on the residuals of standardized scores corrected for average effects of age and sex . This is2.4http://www.R-project.org; Although the results of twin analyses can easily be gleaned from the simple twin correlations, structural equation model-fitting tests alternative models and provides confidence intervals for estimates of the proportion of variance within a trait that can be attributed to genetic (A), shared (C) and non-shared (E) environmental influences. In twin model fitting, the estimate for non-shared environment incorporates the measurement error — as unsystematic error can only contribute to the twins' dissimilarity in the measured trait. We employed standard model-fitting procedures. Parameters were estimated using OpenMx software  conducte2.5Alternate models can be used to estimate two types of sex differences in the A, C and E parameters . Quantitg) for DZ same-sex pairs  is 0.5 as DZ twins share half of their segregating genes. If different genes affect males and females, rg for DZos will be less than 0.5. If sex differences are quantitative, the same genetic factors influence males and females, therefore the rg for DZos will be 0.5, but A, C and E estimates for males and females will be significantly different. The same logic applies to the coefficient indicating relatedness due to shared environmental factors (rc), which should be equal to 1 as twins in the same family share the same environments. It is not possible to estimate rg and rc at the same time , so qualitative and quantitative differences in genetic influences have to be modelled separately from shared environmental influences.To investigate qualitative and quantitative gender differences, ACE parameters and their 95% confidence intervals were estimated with the data divided into MZ male (MZm), MZ female (MZf), DZ male (DZm), DZ female (DZF), and DZ opposite-sex (DZos) twin pairs. Quantitative sex differences are tested by running a model that estimates A, C, and E parameters separately for males and females, this full model is then compared with reduced models that equate A, C, and E parameters for males and females. Qualitative sex differences are tested by comparing the variance and covariance for DZos and same-sex DZ (DZss). The genetic relatedness coefficient  using the massive information available from a large sample of individuals, each compared pair by pair with every other individual in the sample.GCTA genetic similarity is not only limited to the additive effects of genotyped SNPs themselves but also includes unknown causal variants to the extent that they are correlated with the SNPs. Mendel's second law of inheritance is that genes  are inherited independently , but Mendel did not know that genes can be on the same chromosome, in which case they are not inherited independently (linkage disequilibrium). This violation of Mendel's second law is complicated by the fact that during meiosis, chromosomes from the mother and father recombine on average once per meiosis, which means that, in the population, genes on the same chromosome are separated by this process of recombination to the extent that the genes are not close together on the chromosome. GCTA provides a lower-limit estimate of heritability because it misses genetic influence due to causal variants that are not highly correlated with the common SNPs on genotyping arrays.Genetic effects on a trait may not just derive from the simple sum of independent genetic actions, they may stem from more complex gene–gene interactions. One of the assumptions of the twin method is that the variance explained by genetic influences is attributed to additive genetic effects. In practice, the method captures both additive and non-additive genetic effects because the DNA sequence of identical twins is virtually identical and thus they share all genetic effects including non-additive effects , corrected for age and sex and scores outside +/− 3 standard deviations were considered as outliers and excluded. The transformation improved normality of both variables. However, even after transformation, Weber Fraction scores did not fully meet assumptions of normality as skewness was 1.09 (SE = .05) and kurtosis 1.27 (SE = .10). Although maximum likelihood estimation assumes normality of the data, the method has been shown to be robust when assumptions of normality are violated c.f. . We alsoDescriptive analyses run on the other half of the sample yielded highly similar results (available from the authors).The model-fitting results confirmed these interpretations . For botAs shown in The results of the Sex-Limitation model fitting are shown in Genome-wide Complex Trait Analysis (GCTA) yielded a non-significant estimate of zero heritability for number sense. Because the sample size was relatively small for GCTA analysis, the standard error of estimate was large (.29). Nonetheless, this GCTA analysis provides support for the relatively low twin study estimate of heritability for number sense.4We performed the first large-scale genetically sensitive analysis on number sense and found that individual differences in this ability at age 16, as indexed by a measure of accuracy in numerosity discrimination and by the Weber Fraction, were only modestly influenced by genetic factors. Most of the variance was explained by non-shared environment (.68 for Weber Fraction and .65 for the accuracy scores). The modest estimate of heritability from the twin study was supported by a zero heritability estimate from the GCTA analysis. Because GCTA estimates are limited to the additive effects of common SNPs included on DNA arrays, GCTA estimates are typically about half of the estimates from twin studies. For example, another TEDS study estimated heritabilities for verbal ability as .40 and .26 for twin and GCTA analyses, respectively .As number sense is linked to other cognitive abilities, which have been found to be at least moderately heritable, its modest heritability may come as a surprise. However, as indicated earlier, evolutionarily useful traits are not necessarily heritable. Fear for example is considered an evolutionary useful trait; but individual differences in acquisition, habituation and extinction of fear in the presence of stimuli such as snakes and spiders, are mostly explained by environmental influences . In termHeritability is a descriptive statistic specific to a particular age and population . For thiOne of the implications of the large environmental component of individual differences is that higher levels of accuracy in estimation of numerosity may be achieved through training or less focused experience. One study involving 6 month-old infants showed that when babies were simultaneously presented with a congruent visual and auditory stimulus they were able to discriminate numerosities with a ratio usually present in 9 month-old infants . One expAlthough we need to understand in more depth the mechanisms through which the environment acts upon numerosity discrimination skills, there are some studies showing how estimation of numerosity skills can be manipulated through exposure to numerical material. It has been suggested that playing numerical board games gives children familiarity about numbers and improves their estimation of numerical magnitudes . HoweverStudies on artificial learning provide further evidence that individual differences in numerosity skills similar to number sense can be taught. Neural network models can be modelled to detect numerosity from visual inputs , with thOur results add a novel perspective on a current debate in the mathematical literature. One theory proposes that the severe mathematical disability of Developmental Dyscalculia emerges from difficulties in numerosity processing. This occurs even in the absence of general cognitive impairments . It has Evidence shows that variation in number sense may also arise under the influences of general cognitive development e.g. . It is pStudies on number sense have reported no mean sex differences in this ability, with the exception of one study that revealed marginal male advantage in numerosity discrimination in 4 year-old children . As ment5The two methods employed in this study, the twin method and the GCTA analysis, showed that individual differences in numerosity estimation are only modestly influenced by genetic factors. One interpretation of these results is that number sense has evolved as crucial for survival . SimilarSex differences in number sense are minimal, both descriptively and etiologically: in addition to finding no mean sex difference, we also find that the same genetic factors influence individual differences in number sense skills in males and females equally.As number sense measured at 16 is associated with general intelligence, we plan to investigate the etiology of the links between number sense, general intelligence, and other cognitive abilities such as spatial ability and other learning abilities such as mathematics. As the environment is a major source of individual variation in number sense, it is particularly important to understand its role in the covariation among these traits and to identify specific environmental factors involved.The authors have declared that no competing interests exist.


Two hallmarks of this complexity are epistasis, meaning gene-gene interaction, and pleiotropy, in which one gene affects multiple phenotypes. Understanding the genetic architecture of complex traits requires addressing these phenomena, but interpreting the biological significance of epistasis and pleiotropy is often difficult. While epistasis reveals dependencies between genetic variants, it is often unclear how the activity of one variant is specifically modifying the other. Epistasis found in one phenotypic context may disappear in another context, rendering the genetic interaction ambiguous. Pleiotropy can suggest either redundant phenotype measures or gene variants that affect multiple biological processes. Here we present an R package, R/cape, which addresses these interpretation ambiguities by implementing a novel method to generate predictive and interpretable genetic networks that influence quantitative phenotypes. R/cape integrates information from multiple related phenotypes to constrain models of epistasis, thereby enhancing the detection of interactions that simultaneously describe all phenotypes. The networks inferred by R/cape are readily interpretable in terms of directed influences that indicate suppressive and enhancing effects of individual genetic variants on other variants, which in turn account for the variance in quantitative traits. We demonstrate the utility of R/cape by analyzing a mouse backcross, thereby discovering novel epistatic interactions influencing phenotypes related to obesity and diabetes. R/cape is an easy-to-use, platform-independent R package and can be applied to data from both genetic screens and a variety of segregating populations including backcrosses, intercrosses, and natural populations. The package is freely available under the GPL-3 license at  PLOS Computational Biology Software ArticleThis is a Advances in genomic and phenotypic technologies have vastly expanded the scope of genetic studies. Multidimensional phenotyping and high density genotyping are routinely combined to produce highly detailed views of biological systems. Translating these data into predictive models of health and disease will require new analytical methods to understand how genetic variants combine to influence multiple phenotypes. Here we present an R-based software application to use the information in multiple phenotypes to interpret statistical epistasis. The method derives models of genetic interactions as quantitative variant-to-variant and variant-to-phenotype influences. By fitting models to multiple phenotypes simultaneously, the method constrains possibilities for how individual variants interact to affect each phenotype. The result is a directed network of quantitative, variant-to-variant influences that represent specific levels of suppression or enhancement . The appr|≤0.8) are ideal.R/cape is designed to detect and interpret pleiotropy and statistical epistasis. The user begins by loading their data in R/qtl format To maximize linear independence of phenotypes, R/cape by default performs singular value decomposition (SVD) on two or more selected phenotypes to derive eigentraits (ETs). R/cape includes functions to visualize the percent variance accounted for by each ET as well as their individual contributions to the phenotypes, thereby aiding the selection of ETs on which subsequent analysis should be performed. After ET selection, the user performs a genome-wide single-variant scan to assess the association of each individual marker. Since very strong main-effect markers can obscure interactions between other marker pairs, the user can assign covariates for the pair-scans that follow. The single-variant scan can also be used to specify marker pairs to test for interactions. The default setting is to test all possible pairs that are not in linkage disequilibrium. However, for data sets with a large number of markers, a subset of loci can be selected based on single-locus significance. Pair-scans are performed for each ET by multivariate linear regression, with an intercept, covariates, main effects, and interaction term for each pair of markers. Interaction (epistasis) coefficients from each ET can be plotted as shown in N interaction coefficients are dimensionally reduced to two variant-to-variant influences . Phenotype contributions to each ET are shown in R/cape analysis was then performed on the selected ETs. Two markers were excluded due to full redundancy with adjacent markers and individual pairs with substantial linkage disequilibrium (LD) were not tested, defined as fewer than six instances of the four possible pair-wise genotypes at the two loci. These pairs are marked by gray dots in the pairscan results . We idenThe interaction between Chromosomes 1 and 12 is illustrative of the R/cape strategy . AlthougThe interacting loci we detected were distinct from those reported in the original analysis By simultaneously anlayzing multiple phenotypes, R/cape can reveal complex interactions between genotype and phenotype not seen when examining phenotypes in isolation. While the original study was based on detecting loci involved in the composite phenotype “diabesity,” R/cape parses how genetic loci both jointly and distinctly affect diabetes and obesity. This is of interest to geneticists because, although these traits are correlated, not all obese individuals are diabetic and not all diabetic individuals are obese Together these findings illustrate how R/cape is designed to find interactions that simultaneously model all phenotypes under the assumption that interactions between variants across multiple contexts represent a single underlying interaction network. We note that in a large-scale, highly-powered study based on genetic interaction screens R/cape analysis yielded significant interactions for 55% of the interactions identified as significant using standard epistasis analysis http://cran.r-project.org/web/packages/cape. The R platform and all R dependencies are similarly available from CRAN.The R/cape package is freely available under the GPL-3 license at the Comprehensive R Archive Network (CRAN), The time required to perform an R/cape analysis depends on multiple parameters including the number of markers, the number of phenotypes, and the amount of linkage between markers since linked marker pairs are not tested. Overall processor time is proportional to Future additions to R/cape will include advanced graphics capabilities and additional features to manage populations. We intend to augment the current graphic with scalable graphs. Application of the software to genome-wide association studies is currently possible but requires quantitative data and care in managing population structure. Corrections for sample relatedness are under development. Finally, we will expand allele encoding to address more than two alleles at each locus, based on precision genotyping or inferred haplotype probabilities. This will enable application of R/cape to data from the Collaborative Cross In conclusion, R/cape is an efficient tool to represent genetic complexity in terms of models of variant action. The program will aid in the interpretation of epistasis without prior knowledge by providing a fast overview and hypothesis generation tool.Software S1R/cape software package. Source code, documentation, test data, and an instructive vignette for R/cape.(GZ)Click here for additional data file.