Haemonchus contortus. Second, to compare the precision of the estimates when using two different relationship matrices: including pedigree information only or including also SNP (single nucleotide polymorphism) information.Haemonchosis is a parasitic disease that causes severe economic losses in sheep industry. In recent years, the increasing resistance of the parasite to anthelmintics has raised the need for alternative control strategies. Genetic selection is a promising alternative but its efficacy depends on the availability of genetic variation and on the occurrence of favourable genetic correlations between the traits included in the breeding goal. The objective of this study was twofold. First, to estimate both the heritability of and the genetic correlations between growth traits and parasite resistance traits, using bivariate linear mixed animal models, from the phenotypes and genotypes of 1004 backcross lambs (considered as a single population), which underwent two subsequent experimental infestations protocols with Haemonchus contortus free environment and negative in a contaminated environment. The standard errors of the estimates obtained by including SNP information were smaller than those obtained by including pedigree information only.Heritabilities were low for average daily gain before infestation (0.10 to 0.15) and average daily gain during the first infestation (0.11 to 0.16), moderate for faecal egg counts during the first infestation (0.21 to 0.38) and faecal egg counts during the second infestation (0.48 to 0.55). Genetic correlations between both growth traits and faecal egg count during the naïve infestation were equal to zero but the genetic correlation between faecal egg count during the second infestation and growth was positive in a The genetic parameters estimates suggest that growth performance can be selected for independently of selection on resistance to naïve infestation. Selection for increased growth in a non-contaminated environment could lead to more susceptible animals with long-term exposure to the infestation but it could be possible to select for increased growth in a contaminated environment while also increasing resistance to the long-term exposure to the parasite. The use of molecular information increases the precision of the estimates. Haemonchus contortus (H. contortus) is a nematode that feeds on blood through the abomasal mucosa of bovine, ovine and caprine species [H. contortus infection or haemonchosis for the production sectors of sheep farming in terms of anthelmintic treatments that are currently the most popular control strategy, and the resulting economic loss have been estimated in different countries to be in the order of several million dollars per year [ species . The cosper year -7. Furthper year ,9, whichBoth the long-term loss of efficacy and the growing public concern for the use of chemicals in food production fostered the research on alternative control strategies or combinations of them , among wH. contortus. The genetic parameters have been estimated both by using pedigree information only and pedigree and SNP (single nucleotide polymorphism) chip information jointly. Computing the relatedness between individuals using pedigree information only is based on expectation and results in an estimate corresponding to the average number of alleles shared by two individuals, for example: all the individuals belonging to the same full-sibs group would have a coefficient of 0.5 between each other, which means that it does not take into account the deviation from this average caused by segregation and recombination. However, including molecular information makes it possible to compute the relatedness between individuals by identifying on a relatively dense map the actual number of alleles they share, which provides a more detailed estimate of the relatedness between individuals [The first objective of this study was to estimate the heritability of average daily gain and faecal egg count from experimental observations, together with the genetic correlations between them. We report the results of an analysis performed on 1004 phenotypic records of growth traits and faecal egg counts collected on genotyped (50 k SNPchip) back-cross lambs (25% Martinique black belly and 75% Romane), following two experimental infestations with ividuals . Since tThe population in which the observations were collected from resulted from a back-cross mating scheme between two pure-bred populations: Martinique Black Belly (MBB) and Romane (ROM). MBB is a tropical sheep breed, which is characterized by adaptation to heat-stress, to parasitism and to extensive raising conditions. The ROM sheep breed features good productive performances (both for meat production and prolificacy) and no selection for resistance to parasites. The pedigree used in the analysis Figure  was thre-6) and (3) quality control of the genotypes resulted in 42 469 SNPs that comply with all quality checks. More details about the genomic information can be found in Sallé et al. [A total of 1044 animals among the population of the back-cross lambs and their four F1 sires were genotyped with the OvineSNP50 Beadchip . Quality control of the SNPs included the following tests: (1) 50 animals were genotyped twice in order to assess the technical reliability of the genotyping, which resulted in a value of 99.9%, (2) individuals with a call rate below 98% and SNPs with a call rate below 97% or with a minor allele frequency below 1% or featuring a deviation from expected heterozygosity or showing mendelian inconsistencies were discarded  and aroT strain . The aveA transformation for scaling the standard deviation to 1 and avoiding 0 s was applied on growth traits as well. The observations outside a range of 2.96 standard deviations around the average of each trait were considered atypical and excluded from the analysis. Finally, only the animals featuring a valid observation both on genotyping and on at least one trait were included in the analysis, which resulted in 40 animals being discarded and 1004 animals being included.Estimation of the genetic parameters was performed by considering the back-cross population as a single breed population. The breed proportions are taken into account by the genetic relationship matrix in the model including SNP information but could not be taken into account in the model including pedigree information only due to convergence failure. The heritability of each trait and both the genetic correlation and phenotypic correlation between each pair of traits were estimated by bivariate animal mixed models, which were solved by the AIREML procedure implemented in AIRemlF90 software . This soy1 and y2 are the vectors of observations of trait one and two, respectively, X1 and X2 are incidence matrices relating each observation to its respective set of fixed effects and b1 and b2 are the vectors of the fixed effects: weight at weaning (for ADG0 only) or weight at first infestation , contemporary group , sex and feeding mode (breast feeding or bottle feeding). a1 and a2 are the vectors of random animals breeding values, with the associated incidence matrices Z1 and Z2. e1 and e2 are the vectors of random residuals. It is assumed that the random effects are normally distributed and feature the following variance-covariance structure:where I is an identity matrix and T is the genetic relationship matrix between the animals.where T matrix used in the PED model was computed according to Quaas [T matrix used in the SNPED model was computed according to VanRaden [T = 0.95G + 0.05A22, where A22 is the relationship matrix between the genotyped animals computed by using the pedigree information only [G is the genomic relationship matrix among genotyped animals. In the software package used for this study [G matrix is computed by default as follows: G = WK− 1W, [W is a rectangular matrix  with elements: wij = fij − 2pj, where fij is a scalar equal to the number of copies of one allele an animal i has at locus j, pj is the frequency of allele j in the population, K is the diagonal matrix of the scaling parameters with elements: Kjj = 2 ∑ pj(1 − pj). The weights of G and are used for bending the genetic relationship matrix and make it positive definite, as required for its inversion [The genetic and phenotypic correlations between each couple of traits were computed using two different genetic relationship matrices: one computed by using pedigree information only (PED model) and one computed by including both SNP and pedigree information jointly (SNPED model). The to Quaas  and withVanRaden , using aion only  and G isis study , the G m WK− 1W,  where W nversion , this isnversion .The precision of the heritability and correlation estimates was computed by estimating their standard error according to the following formulas :where n matrix :SE(rg) is the standard error of the estimate of the genetic correlation, rg is the estimated value of the genetic correlation; are where 1/2 < d < n  [The parameters under study are the ratio of two normally distributed variables (heritability) and the ratio between a gaussian variable over the square root of the product of two gaussian variables (correlations). The sampling distribution of heritability can be approximated to a gaussian distribution under certain specific conditions only . When thdataset) . One thoPedi is the vector containing the n realizations of the estimate obtained from the PED model, Snpedi is the vector containing the n realizations of the estimate obtained from the SNPED model, di the vector of the differences between and each element of Pedi and Snpedi. The confidence interval of the distribution of the difference was computed as above. The null hypothesis “the difference between the estimate obtained from the PED model and the estimate obtained from the SNPED model is 0” was tested against the alternative hypothesis “the difference between the estimate obtained from the PED model and the estimate obtained from the SNPED model is not 0” as above as well.where Table H. contortus free environment was unrelated to the parasite resistance phenotype. However, the estimates of both the phenotypic correlations between ADG1 and FEC1 and between ADG1 and FEC2 were negative: -0.24 (SE = 0.15) and −0.20 (SE = 0.19) for the PED model and −0.23 (SE = 0.11) and −0.19 (SE = 0.11) for the SNPED model. These results suggest an inverse proportionality between the growth rate and the parasite burden, in accordance with the finding that contaminated animals had a slower growth than non-contaminated animals. The average faecal egg count during the second infestation was significantly lower than the faecal egg count during the first infestation , which suggests that the development of a specific immune response was triggered by the first infestation, that enhanced the intrinsic resistance of the animals to subsequent infestations [Table stations . FurtherTable Concerning previous estimates of the genetic correlation between growth traits and faecal egg counts, no other estimations of these parameters based on experimental infestation were found in the literature. Furthermore, previous studies on similar traits based on natural infestations do not show consistency among them -32, whicConcerning the genetic correlations between growth traits (ADG0 and ADG1) and parasite resistance during the naïve infestation (FEC1), the following picture can be drawn. In accordance with the estimates of the phenotypic correlation between ADG0 and FEC1, the genetic correlations between these traits were also not significantly different from 0: -0.52 (SE = 1.06) for the PED model and 0.11 (SE = 0.47) for the SNPED model. The same results were obtained with the estimate of the genetic correlations between ADG1 and FEC1 : -0.19 (SE = 0.80) for the PED model and −0.12 (SE = 0.58) for the SNPED model. These results suggest that during the naïve infestation, the genotype for growth (ADG0 and ADG1) could be expressed independently from the genotype for parasite resistance (FEC1).The results obtained for the genetic correlations between growth traits and the long-term resistance (FEC2) were on the contrary significantly different from 0. On the one hand, the genetic correlation between ADG0 and FEC2 was positive but not significantly different from 0 according to the PED model  and positive  according to the SNPED model. According to the approximate standard errors, the positive estimate obtained with the SNPED model is more reliable and suggests that if animals were selected for growth in a parasite-free environment, a correlated selection response for lower long-term resistance to gastrointestinal parasites could occur as well. This estimate supports the hypothesis that enhancing growth traits could come to a cost to the sheep's long-term susceptibility to parasite infestations, and vice versa ,50. On tThe genetic correlation between FEC1 and FEC2 was 0.46 (SE = 0.43) and 0.62 (SE = 0.20) for the PED and SNPED models, respectively, which suggests that these traits have different determinisms. While FEC1 represents a measure of the parasite resistance expressed by a naïve lamb, FEC2 is a measure of the parasite resistance expressed by an immunized lamb, and indeed the mechanisms by which these types of animals respond to the infestation are different .The estimate of genetic correlation between ADG0 and ADG1 was unstable due to its sensitivity to the starting values used for its estimation and is not reported.Table The results obtained show that including SNP information in the computation of the relationship matrix between individuals can increase the precision of the genetic parameter estimates up to twice the precision obtained by using pedigree information only . The incH. contortus free environment could result in more susceptible animals, whereas growth performance in a contaminated environment can be increased while enhancing long-term resistance to H. contortus. The two results taken together can also be interpreted as an indication of genotype by environment interaction affecting growth expressed across the two environments [According to the results obtained by the model including both pedigree and molecular information, the genotypes for growth and for resistance to naïve infestation can be selected for independently. However, the genetic correlations between long-term parasite resistance traits and growth traits were different from 0 and suggest that increasing growth performance in a ronments . The modThis study shows that, when the pedigree is poorly informative using molecular information and pedigree information jointly result in more precise genetic parameters than using pedigree only.The authors declare that they have no competing interests.FA edited the observations on average daily gain, estimated the genetic parameters and their standard errors, interpreted the results and wrote the manuscript. CR supervised the PhD project, helped with performing the statistical analysis and with the interpretation of its results and revised the manuscript. JME, AL and CRG helped with performing the statistical analysis, interpreting its results and revising the manuscript. CC programmed the Jackknife procedure in parallel and helped with the interpretation of the results and manuscript review. GS performed the data editing of both the observations on faecal egg counts and of SNP data, of results interpretation and manuscript review. All authors read and approved the final manuscript.


KITLG) and Sex Hormone Binding Globulin (SHBG) using 210 families . This finding supports a role of maternal hormones in offspring testicular cancer and warrants confirmation in a larger dataset.The analyses of genetic interaction between maternal and offspring genotypes are usually conducted considering a single locus. Here, we propose testing maternal × offspring (M×O) and maternal × maternal (M×M) genotype interactions involving two unlinked loci. We reformulate the log-linear approach of analyzing cases and their parents (family trios) to accommodate two loci, fit fuller models to avoid confounding in a first analysis step and propose that the model be reduced to the most prominent effects in a second step. We conduct extensive simulations to assess the validity and power of this approach under various model assumptions. We show that the approach is valid and has good power to detect M×O and M×M interactions. For example, the power to detect a dominant interaction relative risk of 1.5 (both M×O and M×M) is 70% with 300 trios and approaches 100% with 1,000 trios. Unlike the main effects, M×O and M×M interactions are conditionally independent of mating types, and consequently, their power is not affected by missing paternal genotypes. When applied to single-locus M×O interaction, our method is as powerful as other existing methods. Applying the method to testicular cancer, we found a nominally significant M×M interaction between single nucleotide polymorphisms from C-Kit Ligand ( It has been suggested that maternal effects play an important role in species adaptation , and theDissecting maternal effects is attracting renewed attention, and novel statistical methods and designs, such as those exploiting Assisted Reproduction Technology (ART), are emerging . ART allKITLG), and the accumulation of this growth factor eventually transforms PGCs into tumorigenic cells [KITLG are the most associated with TGCT in genome-wide association studies (GWAS) [KITLG. The purpose of this study is to investigate the application of log-linear methods to test the interactions between a maternal genotype at one locus and a child genotype at another unlinked locus and between two unlinked maternal loci. We use a method originally proposed to analyze gene-environment interactions [KITLG.The risk of developing testicular germ cell tumor (TGCT) has long been associated with antenatal exposure to maternal hormones. When exposed to high levels of estrogens in vitro, primordial germ cells  start transcription of C-kit ligand (ic cells . We and s (GWAS)  and it cractions , consideM2), the 15 categories can be stratified on the second locus genotypes and the proportions of family trios falling in each of the new defined categories is given in iS and main offspring effects jR, where i, j represent the number of susceptibility alleles carried by mother and child, respectively. iS and jR are the relative excess of disease risk in the offspring conferred by maternal and offspring disease alleles. Scaling factors μm are population frequencies of the mating types.We assume that family trios have been genotyped at a first locus and that the mothers are additionally genotyped at a second locus, both unlinked and in linkage equilibrium with the first locus. Although some of the tests we propose rely only on the assumption of linkage equilibrium, M×O and M×M cannot be tested simultaneously without imposing the absence of both linkage and linkage disequilibrium (LD). Therefore, throughout this article when we say “unlinked” we also imply absence of LD. Originally, the log-linear method was designed to analyze genetic effects from one locus in cases, their mothers, or both, and possibly subject to imprinting . Under tik coefficient for M×M interaction between the two loci and Φjk for M×O interaction . Stratifying the expected cell counts by the second locus implies that each of the new cells must be weighted by the population genotype frequencies kp at this locus. Using Mni,F = f,C = j,M2 = k=  is the expected number of family trios with specific maternal (M), paternal (F), and child (C) genotype at locus 1 and maternal (M2) genotype at locus 2. Mating type index m on the right-hand side of the equation ranges from 1 to 6 as  on the left-hand side takes the values , , , , , and . Regression coefficients αi, βj, ϕik, and φjk are, respectively, the natural logarithm of disease relative risks iS, jR, ϑik, and Φjk of 0 = β0 = 0, ϕik = 0 if i = 0 or k = 0, and φjk = 0 if j = 0 or k = 0. The last term of the equation represents a “weight” or “offset.” There are two different ways giving rise to a heterozygous offspring when both parents are heterozygous, and they are equally likely in the absence of imprinting. In this case, the cell count is weighted by a factor of 2, which is reflected in the “offset” . In addition, with a second locus, each cell count is proportional to the genotype frequencies kp as shown in We introduce two new sets of parameters: ϑk is the natural logarithm of τk shown in k at locus 2, assuming the absence of effect from the baseline genotype (δ0 = 0). Although we are not interested in this effect, its inclusion in the model is necessary to ensure that the interaction test does not depend on genotype frequencies. On closer inspection, it can be seen that each of the new cells created by stratifying on locus 2 is weighted by genotype frequencies independently of coefficients ϑik and Φjk (k itself when the genotype frequencies are correctly specified in the offset term (see Results).Parameter δ and Φjk . It foll and Φjk  is validk, k = 1,2) instead of one level. Their term, which is the natural logarithm of the ratio of exposed to unexposed individuals in the population, would be written as ln(k/pp0) in our notation. The i is then equivalent to ln(μm) + ln(p0) in our notation.Equation  requiresk equals ln(k/pp0), k = 1,2. Using the ratio of frequencies in the model ensures that their sum is constrained to equal to 1.The locus 2 genotype frequencies could be estimated, instead of specifying them in an offset term as in Equation . Howeverm can be expressed as a function of one allele frequency if HWE is assumed at locus 1 [iS and jR that depend directly on the mating types, estimates of the interaction terms ϑik and Φjk are conditionally independent of these nuisance parameters when iS and jR are fitted in the model. To see this, note that in ik and Φjk are defined in columns where M2 = 1 and M2 = 2 and they are tested against the baseline column M2 = 0. They are independent from mating type strata because the same coefficient μm multiplies cell counts in these three columns on each row of An alternative approach to reduce the number of parameters is to use Equation  and assu locus 1 . Howeverij in Equation . A likelihood ratio test of the interaction term ρij is equivalent to the multinomial test used in the EMIM software [Using Equation  or EquatEquation , definedsoftware  and to tsoftware  implemensoftware . ik and Φjk are conditionally independent of the mating type strata when iS and jR are included in the model. In other words, missing fathers cause an ambiguity of mating types that could have an impact on the power to detect main effects iS and jR, but the interactions ϑik and Φjk are completely unaffected. However, when the missing parent is the mother, it is reasonable to assume that her genotypes will be unavailable for both loci, in which case the data contain little information about locus 2. We investigate the method's efficiency for M×M and M×O interactions when the EM algorithm is used to analyze trios with missing parental genotypes. More information on the application of the EM algorithm to incomplete family trios can be found in The EM algorithm is a powerful approach to jointly analyzing full trios and incompletely genotyped trios in the log-linear framework . For theWe designed simulations to evaluate the type 1 error and power of Equation  to test iS and jR of similar magnitude; and  and rs6259, a missense SNP from Sex Hormone Binding Globulin . The two KITLG SNPs were the most significant in a TGCT GWAS and showed independent signals in multivariable analysis (KITLG SNPs was estimated using 640 subjects (370 case parents and 270 unrelated controls) with STATA add-on command pwld.We applied two-locus log-linear models to 210 families (147 full trios and 63 case-mother pairs), testing the interactions between analysis . The LD P-value for the difference is 0.14). Results also confirm that the interaction tests based on Equation , the power is substantially increased, as can be seen in The validity, power, and estimation accuracy of the M×M and M×O interaction tests using Equation When the correct model for the interactions is codominant, assuming a dominant model in the analysis does not invalidate the test, nor does it cause a power loss . However, as expected, the parameters estimation becomes inaccurate when the effects exist .KITLG rs1508595 and rs995030 are in strong LD . In the original results from the GWAS [KITLG genotypes. Whichever SNP analyzed showed a protective dominant minor allele relative risk R≍0.40 (P ≤ 4 × 10−4), confirming the GWAS results. The second most significant effect was the M×M interaction involving rs1508595 and rs6259 . In a model restricted to the two most prominent effects, the effect of inherited genotypes in KITLG is maintained, and the M×M interaction is also nominally significant between rs1508595 and rs6259 (P = 0.03). In an analysis assuming a multiplicative model for the main effects and dominant interaction terms, the results were very similar, reflecting the low number of minor homozygotes (results not shown).Despite showing independent effects on TGCT in GWAS, we found that the GWAS , where mMaternal risk factors for child disease include environment exposures and genetic effects  and it has been suggested that maternal genetic effects might contribute to the so-called “missing heritability” . DiffereKITLG and SHBG.Overall findings of this study are that: (1) two-locus log-linear modeling is effective in detecting M×O and M×M interactions; (2) similar power can be achieved to detect M×O and M×M interactions of similar strengh; this power can be substantial, although lower than the power to detect the main effects; (3) M×O and M×M interactions are conditionally independent from paternal genotypes but are inflated by missing maternal genotypes; (4) in scenarios where HWE applies, its assumption increases the power to detect main effects but does not affect the interaction tests; (5) M×O and M×M tests are robust to the violation of mating symmetry assumption but, as has already been observed in single-locus models, the main maternal effect needs to be included in the model, although it is not itself robust to the violation of this assumption ; (6) in The main limitation of the approach is sample size requirements, which is a common feature of all interaction analyses. However, the power to detect intergenerational interactions is reasonable and comparable to the power to detect intragenerational interactions or single-locus M×O interaction. With 300 trios, a dominant relative risk of 1.5 is detectable with 70% power in the restricted model, while power approaches 100% in samples of 1,000 trios . The fulKITLG variants  have a strong protective effect against the disease . The method further detected a nominally significant M×M interaction between KITLG rs1508595 and SHBG rs6259 . A simulation of similar conditions to this real dataset shows that we had 60% and 77% power to detect this interaction in full and reduced models, respectively. There was no significant main maternal effect, and a model including imprinting showed that it was not significant. TGCT has an early age of onset and a good response to chemotherapy [KITLG is one of the largest published so far and it has been replicated within and across studies. These unusual characteristics may derive from complex biological mechanisms such as the one reported here, although a larger sample is needed for confirmation. Both KITLG and SHBG have been previously implicated in TGCT, and if their interaction could be verified it would consolidate the observation made in in vitro studies that KITLG is activated by estrogen.The application of the method to TGCT confirmed the finding of two genome-wide studies: inherited otherapy . Its famIt has been demonstrated for single-locus analysis that power and the ability to check assumptions can be enhanced by supplementing the case-parent trios with unrelated controls and one or both parents [see e.g.,


U-test for gene-based association analysis. The method considers both rare and common variants. It adaptively searches for potential disease-susceptibility rare variants and collapses them into a single “supervariant.” A forward U-test is then used to assess the joint association of the supervariant and other common variants with quantitative traits. Using 200 simulated replicates from the Genetic Analysis Workshop 17 mini-exome data, we compare the performance of the proposed method with that of a commonly used approach, QuTie. We find that our method has an equivalent or greater power than QuTie to detect nine genes that influence the quantitative trait Q1. This new approach provides a powerful tool for detecting both common and rare variants associated with quantitative traits.We propose a novel aggregating  In the past decade, extensive genome-wide association studies (GWAS) have been conducted to understand the genetic etiology of complex diseases. However, the common variants identified so far explain only a small fraction of the variations of complex diseases . It now The most commonly used approach for detecting the association between rare variants and disease outcome is to collapse multiple rare variants into a single “supervariant,” which is tested further as a common variant. Based on this idea, Li and Leal developed a combined multivariate and collapsing (CMC) method for rare variant analysis . This meU-test to examine the association between the quantitative traits and multiple genetic variants, including both rare and common variants. First, the method adaptively collapses the disease-susceptibility rare variants into a supervariant; it then searches the supervariant and the remaining common variants for the best multi-SNP (single-nucleotide polymorphism) combination, using a forward selection. We have applied our method to the Genetic Analysis Workshop 17 (GAW17) mini-exome data and compared its performance with QuTie.In this paper, we propose an aggregating U-test to detect gene-gene interactions using general multisample U statistics [U statistics and then explain the aggregation of rare variants and the forward selection of multi-SNP combinations using U statistics.We have recently developed a forward atistics . Here, wN subjects. Let iY denote the observed value of the quantitative trait for the ith individual ; and let iX = i1X, i2X, … , iKX, denote the genotypes of K SNPs for the ith individual, each taking its value from one of the three possible genotypes, ijX ∈ {AA, Aa, aa}, j = 1, 2, …, K. Without loss of generality, we assume that a is the minor allele, and the first r SNPs  are rare variants.Suppose that we have a study population of L multi-SNP genotypes formed by k SNPs of interest, denoted as G1, G2, …, LG. A multi-SNP genotype lG is defined here as a vector of the k genotypes that an individual carries . The k SNPs and L multi-SNP genotypes are selected sequentially out of a total of K genotyped SNPs (see the “Forward U-test” section for details). Let:Suppose that we have lG, and let lm = |lS| be the number of subjects in lS. We measure the trait difference between two sets of subjects lS and l′S as:be the set of subjects carrying multi-SNP genotype where the kernel function is chosen as:l,l′U, is the summation of all possible pairwise trait comparisons for any two subjects from lS and l′S. In the presence of an association, we would expect individuals carrying different multi-SNP genotypes to have different trait values . Based on l,l′U, we can form the global U statistic. We assume that the expected quantitative trait value of the L multi-SNP genotypes decreases with l  when L = 2.Here, the weight parameter g1 = Aa | aa, g2 = AA}, for which a U statistic can be calculated by using Eq. (2). We rank the U statistics in decreasing order as U(1), U(2), …, U(r). Assume that V(1), V(2), …, V(r) are the corresponding rare variants in a candidate gene and that iX(1), iX(2), …, iX(r) are their observed genotypes for individual i. We start from variant V(1), and define a supervariant as:When dealing with a large number of rare variants, it is likely that a significant proportion of the rare variants will not be associated with a disease; thus collapsing on a selected subset of rare variants will be necessary. Each rare variant can form two single-SNP genotypes, {U statistic to the supervariant. Accordingly, we redefine the supervariant as:At each step of the aggregation process, we add a rare variant with the largest ijR = 1, ijR = 0}, for which a U statistic can be calculated using Eq. (2). The collapsing procedure stops at step t, where the U statistics start to decrease  always forms two different genotypes, {U-test is used to evaluate the supervariant and other common variants for their joint association with the trait [j can form two single-SNP genotypes, K − r) + 1 possible grouping strategies, which can be represented by lth multi-SNP genotype at step s. A U statistic can be calculated for two sets of subjects grouped by U statistic value is selected, and the corresponding grouping is recorded.A forward he trait . We starj′ is chosen to form four two-SNP genotypes, denoted by {U statistics for each of these grouping strategies using Eq. (4) and choose the one with the largest U statistics. It should be noted that, if the same SNP from step 1 is chosen in step 2, then only three single-SNP genotypes will be formed, denoted by {U statistic is expected to increase until groups cannot be split further. This results in a series of models with different numbers of groups. The best model, with the appropriate number of groups, can be determined by using a 10-fold cross-validation procedure. The U statistic of the best model is calculated using the whole data set, and the significance of its association can be obtained using permutation. For each permutation replicate, the same procedure (including the aggregation process and model selection) is applied to calculate the U statistics. An empirical p-value, which accounts for inflated type I error resulting from model selection, can be calculated by using a large number of permutations.In the second step, based on the first selected SNP, a second SNP U statistics. We then evaluated the power of our method by counting the number of replicates whose U statistics exceeded the 95th percentile of the null distribution. A similar analysis was also conducted using QuTie, version 0.2. The threshold for rare variants was chosen as a minor allele frequency (MAF) less than 0.01. The performance of the two methods varied according to the number of SNPs within the genes, the number of causal SNPs, and their effect sizes. We divided the nine genes into five groups accordingly , were mitigated by collapsing them with other SNPs, which led to low power by QuTie. In addition, both methods had high power to detect the association for genes in group 4, whereas QuTie attained higher power than the aggregating U-test. Because most of the rare variants in the gene were causal and their effects were relatively large, it would be ideal to collapse all the variants. In such a case, the aggregation would have a smaller advantage because the selection process would introduce additional variations. However, we believe that this is not a common scenario in a real data application. Finally, both methods had low power to detect the association for genes in group 5. For both genes, the selection of rare variants did not show any advantage because of the low effect of each functional rare variant.We found that both methods had a high power to detect the association for genes in group 1 and group 2. As a special case, gene U statistic has a mean of zero and follows a normal distribution.Our method has two major advantages: (1) It can substantially improve the testing power when only a small proportion of rare variants under examination are functional; and (2) it collapses only a subset of selected rare variants that are potentially trait related. Therefore it can help to identify disease-susceptibility rare variants, which makes our results easier to interpret and replicate in follow-up studies. The existing methods, such as QuTie, collapse all rare variants within the same genomic region and analyze the rare and common variants without differentiating the functional and nonfunctional variants. As illustrated by our analysis, such methods are subject to low power when a significant proportion of the variants in the genomic region are not trait related. Nevertheless, they could have comparable or even higher power over our method when most  of the variants within a gene are trait related. Moreover, because existing methods do not adopt a model selection algorithm to eliminate the noise loci, they are computationally faster than the proposed method. However, if we make the same assumption as existing methods , then we could also use the asymptotic result of the proposed method to test for association without model selection and permutation . Under tU statistic first increases by adding the risk rare alleles and then decreases when nonrisk rare alleles are added. On the contrary, the number of genotype groups in the forward selection process keeps increasing as the algorithm moves forward, which results in an increasing global U statistic . Therefore the cross-validation procedure is necessary for forward selection to avoid overfitting the data. The cross-validation procedure, however, is not practical for the aggregation of rare variants, because their low MAFs may cause the absence of rare alleles in the testing set.We also note that the aggregation of rare variants is different from the forward selection of multi-SNP genotypes. During the aggregation process, the supervariant always forms two genotype groups: one with all rare alleles and the other without any rare allele. The corresponding U-test provides a powerful tool for genetic association studies with both common and rare variants. Our method could also be useful for identifying disease-susceptibility variants underlying quantitative traits.The proposed aggregating The authors declare that there are no competing interests.ML developed the method, carried out the analysis and drafted the manuscript. WF participated in the design and helped to draft the manuscript. QL conceived of the study, developed the method, participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.


Family history and African-American race are important risk factors for both prostate cancer (CaP) incidence and aggressiveness. When studying complex diseases such as CaP that have a heritable component, chances of finding true disease susceptibility alleles can be increased by accounting for genetic ancestry within the population investigated. Race, ethnicity and ancestry were studied in a geographically diverse cohort of men with newly diagnosed CaP.Individual ancestry (IA) was estimated in the population-based North Carolina and Louisiana Prostate Cancer Project (PCaP), a cohort of 2,106 incident CaP cases (2063 with complete ethnicity information) comprising roughly equal numbers of research subjects reporting as Black/African American (AA) or European American/Caucasian/Caucasian American/White (EA) from North Carolina or Louisiana. Mean genome wide individual ancestry estimates of percent African, European and Asian were obtained and tested for differences by state and ethnicity (Cajun and/or Creole and Hispanic/Latino) using multivariate analysis of variance models. Principal components (PC) were compared to assess differences in genetic composition by self-reported race and ethnicity between and within states.p = 0.03) and EA (p = 0.001). This geographic difference attenuated for AAs who answered “no” to all ethnicity membership questions  but not EA research subjects, p = 0.002. Mean ancestry estimates of self-identified AA Louisiana research subjects for each ethnic group; Cajun only, Creole only and both Cajun and Creole differed significantly from self-identified non-ethnic AA Louisiana research subjects. These ethnicity differences were not seen in those who self-identified as EA.Mean individual ancestries differed by state for self-reporting AA (Mean IA differed by race between states, elucidating a potential contributing factor to these differences in AA research participants: self-reported ethnicity. Accurately accounting for genetic admixture in this cohort is essential for future analyses of the genetic and environmental contributions to CaP. Prostate cancer (CaP) is the most common cancer diagnosed in men and the second leading cause of cancer death among men in the US, with African American (AA) men having substantially higher CaP incidence and mortality rates than men self-reporting as European American (EA). CaP is a multifactorial disease with both genetic and environmental components. Familial aggregation has been demonstrated in both AA and EA Two main methods have been used to summarize individual ancestry in population-based studies: (a) self-identified race and ethnicity and (b) Ancestry Informative Markers (AIMs) genotyped in the population under study The PCaP cohort is racially and ethnically diverse and was designed to investigate the contribution that social, biological, and environmental factors make to observed racial differences in CaP mortality in AAs and EAs in the United States. Specifically AAs from North Carolina and Louisiana have, respectively, some of the highest and lowest AA CaP mortality rates in the United States, while EA men in the two states have similar CaP mortality that is less than either AA group. One PCaP research hypothesis was that the higher CaP mortality rates could, in part, reflect a higher proportion of African ancestry in AAs from North Carolina vs AAs in Louisiana. However, the genetic background of PCaP research subjects must be carefully characterized in order to examine the molecular and genetic factors associated with susceptibility to aggressive CaP and other CaP phenotypes. The proportion of African, European and Asian genetic ancestry was measured to evaluate differences in African ancestry across and within regions by ethnicity using individual ancestry estimates Written informed consent was obtained from all research subjects prior to blood and questionnaire collection. The study was approved by the University of North Carolina at Chapel Hill (UNC-CH) and Louisiana State University Health Sciences Center (LSUHSC) Institutional Review Boards and the Department of Defense Human Subjects Research Review Board. PCaP is a multidisciplinary study of racial/ethnic differences in social, host, and tumor-specific factors on CaP aggressiveness and outcome http://hapmap.ncbi.nlm.nih.gov) from three populations: Yoruba individuals in Ibadan, Nigeria (YRI) represented African ancestry, Utah residents with Northern and Western European ancestry collected by the Centre d'Etude du Polymorphisme Humain (CEU), represented European ancestry and Japanese individuals from Tokyo, Japan (JPT) and Han, China (CHB), the latter two groups collectively represented Asian (ASI) ancestry. SNPs were selected as follows: twenty-five SNPs had a variant allele frequency (VAF) = 0 in CEU, were rare in ASI, VAF<0.01, but common in YRI with VAF>0.65 and AA VAF>0.25. The other half of the selected SNPs had a VAF = 0 in YRI, were rare in ASI (VAF<0.05), but common in CEU (VAF>0.5). Selected SNPs were at least 10 million base pairs apart.IA estimates obtained using as few as 30 AIMs shows a correlation of approximately .9 with true individual ancestry estimated using a much larger genome wide panel http://hapmap.ncbi.nlm.nih.gov). Forty SNPs passed quality control and greater than 98.6% genotyping success was achieved for all research subjects.DNA was extracted from fresh peripheral blood mononuclear cells (PBMCs), or immortalized lymphoblasts. Genotyping was performed on an Illumina platform at the Center for Inherited Disease Research (CIDR) at Johns Hopkins University as part of a larger genotyping effort Allele frequencies were estimated using maximum likelihood methods. IA proportions for self-reporting AA and EA research subjects were estimated using a Bayesian Markov Chain Monte Carlo (MCMC) clustering algorithm implemented in STRUCTURE 2.3.1 http://cran.r-project.org). Tests of mean CEU and YRI ancestry estimate differences between states were performed using one-way multivariate analysis of variance (MANOVA). As follow-up to the multivariate model, Welch's t-tests were used to test the null hypothesis that there was no difference in mean YRI estimates by state. Within race MANOVA models were constructed to compare ancestry estimates of individuals reporting no ethnicity with those of each ethnic group . All ethnic analyses were limited to research subjects from Louisiana because only 2% of research subjects reporting ethnicity membership were from North Carolina.R Statistical software was used for all analyses comparing research subjects between and within North Carolina and Louisiana . However, there was no significant mean difference in IA by state for either CEU or YRI (YRI only) in non-ethnic AA research subjects, p = 0.87 (p = 0.78). CEU and YRI ancestry estimates differed by state, in both all and non-ethnic self-reporting EA research subjects, p = 0.001 and p = 0.002, respectively. When comparing only the mean proportion of YRI between states in all men self-reporting as EA, there were also significant differences (p<0.0006). However, as with self-reporting AA men, these geographic differences attenuated when comparing mean YRI estimates in non-ethnic EA research subjects, p = 0.06.The multiple STRUCTURE runs yielded likelihood and IA estimates that were very close in value. Likelihood estimates consistently favored a model with two populations (CEU and YRI); these two populations were used in all subsequent statistical calculations. p<0.00001, p<0.00001, p = 0.03, respectively. Mean CEU and YRI ancestry differences in self-reported EA were seen when comparing Hispanics only to non-ethnic EA research subjects (p<0.0001).Mean IA estimates by race and ethnicity are shown in PCs 1–4 sequentially (cumulatively) accounted for 73.2, 6.6 (79.8), 2.8 (82.6) and 2.7 (85.3) percent of the total genetic variation. The amount of variation explained after PC2 appears minimal with constant scree plots  of the eigenvalues becoming essentially constant at PC3 onward Due to the self-reported and genetic diversity in Louisiana, the Wilcoxon rank sum was used to assess differences in the top two PCs by ethnicity. PC1 and PC2 differed significantly  between Creole and non-ethnic AA. As with IA estimates other significant differences in CEU were observed in a series of exploratory analyses. For example, when comparing subjects who reported both Cajun and Creole to the non-ethnic AA, PC1 (p<0.05) and PC2 (p<0.015) were significantly different as was PC1 when comparing Cajun alone and non-ethnic African American men, p<0.0001. PCs 1 and 2 were differentially distributed (p<0.0001 for both PCs) between Hispanic and non-ethnic EA.The ethnicity specific PCs for Louisiana AA and EA are shown in Ancestry estimates, derived using MCMC clustering algorithm implemented in STRUCTURE, were used to evaluate genetic composition between and within self-reported races in North Carolina and Louisiana. Mean IA differences by race between states were small (<3%) but statistically significant. Given the large mean IA differences across ethnic groups within Louisiana genetic heterogeneity appeared to be the main contributing factor to the mean IA differences by state. However, it is possible that other factors are driving these differences. For example, a similar effect would be seen if research participants self-reporting as AA derive ancestry from different areas in Africa and the chosen reference population, YRI, is more similar to that found in one state versus the other.The limitations in using self-reported race to reveal population genetic substructure have been shown repeatedly Characterization of the genetic background that exists at both the population and individual level offers the promise of an improved understanding of the underlying factors leading to differential disease susceptibility and differential response to pharmacological agents, and to disentanglement of the complex interaction between genetic and environmental factors in the disease phenotype. The topics of race and ethnicity continue to be of considerable interest and debate with respect to scientific and medical research Table S1Allele frequencies for AIMs in PCaP cohort by self-reported race.(DOC)Click here for additional data file.


Association tests that pool minor alleles into a measure of burden at a locus have been proposed for case-control studies using sequence data containing rare variants. However, such pooling tests are not robust to the inclusion of neutral and protective variants, which can mask the association signal from risk variants. Early studies proposing pooling tests dismissed methods for locus-wide inference using nonnegative single-variant test statistics based on unrealistic comparisons. However, such methods are robust to the inclusion of neutral and protective variants and therefore may be more useful than previously appreciated. In fact, some recently proposed methods derived within different frameworks are equivalent to performing inference on weighted sums of squared single-variant score statistics. In this study, we compared two existing methods for locus-wide inference using nonnegative single-variant test statistics to two widely cited pooling tests under more realistic conditions. We established analytic results for a simple model with one rare risk and one rare neutral variant, which demonstrated that pooling tests were less powerful than even Bonferroni-corrected single-variant tests in most realistic situations. We also performed simulations using variants with realistic minor allele frequency and linkage disequilibrium spectra, disease models with multiple rare risk variants and extensive neutral variation, and varying rates of missing genotypes. In all scenarios considered, existing methods using nonnegative single-variant test statistics had power comparable to or greater than two widely cited pooling tests. Moreover, in disease models with only rare risk variants, an existing method based on the maximum single-variant Cochran-Armitage trend chi-square statistic in the locus had power comparable to or greater than another existing method closely related to some recently proposed methods. We conclude that efficient locus-wide inference using single-variant test statistics should be reconsidered as a useful framework for devising powerful association tests in sequence data with rare variants. The advent of high-throughput sequencing technologies is providing an unprecedented opportunity to examine the association of both common and rare variation with disease on an exome-wide, and soon genome-wide, scale. In this study, we consider the problem of using sequence data from a case-control sample to perform a test for association between a disease and a locus, which we define as a region of contiguous sequence including many variants . These variants may be either common or rare; those with minor allele frequencies (MAFs) <2–3% To mitigate the power loss due to allelic heterogeneity D<0), even at very high levels of recombination The power of pooling tests will depend on the linkage disequilibrium (LD) patterns in sequence data. Simulations using a coalescent approximation to a neutral two-locus Wright-Fisher infinite allele model have shown that a substantial proportion of the pairwise LD between biallelic variants can be expected to be negative  and accounts for LD-induced correlations between test statistics Nonetheless, many novel methods have not departed from the pooling test framework but rather attempted to devise improved weighting schemes and adaptive thresholds that reduce the influence of neutral and protective variants In this study, we compared two existing methods for locus-wide inference using nonnegative single-variant test statistics to the two originally proposed pooling tests CSRP3, LDB3, MYH7, SCN5A, TCAP, and TNNT2) previously obtained for a study of candidate genes for dilated cardiomyopathy P≥0.001), including both SNPs and small insertions/deletions, in calculating the MAF and LD distributions. Pairwise LD measured by the correlation coefficient between major alleles, r, was calculated only between variants within the same gene under the assumption of HWE using the method of Weir and Cockerham To ground our interpretation of analytic power approximations in actual data and provide a basis for evaluating the realism of our simulated data, we estimated the MAF and LD distributions of variants in resequencing data from six genes  in the set of all w = 3v possible combinations at each of the v variants . If we consider a single variant and use scores equaling the number of minor alleles at that variant, we have k = 1, 2, or 3, and T in (1) reduces to the single-variant Cochran-Armitage trend chi-square statistic.We began by examining the relative performance of pooling tests and locus-wide inference using nonnegative single-variant test statistics in a simple model of a locus comprising one rare risk variant and one rare neutral variant. Specifically, we compared the locus-wide analytic power of collapsing and summing to that of performing Bonferroni-corrected inference on the maximum single-variant Cochran-Armitage trend chi-square statistic in the locus (the BC-CA test) under our model at varying levels of LD. The Cochran-Armitage test for a single variant is well-known in the field of genetics j = 1) and one rare risk (j = 2) biallelic variant with the same MAF (p) was used. In this model, frequencies of haplotypes  for a given level of LD (D) were calculated as (1-p)2+D for haplotype , (1-p)p-D for haplotypes  and , and p2+D for haplotype kG = , denoted by kp, were then determined under HWE. A multiplicative risk model, A or C denoting affection status (affected case or control), 0f denoting the penetrance for the variant 2 major allele homozygote, and γ denoting the relative risk for an additional minor allele at variant 2. The conditional frequencies of each kG in cases (k|Ap) and controls (k|Cp) were determined based on this risk model and population multi-variant genotype frequencies using Bayes' rule:k. Under the alternative hypothesis, at least one variant in the locus influences disease risk, implying that k.A simple model of a locus comprising one rare neutral (U in (1) with arbitrary scores kX when v = 1. These formulas immediately generalize to multinomial genotype distributions with more than 3 possible genotypes (w>3):U is T has the same asymptotic U is T with type I error rate α is:χ2 CDF with 1 degree of freedom and noncentrality parameter Freidlin et al. kX in the statistic T yield a single-variant Cochran-Armitage trend test, a collapsing test, and a summing test. This correspondence allows us to use the generalized formulas presented above to approximate the locus-wide asymptotic power of each approach to association testing in our model. The single-variant trend test uses j; collapsing defines I(E) = 1 if E is true and 0 otherwise; and summing defines We show in Appendix A  that parpendix B , we estaf0 = 0.05 and two types of rare variant pairs . Balanced case-control samples with total sizes of N = 500, 1,000, and 2,000 were considered. Values of r were chosen by taking 100 evenly spaced increments of D starting from Dmin = −p2 and ending at Dmax = p(1-p).We examined power as a function of We used Monte Carlo simulations to extend the conclusions drawn from the two-variant locus model to a larger locus with heterogeneous risk alleles, extensive neutral variation, realistic LD patterns, and randomly missing genotype data. We begin by outlining the major components of the simulation procedure and subsequently provide more detailed exposition for each component. One thousand populations of haplotypes at a 100 kb locus with realistic MAF and LD spectra were first generated based on a neutral coalescent model. Using the same 1,000 haplotype populations, a separate simulation was then conducted for each combination of user-specified risk variant parameters and sample size. Within each simulation:The disease model for each haplotype population was generated by randomly selecting risk variants for inclusion based on user-specified parameters;A case-control sample was drawn from each haplotype population according to the disease model and a user-specified sample size;Data sets with randomly missing genotypes were generated from each sample for each user-specified call rate; andAll association testing techniques were applied to each data set.N = 500, 1,000, and 2,000. The disease risk for the multi-variant genotype with no minor alleles at any risk variant was 5% for all simulations. For power, 50 risk variants with independent effects were randomly selected in each haplotype population under three different disease models:Type I error and power for each technique were estimated for balanced case-control samples of total sizes Multiple rare risk variants  = 3);Multiple rare risk variants ;Combinations of multiple rare risk variants , low-frequency risk variants , and common risk variants .The first two models represent situations in which pooling tests are expected to perform best, and the third model is included to consider cases where both common and rare variants might contribute to disease susceptibility. We chose a number risk variants that represented ∼5% of all variants at the locus in the average haplotype population to reflect a situation in which associations between the locus and disease were driven by heterogeneous risk alleles characterized by a small number of risk variants among a much larger number of neutral variants. Per-position genotype call rates of 100% (complete data), 99.5%, and 95% were simulated. In the following subsections, we provide a detailed description of each component of the simulation procedure.−8 estimated assuming an effective diploid population size of eN≈10,000 −8 between adjacent nucleotides derived by using the approximation 1 cM≈1 Mb eN = 10,000 . We used the program MS One thousand populations of 10,000 haplotypes each were generated at a 100 kb locus, which is representative of a larger human protein coding gene based on recent data  kG was determined according to a logistic penetrance model of the form:B = T is the vector of log odds ratios for the haplotype population. The odds ratio j. This model implicitly assumes that (1) each additional minor allele has a multiplicative effect on the odds of disease and (2) this effect at variant position j is independent of the effects at other variant positions.Disease risk for the multi-variant genotype c, where class c was defined by a half-open MAF range, B was populated by repeating the following steps until the specified number of risk variants was selected:To parameterize this model, we specified the desired number of risk variants as well as a set of risk variant classes indexed by j was randomly selected from among the variants in the haplotype population;A variant j had a population MAF in the interval c and had not already been designated a risk variant, then it was labeled a risk variant and assigned a coefficient If the randomly selected variant jβ = 0 (cθ = 1).This procedure effectively randomly samples risk-variant classes from the haplotype population in proportion to the occurrence of each MAF range in the population. All neutral variants had B was populated for a haplotype population, a case-control sample was generated according to the disease model. To generate a case-control sample, the following procedure was repeated until the user-specified numbers of case and control subjects were selected:After the vector of log odds ratios kG;Haplotypes were randomly drawn with replacement to form an individual's multi-variant genotype P(A|kG), was then calculated according to the logistic model in (2);The disease risk of the individual's multi-variant genotype, A with probability P(A|kG) or C with probability 1-P(A|kG).The individual was randomly assigned affection status kG by randomly drawing haplotypes with replacement implicitly assumes random mating in the population.Our method of forming O) or “missing” (M) at each position defined by a single base pair. Given a call rate of λ<1 per base pair, the number of base pairs that a chain remains in M before a genotype is called is distributed Exp(λ), assuming a sufficiently long sequence for continuous measurement of base pair position to be reasonable. At the position of this called genotype, the state of the chain changes from M to O with probability 1. Because genotypes are missing at a rate of 1-λ per base pair, the number of base pairs that the chain remains in O before a genotype is missing is distributed Exp(1-λ), and the chain transitions from O to M with probability 1 at this position. If we rescale position to  by measuring in units of L base pairs, where L is the total sequence length, the distance the chain remains in M is distributed Exp(Lλ), and the distance the chain remains in O is distributed Exp(L(1-λ)). It can be shown that the expected proportion of the sequence length that a Markov chain with these transition rates and probabilities spends in O is simply the call rate, λ. Thus, for each call rate, the following steps were performed in each individual to generate the observed genotype data:For each case-control sample, data sets with different rates of randomly missing genotypes were generated based on user-specified per-base-pair call rates. The observation process over the sequence was modeled as a two-state Markov chain with states “observed”  in the locus was greater than or equal to the Bonferroni-corrected quantile of the asymptotic null v is the number of polymorphic variants in the data set.The most widely known method is the BC-CA test presented above, which uses a conservative approximation that does not make efficient use of the single-variant information for joint inference TT values T was obtained by repeatedly randomly shuffling affection status labels, calculating all single-variant T values, and recording the resulting value of the max T statistic. Letting tQ denote the value of the max T statistic in permutation t and obsQ denote the observed value in the sample, the two-sided p-value is estimated from m permutations as m = 10,000 permutations. We will refer to the inferential procedure based on the permutation p-value of the max T statistic as the CA max test.A second popular method involves performing locus-wide inference based on the permutation null distribution of max T) over the locus, which is also efficient because LD-induced correlations between single-variant T values are fully taken into account in the permutation null distribution. Variations on this theme have been proposed for candidate gene association studies T statistic was obtained using the same permutation procedure as for the CA max test. We will refer to the inferential procedure based on the permutation p-value of the sum T statistic as the CA sum test.A final method involves performing locus-wide inference based on the permutation null distribution of the sum of Cochran-Armitage trend chi-square statistics (sum jU be the score statistic U in (1) for a single variant j with additive scores j. In the absence of covariates and with complete genotype data, the SKAT statistic can be expressed as j, SKATQ is equivalent to the C-alpha statistic SKATQ is equivalent to the sum T statistic  or was performed with only a very small number of effective ddf. We considered only results from F tests with effective ddf>4 in our Type I error and power estimates because (1) our testing indicated that algebraically identical generalized inverses could yield different numerical results with effective ddf≤4 and (2) the expectation and variance of the F distribution only exist for ddf >2 and ddf >4, respectively Only individuals with complete genotype data at common variants could be used in calculating Hotelling's W. Second, we used a two-sided p-value. A one-sided p-value will only be well-powered for a deviation from the null in which the cumulative number of minor alleles at lower-frequency variants is higher in cases than controls. However, any departure from the null of equal genotype frequencies in cases and controls at the locus is of interest in association testing, which is why the BC-CA, CA max, CA sum, and CMC tests all use two-sided p-values. Therefore, one would also want to be able to detect deviations in which controls have a higher cumulative number of minor alleles at lower-frequency variants, which is not possible with a one-sided WSS p-value. Such deviations could arise in plausible situations, such as one in which the minor allele of a rare risk variant with a strong effect appears exclusively on a haplotype with few other minor alleles. Third, we estimated the two-sided p-value directly from the permutation distribution of W. Letting tW denote the value in permutation t, obsW denote the observed value in the sample, and W over all m permutations, the two-sided p-value was estimated from m = 10,000 permutations as Our implementation of the WSS method P≥0.001) at 211 of these variants, which were carried forward to the analysis of MAF and LD distributions.We began by analyzing the MAF and LD distributions of variants in actual sequence data from six candidate genes for dilated cardiomyopathy. These genes spanned a total of 236,059 base pairs (bp), of which 53,466 bp were scanned for variation. A total of 215 biallelic variants were identified in 184 Coriell samples of white European ancestry, yielding approximately 4 variants per kb scanned. We found no evidence against HWE , the worst-case power of the BC-CA test was greater than that of the collapsing and summing tests for all r≤0.08 under both models considered  and large (r≤0.52) samples  or kX = 0, 2, or 4 (summing) when 2kG = 0, 1, or 2, respectively. Under these circumstances, the collapsing and summing tests are equivalent to level α single-variant Cochran-Armitage tests for the risk variant using scores for a dominant model and an additive model, respectively. Because these tests are not Bonferroni corrected, their power when r = 1 is substantially above the lower bound for the BC-CA test, which is based on a level α/2 single-variant Cochran-Armitage test for the risk variant using scores for an additive model.We can explain the relationship between the power of the three tests and r decreases, the amount of noise introduced into the collapsing and summing test statistics by including the neutral variant increases and results in a concomitant decrease in power. Moreover, the problem of masking by the rare neutral variant further reduces power when r<0. The worst-case power of the BC-CA test, which inefficiently combines single-variant test statistics, was substantially greater than the power of the collapsing and summing tests for r<0, which is where over 95% of the r values between variants with MAF≤0.01 fell in our actual sequence data. These results suggest that, by eliminating the problems of noise and masking, even inefficient techniques for locus-wide inference using nonnegative single-variant test statistics can yield more powerful tests for association than pooling minor alleles in the presence of rare neutral variants.However, as Monte Carlo simulations were performed to extend the analytic power results to more realistic situations. These simulations were based on case-control samples generated at a hypothetical 100 kb disease locus with heterogeneous risk alleles, extensive neutral variation, realistic patterns of LD, and randomly missing genotypes.The variants in our populations of haplotypes simulated based on a coalescent model closely resembled those analyzed in our actual sequence data. First, the rates of variants per kb were compatible when the sampling process that generated our actual sequence data was taken into account. Each simulated population had an average of 981 variant sites (range: 805–1193) over the 100 kb locus, or approximately 10 variants per kb. While this rate was somewhat higher than the observed rate of 4 variants per kb scanned in our actual sequence data, it was not inconsistent with this observation because fewer variants are expected to be observed in any small sample from a large population. In fact, an average of only 747 variants, or 7.5 per kb, appeared in samples of 500 individuals drawn from these haplotype populations under a null disease model with complete genotype data. A further reduction in the number of variants per kb would be expected in a sample of the same size as our actual sequence data, which was about one-third the size of our smallest simulated samples.Second, the variant MAF and pairwise LD distributions across the populations of simulated haplotypes  closely N = 500, N = 1,000, and N = 2,000; call rates of 100% (complete data), 99.5%, and 95%; and α = 0.05 and 0.01.We evaluated the performance of the BC-CA, CA max, CA sum, CMC, and WSS tests in samples drawn from these haplotype populations according to different disease models. For each disease model, we considered balanced case-control samples with α level under nearly all conditions, reflecting an observed distribution of p-values extremely close to the uniform expected under the null hypothesis. The CMC based on Hotelling's T2 was often conservative in complete data, with 95% upper confidence limits below the nominal α level. Our result agreed with that of Li and Leal T2 to a random vector containing between 5 and 20 rare variants in balanced case-control samples of sizes 500 and 2,000. In our null simulations, Hotelling's T2 was applied to a much larger vector containing at least 110 effectively linearly independent elements in all data sets with complete genotype data. Finally, the BC-CA test nearly always had the lowest Type I error of all tests considered, reflecting conservatism due to failure to account for LD-induced correlations between single-variant test statistics.Under a null model with no risk variants, all techniques controlled type I error at the nominal level . The 95%F tests with ddf>4 for N = 500, and the type I error rate estimates for both α levels had wide 95% confidence intervals.The type I error results for the CMC with missing genotypes also reflect the substantial loss of sample information resulting from having to exclude all individuals with incomplete genotype data at common variants. With a call rate of 95%, no type I error rate could be estimated because no individual had complete data in any of the 1,000 samples and the CMC could not be performed. With a call rate of 99.5%, about 92% of individuals in the average sample were unusable due to missing genotype data for each sample size. For this reason, only 23 of the 1,000 samples had reliable N≥1,000. The CA sum test was always more powerful than the WSS test. Although the CA sum test was more powerful the BC-CA test under all scenarios with α = 0.05, its power deteriorated to below that of the BC-CA test in larger sample sizes with α = 0.01.Under a disease model with 50 rare risk variants , which represent ∼5% of all variants in the locus in the average haplotype population, the CA max test had higher power than the CMC and WSS tests under all conditions . It alsoN = 500, it began to lag the BC-CA test for N≥1,000, sometimes substantially. This observation suggests that, when summing over minor alleles to reduce the number of tests performed, the power gain from reducing the multiple-testing penalty was rapidly outweighed by the power loss due to increased noise and masking as the sample size grew. The WSS test also had lower power than the CMC test in most scenarios with complete data. Because the CMC test collapses over only variants with MAF≤0.01 and analyzes common variants in a manner not subject to masking, it may perform better than the WSS test, which sums over all variants.Although the WSS test was more powerful than the BC-CA test when N = 2,000 with complete data. The CA sum test was more powerful than the BC-CA test under all conditions other than N = 2,000 and α = 0.01. The WSS test also exhibited a similar pattern of performance relative to the CMC and BC-CA tests.N = 2,000 and complete data, the CMC test had comparable power to the CA max test but was still less powerful than the CA sum test. The WSS test showed the same pattern of having higher power than the CMC and BC-CA tests for N = 500 but beginning to lag these tests for N≥1,000.In the disease model with 50 total risk variants randomly allocated between rare variants , low-frequency variants , and common variants , the CA max and CA sum tests were both more powerful than the CMC and WSS tests under nearly all conditions . Under tWe have compared existing methods for efficient locus-wide inference using nonnegative single-variant test statistics to two widely cited pooling tests in terms of their ability to detect associations between rare variants and disease. We began our analysis by exploring the characteristics of variants likely to appear in actual sequence data. Based on these characteristics and a simple model of a locus with one rare risk and one rare neutral variant, we were able to demonstrate that even using Bonferroni-corrected single-variant tests for locus-wide inference may have higher power than collapsing or summing rare variant minor alleles in the presence of a neutral variant. We then simulated populations of haplotypes at a hypothetical 100 kb locus with MAF and LD spectra closely matching those of our actual candidate gene sequence data. We examined power in balanced case-control samples drawn from these simulated haplotype populations according to a disease model with heterogeneous risk alleles and extensive neutral variation. In these simulations, one or more of the existing approaches for efficient locus-wide inference using nonnegative single-variant test statistics, the CA max test or CA sum test, had power comparable to or greater than the CMC and WSS tests under the scenarios considered. Moreover, the type I error and power of the CA max and CA sum tests were robust to randomly missing genotype data, which was not observed with the CMC test. Finally, the CA max test was nearly always more powerful than the CA sum test for disease models with only rare risk variants, suggesting that the CA max test may outperform the class of techniques represented by the SKAT and C-alpha test in these scenarios.Our results contradict those of the original studies Second, we used efficient methods for locus-wide inference based on nonnegative single-variant test statistics that reduce the multiple-testing penalty by accounting for LD-induced correlations between the single-variant test statistics. However, the CMC and WSS tests were both compared in the original studies to the Bonferroni and Dunn-Sidak corrections T2, which rapidly became unreliable with as little as 0.5% randomly missing genotypes. Other multivariate techniques that rely on a generalized linear model framework, such as the SKAT, will also be subject to the same problem because generalized linear models can only use individuals with complete data. Although all individuals' data could be made complete by imputing missing genotypes, low-frequency or rare variants may be difficult to impute with high accuracy. One caveat to our robustness result is that any method relying on the permutation null distribution for inference, which includes the CA max, CA sum, and WSS tests, will only be valid if genotype missingness does not depend on either affection status or the unknown value of the underlying genotype. In other words, genotypes must be missing completely at random in the sense of Little and Rubin These methods also make use of all available genotype data and are therefore robust to randomly missing genotypes. This robustness stands in stark contrast to our observations for the CMC test using Hotelling's The problem of neutral and protective rare variants masking case-control differences in pooling tests has been recognized by other authors An additional advantage of applying existing methods for locus-wide inference using nonnegative single-variant test statistics is their adaptability. Although we have focused on single-locus inference for concreteness, test statistics can be combined over any relevant grouping of variants, including single exons, pathways, or the entire exome, to perform joint inference. Pooling tests can also be applied to arbitrary groupings, but they are not inherently robust to the inclusion of neutral and protective variants. Moreover, although we focused on case-control association testing in the absence of confounding and population stratification, existing methods using nonnegative single-variant test statistics can be readily extended to multi-variant joint inference in more complex case-control or family-based designs by simply changing the test statistic and permutation strategy. As long as the new test statistic has a nonnegative value that depends only on the magnitude of the deviation from the statistical null hypothesis at each variant, the locus-wide test is inherently robust to the inclusion of neutral and protective variants. The permutation strategy would then need to be adapted to ensure exchangeability under the model implied by the new single-variant test statistic . Analytic locus-wide power at α = 0.05 of the BC-CA (lower bound), collapsing, and summing tests at a locus comprising one neutral and one risk variant as a function of the pairwise correlation coefficient between major/minor alleles (r). The variants had the same MAF = 0.005 (Panel A) or MAF = 0.01 (Panel B), and the relative risk was 3 (Panel A) or 2 (Panel B) for each additional minor allele at the risk variant. Both panels assume penetrance of 0.05 for the major allele homozygote at the risk variant and a balanced case-control sample with N = 1,000 total subjects.(PNG)Click here for additional data file.Figure S2Analytic power comparisons in a large sample . Analytic locus-wide power at α = 0.05 of the BC-CA (lower bound), collapsing, and summing tests at a locus comprising one neutral and one risk variant as a function of the pairwise correlation coefficient between major/minor alleles (r). The variants had the same MAF = 0.005 (Panel A) or MAF = 0.01 (Panel B), and the relative risk was 3 (Panel A) or 2 (Panel B) for each additional minor allele at the risk variant. Both panels assume penetrance of 0.05 for the major allele homozygote at the risk variant and a balanced case-control sample with N = 2,000 total subjects.(PNG)Click here for additional data file.Figure S3MAF and pairwise LD distributions in simulated sequence data. Distributions of MAFs (Panel A) and pairwise LD (Panel B) for biallelic variants in 1,000 populations of 10,000 simulated haplotypes each at a 100 kb locus. Pairwise LD was measured by the within-gene pairwise correlation coefficient (r) between major/minor alleles. Because it was computationally infeasible to summarize hundreds of millions of pairwise LD values, a 0.1% simple random sample of these values was taken from each haplotype population. We repeated this sampling procedure several times and obtained similar results. The vertical dashed line in Panel B indicates r = 0.(PNG)Click here for additional data file.Text S1Appendices.(DOC)Click here for additional data file.


Phenotypic traits may exert causal effects between them. For example, on the one hand, high yield in dairy cows may increase the liability to certain diseases and, on the other hand, the incidence of a disease may affect yield negatively. Likewise, the transcriptome may be a function of the reproductive status in mammals and the latter may depend on other physiological variables. Knowledge of phenotype networks describing such interrelationships can be used to predict the behavior of complex systems, e.g. biological pathways underlying complex traits such as diseases, growth and reproduction. Structural Equation Models (SEM) can be used to study recursive and simultaneous relationships among phenotypes in multivariate systems such as genetical genomics, system biology, and multiple trait models in quantitative genetics. Hence, SEM can produce an interpretation of relationships among traits which differs from that obtained with traditional multiple trait models, in which all relationships are represented by symmetric linear associations among random variables, such as covariances and correlations. In this review, we discuss the application of SEM and related techniques for the study of multiple phenotypes. Two basic scenarios are considered, one pertaining to genetical genomics studies, in which QTL or molecular marker information is used to facilitate causal inference, and another related to quantitative genetic analysis in livestock, in which only phenotypic and pedigree information is available. Advantages and limitations of SEM compared to traditional approaches commonly used for the analysis of multiple traits, as well as some indication of future research in this area are presented in a concluding section. In animal breeding and quantitative genetics, relationships among phenotypic traits are traditionally studied via probabilistic relationships between them, using standard Multiple Trait Models (MTM) - see, for example, ,2. Althoy1 and y2 can be due to a direct effect of y1 on y2 (or y2 on y1) or to extraneous variables that jointly affect y1 and y2. Knowledge about the causal structure underlying phenotypic relationships is necessary to predict the effect of interventions  applied to trait y1 or y2. For example, if trait y1 affects y2, and y2 has no effect on y1, an intervention on y1 will cause changes on y2, but the reverse would not hold true.Information regarding phenotype networks describing such interrelationships can be used to predict the behavior of complex systems, e.g. biological pathways underlying complex traits such as diseases, growth and reproduction, and ultimately it can be used to optimize management practices and multi-trait selection strategies in livestock. For instance, a correlation between traits g) on two traits (y1 and y2), as illustrated in Figure y1, and the phenotypic change on trait y1 affects trait y2; (b) the expression of the gene acts on trait y2, and the phenotypic changes on trait y2 modify trait y1; or (c) the expression of the gene changes both traits directly, which may or may not have a phenotypic causal effect between them. Knowledge about these different sources of genetic correlation between traits could be used to further improve selection decisions and increase the genetic progress of breeding programs.Similar situations can be considered from a genetic improvement standpoint. Conventionally, genetic correlation is defined as the proportion of variance that two traits share due to genetic causes, and it indicates how much of the genetic influence on two traits is common to both, e.g., due to pleiotropism. However, different scenarios can cause a pleiotropic effect of a specific gene  can b can b6])In the last few years, genetics has been used as a means to infer phenotype networks, including causal relationships among them , and SEMx and some additional  variables e directly affect variables y, which have also some causal relationships among them.Structural Equation Models ,4 providThe graph in Figure β's are model parameters representing the "fixed effects" of the x covariates on y's, and λ's are structural coefficients representing the magnitude of the casual effects among y's. Hence, in matrix notation, a SEM can be represented as y = Λy + Xβ + e, where Λ is a quadratic matrix with zeroes in the diagonal and with structural coefficients λ or zeroes in the off-diagonal, and y, X, β and e are appropriate vectors or matrices with the observations y's, exogenous variables x's, model parameters β's and residuals e's, respectively. Competing networks representing different causal structures among y's may be compared using some model selection criteria, such as likelihood ratio tests (LRT), Akaike information criterion , Bayesion , or Bayon .Structural equation models have been intensively used in many fields, such as economics, psychometrics, social statistics, and biological sciences. In genetics, they have been used, for example, to study the relationships between phenotypic traits in humans, especially in the context of twin designs e.g. ,21). Mor. Mor21])Thomas and Conti  have poig, y1 and y2 denote the cis-acting eQTL, the transcriptional activity of the gene, and the complex trait, respectively. Model C depicted in Figure y1, in which allelic variations in g change y2 by changing the transcriptional activity y1. Model R , g and y1 are unconditionally correlated, and by conditioning on y2 their dependence vanishes. When the QTL g and both traits y1 and y2 are causally connected, as in model (1)-(3), the raw and partial correlations between them will all be nonzero, but they will change in magnitude depending on the signs of the path coefficients  y = Xβ + Zu + e, so that an equivalent reduced model can be obtained as [where ained as :y given the location parameters and the residual covariance matrix is:The resulting sampling distribution of Ψ = Ψ0⊗In.where By reducing the SEM, the location and dispersion parameters are transformed into parameters of a standard MTM ,25, as iwhere with Here, y. Conversely, SEM carries extra parameters in Λ, resulting in an unidentifiable likelihood function. Nevertheless, it is possible to introduce constraints in SEM to achieve parameter identifiability [Ψ0 to be diagonal, as in the examples discussed below. After defining the causal structure and achieving parameter identifiability, one may apply standard statistical methodologies  to makeSEM models have been used to study simultaneous and recursive relationships between phenotypes in various species and breeds, such as dairy goats , LandracDe los Campos et al. ,27 have Also working with MY and SCS data in dairy cattle, Wu et al.  have extAnother study concerning the relationships between MY and SCS has been conducted by Jamrozik et al.  with CanTo infer simultaneous and recursive relationships between binary and Gaussian characters, Wu et al.  have proA related application of Bayesian linear-threshold SEM has been presented by König et al. , who havOther applications of Gaussian-threshold SEM with heterogeneous structural coefficients have been presented by de Maturana et al. ,29 to exWorking with health and fertility traits in dairy cows, Heringstad et al.  have empFinally, Varona et al.  have preΛ of coefficients defining the causal structure must be specified. In all applications of SEM in quantitative genetics so far, the causal structure was assumed known a priori , or , or Λ ofi  in the path. Colliders block the flow of dependency in a path, which makes a and b independent in the structure above. Conditioning on a variable that is not in the extremes of the path switches its status regarding the flow of dependence through it, i.e. if the variable is a collider it allows the flow, whereas if it is a non-collider it blocks the flow. Two variables a and b in a DAG are said to be d-separated conditionally on a subset S of remaining variables if there are no path between a and b such that all its nodes allow the flow of dependence . Under some assumptions, d-separations in the causal structure of a SEM result in conditional independencies in the joint probability distribution of y. This is used to guide the selection of a causal structure or a class of equivalent causal structures  that is compatible with the joint distribution of the data [A recursive causal structure can be represented by a Directed Acyclic Graph (DAG), which is a set of variables (or nodes) connected by directed edges (arrows). Pairs of connected nodes represent direct causal relationships. A path in the causal structure is a sequence of connected variables. Unconditionally, flows of dependence between variables in the extremes of paths may take place, unless there is a collider , search for a set Sab that contains c such that a is independent of b given Sab. If this set does not exist, then add arrowheads pointing at c (a → c← b). If this set exists, then continue.2. For each pair of non-adjacent variables 3. In the resulting partially-oriented graph, orient as many undirected edges as possible in such a way that it does not result in new colliders or in cycles.The goal of the first step of the algorithm is to obtain a graph that specifies pairs of traits that are directly connected by an edge, because variables that are adjacent in the underlying causal structure are not d-separated  given any possible set of variables. The second step aims to orient edges by searching for unshielded colliders (structures where a collider is directly caused by two non-adjacent variables). Non-adjacent parents of a collider variable are d-separated given at least one set of variables, but not if conditioned to any set of variables that contains the collider. The observational consequence of this is the probabilistic dependence between the non-adjacent parents conditionally on every possible set of variables that contains the common child. The third step performs every further edge orienting that does not result in a new collider or in a cycle. Additional constraining of the output may be achieved by incorporating background knowledge like time precedence or other prior beliefs ,23.The decisions about declaring pairs of variables as conditionally dependent or not are based on partial correlations inferred from a sample, which involves some degree of uncertainty. To account for that, decisions may be made by testing null hypotheses of vanishing partial correlations or, in a Bayesian approach, using highest posterior density (HPD) intervals for the partial correlations.Ψ0 to be diagonal) is already adopted in recent applications of SEM in animal breeding in order to achieve model identifiability. Therefore, the assumptions of the IC algorithms are not stronger than the assumptions considered in recent application of SEM in quantitative genetics. In those applications, not only covariance matrices of random variables are assumed to be structured , but the causal structure itself is assumed to be known.The IC algorithm was developed based on the connection between causal structure and joint distribution, which requires some assumptions . Maybe tValente et al.  have adoy1 through y5, with uncorrelated residuals  and correlated additive genetic effects . The connection between the causal structure among phenotypes and their joint probability distribution does not hold in a model where genetic effects are uncontrolled hidden variables. For example, given such causal structure y1 would be expected to be independent of y3 given y2, but this may not hold because of the correlation between u1 and u3.Take as an example the causal structure depicted in Figure yi can be expressed as:Nonetheless, as indicated by , genetic(It - Λ)-1 G0 (It - Λ)-1 and (It - Λ)-1 Ψ0 (It - Λ)-1 are the covariance matrices of additive genetic effects  and of residuals  obtained from a standard multiple trait mixed model that accounts for covariance between genetic effects and residuals from different traits, but not for causal relationships between phenotypes [yi can be then rewritten as Note that enotypes ,25. The In Valente et al. , the  Obtain the posterior distribution of residual partial correlation ρa,b|S.b) Compute the 95% HPD interval for the posterior distribution of ρa,b|S as null. Otherwise, declare a and b as conditionally dependent.c) If the HPD interval contains 0, declare 3. Lastly, a SEM using the selected causal structure  is fitted, as in , such thValente et al.  have valStructural equation models are able to express causality among traits. However, one may fit a SEM with causal structures that do not express the actual causal relationship among traits. The inference of the causal structure is a much harder task than just describing data by a stochastic model. As discussed in this review, using the IC algorithm and related techniques involves accepting specific assumptions, from which the causal sufficiency seems to be the strongest one. In this regard, applying the IC algorithm may be regarded as a causal structure inference only if one is willing to accept the causal assumptions. Otherwise, the application of such algorithms can be viewed simply as a causal structure selection for SEM constructed with diagonal residual covariance matrices.Nonetheless, the latter applications may still produce interesting and useful results such as the generation of causality hypotheses for further research and investigation. Such hypotheses can then be supported or dismissed by additional data collected from other studies, or they might be tested experimentally through controlled interventions. In genetics, for example, a putative causal mutation could be ultimately tested using gene knockout or knockdown methodologies. However, quite often, randomized experiments are not an alternative due to logistic or ethical constrains, and one is restricted to the analysis of observational studies. In this context, SEM and causal search tools like the IC algorithm are handy. Moreover, in genetics and genomics studies, causal inference is aided by the concept of Mendelian randomization , in whicIn addition to DNA polymorphism information and knowledge about genes or QTL that can be used as parent nodes in phenotype network reconstruction, the joint analysis of multilayer large-scale "omics" data such as transcriptome, metabolome and proteome can certainly provide added information and enhance the ability to infer causal phenotype relationships, although it also brings another level of statistical, computational and data mining challenge . MoreoveA constructed from pedigree information has been considered for such task. As an alternative, if high density molecular marker data is available , more efficient genetic merit prediction approaches can be employed such as Bayesian regression techniques [SEM have also been used in the context of quantitative genetics analysis of multiple phenotypic traits when QTL or genomic information is not available , allowinchniques  or kernechniques . This isSome other areas related to phenotype network inference that would also warrant additional research refers to the development of (parametric or non-parametric) methods to deal with non-Gaussian traits, as well as search algorithms and software suitable to handle huge number (on the level of thousands) of variables. Lastly, and specifically in the context of animal and plant breeding, extra research is required to study how knowledge regarding causal effects between traits could be explored for the development of more efficient breeding programs and agricultural production enterprises.In summary, SEM provide a flexible and insightful approach for the genetic analysis of multiple traits, allowing the characterization of pleiotropic and heterogeneous genetic effects of multiple loci on multiple traits, as well as causal relationships among phenotypes, which can be used to predict behavior of complex systems, e.g. biological pathways underlying disease traits. More specifically with livestock, SEM can be used to infer phenotype networks in the genetic analysis of quantitative traits, such that the effect of external interventions can be better predicted. This may foster the development of more efficient breeding programs and optimal decision-making strategies regarding farm management practices.The authors declare that they have no competing interests.GJMR and BDV wrote the manuscript; GC, XLW, DG and MAS provided critical insights and helped revising the manuscript. All authors read and approved the final manuscript.