The relative proportion of additive and non-additive variation for complex traits is important in evolutionary biology, medicine, and agriculture. We address a long-standing controversy and paradox about the contribution of non-additive genetic variation, namely that knowledge about biological pathways and gene networks imply that epistasis is important. Yet empirical data across a range of traits and species imply that most genetic variance is additive. We evaluate the evidence from empirical studies of genetic variance components and find that additive variance typically accounts for over half, and often close to 100%, of the total genetic variance. We present new theoretical results, based upon the distribution of allele frequencies under neutral and other population genetic models, that show why this is the case even if there are non-additive effects at the level of gene action. We conclude that interactions at the level of genes are not likely to generate much interaction at the level of variance. Genetic variation in quantitative or complex traits can be partitioned into many components due to additive, dominance, and interaction effects of genes. The most important is the additive genetic variance because it determines most of the correlation of relatives and the opportunities for genetic change by natural or artificial selection. From reviews of the literature and presentation of a summary analysis of human twin data, we show that a high proportion, typically over half, of the total genetic variance is additive. This is surprising as there are many potential interactions of gene effects within and between loci, some revealed in recent QTL analyses. We demonstrate that under the standard model of neutral mutation, which leads to a U-shaped distribution of gene frequencies with most near 0 or 1, a high proportion of additive variance would be expected regardless of the amount of dominance or epistasis at the individual loci. We also show that the model is compatible with observations in populations undergoing selection and results of QTL analyses on F2 populations. Complex phenotypes, including quantitative traits and common diseases, are controlled by many genes and by environmental factors. How do these genes combine to determine the phenotype of an individual? The simplest model is to assume that genes act additively with each other both within and between loci, but of course they may interact to show dominance or epistasis, respectively. A long standing controversy has existed concerning the importance of these non-additive effects, involving both Fisher An understanding of the nature of complex trait variation is important in evolutionary biology, medicine and agriculture and has gained new relevance with the ability to map genes for complex traits, as demonstrated by the recent burst of papers that report genome-wide association studies between complex traits and thousands of single nucleotide polymorphisms (SNPs) VG can be partitioned into additive (VA), dominance (VD), and a combined epistatic component (VI), which itself can be partitioned into two locus  and multiple locus components  The genetic variance VAA/16 for half-sibs), so estimates have high sampling error and there is little power to distinguish VA from, say, VAA. Selection, assortative mating, and non-genetic covariances also confound estimates. Consequently, there are few accurate estimates of non-additive variance components but there is indirect evidence. For instance, a narrow sense heritability value (h2 = VA/VP) of one-half, typical for many traits, implies that dominance, all the vast number of epistatic components, and the environmental component, collectively contribute no more than VA. Similarly if the heritability is only a little less than the repeatability (the phenotypic correlation of repeated measures), all non-additive genetic variances and the permanent environmental variance together comprise this small difference. With these caveats we summarise data of various types.In an extremely large data set with very many different kinds of relationships present, it is possible in principle to partition variation into many components using modern statistical methods such as residual maximum likelihood The extensive data on experimental organisms show a range of heritability, higher for morphological than fitness associated traits, averaging as follows There have been extensive estimates of heritability for traits of livestock. For example, for beef cattle, these averaged: post-weaning weight gain 0.31, market weight for age 0.41, backfat thickness 0.44 VA contributes almost all of VG and indeed almost all of VP. For bristle number in Drosophila melanogaster, the phenotypic correlation between abdominal segments, which, assuming they are influenced by the same genes, estimates VG/VP, is only a little higher than the heritability, indicating that VA/VG∼0.8 VA/mean) is higher for fitness than morphological traits There are a number of cases where it can be shown directly that There are rarely good direct estimates of epistatic or dominance variance because these variance components are usually estimated from full-sibs and therefore confounded with the common environment shared by full sibs. However, if the heritability is high, the space for them is limited.Experiments on inbreeding depression provide some evidence on the importance of non-additive effects. Inbreeding depression implies directional dominance in gene effects but, for a given rate of inbreeding depression, as the number of loci increases and the gene frequencies move toward 0 or 1.0, the dominance variance decreases towards zero. Consequently, the importance of inbreeding depression for traits related to fitness is not evidence that the dominance variance is large. The observed linearity of inbreeding depression with inbreeding co-efficient is easiest to explain with directional dominance but not with DD or higher order epistatic effects because these would cause non-linearity unless they happened to exactly cancel each other out.VG. The classical twin design of samples of monozygotic (MZ) and dizygotic (DZ) twin pairs has been used extensively to estimate variance components for a wide range of phenotypes in human populations. The primary statistics from these studies are the correlations between MZ pairs (rMZ) and between DZ pairs (rDZ). If twin resemblance due to common environmental factors is the same for MZ and DZ twins then rMZ>rDZ implies that part of the resemblance is due to genetic factors and rMZ>2rDZ implies the importance of non-additive genetic effects. Conversely, rMZ<2rDZ implies that common environmental factors cause some of the observed twin resemblance. Sophisticated variance component partitioning methods to estimate components of additive, non-additive and common environmental effects are used widely In contrast to studies of sibs and more distant relatives, identical twins can provide estimates of rMZ−2rDZ across all 86 phenotypes is shown in We summarised the MZ and DZ correlations for a wide variety of phenotypes from published twin studies from a single productive laboratory in Australia (genepi.qimr.edu.au). The criteria were that each study must have more than 100 MZ and more than 100 DZ pairs and that the study subjects were Australian twins. For non-continuous traits, studies were included only if they reported polychoric or tetrachoric correlations. In total, 86 phenotypes qualified of which 42 were clinical measures of quantitative traits . The MZ and DZ correlations are summarised in In view of the apparent conflict between the observations of high proportions of additive genetic variance  and the recent reports of epistasis at quantitative trait loci (QTL) p) of the mutant allele is f(p)∝1/p, i.e. approximately L-shaped p)∝1/p+1/(1−p) = 1/[p(1−p)]. As we shall use it often, we define the ‘U’ distribution explicitly by this formula. For loci at which the mutants are generally deleterious, the frequency distribution will tend to be more concentrated near p = 0 or 1 than for this neutral reference point. As another simple reference point we use the uniform distribution, f(p)∝1, 1/(2N) ≤ p ≤ 1−1/(2N), with N the population size. This approximates the steady state distribution of a neutral mutant gene which has been segregating for a very long time p = 0.5, as in populations derived from inbred crosses, and is the extreme case of central tendency of gene frequency.Genetic variance components depend on the mean value of each genotype and the allele frequencies at the genes affecting the trait These analyses assume a gene frequency distribution which is relevant to no selection. For a more limited range of examples we consider the impact of selection on the partition of variance. We consider a limited range of genetic models, some simple classical ones and others based on published models of metabolic pathways or results of QTL mapping experiments.Uniform: f(p) = 1, assuming N is sufficiently large that the discreteness of the distribution and any non-uniformity as p approaches 1 or 0 can be ignored, i.e. integrated over 0 to 1. This and the ‘U’ gene frequency distributions are, for simplicity, assumed to be continuous.Neutral mutation model (‘U’): f(p)∝1/[p(1−p)]. To standardise the distribution, with population size N assumed to be large, note thatK∼ln(2N).Thus p for a specific model of the gene frequency distribution. For multiple locus models the distribution of all loci is assumed to be identical and there is no linkage disequilibrium. We focus on the contribution of additive genetic variance (VA) to genotypic variance (VG).Genetic variance components are obtained by integration of expressions for the variance as a function of a, d and −a, respectively  = a2/3 +d2/15 and E(VD) = 2d2/15, giving E(VA)/E(VG) = 1−2d2/(5a2+3d2).Hence E, the integrals simplify to E(VA)∼(a2+d2/3)/K, E(VD)∼d2/(3K) and E(VA)/E(VG) = 1−d2/(3a2+2d2).For the A simple additive × additive epistatic model has these genotypic values:p and of C is q, with linkage equilibrium:Assuming the frequency of B is M = 2a[pq+(1−p)(1−q)]Mean = The average effect of substitution of allele B is given by VA = 2a2[p(1−p)(1−2q)2+q(1−q)(1−2p)2] = a2(pH+qH−4pHqH), where H is heterozygosityαα)BC = ¼ d2M/dpdq = a.The AA epistatic effect is given by (VAA = 4a2p(1−p)q(1−q)a2 = a2pHqH and GV = a2(pH+qH−3pHqH),Hence Uniform: simple integration gives E(VA) = 2a2/9, E(VAA) = a2/9, E(VG) = a2/3‘U’: VAA) = a2/(4K). Hence E(VA)/E(VG) = (2−4/K)/(2−3/K) = 1−1/(2K−3), which → 1 for large K. The residue, if any, is VAA.Similarly E of loci (i), the genotypic value is a except for the multiple ‘recessive’ homozygote, and for one locus it is complete dominance.For an arbitrary number L2−(¼)L2], VA = a2L(½)L−14 and VA/VG = 2L/(2L2−1). For two loci, VA/VG = 4/15.For Uniform:VA)/E(VG) = 9/16 and declines to 0 as L increases.For two loci, E /E(VG) → 4/5 and for very many loci E(VA) /E(VG) → 0For large ip = 0.5, VA/VG = 0.56; with the uniform distribution, E(VA)/E(VG) = 2/3; and with the ‘U’ distribution Another simple epistatic model involving all components is the following:VA and VG calculated as functions of the gene frequencies p and q. Bivariate numerical integration was undertaken using Simpson's rule by computing e.g. VAf over an (m+1) × (m+1) grid of equally spaced p and q values, taking m = 210 or higher power of 2 as necessary for adequate convergence. Results were computed for some models of metabolic pathways For two-locus models in which the genotypic values were not functions of simple parameters, the genotypic values were entered as data, and a, d and −a, respectively, VA = 2p(1−p)[a+d(1−2p)]2 and VD = 4p2(1−p)2d2. For d = a, i.e. complete dominance of C, VA = 8p(1−p)3a2 and VD = 4p2(1−p)2a2 and thus: at p = 0.5, VA = (2/3)VG; if the dominant allele is rare (i.e. p → 0), VG → 8p and VA/VG → 1, and if it is common, VG → 4p2 and VA/VG → 0. Note, however, that VG and VA are much higher when the dominant allele is at low frequency, e.g. 0.1, than are VG and VD when the recessive is at low frequency, e.g. p = 0.9. Even for an overdominant locus (a = 0), all genetic variance becomes additive at extreme gene frequencies. Considering now expectations (E) over the frequency distributions, let η2 = E(VA)/E(VG), an equivalent to narrow sense heritability if VE = 0. For the ‘U’ distribution, η2 = 1−d2/(3a2+2d2) and for the uniform distribution, η2 = 1−2d2/(5a2+3d2). Hence, for a completely dominant locus, η2 = 0.8 and η2 = 0.75 respectively; whereas VA/VG = 0.67 for p = 0.5. In summary, the fraction of the genetic variance that is additive genetic decreases as the proportion of genes at extreme frequencies decreases . With linkage equilibrium, VA/VG = 1−pHqH/[pH+qH−3pHqH], where the heterozygosities are pH and qH at loci B and C. Thus VA/VG → 1 if either locus is at extreme frequency (i.e. p or q → 0 or 1), and equals 0 when p = q = 0.5. If p = q, for gene frequencies 0.1, 0.2, 0.3 and 0.4, VA/VG = 0.88, 0.69, 0.43 and 0.14. For the uniform distribution η2 = 2/3, and for the ‘U’ distribution, the variances are a function of the population size, because more extreme frequencies are possible at larger population sizes. Thus η2 = (2−4/K)/(2−3/K), where K = ln(2N), so η2 → 1 for large K. Any residue is VAA.The genotypic values (see Theory section) for the simple AA model for double homozygotes BBCC and bbcc are +2VG is accounted for by VA.These two examples, the single locus and A × A model, illustrate what turns out to be the fundamental point in considering the impact of the gene frequency distribution. When an allele (say C) is rare, so most individuals have genotype Cc or cc, the allelic substitution or average effect of C vs. c accounts for essentially all the differences found in genotypic values; or in other words the linear regression of genotypic value on number of C genes accounts for the genotypic differences see , p 117. VA for the ‘U’ distribution for a few loci but the proportion of the variance which is additive genetic declines as the number increases. With many loci, however, such extreme models do not explain the covariance of sibs (i.e. any heritability) or the approximate linearity of inbreeding depression with inbreeding coefficient, F, found in experiments With the ‘U’ distribution, most genes have one rare allele and so most variance is additive. Further examples  illustraVG that is accounted for by VA is large  have been published in which particular pairs (or more) of loci have been identified to have substantial epistatic effects VA), E(VG) and the ratio E(VA)/E(VG). The quantity actually observed is VA/VG = ΣiViA/ΣViG where the expression denotes the sums over loci (i) of the additive and total genetic variance contributed by each in the absence of epistasis or linkage disequilibrium, or in the presence of these, sums over relevant sets of loci. As, for any locus, or for their sum, in general E(VA/VG) ≠ E(VA)/E(VG), we need to consider the relevance of the quantities calculated. Whilst it would be possible to obtain approximations using statistical differentiation The formulae we have given have been for the quantities E(a for heterozygotes (‘pure’ overdominance), or only BC and bc haplotypes, with genotypic values 2a for homozygotes and a for heterozygotes (‘pure’ underdominance). In either case variances are the same as for the dominance case with a = 0. Thus LD would lead to attribution of real epistatic variance to additive or dominance variance, and would exacerbate the results obtained from discussions of gene frequency distribution.In this analysis we have assumed there is Hardy-Weinberg equilibrium (HWE) and linkage equilibrium among the loci. As departures from HWE are transient with random mating, they can be ignored, but LD can persist, and hence the estimated effects at locus C depend on those fitted at B and vice versa. The effect of LD is to reduce the number of haplotypes that segregate in the population so what would be epistatic variance becomes additive or dominance variance. For example, consider the A × A model and complete LD, i.e. equal frequencies at B and C loci and both loci segregating but with only two haplotypes present. Then only Bc and bC haplotypes are present, and genotypic values are 0 for homozygous classes and k(k−1)/2 genotypic values at a k allelic locus, and many further epistatic terms, so we consider two extreme cases. If the alleles all have similar effects, for example due to a knock-out, the effective mutation rate is increased, but it would require very many such sites for the distribution of frequencies of the trait alleles to differ greatly from proportionality to 1/[p(1−p)]. Such segregation of multiple alleles will be more common in large populations, where in any case the frequency distribution is most extreme, and so the impact is unlikely to be large. A second case is where all alleles have different effects and dominance interactions. Any allelic substitution then produces a change in the mean and so additive variance is present and for example, contributes more VA than does the overdominance model at p = 0.5.In these models we have considered solely biallelic loci, appropriate for low mutation rates. Multiallelic loci, in terms of their effects on the trait, can arise from mutations at different structural or control sites. Predictions are complicated by the need to consider The analysis we have given for estimating effects of dominance and epistasis is for the classical method using simple averages over genotypes weighted by their frequencies, which are the least squares estimates in the balanced case and the basis for the analysis of variance VA/VG to be small also cause VG to be small. Consequently, when VA and VG are summed over a full range of frequencies, VA/VG is large. This conclusion is dependent on the distribution of gene frequencies being symmetrical, so that cases with large VG and large VA/VG are as common as cases with small VG and small VA/VG. The impact of selection will depend on how it acts on the trait or traits analysed and also on other aspects of fitness, so we need to consider whether the findings are robust to selection.The ‘U’ and indeed uniform gene frequency distributions are limiting cases applying in the absence of selection on loci affecting the quantitative trait. The results for a wide range of models can be summarised as follows: gene frequencies that cause Stabilising selection on the trait, such that individuals with phenotype closest to an optimum are most fit, leads to maintenance of the population mean at or close to the optimum, so that mutants are at a disadvantage if they increase or decrease trait values. Consequently the gene frequency distribution is still broadly U-shaped, but with much more concentration near 0 or 1 Under the neutral mutation or stabilising selection models where gene frequency distributions have extreme U shape, subsequent directional selection will lead to either rapid fixation or increase to intermediate frequency of genes affecting the trait. Even if the distribution of allele frequencies is initially symmetric, a net increase in variance over generations might thus be expected N) 100 and selective values (s) of the homozygote of 0.05 (Ns = 5), but the qualitative result is not affected by using weaker or stronger selection. Deleterious, recessive mutations show the lowest VA/VG but even here it is 0.44 and these cases also show the lowest total variance. Consequently, in a trait affected by a mix of genes with varying types of gene action, VA/VG is likely to be well above 0.5.Other types of selection do lead to an asymmetrical distribution of allele frequencies because the unfavourable allele will typically be at a low frequency. We have considered the case of genes whose effect on both the trait measured and on fitness shows complete dominance. Thus recessive and dominant favourable and unfavourable mutants were considered, and their expected contribution to variance computed during their lifetime to fixation or loss, using transition matrix methods. Results are given in Thus if the highest and lowest genotypic values correspond to multiple homozygous classes, it is clear that a high proportion of the variance is expected to be additive genetic even with selection. The potential exceptions occur when there is a maximum at intermediate frequencies, such as with an overdominant locus or some of the cases shown in VA for recessives at low frequency VA/VG. For example, for the additive × additive two locus model, the ratio of variances at inbreeding level F in terms of values at F = 0 is VA(F)/VG(F) = (VA+4FVAA)/(VA+VAA+3FVAA) for any gene frequency (using results of VA(F)/VG(F) depends on gene frequency. This occurs because the bottleneck leads to the dispersal of gene frequencies and the reduction in mean heterozygosity, so for the AA model, if frequencies are initially intermediate (e.g. 0.5) there is a substantial increase in VA/VG, whereas if frequencies initially follow the ‘U’ distribution, there is little VAA initially, total variance falls and the level of dispersion and VA/VG do not increase appreciably. Indeed, for a population that starts with the gene frequency distribution U-shaped, the loss of heterozygosity is due to fixation. Among the genes that remain segregating the distribution of gene frequencies flattens considerably, and in the absence of new mutation approaches the uniform distribution which has a lower ratio of VA/VG than the ‘U’ distribution. However, despite this, VAA declines faster than VA because, as loci become fixed, the number of pairs of segregating loci declines faster than the number of segregating loci. Thus it is not obvious what effect the bottlenecks in livestock, laboratory or human populations have had on the ratio VA/VG. We suspect it has not been large because, if a large reduction in heterozygosity had occurred, these populations would show low genetic variance and there is no indication that this is the case. In any case, the results show that the conclusion that most genetic variance is additive is fairly robust to assumptions about the distribution of gene frequencies, for instance the ‘U’ and uniform distributions both lead to qualitatively the same conclusion.The theoretical analysis has been undertaken for large populations but much of the experimental data comes from livestock, laboratory animals and humans, all of which have experienced bottlenecks of reduced effective population size. As has been much explored, bottlenecks of population size are likely to change the proportion of variation that is additive, and for example to increase levels of 2 and later inter se generations, however, so it is difficult to partition variation between single locus and epistatic components  have been summarised h2 were 0.19 for open-pollinated populations, 0.23 for synthetics from recombination of many lines, 0.24 for F2 populations, 0.13 for variety crosses and 0.14 for composites. Estimates of VA/VG  or recessive (high VA). The equivalent case for epistasis is that all genotype combinations except one is favourable (low VA) vs. only one genotype combination is favourable (high VA).The theoretical models we have investigated predict high proportions of additive genetic variance even in the presence of non-additive gene action, basically because most alleles are likely to be at extreme frequencies. If the spectrum of allele frequencies is independent of which are the dominant or epistatic alleles, VA/VG. However, neither the empirical evidence nor the theory supports this expectation. There seems to be substantial additive genetic variance for fitness associated traits VA/VG is one-half or more. In agreement with this, when the life history of deleterious, recessive mutants was modelled, VA/VG was found to be 0.44 (VD, in non-inbred populations.If genetic variation in traits associated with fitness is due almost entirely to low frequency, deleterious recessive genes which are unresponsive to natural selection, these traits would show low  be 0.44 , basicalVA is important so we should be able to detect and identify alleles with a significant gene substitution effect within a population. Such variants have been reported from genome-wide association studies in human population and the genes have intermediate frequency. Thus we expect that the success in replicating reported epistatic effects will be even lower than it is for additive or dominance effects, both because multi-locus interactions will be estimated less accurately than main effects and because they explain a lower proportion of the variance. Finally, if epistatic effects are real, gene substitution effects may vary widely between populations which differ in allele frequency, so that significant effects in one population may not replicate in others.We believe we have a plausible gene frequency model to explain the minimal amounts of non-additive genetic and particularly epistatic variance. What consequences do our findings have? For animal and plant breeding, maintaining emphasis on utilising additive variation by straightforward selection remains the best strategy. For gene mapping, our results imply that


LRTS) for case-control sampling design based on the underlying continuous quantitative measurement, and to assess its power and relative efficiency (as compared to the chi-square test of independence on CNP counts). The sample size and power formulas of both methods are given. For the latter, the CNPs are classified using the Bayesian classification rule. The LRTS is more powerful than this chi-square test for the alternatives considered, especially alternatives in which the at-risk CNP categories have low frequencies. An example of the application of the LRTS is given for a comparison of CNP distributions in individuals of Caucasian or Taiwanese ethnicity, where the LRTS appears to be more powerful than the chi-square test, possibly due to misclassification of the most common CNP category into a less common category.Recent studies suggest that copy number polymorphisms (CNPs) may play an important role in disease susceptibility and onset. Currently, the detection of CNPs mainly depends on microarray technology. For case-control studies, conventionally, subjects are assigned to a specific CNP category based on the continuous quantitative measure produced by microarray experiments, and cases and controls are then compared using a chi-square test of independence. The purpose of this work is to specify the likelihood ratio test statistic ( CCL3L1 gene [MIM 601395] is associated with HIV/AIDS susceptibility FCGR3 genes [MIM 146740] is a determinant of susceptibility to immunologically mediated glomerulonephritis Large-scale copy number polymorphisms (CNPs) are a recently discovered feature of human genomic architecture Case-control genetic association designs can be a powerful way to map disease susceptibilty genes, particularly for diseases with smaller effect sizes Compared to SNP genotyping technologies, procedures for calling CNPs are less developed and less accurate One reason for the relative difficulty of CNP classification is that such classification is determined by relative intensity of a signal at a probe (or probe sets). In contrast, the two alleles of a SNP have two distinct nucleotides that can be represented by two distinct probes. Moreover, for multi-allelic CNPs, only the total number of copies (or categories) and not the alleles are observed for each individual. As an example, for a CNP locus of three alleles, with 1 copy, 2 copies and 3 copies respectively and probe intensity proportional to the number of copies, an intensity observation of 4 can be a genotype of 2/2 copies or a genotype of 1/3 copies.i, 1≤i≤4 comes from a univariate normal distribution with mean i and variance i as having CNP category j≠i is much higher. It has been reported that the chi-square test of independence loses power as the misclassification rate increases Consider the pictorial examples in Is there a more  powerful method of using CNP data when testing for association with a complex trait than the usual chi-square test of independence?An additional concern is that the CNP category that increases risk may occur with low frequency, as is often the situation with Mendelian diseases LRTS) comparing the mixing proportions of cases and controls estimated from the underlying quantitative measures for CNPs. Rather than assign classifications to each individual's CNP, we perform a test of association on the CNP quantitative measure. We present an analytic solution to computation of power and sample size calculations for genetic association with CNP quantitative measures. We then calculate the efficiency of the chi-square test of independence using Bayesian classification compared to the LRTS to examine which test statistic has greater power for a wide range of trait specifications. By efficiency, we mean the ratio of sample size requirements for the chi-square test of independence and LRTS, respectively, for a fixed power and type I error rate. Finally, we demonstrate the use of the LRTS for differences in the mixing proportions of the CNP categories between two ethnic groups for a CNP with relevance to a genetic disease.To answer this question, we propose use of the likelihood ratio test statistic  or case (α = 2) status.n1, and cases n2, with N = n1+n2 and The number of controls is d = The number of CNP categories; the subscript i indexes the category, 1≤i≤d.f = The probability density function (pdf) of the continuous random variable X = x, conditional on the CNP category. This pdf is a function of the parameters iθ and η, where η is a parameter that is constant for all component distributions. For example, if f is a normal pdf, then θi is the mean and η is the variance.αip, 1≤i≤d, are the proportions of CNP category i in the α affection status class . Under the null hypothesis, pi1 = pi2.pi0 = Q1pi1+Q2pi2. Since, under the null hypothesis, pi1 = pi2, then pi1 = pi2 = pi0, under the null.iq = The CNP category frequencies in the population from which cases and controls are drawn.f. In the examples used here, iθ is the mean of the CNP category i distribution. Also, in the efficiency calculations reported later, iθ+1−iθ = 1, i = 1,…, d−1. The separation αX is given by d is known and equal in both cases and controls. Given a CNP category i, the underlying pdf f(·) is the same for cases and controls. When f is a normal distribution, we specify that the variance (η) is equal across all CNP categories i and affection statuses α. While these specifications are not critical for performing power and sample size calculations, they may be advantageous when performing mixture analyses of real data. For instance, there may be convergence problems for the computed maximum likelihood of a univariate normal mixture if one allows the category variances to be unequal. Methods such as those proposed by Hathaway The probability density function of the random variable L0 and L1 are calculated by using the maximum likelihood estimates (MLEs) of the parameters.The likelihood function under the null hypothesis is given by:LRTS applied directly to the CNP quantitative measures for cases and controls; and (2) the chi-square test of independence applied to 2×d tables after the CNP quantitative measures have been classified into one of d categories for cases and controls using a Bayesian classification rule (see section immediately following). The LRTS (1) is defined asIn this work, we consider two test statistics: (1) the x is assigned to CNP category i if and only if pi0 = Q1pi1+Q2pi2, (defined above – Notation). For an example with d = 3 copy number categories and a normal CNP category distribution, application of the Bayes rule yields:To categorize CNP quantitative measures into a CNP category, we consider a classification formula based on Bayes rule x is placed in the left-most component if x<min,x is placed in the middle component if γ12<x<γ23,x is placed in the right-most component if max<x,i<j≤3. In applications,  are estimated using the MLEs of the parameters.where iθ+1−iθ = 1 unit apart. Separations are fixed to be n1( = n2) = 200 or 500, and mixing proportions We perform simulation studies to verify the accuracy of the asymptotic null distribution of the chi-square test of independence applied to CNP counts after classification using the Bayesian classification rule (described above). We consider two settings each of sample size and mixing proportion vectors . Our mixture model is a mixture of four univariate normal distributions with consecutive mean distances LRTS under the null hypothesis follows a LRTSλ given in The asymptotic distribution of the d test under an alternative hypothesis NH asymptotically follows a non-central chi-square distribution p* can be written in terms of a matrix of classification probabilities ε = (ijε), where ijε = Pr (subject's observed genotype = i|subject's true genotype = j). The power for the chi-square test of independence with misclassification errors can be calculated from the NCP λCSThe 2×d contingency tables with respect to the LRTS on CNP quantitative measures for two genetic models of inheritance (MOI) associated with CNPs that have been documented as a possible MOI for CNPs >φ, the population frequencies iq for CNP category i,1≤i≤d, and the relative risks iR of becoming affected, given that an individual has CNP category i. We then compute the penetrances ig = Pr(affected|CNP_category = i), where we specify iR = ig/g1, so that the reference CNP category relative risk is 1. The reference CNP category may be chosen arbitrarily without loss of generality. The penetrances are given by ig = igR1. Using Bayes Theorem, the CNP category mixing proportions conditional on affection status are:d = 4, q1 = 0.4, q2 = 0.35, q3 = 0.2, q4 = 0.05, and >φ = 0.05. In the first (Dosage) model, the risk of becoming affected increases geometrically with increase in CNP category. We specify R2 = 1.8, R3 = 1.82 = 3.24, and R4 = 1.83 = 5.83, so that risk increases by a factor of 1.8 for each increase in CNP category.We calculate the efficiency of the chi-square test on 2×d and decreases for all other categories. For this work, we specify R2 = 0.3, R3 = 0.3, and R1 = R4 = 1. Finally, we set the means to be equally spaced for all components. Specifically, iθ = i for comparative analyses so that separation is given by In the second (Extremes) model, risk of becoming affected increases for CNP categories 1 and LRTS. For the null distribution simulations, we consider two settings each of sample size and mixing proportion vectors . For the alternative distribution simulations, we consider one setting of sample size and two different MOIs . Also, for both sets of simulations, our mixture model is a mixture of four univariate normal distributions with consecutive mean distances iθ+1−iθ = 1 unit apart. Separations are fixed to be We perform simulation studies to verify the accuracy of the asymptotic null and alternative distributions of the n1( = n2) = 200 or 500, and mixing proportions n1 ( = n2) = 200, and mixing proportions are determined using equations (4) with the specified parameters (including CNP population frequencies) for the Dosage and Extremes MOIs, given above .For the null distribution simulations, we specify sample sizes −5 and the maximum iteration number is set at 300.To find the global maximum (equations (1) and (2)), we use Expectation-Maximization algorithms (EM). A small pilot study found that there were typically three relative maxima under the null specification and two under the alternative. Consequently, we use 100 random starting points (RSPs) for parameter estimation under the null distribution simulations and 50 RSPs for the estimation under the alternative distribution simulations. EM algorithm computations are performed using MCLUST in the R programming environment d test relative to the LRTS is denoted Eff and is the ratio LRTS. When the relative efficiency is less than 1, the chi-square test requires a larger sample size to achieve the same power as the LRTS, given that both tests have the same level of significance. For example, if the relative efficiency of the 2×d test is 0.8, the 2×d test requires 100 observations to have the same power as the LRTS using 80 observations.The efficiency of the 2×LRTS to test for differences in mixing proportions of CNP categories between two groups of individuals (Caucasian and Taiwanese) using probe ratio data for a multi-allelic CNP probe in the FCGR3 gene on Chromosome 1. We also apply the chi-square test of independence to the probe ratio data after the individuals are classified into categories using the Bayesian classification rule described above. Oligonucleotide probes are designed as described previously Since recent work documents different CNP distributions in different ethnic populations To be consistent with notation used throughout this work, from this point forward we label the Taiwanese samples as “controls” and the Caucasian samples as “cases”, although individuals in this study were not ascertained for any particular disease phenotype.empirical type I error rates at the 0.975, 0.10, 0.05, 0.025, and 0.01 significance levels for each set of parameter settings. For each simulation, these type I error rates are the proportion of replicates for which the computed LRTS exceeds 0.2157, 6.25, 7.81, 9.348 or 11.34, which correspond to the 0.975, 0.10, 0.05, 0.025 and 0.01 significance level cutoffs for a central chi-square distribution with 3 degrees of freedom (the asymptotic null distribution for each simulation). For each empirical type I error rate, we report the 95% confidence interval, based on 1000 replicates. As an additional confirmation, we apply the Kolmogorov-Smirnoff (KS) goodness of fit test LRTS values , and report the p-values in In In each simulation, the target type I error rate is contained in the 95% confidence interval for the corresponding empirical type I error rate. In addition, the smallest KS test p-value is 0.36, indicating that we do not reject the null hypothesis that the data are drawn from a central chi-square distribution with 3 degrees of freedom.LRTS may be written in a quadratic form as:J0 is the (d−1)×(d−1) symmetric matrix specified in When the alternative hypothesis LRTS exceeds 6.25, 7.81, or 11.34, which correspond to the 0.10, 0.05, and 0.01 significance level cutoffs for a central chi-square distribution with 3 degrees of freedom (the asymptotic null distribution for each simulation). For each empirical type I error rate, we report the 95% confidence interval, based on 1000 replicates. As an additional confirmation, we apply the Kolmogorov-Smirnoff (KS) goodness of fit test LRTS values , and report the p-values in As in In each simulation, the target type I error rate is contained in the 95% confidence interval for the corresponding empirical type I error rate. In addition, the smallest KS test p-value is 0.34, indicating that we do not reject the null hypothesis that the data are drawn from a central chi-square distribution with 3 degrees of freedom.simulation power at the 10−3, 10−4, and 10−5 significance levels for each set of parameter settings. For each simulation, these powers are the proportion of replicates for which the computed LRTS exceeds 16.27, 21.11, or 25.90, which correspond to the 10−3, 10−4, and 10−5 cutoffs for a central chi-square distribution with 3 degrees of freedom (the asymptotic null distribution for each simulation). More stringent significance level cutoffs are chosen for the power analyses since power at the 0.10, 0.05, and 0.01 levels is close to or equal to 100% for these parameter specifications. As with the empirical type I error rates in LRTS values , and report the p-values in In −3 and 10−4 significance levels, the simulation power is contained in the 95% confidence interval for each simulation. The results of this table suggest that our simulation results are consistent with asymptotic results for at least the 10−3 and 10−4 significance levels.While the KS p-values are much smaller, we see that, for the 10LRTS,LRTS for the Extremes and Dosage models against the separation between successive category means. In all models, the relative efficiency is less than 1; that is, the LRTS is more powerful. When the separation is 5 standard deviations or greater, both tests have essentially the same power. The relative efficiency steadily declines as the separation between category means decreases, with less efficiency for the Extremes model.Using the result for the NCP of the LRTS applied to P4077 probe ratio data for the Caucasian and Taiwanese samples are presented in LRTS p-value for the P4077 probe is 0.014. In comparison, the chi-square test of independence p-value based on the asymptotic null distribution for the P4077 probe data with classification by the Bayesian rule is 0.03. The p-value based on Fisher's Exact Test is 0.0175. The numbers of Caucasian and Taiwanese individuals in CNP categories 1, 2, and 3 are: 229, 31, and 1; and 67, 20, and 1, respectively, as determined by the Bayesian classification rule. Additionally, we report the estimated classification rates as follows:ijε = Pr(reported CNP classification = j|true CNP classification = i). The LRTS method provides a slightly more significant p-value.Results for the ε along with the estimated mixing proportions under the alternative hypothesis (Power for Association With Error (PAWE) webtool, the power at the 5% significance level for the sample sizes specified in our example is 98% with error-free data, and is 76% with error rates given in ε, a power loss of 22%. From the perspective of power loss, Kang et al. When we use the estimated misclassification parameters in the matrix pothesis  in the Pi = 1, 2, using the estimated parameters from LRTS.Additionally, if we compute the separation values LRTS due to the relatively high misclassification rates that are consistent with the estimated parameters.While one cannot use parameters estimated from data collected to calculate actual power, we present these calculations as indications of the source of the greater power of the LRTS of the mixture proportions applied to the CNP quantitative measurements. The relative efficiency of the 2×4 chi-square test is less than 1 for the example disease MOIs considered here, with greater decreases as the separation between category-means decreases. That is, for the models considered, the LRTS is more powerful than the chi-square test. In the example, power may have been lost for the chi-square test because of relatively high estimated misclassification rate from the most common category to the second most common category. The chi-square test of independence can lose substantial power under such misclassification We have derived the non-centrality parameter for the LRTS is that it can be computed on any CNP data, whether or not that data can be categorized. While the example presented . Additionally, extensions of a multivariate procedure can incorporate more complex modeling of the mixture mechanism, for example, including a Hidden Markov Model approach.A key advantage of the resented  used onlThe results indicated in LRTS to avoid specification of a mathematical model of association. That is, the LRTS presented here only tests whether mixing proportions are different in two groups. There are mixture models that examine whether covariates are associated with CNP category membership LRTS is similar in spirit to the commonly used chi-square test of independence for genotype data on cases and controls. That statistic similarly tests for differences in allele or genotype frequencies among different categories . We further note that there is literature on power and sample size for logistic regression As an alternative analysis, one might consider a logistic regression model with case/control status as the dependent variable and CNP quantitative measure as the independent variable. One potential advantage of this method is that determination of optimal estimates is less computationally intensive than the LRTS procedure documented in this work. Another potential advantage of logistic regression is that it allows for the possible inclusion of covariates. In this work we focus on the The recent work documenting differences in CNP distributions for different ethnic populations is consistent with the frequently replicated results that there are different allele and genotype frequency distributions in different ethnic populations An additional limitation in the data analysis of our example is our assumption of equal variances among the component distributions. While this assumption appeared to be true for this example, it will not hold in general. In that event, methods such as those proposed by Hathaway The power and sample size calculations presented here are based on asymptotic theory; that is, our results should hold when sample sizes are sufficiently large. When sample sizes are smaller, one can use simulation methods to estimate power. Of course, p-values should be based on permutation tests in such instances.http://www.ncbi.nlm.nih.gov/Omim)Online Mendelian Inheritance in Man (http://linkage.rockefeller.edu/pawe/)Power for Association With Error (Appendix S1(0.02 MB PDF)Click here for additional data file.


Construction of precise confidence sets of disease gene locations after initial identification of linked regions can improve the efficiency of the ensuing fine mapping effort. We took the confidence set inference, a framework proposed and implemented using the Mean test statistic (CSI-Mean) and improved the efficiency substantially by using a likelihood ratio test statistic (CSI-MLS). The CSI framework requires knowledge of some disease-model-related parameters. In the absence of prior knowledge of these parameters, a two-step procedure may be employed: 1) the parameters are estimated using a coarse map of markers; 2) CSI-Mean or CSI-MLS are applied to construct the confidence sets of the disease gene locations using a finer map of markers, assuming the estimates from Step 1 for the required parameters. In this article we show that the advantages of CSI-MLS over CSI-Mean, previously demonstrated when the required parameters are known, are preserved in this two-step procedure, using both the simulated and real data contributed to Problems 2 and 3 of Genetic Analysis Workshop 15. In addition, our result suggests that microsatellite data, when available, should be used in Step 1. Also explored in detail is the effect of the absence of parental genotypes on the performance of CSI-MLS. With the advent of high-throughput genotyping technologies, traditional gene mapping methods, including linkage approaches, may be improved upon to realize the full potential of the wealth of genotype data available today. In particular, there has been considerable development in recent years in "intermediate fine mapping" approaches  in whichA recent study on several competing approaches to intermediate fine mapping shows that the confidence set inference (CSI) framework is particularly promising . In CSI,τ) in a broad region with linkage evidence is tested to see whether it is the putative disease causing locus (τ*). In contrast to the traditional null hypothesis of no linkage, a new hypothesis is tested:In the CSI framework using affected-sibling pair (ASP) data, each genomic location (zi = P(τ*IBD = i), i = 0, 1, 2, where τ*IBD is the identical by descent (IBD) allele sharing between two affected siblings at the disease locus. CSI-MLS provides more precise confidence sets compared to CSI-Mean when zi values are known [A 95% confidence set for the disease gene location is constructed to contain all the genomic locations where the above null hypothesis is not rejected at the 5% level. Any traditional linkage test statistic can be reformulated to test hypothesis (1), including the mean test statistic  and there known .a priori, a two-step procedure that relies on the availability of two sets of marker data on the same set of ASPs can be adopted [zi parameters. 2) Using a set of denser markers, termed the fine map  markers), and restricted to a region of, say, 25 cM on either side of the linkage peak, we constructed confidence sets employing CSI-MLS or CSI-Mean with the parameter estimates from Step 1. Because confidence sets so obtained may not be contiguous, we employed the smoothing scheme suggested by Papachristou and Lin [In the realistic case in which these parameters are not known  adopted : 1) iden adopted  above 2.r2 < 0.02), as linkage disequilibrium between markers can lead to erroneous estimates of multipoint IBD sharing probabilities [Traditionally, a whole-genome linkage scan is often pursued with microsatellite markers, and then the preliminary linkage signals are followed up using a denser, usually SNP, map. It was this practice that motivated the two-step CSI procedures. However, with the advent of high-density genome-wide SNP chips, this practice may soon be replaced by a single, dense SNP map on which all individuals are genotyped. The 5 K and 10 K chips have already been successfully employed in linkage analysis. In such situations, we propose using a subset of the SNP markers as the coarse map and another mutually exclusive subset of the SNP markers as the fine map. Within each of these two maps, markers are chosen to be in linkage equilibrium with each other  per replicate taken to be the first n families of each replicate. Estimates of means, standard deviations, etc., of the parameters of interest are based on the 100 replicates of the respective sample size. "Answers" were known.We apply our methods to Genetic Analysis Workshop 15 simulated data and the North American Rheumatoid Arthritis Consortium (NARAC) data, with the phenotype being the binary trait of the affection status of rheumatoid arthritis (RA). The microsatellite and SNP data (excluding the dense SNPs for the simulated data) on chromosome 6, containing the HLA-DRB1 locus, are used. Three strategies/settings, two using the microsatellite markers and one using the SNP markers, were compared in terms of their effects on the estimation of the disease-model-related parameters and the construction of the confidence sets. The two-step CSI-MLS and CSI-Mean were compared in terms of the precision of the confidence sets that include the DRB1 locus with a pre-determined probability. In what follows, we consider all of the 100 replicates, with a sample size of z0, z1, and z2. While these are not available from the "Answers" to the simulated data, a good guess is the MLE at the HLA-DRB1 locus obtained from the chromosome 6 SNP data on all families from all available replicates. The MLE  will be considered to be the true values of the corresponding parameters. The information content [Central to the CSI framework is the knowledge of the IBD probabilities at the trait locus, zi values, with and without parental genotypes. All estimates improve with increasing sample size. With parental genotypes, SNP1 yields slightly better estimates of zi values than MS1 and MSINT. This finding agrees well with the information content of the maps. However, when parental genotypes are not available, MS1 yields much more precise estimates than SNP1 and MSINT. This is also reflected in the density plot of the estimates given in Figure z1 being underestimated  and z2 being overestimated .Three estimation strategies/settings have been explored in Step 1 of CSI: 1) the MLE is evaluated at the microsatellite marker with the largest KAC score; 2) the MLE is evaluated at the location, possibly between two microsatellite markers, with the highest KAC score; 3) the MLE is evaluated with a set of SNPs that is mutually exclusive with the SNPs used in the second step as the coarse map. The three strategies are referred to as MS1, MSINT, and SNP1, respectively. The microsatellite and the SNP coarse maps contain 41 microsatellites and 292 SNPs, respectively. With parental data, the information content at the disease gene locus is 0.85 for the microsatellite map and 0.88 for the SNP map; without parental data, it is 0.56 for both. The precision of the parameter estimates were evaluated in terms of root mean squared errors (RMSEs). Table zi values from Step 1, CSI-Mean and CSI-MLS confidence sets are constructed with a fine map of 350 SNPs on the chromosome. Besides the three settings MS1, MSINT, and SNP1, we added the group TRUE, where the CSI confidence sets were constructed assuming the true parameter values. Mean lengths of the confidence sets, together with the empirical coverage levels, are given in Table Using the estimates of The MSINT strategy appears to be the best in terms of providing precise confidence intervals for the underlying disease locus. Hence, we used this strategy with the CSI-MLS and the CSI-Mean to measure their ability in localizing the HLA-DRB1 locus on chromosome 6, a known causal locus for RA. Our sample comprised 308 of the smallest families (<9 pedigree members each). This was done to have a sample size more representative of most linkage studies, and to have nuclear families with ASPs, the setting for which the CSI-Mean and CSI-MLS have been proposed. There were 363 ASPs in the chosen families . Figure z1 is 0.5. The value of z1 for RA at the HLA-DRB1 locus is 0.441, significantly different from 0.5. This difference suggests a reasonable contribution of dominance genetic variance to the total variance. CSI-MLS performs better than CSI-Mean in most situations, with the gain most substantial when dominance variance constitutes a substantial portion of the total variance of the disease [We demonstrate here that CSI-MLS performs substantially better than CSI-Mean, with the advantage more apparent when parental genotype data are missing. This is not surprising because the trait model represents a favorable situation for CSI-MLS . In a di disease . The betzi values under both MSINT and SNP1 settings. However, parental data is also absent on the fine map, which leads to a similar bias in Step 2. As long as these two biases are consistent (cancel each other out), which is reasonable, the performance of MSINT and SNP1 in terms of precision of the localization can be superior to MS1. Another striking observation is that CSI-MLS confidence sets are shorter when parental genotypes are missing rather than available. For example, using MSINT with 750 ASPs, the mean length of CSI-MLS intervals is 9.1 cM with parental data compared to 12.2 cM without parental data. This is due to the former having a lower empirical coverage (95%) than the latter (100%). When microsatellite and SNP genotypes are available on the same set of families, we suggest using MSINT to estimate the parameters in Step 1 and then constructing the confidence sets using CSI-MLS on a dense SNP map in Step 2. When microsatellite data are not available, use two independent dense SNP maps in the two steps. Even though the use of SNPs in Step 1 leads to some loss in efficiency, this loss is limited. This loss can presumably be reduced by using a denser SNP map in Step 1, though one has to be careful about the LD between markers affecting parameter estimates.When parental genotypes are absent, estimates of the disease locus IBD sharing probabilities using the MS1 strategy are superior to those using MSINT or SNP1. However, when the precision of the confidence sets are considered, MSINT provided the shortest sets while maintaining the nominal coverage. This apparent contradiction could be explained as follows. When parental information is not available, there is a bias in the estimates of CSI-MLS provides much more precise confidence intervals than CSI-Mean, yielding a reduction in length of 4.4 to 11.3 cM (16 to 49%) for the simulated data and 4.6 cM for the real data. A chromosomal region of such sizes may house hundreds of genes, and their exclusion reduces the number of candidate genes that need to be followed up by fine mapping. The problem of multiple testing is thus alleviated, and hence the power of detection of the causal locus is increased. Furthermore, budgetary constraints often limit genotyping to a fixed number of SNPs in the fine mapping stage, a smaller chromosomal region leads to better coverage of the region by the SNPs, and the causal mutations can thus be better interrogated.The author(s) declare that they have no competing interests.


The restricted partition method (RPM) provides a way to detect qualitative factors  associated with variation in quantitative or binary phenotypes, even if the contribution is predominantly an interaction displaying little or no signal in univariate analyses. The RPM provides a model (possibly non-linear) of the relationship between the predictor covariates and the phenotype as well as measures of statistical and clinical significance for the model.Blind to the generating model, we used the RPM to screen a data set consisting 1500 unrelated cases and 2000 unrelated controls from Replicate 1 of the Genetic Analysis Workshop 15 Problem 3 data for genetic and environmental factors contributing to rheumatoid arthritis (RA) risk. Both univariate and pair-wise analyses were performed using sex, smoking, parental DRB1 HLA microsatellite alleles, and 9187 single-nucleotide polymorphisms genotypes from across the genome. With this approach we correctly identified three genetic loci contributing directly to RA risk, and one quantitative trait locus for the endophenotype IgM level. We did not mistakenly identify any factors not in the generating model. All the factors we found were detectable with univariate RPM analyses. We failed to identify two genetic loci modifying the risk of RA. After breaking the blind, we examined the true modeling factors in the first 50 data replicates and found that we would not have identified the additional factors as important even had we combined all the data from the first 50 replicates in a single data set. Many diseases important to public health are not due solely to a single mutation or environmental insult. Instead, complex interactions among multiple genes and environmental exposures likely play crucial roles in the etiology of diverse phenotypes from schizophrenia to chemotherapy response.The restricted partition method (RPM)  is an alWe applied the RPM to the Replicate 1 of the Genetic Analysis Workshop 15 data, blind to the generating model, in order to test the utility of the method for genome-wide association data for a complex trait.R2, is calculated and statistical significance is evaluated using empirical null distributions generated by permutations of the data. Details of the RPM can be found in Culverhouse et al. [The RPM merges multivariate genotype and environmental strata with similar mean trait values (proportions of cases) until all of the remaining groups have significantly different means or all the genotypes are merged into a single group (indicating that the particular loci are not jointly correlated to variation in the trait). If distinct subgroups are found, the proportion of variation attributable to the partition, e et al. . Originae et al. .N = 1500) by selecting the most severely affected sib. If both sibs were equally affected, we selected the first sib. This case sample was compared to the N = 2000 unrelated controls supplied by the data providers. Potential predictors of phenotype consisted of genotypes from the sparse SNP map (N = 9187) along with the covariates sex, smoking, and the DRB1 HLA (henceforth "DR") allele inherited from each parent. This resulted in a total of 9191 predictor variables.We initially restricted our attention to data from a single replicate (Rep 1). We formed a group of unrelated cases . An analysis of LD by Suarez et al. [For the SNPs on chromosome 6, the sex-averaged genetic distances from SNP 128 are listed to give an idea of the recombination rates between the markers. We compared these results to what would have been obtained by a traditional χR2 values over the univariate analyses, decreased p-values (after correction for the increased number of tests), and by the structure of the resultant multivariate model. We do not currently have a formal test for interactions. In part, this is because the main goal of the RPM is to detect factors contributing to phenotype, even if their effects are primarily seen in interactions, rather than to estimate the variance components accurately.Interactions in multivariate RPM analyses are indicated by increased R2. Because the R2 in these analyses are estimated from a highly ascertained data set, they likely overestimate the proportion of variation explained by these factors in the whole population.We began our analyses by searching for increases in R2. Individually, these factors each account for approximately 37% of the trait variation . We evaluated the correlation between haplotypes at these two loci (since haplotypes were available). We found that in cases, allele 1 at SNP 153 was always on the same haplotype as allele 2 or 3 at the DR locus, while allele 2 at SNP 153 was on the same haplotype as allele 1 from the DR locus 98.9% of the time (χ2 = 2962.3). Only four of the six possible haplotypes were observed, and one was extremely rare. In controls, the correlation is similarly high (χ2 = 3990.6). This suggests that chromosome 6 SNP 153 is merely a surrogate for DR.The addition of chromosome 6 SNP 153 to the model for the parental DR alleles did not substantially increase R2. The RPM gave evidence of sex and chromosome 11 SNP 389 being contributing factors independent of the DR locus. Both increased the explained variation from 56% to 58%. The model including sex and DR resulted in slightly different groups for the two sexes, suggesting a possible small, non-additive interaction.In contrast, other factors in combination with DR could produce a modest increase in After we performed these analyses, we combined the two DR alleles into a single DR genotype, thereby reducing analyses involving DR by one dimension. Using DR genotypes produced almost identical results as treating the alleles as separate factors, so we will only present results from using DR genotypes in Table Once the model was known, we discovered that the factors we identified from our univariate analyses (see Table R2) in addition to a measure of statistical significance  and that it proposes a model of how the predictive factors affect the phenotype.Two related and appealing properties of the RPM are that it provides a measure of effect size (R2 in two ways: i) as a threshold to filter out factors of little clinical interest and ii) as a method to localize a signal. In practice, the decision of how small an effect of clinical interest is depends on the phenotype, the researcher, and funding, and should be made in consultation with a skilled clinician. In this case, we picked a low threshold, R2 ≥ 0.005. Because this only identified a small number of markers of interest, we stayed with that threshold. As a result, we were able to identify a QTL related to our phenotype of interest without picking up any false-positive signals. In this we were lucky. Our second use of R2 was to localize a signal within regions of LD on chromosome 6. Because of the LD, several markers  declare that they have no competing interests.


To identify genetic loci influencing BMI, we carried out a pooled analysis of genome-wide admixture mapping scans in 15,280 African Americans from 14 epidemiologic studies. Samples were genotyped at a median of 1,411 ancestry-informative markers. After adjusting for age, sex, and study, BMI was analyzed both as a dichotomized (top 20% versus bottom 20%) and a continuous trait. We found that a higher percentage of European ancestry was significantly correlated with lower BMI . In the dichotomized analysis, we detected two loci on chromosome X as associated with increased African ancestry: the first at Xq25 ; and the second at Xq13.1 . Quantitative analysis identified a third locus at 5q13.3 where higher BMI was highly significantly associated with greater European ancestry . Further mapping studies with dense sets of markers will be necessary to identify the alleles in these regions of chromosomes X and 5 that may be associated with variation in BMI.The prevalence of obesity (body mass index (BMI) ≥30 kg/m Obesity is about 1.5-fold more prevalent in African Americans than European Americans. To determine whether genetic background may contribute to this observed disparity, we scanned the genomes of African Americans, searching for genomic regions where obese individuals have a difference from the average proportion of African ancestry. By examining genetic data from more than 15,000 African Americans, we show that the proportion of European ancestry is inversely correlated with BMI. In obese individuals, we detect two loci with increased African ancestry on chromosome X (Xq13.1 and Xq25) and one locus with increased European ancestry on chromosome 5 (5q13.3). The 5q13.3 and Xq25 regions both contain genes that are known to be involved in appetite regulation. Our results suggest that genetic factors may contribute to the difference in obesity prevalence between African Americans and European Americans. Further studies of the regions may identify the causative variants affecting susceptibility to obesity. Obesity is a highly prevalent condition that increases the risk of many illnesses such as cardiovascular disease, diabetes, and some cancers. Familial aggregation studies have shown that both genetic and environmental factors are involved in the development of common forms of obesity, and heritability estimates suggest that approximately 40% of variation in body mass index (BMI) can be attributed to genetic factors 2) than European Americans even in homogeneous socioeconomic groups The current increase in prevalence of obesity in the U.S. has been hypothesized to be the result of genetic susceptibility in an environment that promotes obesity FTO) are significantly associated with obesity in populations of European origin 2 rise in BMI is associated with each copy of the A allele at rs9939609 in populations of European descent FTO may be different from that in other populations, although an alternative explanation for these observations is that the causal variant has not been identified, and that the linkage disequilibrium patterns to the causal variant are different in African and non-African populations.Recent genome-wide association studies have shown that variants in the fat mass and obesity-related gene . Altogether 4,372 markers were genotyped in the present study, with a median of 1,411 markers genotyped per sample. We found in practice that all marker panels provided at least 60% of the maximum possible information about ancestry.Our analysis was carried out in 15,280 African Americans. Samples were scanned with at least one of three iteratively improved and partially overlapping panels of ancestry-informative markers. The Phase 1 panel was published in Smith et al. 2004 The samples were assembled from 14 studies . Of thesIn the six prospective cohort studies, anthropometric measurements were performed using study-specific standardized protocols, and BMI was calculated as weight (in kg) divided by height (in meters) squared. In the BCFR, SFBABCS and WCHS, BMI was also calculated from height and weight measures taken at the time of study interview by trained research staff. In the remaining studies, BMI was calculated using self-reported weight and height.We used previously published genotyping data to estimate the frequency of each SNP in West Africans and European Americans To decrease the likelihood of false-positives in our admixture scans, we applied a series of filters that had the goal of detecting and removing any SNPs with problematic genotyping, as described previously We required all individuals included in the study to have complete phenotypic information, including BMI, age at the time of measurement, and gender. We also required all individuals to have a full admixture scan, and we removed samples that were outliers with respect to others in the same cohort in the sense of having many fewer genotypes, as we found that this predicts less reliable data. The data for the great majority of the samples we analyze in this study was reported previously We estimated the European and African ancestry at each locus and genome-wide using the ANCESTRYMAP software Body mass index was defined as described above. For most of our admixture analysis runs, BMI was adjusted for age, age-squared, sex and study, using multivariate linear regression analyses, and the residuals that emerged from this regression analysis were used for subsequent analysis.ANCESTRYMAP For the dichotomous admixture scans, we defined the top 20% of samples with the highest residuals of BMI as cases and the bottom 20% as controls. Because a prior distribution on risk models is required for the Bayesian statistical analysis in ANCESTRYMAP locus-specific score and a case-control score. A locus-specific score is obtained in cases  by calculating the likelihood of the genotyping data at the SNPs at the locus under the risk model and comparing it to the likelihood of the genotyping data at the SNPs at the locus assuming that the locus is uncorrelated to the phenotype To assess statistical significance, the ANCESTRYMAP software provided two scores: a To obtain an assessment of the evidence for a risk locus anywhere in the genome—which we call the “genome-wide score”—we averaged the likelihood ratio for association across all loci in the genome, and took the log10 to obtain a genome-wide score. We interpret a genome-wide score>2 as significant and >1 as suggestive as previously recommended case-control score was calculated by comparing locus-specific deviations in European ancestry in cases versus controls at each locus across the genome. This score tests whether any deviation in ancestry from the genome-wide average is significantly different comparing cases with controls P<5×10−5, or a corrected nominal P<0.05 after conservatively correcting for 1,000 hypotheses tested, corresponding to independent chromosomal chunks assigned to either African or European ancestry). The case-control score is particularly important for X chromosome analyses. Case-only admixture analyses of the X chromosome are complicated by the fact that African Americans tend to have lower proportions of European ancestry on the X chromosome than on the autosomes, and thus an X-chromosome-wide-specific estimate of ancestry is required A We have now extended ANCESTRYMAP to also allow for association analyses of quantitative traits . BrieflyTo calculate a 95% credible interval (CI) for the position of a locus, we obtained the likelihood ratio for association at each marker across the chromosome where we found an association. This provided a Bayesian posterior probability for the position of the underlying causal variant assuming a flat prior distribution across the region for the position of the disease locus. The central region of this peak containing 95% of the area was used as the CI.I2 inconsistency metric Local estimates of ancestry at the admixture peak were obtained using the ANCESTRYMAP software This study was conducted according to the principles expressed in the Declaration of Helsinki. All sample collections were carried out according to institutionally approved protocols for study of human subjects and written informed consent was obtained from all subjects.http://www.broad.mit.edu/node/549). Combined across all studies, the mean BMI was 29.8±6.6 kg/m2, and 40.2% of the population had BMI ≥30 kg/m2. Mean BMI differed significantly across studies (P<0.001).The demographic and phenotypic characteristics of the 15,280 African Americans included in the admixture scan are summarized in P<0.001).The average percentage of genome-wide European ancestry in these samples was 19.3±11.5% based on estimates from the autosomes, and the average percentage of European ancestry on the X chromosome was 15.5±8.4%. Because the study samples came from different resources and locations across the U.S., there was significant variation in average European ancestry, either estimated from autosomes or the X chromosome, across studies (P = 1.6×10−7) given the large sample size. It was also significantly correlated with European ancestry as estimated from the X chromosome .The relationship between BMI and percentage of European ancestry is shown in P = 8.1×10−5), which also supported an association at this locus, with obese cases having lower European ancestry than non-obese controls. Interestingly, we found another admixture peak at Xq13.1. Although the LOD score at Xq13.1 was far from significant (locus-specific LOD = 2.22), this locus had the strongest case-control Z score, −4.62 , anywhere in the genome. The associations at Xq13.1 was nominally genome-wide significant (P = 3.8×10−3) after conservatively correcting for 1,000 hypotheses tested.The dichotomous admixture scans detected evidence of genome-wide significant associations between markers on the X chromosome and higher BMI . By compTo examine the potential impact of heterogeneity across the studies on our admixture-generated signals, we carried out a series of subgroup analyses . When BMP<0.026; see below for details).Because the admixture peaks we identified were located on chromosome X, which has a different copy number in men and women, we also performed analyses for each gender separately to explore whether the strength of association differed significantly between males and females. We found that the evidence of association at Xq25 was stronger in females  than in males , and that the association signal at Xq13.1 in males grew stronger with the local LOD score rising to 4.40 (Run 6 and 7 in P = 5.6×10−5). The case-control score was marginally significant at genome-wide level, suggesting that higher BMI was, though counter-intuitively, associated with greater European ancestry at 5q13.3 locus.In addition to the two peaks on chromosome X, using dichotomous admixture scans we observed a few interesting regions , particuBy including all African-American samples and using BMI as a continuous trait, our quantitative admixture scan supported and strengthened the evidence of association at 5q13.3 locus . The peaP = 2.2×10−11 and P = 4.5×10−10, respectively). To examine whether these associations could be fully explained by the significant association between BMI and genome-wide ancestry (discussed above), we further adjusted for genome-wide European ancestry in the multivariate analysis. The residual association of local ancestry with BMI after adjusting for genome-wide ancestry remained significant at both Xq13.1 (P = 1.9×10−7) and Xq25 (P = 4.1×10−6) (Model 2 in P = 1.9×10−4 and P = 4.1×10−3) after conservatively correcting for 1,000 hypotheses tested. A naive analysis suggests that each additional copy of a European ancestral allele at either the Xq13.1 or the Xq25 peak is independently associated with a BMI decrease of ∼0.1 Z-score units on average . The true genetic effects are expected to be somewhat weaker because of discovery bias.The local estimate of European ancestry was also extracted for each individual at each of the three admixture peaks and analyzed for association with continuous BMI (Model 1 in P = 5.8×10−7) associated with an increase in BMI of 0.09 Z-score units , which was nominally significant (P = 5.8×10−4) after correcting for the approximately 1,000 independent hypotheses tested.The association at the 5q13.3 peak was particularly interesting in that it did not achieve statistical significance until the genome-wide estimate of European ancestry was added into the analysis. This was presumably because the locus effect was in the opposite direction to the genome-wide ancestry effect . Each additional copy of a European ancestral allele at 5q13.3 was significantly  between the two gender groups, either before or after adjusting for genome-wide European ancestry. A potential mechanism for the difference in the strength of association in men and women at the Xq13.1 locus is that women carry two copies of chromosome X whereas men carry only one, and hence this may simply reflect a difference in the genetics of the two genders on chromosome X.For the two peaks on chromosome X, we further examined whether the effects of the local ancestry on BMI were modified by gender. The local ancestry at Xq13.1 tended to be more strongly associated with BMI in males than in females. After adjusting for genome-wide European ancestry, the gender difference at Xq13.1 was significant  at any of the three peaks  on build 35 of the human genome reference sequence. The 95% CI for the higher admixture peak on chromosome X spanned from 114.4 to 124.4 Mb (an ∼10 Mb region), and then 95% CI for the other chromosome X admixture peak spanned from 47.8 to 89.2 Mb, a much broader region (∼40 Mb).P = 5.8×10−7), and two regions on chromosome X where greater local European ancestry was associated with lower levels of BMI (both P<5.0×10−6). Each of these three associations was above and beyond the contribution of genome-wide European ancestry, and each reached genome-wide significance.We have carried out admixture mapping analyses to search for genomic regions associated with BMI. This pooled analysis of samples from 14 studies is the largest admixture scan reported to date. In more than 15,000 individuals, we identified a locus on chromosome 5 where greater local European ancestry was associated with higher levels of BMI  and found the two sets of results to be largely comparable. Additional subgroup analyses, as shown in The inverse correlation between BMI and percentage of European ancestry estimated on the genome-wide scale confirmed the results from previous studies of smaller sample size and fewer markers CART) gene, which is a candidate for modulating obesity. CART is a hypothalamic neuropeptide that transmits a physiological anorexigenic signal and is involved in appetite regulation CART knock-out mice have increased body weight compared with wild type mice CART gene have also been linked to both BMI and serum leptin levels in a study of French Caucasian families CART gene in these populations, and to our knowledge no published studies have studied CART variants in African Americans. Further mapping work is needed to determine whether the CART gene or other genetic variants in the interval may influence the risk of obesity.The 95% CI for the chromosome 5 peak harbors a number of genes, including the cocaine and amphetamine regulated transcript (solute carrier family 6 member 14 (SLC6A14) is involved in serotonin synthesis and serotonergic receptor mechanisms that have been implicated in appetite control and body weight regulation SLC6A14 was observed in ∼1,800 samples from Finland and Sweden (P = 0.003), and females were found to contribute most to this particular observed association CUL4B) gene. CUL4B was recently identified as a causative gene for an X-linked mental retardation syndrome, which was associated with several clinical features, including central obesity There have been very few studies reporting linkage of obesity with markers on the X chromosome FTO gene, we noticed that the second highest admixture peak (locus-specific LOD = 3.68) identified in our quantitative scans was on chromosome 16, about 5.6 Mb away from the FTO gene, and its 99% CI spanned 51.5 to 66.8 Mb (on build 35 of the human reference sequence), which is a region that includes the FTO gene.  Further fine-mapping analysis may determine whether additional variations in FTO may explain the intriguing admixture signal in this region. The melanocortin-4 receptor (MC4R) gene, located on chromosome 18q21.32, is the second obesity-susceptibility gene discovered by genome-wide association studies in individuals of European origin Although we did not detect a significant association in the region of the In summary, we have carried out a genome-wide admixture mapping scan in 15,280 African Americans and have identified three loci, 5q13.3, Xq13.1 and Xq25, that may harbor genetic variants associated with variations in BMI. The local ancestry associations to BMI at each of the three admixture-generated peaks were statistically significant, suggesting the presence of a genetic effect at these loci above and beyond the effects of genome-wide ancestry. Follow-up fine mapping and focused analysis of each locus using data that emerge from genome-wide association studies in African Americans with measured BMI will be crucial to determine whether these regions harbor genetic variants predisposing to obesity.The present study is also methodologically significant in illustrating how searches for genes in African Americans and diverse populations can result in the detection of genetic loci that have eluded discovery in European-derived populations, perhaps because the underlying variants are too rare in the latter populations. However, there is no reason to think that the three loci we have identified are biologically important only in African Americans. Replication and fine-mapping studies in other ethnic groups, including Hispanic Americans and Pacific Islanders, with a similar risk of obesity to African Americans, and even European Americans and East Asians with a lower but still important rate of this condition, may further elucidate these regions of the genome. Studying multiple populations to fine-map a locus highlighted in an admixture scan can be more informative than studying any one population, as was previously demonstrated by our use of a multi-ethnic cohort to fine-map prostate cancer risk factors at 8q24 Table S1Interesting chromosome regions, other than Xq25, Xq13.1 and 5q13.3, observed in the dichotomous admixture scans for BMI.(0.01 MB PDF)Click here for additional data file.Table S2Correlations between local European ancestry and BMI by study.(0.14 MB PDF)Click here for additional data file.Text S1Description of each study.(0.04 MB PDF)Click here for additional data file.Text S2Details of quantitative trait analysis.(0.07 MB PDF)Click here for additional data file.


The availability of a large number of dense SNPs, high-throughput genotyping and computation methods promotes the application of family-based association tests. While most of the current family-based analyses focus only on individual traits, joint analyses of correlated traits can extract more information and potentially improve the statistical power. However, current TDT-based methods are low-powered. Here, we develop a method for tests of association for bivariate quantitative traits in families. In particular, we correct for population stratification by the use of an integration of principal component analysis and TDT. A score test statistic in the variance-components model is proposed. Extensive simulation studies indicate that the proposed method not only outperforms approaches limited to individual traits when pleiotropic effect is present, but also surpasses the power of two popular bivariate association tests termed FBAT-GEE and FBAT-PC, respectively, while correcting for population stratification. When applied to the GAW16 datasets, the proposed method successfully identifies at the genome-wide level the two SNPs that present pleiotropic effects to HDL and TG traits. Recent technological advances in genotyping along with the capacity to detect increasingly large numbers of single nucleotide polymorphisms (SNPs) have created great demand for developing new strategies to identify genes that underlie phenotypic variation. The availability of high-throughput SNP genotype data is prompting the development of genetic association analyses, including family based association tests (FBAT).For family data sets, such as the Framingham heart study Both the methods FBAT-GEE and FBAT-PC possess the property of protection against population stratification by a transmission disequilibrium test (TDT)-like strategy. Despite its potential merit, this property comes at the cost of a substantial loss of statistical power by the fact that genotypes at each single marker need to be decomposed in order to correct for population stratification and test association simultaneously. The loss of power may become problematic in the context of genomewide association studies (GWAS) where it is critical to achieve a genomewide significance level in order to judge a positive finding.Alternatively, the issue of population stratification can be handled at the population level by studying population structures from genotype data of multiple markers In this study, under the framework of the variance-components (VCs) model We describe the problem in the variance-components (VCs) N nuclear families with in individuals in the i-th family . Let yij = ′ be the vector of m (m = 2 for bivariate) phenotypes for individual j  in family i. Further, let Yi = ′ be the vector of phenotypes for all members in family i. Under the additive mode of inheritance, the genotype score ijg is defined as 0, 1 and 2 for genotypes “11”, “12” and “22”, respectively. In the variance-components model, genetic components contributing to phenotypes are decomposed into the fixed effects, e.g., the effects at the specified locus, and the random effects, e.g., the effects of unknown polygenes. Specifically, the phenotype vector yij can be described by the following multivariate variance-components modelm phenotypes; xij is a m×t design matrix for t covariates, e.g., age, sex, and known environment factors, to the m phenotypes, and xβ is the vector of corresponding covariate effects; gij is a m×m design matrix for genotype scores with the m principal diagonal elements being ijg and the other elements being 0, and gβ the corresponding additive major gene effects. At last, αij and εij are vectors of length m denoting, respectively, the additive polygenic effects and the residual effects.Assume that there are A and E are the m×m variance-covariance matrices accounting for polygenic  matrix at the tested locus (estimated from the genotype data) and in×in kinship coefficient matrix , both of which can be calculated from pedigree structures and available genotype data. Finally, I is the in×in identity matrix and The phenotype vector for the family When the issue of population stratification exists, the model described above may not provide a valid test. We previously proposed to extend the principal component analysis to include familial data H0: H1: i. Under the null hypothesis, the score vector with respect to gβ is defined asm unless there are linear dependences between the phenotypes. The statistic T is valid regardless of the presence of population stratification.With the assumptions of independent families and of within-family multivariate normality distributed phenotypes, the likelihood function of the adjusted genotype and phenotype data is written asStatistical properties and performances of the proposed method were investigated via extensive simulation studies. For genotype data, we simulated 998 SNP markers, with the allele frequency for each marker being drawn from the Uniform distribution U. We also simulated two additional SNPs, both with MAF 0.3, as the causal and the test SNPs, respectively. Two hundred nuclear families were simulated by sampling parental genotypes according to allele frequencies, and then by randomly selecting two parents to produce children. Unless otherwise specified, the number of children per family was drawn from a Poisson distribution with mean 2.aρ) and environmental (eρ) correlations between the traits were set to 0.4 unless otherwise specified.Two quantitative traits were simulated by the equations (3) and (4). To each trait, the causal SNP was assumed to explain a specific proportion of phenotypic variation, which was set to 2% by default, and the background polygenic effects were assumed to explain additional 40% of phenotypic variation. The polygenic . The allele frequencies for populations A and B were then drawn from a Beta distribution with parameters p(1−STF)/STF and (1−p)(1−STF)/STF, where STF is the measure of genetic distance between the two populations. We set STF to 0.05 to simulate moderate population stratification. The two populations were generated separately with different phenotypic means (Aμ and Bμ) and different causal and test SNP MAFs . The values of Aμ and Bμ were selected such that 20% of the phenotypic variance of each trait in the combined population was explained by stratification.When needed, population stratification was generated by mixing equal numbers of families from two discrete populations A and B. Marker allele frequencies in the two populations were generated using the Balding-Nichols model r2 between the causal and the test sites to 0.0, whereas the alternative hypothesis was produced by setting a certain value of r2 between the two sites. Unless otherwise specified, the value of r2 under the alternative was set to 1.0 to produce a perfect association between the two sites.We evaluated the statistical properties, including type I error rates and power, of the proposed method. In all the simulations, the null hypothesis was produced by setting the LD measure We also studied the effects of various influential factors, including locus effect, correlation coefficient between traits, the level of LD, and family structure, on the performance of the proposed method. For comparison, we also included extensive popular univariate and bivariate methods into analysis. For univariate method, we selected the commonly used method QTDT proposed by Abecasis et al. As an application, we analyzed the Genetic Analysis Workshop 16 (GAW16) Problem 3 simulated data sets with the proposed method. The access and analyses of the GAW16 simulated data sets have been approved by the Institutional Review Board (IRB) of the University of Missouri-Kansas City (UMKC). The GAW16 data sets consist of 6476 subjects from the Framingham Heart Study (FHS), where each subject has real genotypes at approximately 550,000  SNPs and simulated phenotypes.Subjects are distributed among three generations and singletons. After dividing large families into smaller nuclear families and applying some quality controls to the data , we finally identified 5456 family members from a total of 1815 nuclear families.rs3200218 and rs8192719) present pleiotropic effects to both traits in the simulation. Phenotype data are simulated at three pseudo-visits with 10 years apart to mimic the context of longitudinal study, and at each visit, 200 simulated data sets are replicated. The dataset from the first replicate of the first visit was analyzed as suggested by the workshop. Both phenotypes were adjusted by age and sex.A total of four correlated quantitative traits, termed HDL, LDL, TG, CHOL, respectively, are simulated in order to mimic the lipid pathway underlying the development of cardiovascular disease aρ) and environmental (eρ) correlations in homogeneous population setting, as listed in We first estimate type I error rates under a variety of polygenic  decrease from +1.0 to −1.0. For example, under homogenous population, the power of the proposed method is 87.6% when both aρ and eρ are +0.8, and increases to 100.0% when both correlations decrease to −0.8. In additional simulations where the major gene correlation between the traits is constrained to −1.0 rather than +1.0, we observe an opposite trend in power change; power increases as aρ and/or eρ increase from −1.0 to +1.0 (data not shown). Therefore, our simulation results indicate that the power of the bivariate approaches increases when the effects of the major-gene and those of the residuals  act in increasingly dissimilar directions, which coincides with previous studies in the literature of linkage analyses Powers of various methods affected by aρ and eρ are +0.8 under homogeneous population, the power for the proposed method is 90.5%, whereas it is only 48.8% for FBAT and 58.0% for PBAT. For the other two methods, PBAT has a higher power than FBAT.Among the three bivariate methods, the proposed one has the highest power under all the parameter settings. The power improvement is remarkably large. For example, when both When comparing the bivariate and univariate approaches, the proposed method has higher power than UT under most conditions, and than QTDT under all the conditions. UT has higher power than QTDT, consistent with our previous study Power when parental information is missing is presented in r2. As the number of children per family increases, the power of the proposed method and UT decreases a little, whereas that of the other methods increases instead. Again, the proposed method has the highest power in all situations.We also study the effects of two factors, including the level of LD and family structure, on power estimations. As presented in rs10820738, with a p-value 9.55e-17. UT has a p-value 4.42e-17 at this SNP. The second most significant signal corresponds to one of its flanking SNPs rs2297398 (7.7kb apart), with a p-value 1.88e-15 from the proposed method. The SNP rs10820738 explains the most phenotypic variation (1.0%) for HDL trait in the GAW16 simulation, but none for TG trait. The third most significant SNP is rs3200218 with a p-value 3.47e-12. This SNP is exactly one of the two SNPs that present pleiotropic effects to both traits. The p-value at the other major pleiotropic SNP rs8192719 is 3.07e-6, which does not achieve the genomewide significance level. However, one of its flanking SNPs rs7249735 (12.8kb apart) presents a p-value 2.77e-8 that is significant at the genomewide level. Generally speaking, the proposed method successfully identifies both pleiotropic SNPs at the genomewide level. We also list p-values at these two SNPs from other methods in As an application, we analyze the GAW16 simulated datasets as described in the preceding section. On a desktop computer with a single 2.8GHz Intel Xeon CPU, the computation from the proposed method for the scan with 550K SNPs takes a total time of approximately 20 hours. In this study, we have presented a bivariate test of association for quantitative traits in families, by the use of the multivariate variance-components model. In particular, the proposed method uses principal component analysis to correct for population stratification. Simulation studies have shown that the proposed method not only outperforms the analysis focused on individual traits when pleiotropic effect is present, but also has increased power compared with the existing bivariate methods, while correcting for population stratification.aρ and eρ are equal to −0.8, corresponding to an approximate phenotypic correlation of −0.76. When pleiotropic effect is present, bivariate analysis is more powerful than univariate analysis unless the residual correlation is high and in the same direction to the correlation of major gene effect, coinciding with the findings in linkage studies rs10820738 that has no pleiotropic effect in the GAW16 simulation, the p-value from the proposed bivariate method, 9.55e-17, is slightly higher than that from the univariate method UT, 4.42e-17. In practical applications in which the existence of pleiotropic effect is unknown in prior, bivariate analysis is not necessarily more powerful than univariate analysis, even when the traits are strongly correlated. Bivariate analysis should thus be processed with caution, and combing the results of bivariate and univariate analyses is warranted.The strength of bivariate analyses is influenced by correlations between traits. Our simulation results show that bivariate analyses are more powerful when the major genes and the residual factors act in more dissimilar ways. For example, by constraining the major-gene correlation to +1.0, bivariate approaches are most powerful when both Thus, our simulations provide a statement in demonstrating that bivariate approaches are more powerful than univariate analyses unless major-gene effects and residual effects are very highly correlated in the same direction, which coincides with the conclusion of Amos et al. Our method that is developed by extending the variance-components model offers a way to powerfully/robustly perform bivariate association analysis in the presence of linkage in general pedigrees. The variance-components model is advantageous in detecting QTL in the following two aspects: first, it combines the analysis of linkage and association and would increase the power of detecting QTL when the marker, itself is not the QTL, is associated with the QTL. Second, the prior evidence on linkage can be incorporated to indicate LD strength between the QTL and the marker Another strengthen of our method is decomposing individual genotype scores by principal component analysis rather than by TDT-like strategy for controlling population stratification. The resulting test statistic provides largely improved power over existing TDT-based methods, where the latter may be prohibitive for application to genome scans due to their poor powers. For example, under the moderate setting where both polygenic and environmental correlation coefficients were set to 0.4 and the locus effect were set to 2%, we observed 164 and 24 significant results over 1,000 replicates for the proposed method and UT, respectively, but only 6, 8, and 4 for FBAT-GEE, FBAT-PC, and QTDT, respectively, at the genome-wide level 1.0e-6.An interesting observation from our simulations is that family structures influence the power of the investigated methods in different patterns. For FBAT-GEE, FBAT-PC and QTDT that control population stratification through TDT, small numbers of large families provide more power than large numbers of small families. This is not surprising, since with these methods the parental information is used to control the stratification, and consequently, only the information of children contributes to test statistics. Contrary, the power decreases slightly as the number of children per family increases for the proposed method. This appears to be caused by the fact that a large number of small independent families provides more information on allele distributions than a small number of larger families can provide.In this manuscript, we focus our attention on data with nuclear families. However, the proposed method is applicable to extended pedigrees as well. The described multivariate variance-components model can be directly applied to extended pedigrees. As for correcting for population stratification, the extension of principal component analysis coupled with TDT strategy to extended pedigrees is also straightforward. For example, all founders can be collected to form an unrelated sample. In cases where there is no founder, one sib can be randomly selected into the unrelated sample, as we described in reference In summary, we develop a novel method for family based bivariate association test. Our method is more powerful than currently available bivariate methods. The proposed method is computationally effective and can complete a typical GWAS scan within hours. The program implementing our proposed method is available upon request to us.