Identification of causal rare variants that are associated with complex traits poses a central challenge on genome-wide association studies. However, most current research focuses only on testing the global association whether the rare variants in a given genomic region are collectively associated with the trait. Although some recent work, e.g., the Bayesian risk index method, have tried to address this problem, it is unclear whether the causal rare variants can be consistently identified by them in the small- Testing the phenotypic association of millions of individual SNPs across the genome has been one of the major goals of the genome-wide association study (GWAS). To date, hundreds of putative disease gene loci have been detected based on the common disease common variant assumption. However, the detected genetic variants typically account for only a small fraction of disease heritability. Nowadays, it has been widely acknowledged that the missing disease heritability may be due to rare variants. Many studies show that the rare variants tend to have larger effects than common variants. As pointed out in The statistical power of genetic variant detection depends on the sample size, the variant effect and the minor allele frequency (MAF). Since the MAF of the rare variant is low, the single variant testing-based methods, such as the Although testing the collective effects of rare variants is challenging, identifications of the rare variants which, if any, are driving the association  is even more challenging and scientifically more interesting. Along this research direction, some methods have been developed, e.g., the RARECOVER method While BRI has made a solid step toward detection of causal rare variants, it is unclear whether it can identify causal rare variants consistently for small- Are there any of the variants associated with the disease? Which variants, if any, are driving the association?The BRVD ensures the causal rare variants to be consistently identified in the small-Assume that Let The Bayes factor (3) depends on the prior distributions, Let Let a priori model size As shown in Let Lγ = 0.4, 0.5, …, 0.9, 0.95, 0.99 or a subset of them.In all simulations of this paper, we fixed the hyperparameters For the global association test, the key step is Bayes factor estimation. As implied by (4), an exact evaluation of the global Bayes factor needs to sum over all models under To facilitate the description of the SAMC algorithm, we define the following notations. Let SAMC seeks to draw samples from each of the subregions with a pre-specified frequency. For the time being, we assume that all the In this paper, we assume that (a) (Sampling) Simulate a sample  by a single MH update with the target distribution as defined in (11):(a. 1) Generate  according to a proposal distribution . Refer to .(a. 2) Calculate the ratiowhere  and  are the indices of the subregions that  and  belong to, respectively.(a. 3) Accept the proposal with probability . If it is accepted, set ; otherwise, set .(b) (-updating) SetIf , set ; otherwise, find a value of  such that  and set , where  denotes a constant -vector of ones.SAMC is an adaptive MCMC algorithm for which the invariant distribution of the MH kernel changes from iteration to iteration. Due to the adaptive change of the invariant distributions, SAMC possesses a self-adjusting mechanism: If a proposal is rejected, then the sample The proposal distribution For global association tests, we set the desired sampling distribution to be uniform, i.e., setting In this section, we describe how to detect rare variants when the global association test shows positive support for the hypothesis Identification of important variables based on the marginal inclusion probability has been widely used in Bayesian variable selection, see, for example, To implement the rule To determine the threshold SKAT. The haplotype dataset is generated by the calibrated coalescent model with a mimicking linkage disequilibrium (LD) structure of European ancestry. To emphasize rare events, the variants with MAF greater than 5% have been removed from the haplotype dataset before resampling. For the 100 datasets simulated under To explore the power of the proposed method versus other alternative methods for the global association tests and rare variant detection, we simulated 200 datasets, with 100 simulated under Bayesian risk index (BRI) for both global association tests and causal variant detection. We also compare BRVD with the commonly used non-Bayesian methods, including CMC, WSS, SKAT, and RARECOVER, for global association tests. Among the four non-Bayesian methods, CMC and WSS belong to the class of variant pooling methods, SKAT belongs to the class of random effect model-based methods, and RARECOVER belongs to the class of variable selection methods. These methods can be briefly described as follows.We compare the BRVD with the competing Bayesian method Bayesian risk index (BRI) Combined multivariate and collapsing (CMC) testWeighed sum Statistic (WSS) testSequence kernel association (SKAT) testRARECOVERThe implementation of BRI is available in the R package BVS, the implementation of SKAT is available in the R package SKAT, and the implementations of CMC, WSS, and RARECOVER are available in the R package AssotesteR. In this paper, all the methods are run under their default settings unless otherwise stated.We first aim to examine the power of the BRVD versus alternative methods for global association tests. The BRVD has a prior hyperparameter Since BRVD, BRI and SKAT are all developed under the regression setting, they are able to adjust for covariates, such as age, gender, race, etc. For this reason, we first compare the powers of these three methods with the simulated covariates adjusted in regression. The CMC, WSS and RARECOVER cannot be adjusted for covariates. To compare with them, we re-run the BRVD, BRI and SKAT methods on the simulated datasets with the covariates omitted. The effect of covariate omission on test power has been discussed in the literature In addition to the power, we also explored the type-I error of the global association test based on the testing statistic Our next aim is to detect rare variants that are associated with the disease, provided that the global association test shows a positive support for the hypothesis a priori knowledge.For global association tests, we suggest to choose the value of The EOMI data (downloaded from dbGaP) is from the NHLBI€s Exome Sequencing Project (ESP), which was designed to identify genetic variants in coding regions (exons) of the human genome that are associated with heart, lung and blood diseases. The dataset consists of 278,263 SNPs in 905 subjects (467 cases and 438 controls) with European origin (EA). After removing the common variants  (Dividing) Divide the variants into subsets that are of an acceptable size in computation.(b)  Apply BRVD to each of the subsets and identify putative associated variants from the subsets for which the hypothesis (c) (Combining) Combine the variants identified at step (b) into a new dataset, the so-called selected subset data; and then apply BRVD to the selected subset data to identify causal rare variants.For each subset, the logistic regression model is potentially misspecified because the causal variants located in other subsets are not included in the regression. If some causal variants are missed, we can expect that the BRVD will find some surrogate variants within the subset for the missing causal variants, and the number of surrogate variants can often be greater than the number of missing causal variants. For this reason, we suggest a high FDR level, say, 0.25 or even higher, to be used for identifying putative causal variants from each subset. For the selected subset data, we can expect that it will include the causal variants, surrogate variants of some causal variants, and some noise variants. It is obvious that Lemma 1 and Lemma 2 are still applicable to the selected subset data. By these two lemmas, the parallel BRVD can also select causal variants consistently.The global association test can also be done on the selected subset of variants. However, a direct application of the BRVD to this subset can lead to a biased test, although for which the power can be very high. This is the same for all other testing procedures. To avoid the bias, a permutation method can be used to evaluate the For the EOMI dataset, we divide the variants into 22 subsets according to the chromosomes where they belong to. The numbers of variants on the 22 chromosomes range from 1,271 (on chromosome 21) to 11,491 (on chromosome 1), which are all acceptable to our current computing facility. BRVD was run 5 times for each subset at each value of Since all chromosomes show positive support for the global association, putative associated variants should be identified from each of them. For illustration, we here work on the first group chromosomes only. The BRVD was then applied to the selected subset of variants with the same setting as described above except for sample space partitioning and Our method is surprisingly successful for this example: A few rare variants identified by it have been verified in the literature. It is reported that SLC1A4 is associated with atherosclerosis For comparison, BRI and SKAT were also applied to this example. BRI was run for 50,000 iterations for each of the 22 subsets. The outputs show that only chromosome 2 provides “substantial” evidence for the global association with a Bayes factor of 7.1. The Bayes factors for all other chromosomes are less than 1. On chromosome 2, BRI identified three SNPs, rs65245292, rs179455352 and rs28827533, whose marginal Bayes factor are all greater than 10. It is interesting to point out that both SNPs, rs65245292 and rs28827533, have been identified by BRVD as shown in SKAT produced a small Similarly, we plot in The computation time for the BRVD depends on the sample size  Are there any of the variants associated with the disease, and (ii) Which variants, if any, are driving the association. The BRVD is developed based on the theory of posterior consistency, under which the causal variants can be identified consistently. The numerical results indicate that the BRVD is more powerful for global association tests than the existing methods, such as CMC, WSS, SKAT, C-alpha, RARECOVER, VT, and BRI, and also more powerful for detection of causal variants than the BRI method. In this paper, we have also developed a parallel version of BRVD based on the strategy of divide-and-conquer. The parallel BRVD can be conveniently used for the datasets for which the number of variants is extremely large.Since the BRVD is developed under the framework of logistic regression, it can be directly applied to identify gene-gene and gene-environment interactions by including in the model some interaction terms of SNP-SNP and SNP-covariates. A gene-gene and/or gene-environment interaction network can then be constructed. This method is very flexible, depending on the specification of interaction terms. For example, to explore complex higher-order interactions, a partially linear tree-based regression model Although BRVD has a high power for both the global association tests and causal variants detection, its power can be further improved by employing a more sophisticated weighting scheme for the variants. The current weighting scheme depends on the MAF only. In the future, one may incorporate other biological information, e.g., the gene information, into the weighting scheme. This may help further to identify the causal variants whose MAFs are extremely low. In the current implementation of the BRVD, the SAMC algorithm is used for sampling from the posterior. At each iteration, a variant is randomly selected to undergo a model update of variant addition, deletion, or exchange. In the future, a SAMC algorithm with an adaptive proposal may be used. The new version of SAMC allows one to select a variant for model update based on the working estimate of marginal inclusion probabilities. In the limit case, the new version of SAMC will update the model according to the marginal inclusion probabilities of all variants. Therefore, it can converge faster than the standard version of SAMC.For global association tests, the BRVD can also be used in conjunction with other frequentist methods, such as SKAT, if one is interested in a The BRVD is general in the sense that it can be used for rare variants, common variants, and also a joint analysis of common and rare variants. In the case of joint analysis, its power for detecting rare variants will not be affected much if In this paper, BRVD is developed for dichotomous phenotypes only. The framework of BRVD can be easily extended to continuous phenotypes. For continuous phenotypes, linear regression can be used to relate the phenotype to the variants, and appropriate prior distributions that lead to the posterior consistency need to be specified for the model and model specific parameters. Alternatively, one can impose a non-local prior on the model parameters as in File S1Supporting Information.(PDF)Click here for additional data file.


Mendelian randomisation analyses use genetic variants as instrumental variables (IVs) to estimate causal effects of modifiable risk factors on disease outcomes. Genetic variants typically explain a small proportion of the variability in risk factors; hence Mendelian randomisation analyses can require large sample sizes. However, an increasing number of genetic variants have been found to be robustly associated with disease-related outcomes in genome-wide association studies. Use of multiple instruments can improve the precision of IV estimates, and also permit examination of underlying IV assumptions. We discuss the use of multiple genetic variants in Mendelian randomisation analyses with continuous outcome variables where all relationships are assumed to be linear. We describe possible violations of IV assumptions, and how multiple instrument analyses can be used to identify them. We present an example using four adiposity-associated genetic variants as IVs for the causal effect of fat mass on bone density, using data on 5509 children enrolled in the ALSPAC birth cohort study. We also use simulation studies to examine the effect of different sets of IVs on precision and bias. When each instrument independently explains variability in the risk factor, use of multiple instruments increases the precision of IV estimates. However, inclusion of weak instruments could increase finite sample bias. Missing data on multiple genetic variants can diminish the available sample size, compared with single instrument analyses. In simulations with additive genotype-risk factor effects, IV estimates using a weighted allele score had similar properties to estimates using multiple instruments. Under the correct conditions, multiple instrument analyses are a promising approach for Mendelian randomisation studies. Further research is required into multiple imputation methods to address missing data issues in IV estimation. In this article we discuss the use of multiple genetic variants as IVs, both for increasing statistical precision and for testing underlying IV assumptions.Recent decreases in genotyping costs and increases in genome-wide association studies (GWAS), have facilitated discovery of a substantial number of genetic variants associated with risk factors and disease-related outcomes, such as adiposityThe structure of the article is as follows: we describe instrumental variable assumptions  and intrG is defined as a variable that satisfies the following assumptions:G is associated with the risk factor (phenotype or intermediate variable) of interest X;G is independent of the (unobserved) confounding factors U of the association between X and the outcome Y;G is independent of outcome Y given X and U.An IV (instrument) 28In the context of Mendelian randomisation, these assumptions can be expressed as: genotype is associated with the modifiable risk factor of interest (assumption 1); genotype is independent of unmeasured confounding factors that could bias conventional epidemiological associations between the risk factor and the outcome (assumption 2); genotype is related to the outcome only via its association with the risk factor (assumption 3). The second assumption can be justified through Mendel’s laws when applied to independent heritable units.29 and its weaker form known as ‘conditional mean independence’ .30 This additional assumption allows causal inferences to be drawn from IV analyses.If we further assume that intervention on the risk factor only affects the value of the risk factor, and hence affects the outcome only through this induced change in the risk factor, then the IV assumptions imply the ‘exclusion restriction’FTO and MC4R genes as IVs.31 The authors concluded that higher fat mass caused increased accrual of bone mass in childhood. We consider whether the IV estimates from the separate instruments are of similar magnitude; whether use of multiple instruments increases the precision of IV estimates; the use of allele scores as IVs; and the impact of missing data on IV estimates.Our example investigates the causal effect of fat mass on bone mineral density (BMD) using four genotypes known to be associated with adiposity from previous GWAS. A previous study found a positive effect of fat mass on BMD using SNPs associated with the 32 ALSPAC is a longitudinal, population-based birth cohort study that recruited 14 541 pregnant women resident in Avon, UK, with expected dates of delivery 1 April 1991 to 31 December 1992 .32 Out of this 13 988 live born infants survived to at least one year of age. Children eligible for inclusion in our analysis: (1) had DNA available for genotyping; (2) attended the research clinic at age 9 and (3) had complete data on height and dual energy X-ray densitometry (DXA) scan-determined total fat mass and total BMD.Our example uses data from the Avon Longitudinal Study of Parents and Children (ALSPAC).a priori to use the four SNPs, namely FTO (rs9939609), MC4R (rs17782313), TMEM18 (rs6548238) and GNPDA2 (rs10938397), that had the strongest associations with adiposity in previous studies.14–16 Functional studies are required to ascertain the specific biological pathways through which these polymorphisms affect adiposity. Whilst most pathways to greater adiposity are likely to involve influences on diet/appetite or physical activity, here for the assessment of the IV assumptions .Fat mass and BMD were positively skewed and were log transformed. To account for sex and age differences in fat mass and BMD, age and sex standardised z-scores of log transformed fat mass and BMD were used in the analysis. Genotypes were incorporated into IV models assuming an additive genetic model for the genotypes coded 0, 1 and 2, as shown in 35–37 The Hausman test of endogeneity38 was used to compare the difference between the ordinary-least-squares (OLS) and TSLS estimates using the user-written Stata command ivendog.35  In models including multiple instruments the Sargan test of over-identification (discussed in 39IV estimation used the two-stage least squares (TSLS) estimator implemented in the user written Stata command ivreg2.FTO, 5,412 (98%) for MC4R, 5,323 (97%) for TMEM18, 5 303 (96%) for GNPDA2 and 4 796 (87%) for all four SNPs. Mean age at the time of the DXA scans was 9.9 years. There was no strong evidence against the FTO, TMEM18 and GNPDA2 genotypes being in Hardy–Weinberg equilibrium. The MC4R genotypes had an Hardy–Weinberg equilibrium p-value of 0.04 in our sample, though in the whole ALSPAC cohort the corresponding p-value was 0.1.FTO, MC4R or GNPDA2 with height, lean mass, mother’s educational achievement and head of household social class. There is some evidence for these data that TMEM18 is associated with lean mass and mother’s educational achievement. Under the IV assumptions, TMEM18 genotypes only affect BMD through fat mass; so for now we view these latter two associations as chance findings similar to baseline covariates found to be associated with treatment group in a randomised controlled trial (RCT).GNPDA2 and 2.33 (1.34–4.05) for MC4R. These four IV estimates generally suggest that BMD has a positive effect on fat mass, although the lower limit of the confidence interval for the TMEM18 estimate and both the lower limit of the confidence interval and point estimate using GNPDA2 as an instrument, were less than 1. For MC4R and TMEM18, there was evidence that the IV estimate differed from the OLS estimate, based on the Hausman test of endogeneity , with both suggesting a stronger positive association than that found in the OLS analysis.R2 and F-statistics for the instruments based on the explained variation in standardised log fat mass show the expected ranking, with FTO genotype explaining the largest proportion of variation followed by MC4R, TMEM18 and GNPDA2 . The variation in standardised log fat mass explained by each SNP was small, ranging from 0.16% to 0.80%, and the TMEM18 and GNPDA2 SNPs were weak instruments, based on their first-stage F-statistic being less than 10 (FTO SNP (0.16) and largest for TMEM18 and GNPDA2 SNPs (0.43 and 0.37). IV estimates using multiple instruments are described in The first stage  than 10 . Consist40 We briefly describe them, and the use of multiple instruments to address issues they raise.Population stratification, linkage disequilibrium and pleiotropy have been identified as factors that could bias Mendelian randomisation analyses.41 Some genetic variants that are potential candidates for use as IVs in Mendelian randomisation studies could have been influenced by such population stratification.42–45 Population stratification therefore has the potential to bias estimates of causal effects in Mendelian randomisation studies.5Population stratification occurs when a sample is composed of a mixture of populations and so contains latent ancestral structure. If there are corresponding differences in the prevalence of the outcome of interest by this structure, then genotype-risk factor associations may result from the presence of ancestrally informative alleles rather than biological function.47 SNPs that are associated with phenotypes in GWAS are unlikely to be functional variants, but rather to be in LD with the unknown functional variant(s).47 IV assumptions are not violated when tag SNPs are used as IVs, providing that they are in LD only with the functional variant(s).11 However, if tag SNPs are also in LD with a variant that affects the outcome of interest via a pathway that does not include the risk factor of interest the IV assumptions will be violated.5Linkage disequilibrium (LD) is correlation between allelic states at different loci on a stretch of the same chromosome when assessed within a population. LD is a function of the frequency of recombination and is subject to regional genomic characteristics as well as more stochastic processes which may be influenced by the physical distance between two loci as well as the relative age of the population in question. Extensive LD can increase the statistical power of a study to detect genotype-risk factor associations and is exploited in GWAS studies where an LD-based set of tag SNPs is chosen to maximise the amount of genetic variation captured per SNP.11 In our example, if any of the adiposity variants had effects on pathways that influence BMD other than through adiposity, for example, if they influenced calcium or vitamin D metabolism, then IV assumptions would not hold.Pleiotropy refers to a single gene having multiple biological functions. In the context of Mendelian randomisation analyses, SNPs in or near genes with pleiotropic effects that directly or indirectly influence the outcome other than through the risk factor of interest violate the IV assumptions.5 Comparison of IV estimates based on multiple genetic variants with independent effects on the risk factor of interest provides an additional way to identify bias resulting from these issues. If IV estimates from different variants are similar, it is less plausible that LD or pleiotropy are present.Population stratification and pleiotropy can to some extent be dealt with by using ethnically homogenous study populations, identifying and incorporating population strata in the analysis and ensuring that the function of the genetic instrument is well understood.Comparison of IV estimates from independent genetic variants is analogous to comparing the results of RCTs of different classes of blood pressure lowering drugs, which lower blood pressure by different mechanisms. If the effect of the drug on stroke risk in each RCT is proportional to the direction and magnitude of its effect on blood pressure, this strengthens the evidence for a causal link between blood pressure and stroke risk, and against the drugs having effects on stroke risk through other mechanisms. Such consistency would also argue against the possibility that the trials were affected by methodological flaws that biased their results.http://www.ncbi.nlm.nih.gov/projects/SNP/) that provide the fixation index STF (a measure of population differentiation), or equivalent information, can be used to examine population stratification..It is possible that separate IV estimates could be identical but biased to a similar extent by population stratification, because stochastic- or selection-driven non-independence that is not predicted by LD profiles could influence more than one genetic variant that affects a given risk factor. Databases such as dbSNP .49Over-identification refers to the situation when there is more than one instrument for a single risk factor of interest or, more generally, when there are more instruments than endogenous variables. In such circumstances testing the ‘over-identification restriction’ checks the joint validity of multiple instruments by testing whether they give the same estimates when used singly or in linear combination. There are two commonly used tests of over-identification; the Hansen test and the Sargan test.50 However, genotypes could still be associated with unmeasured confounders.Verifying that the genotypes are independent of the measured confounding factors  is an in51 Two closely related measures of this are the first-stage regression F-statistic and coefficient of determination R2. It is important to report these. If measured confounders are included then the partial R2 and F-statistics for the instruments should be reported.52IV estimators such as TSLS are asymptotically unbiased but biased in finite samples, with such bias inversely proportional to the amount of phenotypic variability explained by the instrument.R2 is the proportion of risk factor variability explained by genotype. The relationship between the F and R2 statistics is given by:k is the number of parameters in the model (in this case instruments). The relative bias of the TSLS estimator to the OLS estimator is related to the inverse of the F-statistic.53 Hahn and Hausman gave a simplified version of the relative bias as approximately the inverse of the F-statistic:54–56In Mendelian randomisation the first stage R2 increases the relative bias of TSLS decreases, but including additional instruments that do not increase the first stage R2 increases the relative bias of TSLS. A first stage F-statistic less than 10 is often taken to indicate a weak instrument, although this is not a strict limit but a rule of thumb drawn from simulation studies.58 Equation  criterion.62 Pierce et al. recently estimated the power of Mendelian randomisation studies in a range of settings, using both single and multiple genetic instruments.13Genotypic effects on phenotypes are typically small, so Mendelian randomisation analyses can require very large sample sizes to obtain adequate power.63 Whilst there has been considerable research into methods of imputation we are not aware of specific research into appropriate multiple imputation models for IV estimation.In studies where genetic data are not obtained from GWAS  there are typically some missing observations for each genetic variant, due to failure of genotyping or ambiguous genotype allocation. Missing data typically occur in different individuals for each variant. They can therefore result in a considerable cumulative reduction in the number of individuals with complete data on all genotypes, and hence reduce the power of multiple instrument Mendelian randomisation analyses. One approach to dealing with missing data is multiple imputation.64–66 To justify the use of an allele score the genotypes should have an approximately additive effect on the risk factor. For an unweighted score they should also have similar per allele effects.An allele score is a weighted or unweighted sum of the number of ‘risk’ alleles across several genotypes: weights are usually based on each genotype’s effect on the phenotype. Use of such scores is becoming more common in gene–disease association studies.F-statistic to increase, since the number of parameters in the model is reduced. Therefore, the relative bias of the TSLS estimator to the OLS estimator will decrease. However, if the weights are estimated from the same data in which the score is used as an instrument then the single degree of freedom for the allele score F-statistic may not be appropriate. When using an allele score the IV estimator is exactly identified, because there is a single instrument and single phenotype, and it is therefore not possible to use an over-identification test for the joint validity of the SNPs.The use of an allele score as a single IV, compared with multiple instruments, will cause the first stage 61 Given appropriate weighting, results from IV analyses using weighted allele scores will be similar to the multiple instruments approach.In general, using an unweighted allele score will have lower power than the multiple instrument approach, since the latter will estimate the efficient linear combination of the genotypes.We investigated the use of multiple instruments through two simulations both based on our example. Specifically, we investigated bias and precision of IV estimates including: (i) additional non-weak instruments and (ii) weak instruments.G1, G2 and G3 are genotype variables coded additively, X is the risk factor, Y the disease outcome, U the unmeasured confounder and subscript i denotes a subject:Data were simulated as follows, where G1 explained the most variability in X, followed by G2 and G3. The value of the causal effect of X on Y,β, was set to 1. We monitored the estimates of β from the following models:Y on X,OLS estimate of the regression of G1 as the instrument,TSLS using G1 and G2 as instruments,TSLS using G1–G3 as instruments,TSLS using G1–G3 as an instrument,TSLS using an unweighted allele score of G1–G3 as an instrument.TSLS using a weighted allele score of The values of the coefficients on the genotypes were chosen so that R2 and F-statistics and average absolute TSLS/OLS bias ratio  had the smallest average bias. The F-statistic was greater for the weighted allele score than for the three instrument model (1105 vs. 368) despite having the same average R2 statistics. This is because the instruments were independent and the weights were derived internally so the weighted score was similar to the linear combination of the instruments derived in the first stage of TSLS.Models 4 and 6, , had almost identical properties and had the smallest MSE. Model 3 , G2i ∼ Bin, G3i ∼ Bin, G4i ∼ Bin, and, iU ∼ N, i = X0.1G1i + 0.1G2i + 0.05G3i + 0.05G4i + Ui and i = Yβi + UiX. The value of the causal effect of X on Y, β, was set to 1. We monitored the estimates of β from the following models:Y on X;OLS estimate from regression of G1 as the IV;TSLS estimate using G1 and G2 as the IVs;TSLS estimate using G1, G2, G3 and G4 as the IVs;TSLS estimate using G1 and G2 as the IV;TSLS estimate using an unweighted allele score of G1 and G2 as the IV;TSLS estimate using a weighted allele score of G1–G4 as the IV;TSLS estimate using an unweighted allele score of G1–G4 as the IV.TSLS estimate using a weighted allele score of Data were simulated with four IVs as follows such that β in the range 0 to 2 .We used 10 000 replications, each with a sample size of 5 000 observations. We also plotted power curves for testing F-statistic is larger for the weighted allele score due to its smaller model degrees of freedom. The lower half of FTO alone (0.16). As expected, given their low first-stage F-statistics, inclusion of the TMEM18 and GNPDA2 SNPs led only to a small decrease in the SE compared with the multiple instrument model using FTO and MC4R (0.12 compared with 0.14). The IV estimate using all four SNPs had the largest first stage R2 and smallest SE.The SE of the IV estimate using all four SNPs was 0.12, approximately 20% smaller than that of the IV estimate using Mendelian randomisation studies using genetic variants as instruments can control for unmeasured confounding and reverse causation, which can bias results from standard epidemiological analyses. However, population stratification, LD and pleiotropy can all affect the validity of the IV assumptions underlying Mendelian randomisation analyses. Obtaining similar IV estimates from separate independent instruments provides evidence against the presence of bias from pleiotropy and LD, though not bias from population stratification. In our example there was no evidence that the estimates for each instrument differed from each other (based on the over-identification test), providing some reassurance that bias from pleiotropy and LD is unlikely. However, we acknowledge in this example our power to detect differences between the estimates was limited.11Mendelian randomisation analyses require large sample sizes unless the instrument is strongly related to the risk factor (phenotype) of interest. Use of multiple genetic variants as IVs increases the power of such analyses and facilitate tests of the IV assumptions that are not possible in single instrument analyses (such as the test of over-identification). However, inclusion of instruments that explain only a small proportion of the variability in the phenotype can increase finite sample bias of IV estimates. We have limited our consideration to the linear IV model. Non-linear models that naturally arise for discrete outcomes require different treatment.31 and suggested that the size of this effect was larger than that estimated by ignoring unmeasured confounding and using ordinary least squares, based on the Hausman endogeneity test. The SE of the IV estimate decreased by around 20% using all four genotypes, compared with the SE of the IV estimate using only the genotype with the strongest effect on risk factor. Such a reduction in SE corresponds to a 56% increase in sample size.Our illustrative Mendelian randomisation analysis confirmed a positive causal effect of adiposity (fat mass) on BMD, in line with previous research 65 Before an allele score is used as an IV the joint validity of the SNPs should be assessed using an over-identification test. The weights used in weighted allele scores may be internal or external to the study: when internally estimated the single degree of freedom used in the F-statistic for instrument strength may not be appropriate. In their simulations Pierce et al. 13 used external weights based on the true effect of the genotypes on the phenotype: such weights should be taken from the overall available evidence. They concluded that unweighted and weighted allele scores, using these external weights, decreased bias when compared to the traditional multiple instruments approach, but that they had less power than the multiple instruments approach. In our simulations, models including all instruments, either as multiple instruments or in a weighted allele score, had the greatest power and lowest MSE but not the smallest bias. Based on these results the use of allele scores as IVs can represent a good trade off in terms of lower bias but possibly less precision compared to the TSLS estimator. It has been shown that for larger numbers of IVs, with differing effect sizes, it is better to use a weighted allele score.13With increasing availability of multiple genetic variants associated with the same risk factor or disease outcome, it is becoming common for genetic association studies to report associations with allele scores.68Another consequence of the large number of genetic variants that are being indentified in GWAS in relation to particular phenotypes is that it is possible to generate many independent combinations of such variants and from these many independent IV estimates of the causal effect of a risk factor on a disease outcome. These independent estimates will not be plausibly influenced by any common pleiotropy or LD-induced confounding, and therefore if they display consistency would provide strong evidence against the notion that reintroduced confounding is generating the effect.69 In the ALSPAC study, maternal genotypes are available, which could also be used to impute missing offspring genotypes.There are typically missing data on each genetic variant, due to failure of genotyping or ambiguous genotype allocation. Thus in multiple instrument analyses, missing genotype data can offset improvements in power compared with single instrument analyses. It may be reasonable to assume that the mechanism causing genetic data to be missing is independent of a particular analysis of interest, so this may not be a cause of bias. There is scope for methodological research into multiple imputation strategies for IV estimators. It might also be possible to impute missing data for single SNPs by exploiting the LD structure between SNPs in LD with them, as is common in GWAS.In conclusion, the use of multiple genetic instruments increases the statistical power of Mendelian randomisation analyses and provides opportunities to test IV assumptions.


Drosophila showing that fly homologs of GPC5, the member of the glypican gene family implicated by both the human and mouse genetic analyses, influence the fly’s response to ethanol; and (4) discovered data from the literature demonstrating that the genetically implicated gene’s expression is not only temporally and spatially consistent with involvement in ethanol-induced behaviors but is also modulated by ethanol. The convergence of these data provides strong support to the hypothesis that GPC5 is involved in cellular and organismal ethanol response and the etiology of alcohol use disorders in humans.Alcohol use disorders are influenced by many interacting genetic and environmental factors. Highlighting this complexity is the observation that large genome-wide association experiments have implicated many genes with weak statistical support. Experimental model systems, cell culture and animal, have identified many genes and pathways involved in ethanol response, but their applicability to the development of alcohol use disorders in humans is undetermined. To overcome the limitations of any single experimental system, the analytical strategy used here was to identify genes that exert common phenotypic effects across multiple experimental systems. Specifically, we (1) performed a mouse linkage analysis to identify quantitative trait loci that influence ethanol-induced ataxia; (2) performed a human genetic association analysis of the mouse-identified loci against ethanol-induced body sway, a phenotype that is not only comparable to the mouse ethanol-ataxia phenotype but is also a genetically influenced endophenotype of alcohol use disorders; (3) performed behavioral genetic experiments in  The etiology of alcohol use disorders (AUD) involves the interaction of many genetic, environmental and behavioral factors . Two appRecent genome-wide association (GWA) studies have highlighted the genetic complexity of AUDs. Large surveys involving hundreds to thousands of subjects identified a single intergenic locus marked by rs7590720 that barely reached statistical significance after correcting for multiple tests . We compared locomotor responses to ethanol: body sway in humans, ataxia in mice, and locomotor activity in flies.Finding a phenotype that is related to alcohol dependence (AD) and comparable across humans, mice, and flies was central to this analysis. While AD cannot be accurately modeled in mice and flies, acute response to ethanol ingestion is similar between these organisms, and in humans, alcohol response is predictive of later AUD development  data and analysis suite  was calculated for each marker using 1000 permutations. LRS values were converted to the log of the odds (LOD) ratio to be consistent with human genetic reporting norms: LOD = LRS/4.61.www.ensembl.org). The proximal- and distal-most markers exhibiting suggestive linkage defined the limits of mouse ethanol ataxia QTL. Using Ensembl’s mouse genome reference NCBIM37, all genes, either wholly or partially within the defined QTL, were identified. Using the Ensembl comparative genomics module, the syntenic human genes (genome reference GRCh37) were identified. Human SNPs were selected for analysis if located within a syntenic gene as defined by its transcript plus 100 kb of flanking sequence. All analyzed genotypes were derived from the Illumina HumanCNV370-Duo DNA Analysis BeadChip.Mouse chromosomal intervals yielding suggestive linkage for an ethanol-induced ataxia phenotype had their human syntenic counterpart identified using data collected and presented by the comparative genomics functions of Ensembl   and is described in greater detail elsewhere (M-IV-TR) ; (4) hadM-IV-TR) . AnterioSubjects were not consented for release to public databases. Data may be available for sharing under a confidential agreement between a requestor and the authors.http://www1.qiagen.com). Extracted DNA was quantified using the Pico Green method (Molecular Probes/Invitrogen), and all DNA solutions were normalized to a common concentration for genotyping assays. Genotyping was carried out using the Illumina HumanCNV370-Duo DNA Analysis BeadChip. Genotype generation and quality control were performed by deCODE Genotyping Service (http://www.decode.com/).DNA was extracted from blood specimens within 5 days of the draw using Gentra Puregene reagents and protocols , it was corrected for non-normality using the Box-Cox transformation . The scores were then scaled to mean = 0 and SD = 1 to make them comparable essentially as Z-scores.Association tests were performed in R  with theThe fixed-effects portion of the model was used for testing the association between a single SNP and BSA. The SNP was treated in R as a factor with three levels (categories), which is similar to coding the major homozygotes as 1/1, the heterozygotes as 0/1, and the minor homozygotes as 0/0. For each SNP, two tests were performed to determine if a given genotype class differed in its average phenotype from the other genotypes. The test compared (a) the major homozygotes (1/1) and the heterozygotes (0/1), and (b) the major homozygotes (1/1) and the minor homozygotes (0/0). Tests (a) and (b) were performed by testing the significance of the regression coefficients from zero, of the heterozygote term and the minor homozygote term, while holding the major homozygote coefficient constant at zero. For rare SNPs, minor homozygote individuals are often not present in the sample, in which case, only test (a) was available for the SNP. In all cases, the Wald test was used to examine the significance of the regression coefficients.P-values be adjusted for multiple testing. For this, false discovery rate (FDR) q-values were calculated using the method described by The large number of tests performed in the analysis required that the nominal Drosophila strains were maintained on standard cornmeal/molasses/yeast media at 25°C and 70% humidity with an approximately 16-hr/8-hr light/dark schedule. All strains were outcrossed for five generations to the Berlin genetic background strain carrying the 1118w eye-color marker. Strain sources were MB950dally (MB00950: Bloomington Stock Center), 80dally (Xinhua Lin), f03537dlp (Exelixis/Harvard), and 1dlp (Scott Selleck).Ethanol-stimulated locomotor activity and sedation kinetics were determined as previously described  on an ABI PRISM 7900 Sequence Detection System, using expression levels of RpL32 as a standard to normalize sample concentrations. TaqMan probesets (Applied Biosystems) used in this study were Ethanol absorption was measured by exposing groups of 25 flies to either ethanol vapor (47%) or humidified air for 30 min. Flies were immediately frozen on dry ice, and the ethanol concentration in whole-fly homogenates was measured with an alcohol dehydrogenase–based spectrophotometric assay .Materials and Methods. Twelve phenotypes comparable to the human body sway phenotype were present in the GeneNetwork database and were analyzed . Using the comparative genomics functions of Ensembl, all human genes syntenic with the mouse loci were identified. Human subjects had been genotyped previously using the Illumina HumanCNV370-Duo genotyping array. Marker SNPs were chosen for analysis based on location within the genomic limits for the gene’s transcript plus 100 kb flanking the transcription start and stop sites. We thus had markers spanning the transcripts of the syntenic genes plus possible regulatory sequences flanking the transcribed DNA. Each marker SNP was tested for association to BSA using a regression model. False discovery rate q-values were calculated by treating each syntenic locus as a separate hypothesis. The results for each marker are presented in supporting information, Table S1, and the top associated markers are presented in All GPC5 (glypican 5) became the gene of interest due to the statistical strength of the linkage data in mice and the association data in humans. The BXD linkage yielded a LOD score of 3.9 (P ≈ 1.25E−05), slightly shy of the genome-wide significance value of 4.3 (P ≈ 5.0E−06). Similarly, the multiple test corrected significance value for association to BSA was q = 0.08  very close to the standard q = 0.05 threshold for significance. Taken together, these data implicate GPC5 as a candidate QTL that influences ethanol-induced ataxia.Drosophila glypican genes dally and dlp were investigated to determine whether they alter Drosophila ethanol response. The six mammalian glypican genes have two homologs in the Drosophila genome. Evolutionarily, dally is thought to be orthologous to the human glypican gene paralogs GPC3 and GPC5, whereas dlp is orthologous to the other four human glypican paralogs ; f03537dlp failed to complement 1dlp lethality, strong evidence that f03537dlp is a dlp allele. MB950dally was found to be homozygous viable and fertile, and it also had no overt morphological or behavioral phenotypes. Heads of homozygous MB950dally flies were shown to have a 27% reduction in dally transcript , but MB950dally complemented the sterility and wing vein morphology phenotypes of the strong loss-of-function allele 80dally. Given the molecular evidence that MB950dally is an allele of dally, a transposon inserted into the gene coupled with reduced transcript production, we conclude that intragenic complementation is occurring because the MB950dally allele provides enough function to complement the phenotype of the strong loss-of-function allele.Partial loss-of-function alleles of t intron . dlpf035MB950dally and f03537dlp alleles were evaluated for alterations in their response to ethanol. Three assays were performed: locomotor stimulation in response to moderate concentrations of ethanol; sedation in response to higher concentrations of ethanol; and the acquisition of rapid tolerance to ethanol sedation. Flies exposed to a continuous stream of moderate concentration ethanol vapor (47%) exhibit a characteristic pattern of locomotor activity. When ethanol is first introduced, flies show a transient burst of locomotor activity—a startle response to the smell of ethanol  are heparan sulfate proteoglycans (HSPG) linked to the cell surface via a glycosylphosphatidylinositol (GPI) anchor. Much of what we know about glypican function in mammals was founded upon dally and dlp are expressed in the nervous system during development and in adulthood  and its binding to an ethanol response sequence element located between exons 1 and 2, Drosophila dally expression was transiently increased following exposure to a sedating dose of ethanol (In addition to being expressed in brain regions thought to play a role in ethanol response, there is evidence that GPC5 transcription can be modulated by ethanol. After discovering that ethanol-induced expression of Glypicans have been shown to participate in organismal development through their interactions with the Wnt, Hedgehog, FGF, and BMP morphogen signaling pathways . These dDrosophila behavioral experiments demonstrated that mutant alleles of the gene identified in mice and humans alter the fly’s ethanol response. In addition to our analyses, the literature on GPC5 reports its expression to be spatially and temporally consistent with a putative involvement in ethanol behaviors, and moreover, its expression can be modulated by ethanol. Finally, the known functions of glypicans as modulators of cell signaling systems that participate in neural development as well as synaptic maintenance and plasticity are also consistent with participation in ethanol-induced behaviors. GPC5 is thus a strong candidate as a gene involved in ethanol response and the development of alcohol use disorders.In summary, we presented evidence that the glypican GPC5 participates in human ethanol response, an endophenotype related to AUD risk. The strength of the evidence is derived from the coalescence of results across different experimental systems: mouse linkage identified the candidate locus, human genetic association identified a single gene contained within the mouse-defined locus, and


The interest of the scientific community in investigating the impact of rare variants on complex traits has stimulated the development of novel statistical methodologies for association studies. The fact that many of the recently proposed methods for association studies suffer from low power to identify a genetic association motivates the incorporation of prior knowledge into statistical tests.In this article we propose a methodology to incorporate prior information into the region-based score test. Within our framework prior information is used to partition variants within a region into several groups, following which asymptotically independent group statistics are constructed and then combined into a global test statistic. Under the null hypothesis the distribution of our test statistic has lower degrees of freedom compared with those of the region-based score statistic. Theoretical power comparison, population genetics simulations and results from analysis of the GAW17 sequencing data set suggest that under some scenarios our method may perform as well as or outperform the score test and other competing methods.An approach which uses prior information to improve the power of the region-based score test is proposed. Theoretical power comparison, population genetics simulations and the results of GAW17 data analysis showed that for some scenarios power of our method is on the level with or higher than those of the score test and other methods. It shoui et al.  were notQ1 causal genes. As can be seen, our method was among the top three most powerful approaches for ANRT, ELAVL4 and HIF1A genes. Panel 1 of Additional file Q2 causal genes. Our method was the top performer for VNN3 gene, and was among the top three performing approaches for LPL and VNN1 genes. Panel 2 of Additional file PIK3C3 and PTK2B genes. This emphasizes that SKAT-O may significantly outperform other tests under some phenotype models. However, this method also may significantly underperform other tests for some phenotype models, for example, ELAVL4 gene with Q1 trait and VNN1 gene with Q2 trait. Our proposed method was among the top three performing methods for SOS2, PTK2B, PRKCA genes and some other genes for which the power of all the methods was low.Panel 3 of Figure In this article we have described a methodology that incorporates prior information into a region-based score test with the purpose of improving power to identify an association. Prior information, such as observed MAF, functional annotation, predicted deleteriousness of non-synonymous variants, may be used to partition variants within a region of interest. Then, this partitioning is utilized to construct a test statistic whose distribution under the null hypothesis has lower degrees of freedom compared with those of the score test. Based on the theoretical power comparison, population genetics simulations and GAW17 sequencing data analysis, we have shown that under some scenarios our method may perform as well as or outperform other methods.Our suggested partitioning is splitting by functionality and if rare variants are present – by MAF with an arbitrary threshold, e.g., 1%. One of the major justifications for considering MAF in a partitioning design is that MAF may distinguish between different evolutionary forces acting on causal variants. Evolutionary theory predicts that variants that confer susceptibility to a disease which in turn reduces fitness are expected to have low MAF due to purifying selection. Empirically, using whole-exome sequencing data, it was found that non-synonymous substitutions are more significantly skewed towards low frequencies compared with synonymous variants. This finding “almost certainly reflects the operation of purifying selection”  acting oV is invariant with respect to permutation of elements of the score vector U . So, for any SNP ordering the Cholesky decomposition A is the same; hence, the test statistic is, in general, dependent on ordering. To clarify the ambiguity the ordering of SNPs according to a position on a chromosome may be assumed.There are a few limitations of the proposed methodology. First, it is unknown in advance which prior information is relevant for a given genomic region. Prior information that does not help to uncover groups of associated variants is likely to have negative impact on statistical power. Second, the test statistic (3) is, in general, not invariant with respect to the ordering of SNPs. This can be seen, for example, when the covariance matrix A novel statistical approach that incorporates prior information in a region-based score test with the purpose of improving power has been proposed. Theoretical power comparison, population genetics simulations and the results of the GAW17 data set analysis showed that under some scenarios our method may perform as well as or outperform other methods.G = {gnl, n = 1, …, N, l = 1, …, L} coded as minor allele counts, where n and l are indices for individuals and variants respectively; N × 1 vector of genotypes for the lth variant gl = {gnl, n = 1, …, N}; mean of genotype for the lth variant Y = {yn, n = 1, …, N}, mean of phenotype g(Ey) = b0 + Gb, where b = {b1, …, bL} is a vector of regression coefficients, and g is a monotone link function. The score test statistic used to test an association of phenotype with genotype is as follows [Consider a rare variants association study of a genomic region with a dichotomous or quantitative trait. Let us introduce the following notations: genotype matrix  follows :L × L the score covariance matrix assumed to be non-singular, and L × L genotype covariance matrix. Under the null hypothesis (none of the variants is associated with phenotype) T0 asymptotically follows chi-squared distribution with rank(V) degrees of freedom (d.f.). Under the alternative hypothesis (at least one variant is associated with phenotype) the asymptotic distribution of (2) can be approximated by a non-central chi-squared distribution.where U to be asymptotically independent under the null hypothesis. If we denote C = (AT)−1, where matrix A is a Cholesky decomposition of V (V = ATA), the vector S = CU, under the null hypothesis, is asymptotically distributed as a standard multivariate normal random vector (since  Cov(CU) = CCov(U)CT = (AT)− 1VA− 1 = (AT)− 1(ATA)A− 1 = I). It should be noted that under the alternative hypothesis S is approximately distributed as a multivariate normal random vector with nonzero mean and unit covariance matrix under the assumption of V being a reasonable approximation for the covariance matrix of U[V of the form V = ATA can be used. We apply Cholesky decomposition because it is fast to compute even for big covariance matrices.Following is the description of our proposed method. First, let us transform the coordinates of score vector trix of U. In geneL SNPs within a region into K disjoint groups Gk, k = 1, …, K of size Lk, k = 1, …, K. This partitioning is done using prior information, for example: by observed MAF (below and above some arbitrary frequency threshold); by functional annotation ; by functional significance for non-synonymous SNPS  etc. For each group let us define sl, l = 1, …, L is the lth coordinate of the vector S. It should be noted that partitioning based on prior to S, since S = (AT)−1U is the vector U in a new basis, which is the set of column vectors of the matrix AT (old basis is a standard basis). Since the vector S is multivariate normal with unit covariance matrix, tk, k = 1, …, K are asymptotically independent. Under the null hypothesis tk asymptotically follows central chi-squared distribution with Lk degrees of freedom, whereas under the alternative hypothesis at least one of the tk, k = 1, …, K is distributed as a non-central chi-squared random variable. Let us denote as Pl and Pl−1 cumulative distribution function (CDF) of central chi-squared random variable with l d.f. and its inverse respectively. Group statistics Hk are defined as follows:Next, let us describe group statistics. Consider partitioning of Hk follows a central chi-squared distribution with 1 d.f. Finally, the test statistic of our proposed method is:It should be noted that under the null hypothesis T1 is distributed as a chi-squared random variable with K d.f. Large value of T1 indicates the deviation from the null hypothesis. Transformation (3) is used to decrease the degrees of freedom of the final test, as some of the groups Gk, k = 1, …, K may contain only neutral variants. If kth group does not contain associated SNPs, group Gk contributes only 1 d.f. to the final test, whereas it adds Lk d.f. to the score test. Thus, under “informative” partitioning our proposed method may gain power compared with the score test.Under the null hypothesis tk may not hold under moderate sample sizes. Hence, in all our simulations and real data application we used the empirical CDF of tk, denoted as Fk, obtained via 1000 permutations under the null hypothesis. Instead of Fk(tk) with a multi-stage plug-in bandwidth of Polansky and Baker [http://cran.r-project.org) package ‘kerdiest’ contains all the necessary algorithms. Second, the covariance matrix V may happen to be computationally or exactly singular when, for example, considering a region with high LD between common SNPs. In general, to avoid singularity issues we apply Monroe-Penrose pseudoinverse in (2) instead of V−1. This pseudoinverse is uniquely defined and equals the standard inverse matrix when V is non-singular. From the same considerations instead of usual Cholesky factoring we used generalized Cholesky decomposition, which is well defined for square symmetric positive semi-definite matrices. If V is non-singular, the generalized Cholesky decomposition equals the usual Cholesky factoring. Also, it should be noted that in general our test statistic is not invariant with respect to the ordering of variants within a region. To clarify the ambiguity we assume the ordering of SNPs according to a position on a chromosome (see the “Discussion” section).There are few notes that should be taken into account when applying our method. First, the theoretical approximation for distribution of nd Baker . R  of the test statistic (2) under the alternative hypothesis as qα,L1− is 1 − α quantile of the chi-squared distribution with L d.f. Let us assume that all the causal SNPs are uncorrelated with other SNPs within a region, and that they are in one of the K groups . Then, the power of the proposed test is:where K − 1 d.f., because all associated SNPs are in the first group. The probability in (6) can be computed using the convolution of two CDFs as follows:since T0 and those of t1 are equal, since both statistics include all the associated SNPs within a region. Thus, the distribution of H1 is Under the alternative hypothesis non-centrality parameter of −8 per nucleotide per generation. The selective disadvantage was additive. Using the forward simulator SFS_CODE (http://sfscode.sourceforge.net) with the parameters described above, we generated 100 haplotype pools for a 3 kb coding genomic region. Each haplotype pool contained 4000 haplotypes. In our simulations we considered only non-synonymous variants. To generate a data replicate we first selected a haplotype pool at random. Within that pool we sampled a pair of haplotypes at random and added those haplotypes to form a multi-site genotype of an “individual”. Dichotomous phenotype was assigned based on this multi-site genotype using a linear logistic model through which we controlled the odds ratios of causal variants. The probability of a disease conditional on the wild type genotype was set at 1% for all phenotype models.Boyko et al.  found a RET gene associated with Hirschsprung’s disease [We considered four phenotype scenarios. For the first model called “Rare” randomly chosen 50% of rare variants (defined as those with MAF<1%) within a haplotype pool were assigned to be causal with uniform odds ratio of 3. For the “Low Frequency” scenario one randomly chosen low frequency SNP (MAF between 1% and 5% in a haplotype pool) was causal with odds ratio of 2.5. If there was no SNP with MAF between 1% and 5% in a haplotype pool, we selected a SNP with the lowest MAF above 5%. For the “Common” scenario one common SNP with MAF>5% in a haplotype pool was causal and had the odds ratio of 1.5. Finally, the “Interaction” scenario models the hypothesized interaction of common and rare variants in  disease ,41. The  disease . BrieflySample sizes were the following: 500 cases and 500 controls for “Rare”, “Low frequency” and “Common” scenarios; 1000 cases and 1000 controls for “Interaction” model. In total 1000 data replicates were generated for each scenario. Power was estimated as a proportion of data replicates significant at a fixed type-1 error of 0.05. For all the statistical tests 1000 permutations were applied to estimate a significance level.The authors declare that they have no competing interests.SZ, AS and AT conceived the study. SZ and AT designed the experiments. SZ and GHKT conducted the experiments and performed the analysis. SZ wrote the manuscript. All authors read and approved the final manuscript.The difference in theoretical power  between the proposed test and the score test as a function of the total non-centrality parameter r  at the genome-wide type-1 error rate α = 0.05/35000. Each curve corresponds to the number of SNPs in the causal group L1 given in the legend (Panels 1 and 2), number of groups K (Panel 3), and number of causal groups m (Panel 4). The parameters for each of the Panels are as follows: Panel 1: L = 10, K = 2; Panel 2: L = 100, K = 10; Panel 3: L = 50, L1 = 5; Panel 4: L = 54, K = 6, equal number of SNPs in each group, and equal non-centrality parameters in all causal groups.Click here for fileThe estimate of empirical type-1 error rate of the proposed method with MAF partitioning, the score test, VT, WSCS, SSUw and SKAT-O for population genetics simulations. The theoretical type-1 error was assumed to be 0.05. The data for the estimate of the type-1 error was generated using the null phenotype model: no association of genotype with phenotype.Click here for fileThe estimate of empirical type-1 error rate of the proposed method with different partitionings and those of the score test for the causal genes in GAW17 data. The theoretical type-1 error was assumed to be 0.05. Panel 1: Q1 and Q2 causal genes and respective quantitative trait ; Panel 2: causal genes and dichotomous trait.Click here for fileThe estimate of empirical type-1 error rate of WSCS, VT, SSUw and SKAT-O tests for the causal genes in GAW17 data. The theoretical type-1 error was assumed to be 0.05. Panel 1: Q1 and Q2 causal genes and respective quantitative trait ; Panel 2: causal genes and a dichotomous trait.Click here for fileSome results of GAW17 analysis. Panel 1: Comparison of the proposed method (MAF partitioning) with other methods on Q2 causal genes; Panel 2: Comparison of the proposed method (MAF partitioning) with other methods on causal genes and a dichotomous phenotype.Click here for fileThe link between the non-centrality parameter and the effect size for the region-based score test. The derivation of the equation connecting the non-centrality parameter (NCP) and the effect size for a region-based score test, and the description of assumptions for illustrating the dependence of NCP on the effect size in Additional file Click here for fileThe non-centrality parameter  as a function of the effect size (relative risk) of each causal variant  under the assumptions described in Additional file 6. Curves within each panel correspond to the number of causal variants within the region. The assumptions are as follows: all variants within a region are independent; all the causal variants have the same MAF and the same effect size; 500 cases and 500 controls. The MAF of the causal variants are as follows: Panel 1 – 1%; Panel 2 – 0.5%; Panel 3 – 0.25%: Panel 4 – 0.125%.Click here for file


Solanum tuberosum) that is crucial for crop quality. As phenotypic evaluation of bruising is cumbersome, the application of diagnostic molecular markers would empower the selection of low bruising potato varieties. The genetic factors and molecular networks underlying enzymatic tissue discoloration are sparsely known. Hitherto there is no association study dealing with tuber bruising and diagnostic markers for enzymatic discoloration are rare.Most agronomic plant traits result from complex molecular networks involving multiple genes and from environmental factors. One such trait is the enzymatic discoloration of fruit and tuber tissues initiated by mechanical impact (bruising). Tuber susceptibility to bruising is a complex trait of the cultivated potato  and robust marker-trait associations were identified.The observed trait correlations and associated marker fragments provide new insight in the molecular basis of bruising susceptibility and its natural variation. The markers diagnostic for increased or decreased bruising susceptibility will facilitate the combination of superior alleles in breeding programs. In addition, this study presents novel candidates that might control enzymatic tissue discoloration and tuber bruising. Their validation and characterization will increase the knowledge about the underlying biological processes. Enzymatic browning is a phenomenon frequently observed in fruits and vegetables upon physical damage. The tissue discoloration results from the oxidation of endogenous phenolic compounds, which is catalyzed by oxidoreductases, notably polyphenol oxidases (PPOs) or tyrosinases in the presence of oxygen -8. BesidPOTP1 and POTP2  × 100, with L, M and S being the fraction of tubers in the L, M or S bruising category, respectively. The bruising index ranged from 0 (resistant to bruising) to 100 (highest bruising susceptibility). Tuber shape (TS), scores ranging from 1 (completely circular) to 9 , and plant maturity (PM), scores ranging from 1 (very early) to 9 (very late) were also evaluated in accordance with the assessments of the German variety office. Tuber starch content (percent fresh weight) was quantified by measuring specific gravity (SG) . Tuber yhttp://solgenomics.net/) and Additional file Genomic DNA was isolated from freeze-dried leaf tissue as described . DNA fraThe fact that each breeding company evaluated their clones together with the common set of 85 varieties was used to perform a joint analysis of all six locations, where each year-location combination was treated as an environment. The phenotypic data were analyzed using the statistical model:ijk ywas the entry mean for the ith clone in the kth replication of the jth environment, μ was an intercept term, i gwas the genetic effect of the ith clone, j lwas the effect of the jth environment, ij (gl)was the effect of genotype by environment interaction, jk rwas the effect of the kth replicate at the jth location, and ijk ewas the residual. For estimation of variance components, all effects were considered as random. Heritability on an entry mean basis was calculated as where ry means .i gwas considered as fixed. Over all environments, an adjusted entry mean i Mwas calculated for each of the 205 clones as: μ and gi, respectively.For calculation of adjusted entry means, Associations among the 205 tetraploid potato clones were revealed with principal coordinate analysis based on the allele frequency matrix. This analysis was performed based on 157 SSR marker alleles using the statistical software R .The PK method described by  was usedijp Mis the adjusted entry mean of the ith potato clone carrying the pth allele, μ an intercept term, u vthe effect of the uth column of the population structure matrix P, p athe effect of allele p, ith entry except for pa, ip ethe residual.where Stich and Melchinger[q principal components of an allele frequency matrix were used as P matrix. q was chosen in such a way that the explained variance of the first q principal components was about 25%.Following elchinger, the fire = {e1, 1,..., e205, p} was assumed to be ×205 matrix of kinship coefficients that define the degree of genetic covariance between all pairs of entries and was calculated using SPAGEDI [The variance of the random effects  SPAGEDI . R was aIn order to take into account the LD between markers, we used in addition to a single marker approach a multiple forward regression approach to identify based on the above described PK model those marker combinations, which explain best the genotypic variation. A P-to-enter criterion of 0.001 was used.All mixed-model calculations were performed with ASReml release 2.0 .CU and LS carried out the molecular genetics studies and the laboratory procedures. CG and CU conceived of the study, designed and evaluated the experiments and drafted the manuscript. BS carried out the statistical analysis and contributed to draft the manuscript. KHN, AB, ET, LuS, HRH, HB, HJ, JL and JS conceived of the study, provided the genetic material, carried out the phenotypic evaluations and coordinated field experiments. All authors read and approved the manuscript.Genotype and sample identification with the corresponding breeder/distributor affiliation.Click here for fileMarker trait associations in the ALL population (P < 0.01).Click here for fileLocus and marker information.Click here for fileElectrophoresis band distribution and identification of significant markers at P < 0.001. In the case of SSR and DALP markers gel pictures show the band pattern and distribution of the corresponding marker fragments. Polymorphic bands are labelled as described in Additional table 3 and listed with their respective size in basepairs. SSRs were analyzed according to the procedures given in the methods section. DALP markers were separated on agarose gels. SSCP analysis was carried out as described elsewhere [StI013; (B) StI024; (C) SSR327; (D) POLOXA; (E) PHO1A; (F) PHO1B-1; (G) LIPIII-27-1; (H) POT32PS1-Hpy; (I) 4CL-1; (J); 4CL-2; (K) HQT; (L); GLDH. Significant marker-trait associations are indicated by the specific trait abbreviation: bruising susceptibility (BI), specific gravity (SG), starch corrected bruising (SCB), and tuber yield (TY).lsewhere  ExemplarClick here for fileExperimental design of the field experiments and data evaluation. Breeders are listed with their corresponding abbreviation. They cultivated and evaluated the common set of 85 potato varieties in two consecutive years (2007/2008) for bruising susceptibility (BI), specific gravity (SG), tuber shape (TS), tuber yield (TY), starch corrected bruising (SCB) and plant maturity (PM). In addition each breeder evaluated 20 own breeding clones, selected based on a specific gravity below 20 percent, a balanced tuber shape, no extreme maturity phenotype, not being closely related, and one half each having a high or a low bruising phenotype.Click here for fileSSR marker information.Click here for file


We adapt a fast permutation procedure into these three methods. The results show that using the fast permutation procedure can produce FWERs and average powers close to the values obtained from the standard permutation procedure on the GAW17 data sets. The standard permutation procedure is computationally intensive.We evaluate four association tests for rare variants—the combined multivariate and collapsing (CMC) method, two weighted-sum methods, and a variable threshold method—by applying them to the simulated data sets of unrelated individuals in the Genetic Analysis Workshop 17 (GAW17) data. The family-wise error rate (FWER) and average power are used as criteria for evaluation. Our results show that when all nonsynonymous SNPs (rare variants and common variants) in a gene are jointly analyzed, the CMC method fails to control the FWER; when only rare variants  are analyzed, all four methods can control FWER well. All four methods have comparable power, which is low for the analysis of the GAW17 data sets. Three of the methods (not including the CMC method) involve estimation of  Genome-wide association studies have become a useful tool for identifying the genetic variants that influence complex diseases or traits . Traditiα = 0.05, we use the Bonferroni procedure to adjust the threshold (α′) for each gene, α′ = α/ngene. Genes with p-values less than or equal to α′ are treated as significant genes.In this study, we analyze only nonsynonymous single-nucleotide polymorphisms (SNPs) from 22 autosomal chromosomes. The SNPs in each gene are analyzed as a functional unit. For a genome-wide significance level of fn /n, where fn is the number of replicates in which at least one unrelated gene is falsely detected and n = 200 is the total number of phenotype replicates of simulated data.When testing association for a single gene, one can use the type I error rate to measure the false-positive discoveries. For genome-wide gene-based association studies that involve multiple hypothesis testing, we use the family-wise error rate (FWER) to measure the false-positive discoveries. The FWER is the probability of falsely rejecting at least one true null hypothesis. When analyzing the 200 simulated replicates of unrelated individuals in the GAW17 data, we estimate the FWER as FWER = For testing a single gene that affects the disease, we can calculate the power of a test. This power is referred to as the per-hypothesis power. For genome-wide gene association studies, we estimate the average power, which is defined as the average of the per-hypothesis powers of tests for the genes that affect the disease . For exaWe describe briefly the four methods used in this study: the CMC method, the two weighted-sum methods, and the variable threshold method.The basic idea of the CMC method  is to emjγ) for individual j as a weighted sum of the genetic scores of all variants in the gene and then calculates the sum of the ranks for affected individuals as:For a case-control data set, weighted-sum method I proposed by Madsen and Browning  tests a A is the population of the affected individuals. For the case-control data set, permuting the affected or unaffected status of the individuals can generate a new data set. Suppose that k permuted data sets are generated from the original data set by permutation. Let jx denote sums of ranks for the jth permuted data set . Let jx. Weighted-sum method I defines a standardized score-sum test statistic:where z has an approximately standard normal distribution under the null hypothesis. The p-value of the test statistic z for the observed data is calculated using the normal distribution N , where:with a 100, if for any number 1,000 ≤ N ≤ mN the corresponding lower bound LP is greater than or equal to α′, then the true p-value for this gene is unlikely reach the Bonferroni threshold α′; therefore we give up the permutation procedure and do not reject the null hypothesis for this gene. This procedure can save computing time significantly. Note that in this paper, for the standard permutation procedure, we use (X + 1)/(N + 1) to estimate the p-value.In the fast permutation procedure for testing a gene, given the maximum number of permutations D1, D2, and D4 for Q1, Q2, and Q4, respectively. We assign iD = 1 (disease) for individuals with iQ  among the top 30% percentile and iD = 0 (control) for the remaining individuals . The phenotypes D1, D2, and disease status are used to calculate the average power, and D4 is used to calculate the FWER because D4 is not related to any genes.The GAW17 data include 24,487 SNPs from 3,205 genes . We testp-values based on analyzing the simulated replicates of unrelated individuals with all nonsynonymous SNPs. The permutation procedures include the normal approximation procedure with both 103 and 106 permutations, the fast permutation procedure with the maximum number of permutations mN = 106, and the standard procedure with 106 permutations. We set the genome-wide significance level to α = 0.05. Here we report only the FWER and average power of weighted-sum method I  approximately follow the standard normal distribution N under the null hypothesis. However, this may not be the case when the two methods are applied to the 200 simulated case-control data sets from the GAW17 data. For these data sets, when we set k = 1,000 or even larger, the quantile-quantile plots show that the score-sum test statistics z often have distributions that deviate from the normal distribution (data not shown).Why do the normal approximation procedures in the two weighted-sum methods generate inflated FWERs? One major reason is that the two methods assume that the standardized score-sum test statistics p-values. To compare the four methods, we apply them to two scenarios: In scenario 1 all nonsynonymous SNPs (rare and common variants) and therefore 2,196 genes are analyzed. Thus ngene = 2,196, and the threshold for a single-gene α′ = 2.27 × 10−5. In scenario 2 only nonsynonymous SNPs with MAF < 0.05 and therefore 1,999 genes are analyzed. The corresponding ngene = 1,999, and α′ = α/ngene = 2.5013 × 10−5. When applying the CMC method to scenario 1, we divide nonsynonymous SNPs in a gene into subgroups as follows: SNPs with MAFs in the interval  form the second subgroup; each remaining SNP with MAF ≥ 0.1 is treated as a subgroup by itself; all subgroups (variants) are analyzed using a Hotelling T2 test. When applying the CMC method to scenario 2, we divide nonsynonymous SNPs in a gene into three subgroups with SNPs having MAFs in the intervals  may be the small number of replicated data sets (200) and the conservative nature of the variable threshold. Tables The FWER of the variable threshold method is 0 in Table ata sets 00 and thT2 test for subgroups of variants in a gene. Under the null hypothesis that no variants are associated with the disease, a key assumption for the Hotelling T2 test is that within each subgroup the proportion of individuals with at least one minor allele in case subjects is equal to the proportion in control subjects. When there is possible population stratification in the data as well as the existence of confounding variables such as Age or Smoking, the assumption may not be true and therefore the FWER of the Hotelling T2 test is inflated. The results also show that common variants are the main sources of the inflated FWERs (the CMC method does not have an inflated FWER when applied to SNPs with MAF < 0.05).The CMC method is based on the Hotelling In this study, we evaluated four association tests for detecting rare variants by using the GAW17 data sets. When the rare variants and common variants were jointly analyzed, two methods (CMC and weighted-sum method I) had an inflated FWER; when only rare variants with MAF < 0.5 were analyzed, the four methods could control the FWER. It seems that including common variants in the analysis can increase the FWER of these four methods.It is natural to ask why the average powers of these four methods are low for the analysis of the GAW17 data sets. There are three possible reasons. First, the phenotypes Q1 and Q2 do not completely depend on the genetic data. They also depend on other factors, such as age, sex, smoking history, and the correlation between Q1 and Q2. These factors may make it hard to detect the associated genes. Using a more general model that could account for these factors might improve the power. Second, the phenotypes are influenced by multiple genes. The effect of each gene may not be large, although the combined effects may be considerably significant. Testing a gene only individually certainly does not consider the aggregation effect of genes in the data. Third, population stratification and admixture may cause loss of power.p-value estimation can produce FWERs and average powers close to those obtained using the standard permutation procedure.For the simulated data sets of unrelated individuals in the GAW17 data, when the rare variants and common variants are analyzed jointly, the CMC method cannot control the FWER and weighted-sum method I has a slightly inflated FWER; when only the rare variants with MAF < 0.05 are analyzed, all methods control the FWER well. In both situations, the four methods have comparable power, which is low, because of the complexity of the GAW17 data. Using the fast permutation procedure for The authors declare that they have no competing interests.WC and XG participated in the design of the study, did the simulation and drafted the manuscript. JW, CS, WW, DZ, NL, XC participated in the discussion of the study and revision of the manuscript. GG conceived of the study, and participated in designing the study, drafting and revising the manuscript. All authors read and approved the final manuscript.


Microarray single-nucleotide polymorphism genotyping, combined with imputation of untyped variants, has been widely adopted as an efficient means to interrogate variation across the human genome. “Genomic coverage” is the total proportion of genomic variation captured by an array, either by direct observation or through an indirect means such as linkage disequilibrium or imputation. We have performed imputation-based genomic coverage assessments of eight current genotyping arrays that assay from ~0.3 to ~5 million variants. Coverage was determined separately in each of the four continental ancestry groups in the 1000 Genomes Project phase 1 release. We used the subset of 1000 Genomes variants present on each array to impute the remaining variants and assessed coverage based on correlation between imputed and observed allelic dosages. More than 75% of common variants  are covered by all arrays in all groups except for African ancestry, and up to ~90% in all ancestries for the highest density arrays. In contrast, less than 40% of less common variants  are covered by low density arrays in all ancestries and 50–80% in high density arrays, depending on ancestry. We also calculated genome-wide power to detect variant-trait association in a case-control design, across varying sample sizes, effect sizes, and minor allele frequency ranges, and compare these array-based power estimates with a hypothetical array that would type all variants in 1000 Genomes. These imputation-based genomic coverage and power analyses are intended as a practical guide to researchers planning genetic studies. Microarray genotyping has been widely adopted as an efficient means to interrogate variation across the human genome. Such arrays are the technological foundation of genome-wide association studies (GWAS), which have uncovered numerous genetic associations for a wide variety of traits and diseases .2r) between discrete allelic dosages at the untyped variant and any of the genotyped variants. Thus one way to describe genomic coverage is as the fraction of variants in a reference set having a maximum 2r with array variants above a given threshold . Genomic coverage for the first generation of SNP microarrays typically was estimated using this pairwise LD paradigm and with the HapMap Project  allelic dosage at a genomic variant and the imputed (continuous) allelic dosage, over a defined set of samples. For a variant with two alleles A and a, the imputed dosage is calculated as 2P(AA) + P(Aa), where P(AA) and P(Aa) are posterior genotype probabilities from imputation. Under this imputation paradigm, genomic coverage can be reported as the fraction of variants in a reference set with imputation 2r above a given threshold   .2r coverage metrics is to estimate the power to detect an association when the causal locus is not observed directly. 2r for observed genotypes at a marker locus, where 2r is the squared correlation between the discrete allele dosages at the causal and marker loci. This relationship has been used to estimate the genome-wide power to detect an association with various SNP arrays, using the maximum pairwise 2r for each variant in a genome-wide reference set such as the HapMap , assuming a linear relationship. We also provide genome-wide power estimates with imputation 2r values, estimated for each ancestry group in a 1000 Genomes Project reference set.An important application of 2 values . Here we2r. Updated coverage assessments are provided here using new arrays, a denser and more robust data release from the 1000 Genomes Project, and imputation-based 2r. New arrays have been developed in response to both the changing needs of the genetic research community and technological advances in genotyping. While the design of the first generation of arrays was informed primarily by the HapMap Project  Infinium HumanCore and the Affymetrix (www.affymetrix.com) Axiom Biobank. Although vendors frequently provide genomic coverage estimates in product documentation, the methods used are often different and/or not well-described, making it difficult to objectively compare across arrays. Here we have used a consistent, imputation-based approach to evaluate genomic coverage and power across eight different genotyping arrays. These analyses are intended to function as a practical guide to researchers who are planning genetic studies. The results may be used to compare the cost efficiency of commercially available arrays, to assess the power to detect trait associations using imputed allelic dosages, and to design custom arrays. They also provide an evaluation of the accuracy of imputation for different categories of minor allele frequency (MAF).Despite a growing focus on rare variants, the tagging, LD-based variant selection scheme is still of prime importance in array design. Many arrays still include a core, or “backbone,” of GWAS markers selected using the same principles as the first generation of arrays. Each of the major commercial vendors has recently released  productsftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/release/20110521/), comprising 1,092 samples from 14 populations  the concordance between observed and most likely imputed genotype, the “genotype concordance”; and (3) the concordance between observed and most likely imputed genotype, when at least one of those two genotypes contains one or two copies of the minor allele, which we call “minor allele (MA) concordance.” Array (observed) variants are included in these metrics summaries and are given imputation 2r, genotype concordance, and MA concordance values of 1. A more detailed account of data preparation, pre-phasing, imputation, and metrics calculation is available in Supporting Information, File S1.To assess imputation accuracy and by extension genomic coverage, we compared imputed results at all the nonarray variants to observed genotypes from the initial VCF files. These comparisons were performed separately in the four different ancestry groups  and restricted to variants with at least two copies of the minor allele. For each imputed variant we calculated three metrics: (1) the squared correlation between observed and imputed allelic dosage, which we call “imputation 2r as the primary coverage metric mainly due to its simple relationship to power, in addition to the following advantages: (1) precedent in the literature for evaluating both imputation accuracy  low MAF . OverallA is the risk allele and a is the alternate allele: genotype relative risk  from 1.1 to 1.4 in increments of 0.1; disease prevalence (K) of 0.05; frequency of the risk allele (p) assumed in each case to be the minor allele; the sample size (N cases and N controls) from N = 1000 to 10,000 in increments of 50; and a significance level of 5 × 10−8 (appropriate for genome-wide testing). The assumption of an additive model is based on how imputed dosages are used in association tests, and the range of GRR is based on a summary of effect estimates from genome-wide association studies (2 with 1 degree of freedom) using the non-centrality parameter in the Appendix. Sample size was N for observed (array) variants and N/(imputation 2r) for imputed variants. A genome-wide power estimate was obtained for each array and parameter set by averaging over all variants in a reference set, as indicated by 2r values) and averaged over all variants in the set. In practice, to reduce computation, we binned all autosomal variants by a combination of MAF and 2r values, calculated power for the mean values of MAF and 2r within each bin, and took the mean over bins weighted by the fraction of variants in each bin, as in 2r was from 0 to 1 in increments of 0.05.The power to detect a variant-trait association was calculated for a case-control study under the assumption of an additive genetic model, using the following parameters, where i.e., at least two copies of the minor allele observed in any one of the four ancestry groups). As may be expected, the overlap with 1000 Genomes for arrays with a high proportion of rare and exome variants is less than for arrays with less content devoted to rare and exome variants. The possible impact of the differential 1000 Genomes overlap across arrays on the estimated genomic coverage appears to be small (see File S1).We have assessed imputation-based genomic coverage for eight commercially available genotyping arrays using the four ancestry groups in the 1000 Genomes Project phase 1 release. The number of assays and unique genomic positions assayed by each array are summarized in 2r threshold of 0.8, by MAF bin and ancestry group, whereas 2r and genotype concordance and the counts of variants in each MAF bin, as this count differs by ancestry group. Line plots of mean imputation 2r and mean genotype concordance are available in Figure S1 and Figure S2. Plots organized by ancestry panel rather than by metric are available for AFR in Figure S3, AMR in Figure S4, ASN in Figure S5, and EUR in Figure S6.2r ≥ 0.8 is greater than 75% in these three ancestry groups, and up to ~90% in all four ancestry groups for the highest density arrays. The most dramatic increase in coverage occurs for common variants in the AFR group: the fraction of common variants passing the imputation 2r threshold of 0.8 increases by almost 40% from the least to most dense array. This finding is likely explained by the genetic diversity and lower LD characteristic of present day African ancestry populations compared to the other ancestry groups  are well-covered by all the arrays in the AMR, ASN, and EUR ancestry groups. The percentage of common variants with imputation However, coverage at common variants eventually levels off with increasing array density, a phenomenon of diminishing returns previously observed by 2r ≥ 0.8 is substantially lower than for MAF greater than 0.05: less than 40% for low density arrays in all ancestries and 50–80% in high density arrays, depending on ancestry. In all ancestry groups there is some improvement in coverage of less common variants when moving from the Omni2.5M to the Omni5M array, most notably in EUR . Rare (MAF < 0.01) variants are generally not well imputed by either low or high density arrays. Imputation accuracy for rare variants is greatest in the AMR ancestry group, which might be a consequence of the lower genetic diversity and greater LD in Native Americans than in other ancestry groups, due to a recent population bottleneck associated with human migration into the Americas , which is equivalent to direct observation of all 1000 Genomes Project variants passing the ancestry-specific MAF filters  and thus does not involve uncertainty introduced by imputation. Power is generally low for the range of parameters considered here (even for “1000 Genomes” array), except at the upper end of genetic effect and sample sizes. This is particularly true for the less common variants, where power is generally < 50% even at the high end of the parameter space for the actual arrays and up to ~60% for the “1000 Genomes” array.Genome-wide power estimates for GRR values of 1.2, 1.3 and 1.4 are displayed in vs. 8900). Generally, the array differences required to achieve a set power are notably larger in regions where the power curves plateau. One array is more cost-efficient than another when the ratio of cost per sample is less than the ratio of sample sizes required to achieve equal power. Of course, this evaluation assumes that the extra samples required are available and that the main interest of the study is to detect effects characterized by the parameters used in the power calculation. All of the data used to generate the power plots are available for download at http://jhir.library.jhu.edu/handle/1774.2/36508, so that users can select parameter sets appropriate to their project goals. We also provide the number of variants per MAF and 2r combinations within each ancestry group so that users can readily calculate power using parameters beyond the set provided here.The cost efficiency of achieving equal power can be compared among array choices using these power estimates. For example, for the variant set with MAF > 0.05 and GRR = 1.3, achieving 50% power requires N = 3000 for the Omni2.5M array and N = 4000 for the HumanCore array in the African ancestry group. A power of 80% requires a larger sample size difference .In addition to planning experiments with existing arrays, our genomic coverage results can be used to design new genotyping arrays. For example, one might wish to design an array focused on specific candidate variants or regions, while also providing a backbone for genome-wide imputation, such as the existing HumanCore or Biobank variant set. In that case, one might begin with the backbone and then add candidate variants that are not already well imputed by the backbone variants in the target ancestry group(s). For this purpose, we also provide the individual imputation i.e., which arrays are currently in use). Our analysis has the advantage of consistently applying the same approach to evaluate several different arrays from different manufacturers. There are, however, some limitations to this study design that may result in either over- or underestimated genomic coverage. First, the genomic coverage reported here might be over estimated by imputing subsets of 1000 Genomes samples using the remaining samples as reference, as the haplotypes are likely to match better than in real world applications where the samples to be imputed are likely from different source populations than the reference. This advantage may be offset to some degree by using a smaller reference panel than is available for independent study populations, although this effect is likely to be small because the reference panel used here is approximately 90% of the full 1000 Genomes phase 1 reference panel. Second, coverage may be underestimated by having pre-phased each batch separately, as small sample sizes may negatively impact phasing accuracy . Finally, some of the arrays assay a substantial number of variants that are not in the 1000 Genomes phase 1 release (see File S1), the 1000 Genomes data set is clearly the best resource available for evaluating complete genomic coverage in multiple ancestry groups. We expect the results presented here to be useful in planning studies with the current generation of human genotyping arrays.In conclusion, these imputation-based genomic coverage and power analyses are intended as a practical guide to researchers planning genetic studies. Deciding among available arrays is often a delicate balance between scientific aims, monetary resources, and genotyping timeframe (accuracy . An increase see . These o