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Paper text:
the combination of genetic association studies in a meta - analysis framework can increase power to make novel discoveries .
large - scale meta - analyses across multiple data sets have become the mainstream approach to genome - wide association scans ( gwas ) .
these studies have been successful in identifying common ( minor allele frequency ( maf ) 0.05 ) genetic variants that underlie a variety of phenotypes and diseases , and have so far mostly focused on similar - ancestry populations , primarily of european descent . as more gwas across diverse populations of asian and african ancestry
start to accrue , examples of allelic heterogeneity at established common disease loci begin to emerge , and the need to develop powerful strategies for meta - analyzing genetically different populations becomes prominent . although large sample sizes can help guard against false - negative and false - positive results , a major potential caveat is that allelic heterogeneity across populations ( figure 1 ) could lead to the dramatic dilution of power to detect true positive signals . in this work ,
we evaluate the power of existing methods for the meta - analysis of data across populations .
allelic heterogeneity is also poised to be a major concern in whole - genome and whole - exome sequencing studies investigating the association of low - frequency and rare variants with complex traits .
it is expected that multiple different causal variants of low frequency will be found to reside within the same functional units , and that different alleles will be carried by different individuals across studies .
locus - wide approaches to detect association and thus increase power have been proposed , but it is not yet clear whether existing meta - analysis approaches will be useful in combining data across data sets to increase power . whether trying to identify common variants of small effect sizes or low - frequency / rare variants with modest effect sizes ,
large samples are required and those are likely to be achieved by synthesizing data across multiple populations . here
we outline and evaluate different strategies for the meta - analysis of genetic association data in the presence of allelic heterogeneity . based on simulated data ,
we provide insights into strategies for analyzing common variant signals across data sets and resequencing studies that aim to identify low - frequency variant associations in the presence of allelic heterogeneity .
case  control data across three populations were simulated under two distinct allelic heterogeneity scenarios , ( a ) common causal variants with small effect sizes ( odds ratio ( or ) 1 .
11.2 ) and ( b ) low - frequency causal variants with larger effect sizes ( or 23 ; table 1 ) .
for each replicate of simulated data , an association analysis was performed in each of the three populations followed by meta - analysis ( figure 2 ) . for each of the two distinct allelic heterogeneity scenarios , we simulated 1400 replicates of genotypic and phenotypic data for a case  control study of 2000 cases and 2000 controls , for three different populations . to mimic allelic heterogeneity ,
causal alleles were set to be population - specific , that is , they were not present in the other two populations .
when we repeated the simulations allowing causal variants to be present , but not associated with the phenotype , in all three populations , our results remained the same qualitatively ( data not shown ) .
all replicates where there was a shared causal allele among populations were excluded from the meta - analysis , yielding 1025 replicates for the common variant scenario and 1163 replicates for the low - frequency scenario . in population 1 , there is one causal variant , whereas populations 2 and 3 each have two causal variants that are not in strong linkage disequilibrium ( ld ) with one another . for each scenario , the frequency and or of the causal alleles
genotypic data were simulated using the hapsim r package and were based on pilot study data from the 1000 genomes project ( august 2009 release , 68 ceu samples ) for an arbitrary 150  kb region from chromosome 1 with a genome - average recombination rate of approximately 1  mb / cm .
we filtered out variants with quality scores below 10 , which resulted in a region consisting of 110 variants .
first , a population of 40  000 haplotypes was simulated such that the allele frequencies and pair - wise ld mimic those of the specified region from the 1000 genomes project data .
this approach produces realistic resequencing data that include variants with mafs below 0.01 ; in our region there were variants with maf as low as 0.0079 .
the causal allele(s ) were chosen randomly from among those with a maf near the setting for the simulation , and when there were two causal alleles the choice for the second allele was restricted such that it was not in strong ld with the first causal variant ( r < 0.4 ) .
control status was generated by using a multiplicative model for the genotype relative risks ( rrs ) to compute the probability of disease given the genotype at the causal variant and its rr .
this probability was then used to generate a bernoulli random variable that ascertains an individual as a case when its value is 1 , and a control otherwise . for this reason , it was necessary to oversample ( say ,
5n ) the number of individuals n to ensure that the desired number of cases was attained .
the association study was performed for each population separately , using three different approaches : ( a ) classical single - variant analysis for each variant in the region , ( b ) a collapsing method and ( c ) an allele - matching association test for the single - variant analysis a -test of association was carried out at each variant , resulting in a p - value and or for independent association of each variant to phenotype .
the latter two methods were locus - based tests , so that only a single test was performed .
there are various versions of the collapsing method , and all combine information across multiple variant sites into a univariate test for disease association with an accumulation of rare variants .
in the version we considered , phenotype is modeled as a function of the indicator variable of rare variants that carry at least one minor allele :   where yi is the case  control status for individual i , ri denotes the number of rare variants that carry at least one copy of the minor allele , i(ri)=1 { ri > 0 } is the indicator variable for the presence of at least one minor allele at any rare variant for subject i , and  is the expected increase when an individual carries at least one minor allele at any rare variant compared with one that has a complete absence of rare minor alleles .
analysis of deviance is used to compare the maximized likelihoods of the null ( =0 ) and unconstrained  models in the construction of likelihood ratio tests of disease association for an accumulation of rare variants .
the allele - matching association test considered is the kernel - based association test ( kbat ) , which tests for a joint association of multiple variants ( correlated or independent ) with a categorical phenotype , without any assumptions on the directions of individual variant effects .
first , similarity scores yl(ij ) between individuals i and j in group l ( eg , 1=cases , 2=controls ) are determined by using a kernel , such as the allele match kernel , which is the count of shared alleles between the genotypes of two individuals .
let gi be the genotype score at a specific variant , which is conveniently defined as the number of reference alleles at the variant , as knowledge of the risk allele is irrelevant . at a given variant , for individuals ij in group l with respective genotypes gl(i ) and gl(j ) , the similarity score yl(ij )
takes on values 4 ( identical genotypes ) , 2 ( one homozygous , other heterozygous ) and 0 ( otherwise ) . by defining the kernel in this way ,
the similarity scores yl(ij ) between individuals i and j in group l are modeled using a one - way anova model at each variant :   where  is the general effect for pairs of individuals , l is the group - specific treatment effect , and to test for disease association the null hypothesis is h0:1=2 . the single - variant test statistic at marker k is the ratio of the between group sum of squares ssbk and the within group sum of squares sswk , and the k - marker kbat test statistic is   clearly the similarity scores yl(ij ) are not independent normal random variables , so that neither the single - variant test statistics nor the kbat test statistic may be approximated by an f - distribution .
thus , permutation is required to obtain the p - value for each locus . for comparison purposes with the collapsing method
, we only include variants with maf < 0.05 in the kbat . after completing the association analyses , sets of results
were obtained for each of the three analyzed populations : 110 single - variant p - values and corresponding ors , one for each tested variant ; collapsing method p - value and or ; and kbat p - value ( and test - statistic , but no or ) .
we then carried out a meta - analysis across the three populations using two different approaches : ( i ) or based ( fixed and random effects ) and ( ii ) p - value based ( fisher 's product method ) .
methods of fixed effect meta - analysis are based on the mathematical assumption that every study in the meta - analysis shares a common ( or  fixed ' ) true effect size , and differences in the observed effect sizes are only due to the random error within in each study . under this assumption ,
if every study were infinitely large , the results of every study would be identical .
a random effects meta - analysis makes the assumption that individual studies are sampled from populations that may have similar , but not identical , true effect sizes .
differences in observed effect sizes arise from random error as in fixed effects , as well as true variation in effect size .
conclusions arising from the random effects analysis could be generalized to a range of populations , whereas those from the fixed effects are restricted to populations that are identical to those used in the analysis .
both approaches are of interest in testing whether sufficiently strong effects in a single population could still emerge as signals upon meta - analysis .
because of the unavailability of an or from the kbat , we only applied a p - value - based meta - analysis for this method .
we used a combination of in - house scripts and the software gwama to conduct meta - analyses . in each replicate
for the single - point method , we report identified loci as the proportion of replicates that detect association with at least one causal variant . for the collapsing and allele - matching methods ,
110 variant markers were tested and the threshold for variant - specific analysis was set to 0.00045 ( 0.05/110 ) .
in the setting of common causal variants with small effect sizes , the detection of causal variants or loci was unsatisfactory based on single - variant tests , and power of the meta - analysis approaches to detect at least one causal variant ranged from 5 .
the p - value - based meta - analysis approach was slightly more successful than the fixed effects or - based approach ( 15.7% vs 11.0% ) , with the random effects or - based approach performing worst ( 5.5% ) .
it is noteworthy that the meta - analysis of common variants with small effects in the presence of allelic heterogeneity does not detect association with any of the causal variants in 84.3% of the replicates , even in the appreciable sample size of 6000 cases and 6000 controls .
conversely , none of the replicates identified all five causal variants across the three studied populations ( table 2 ) . in the low - frequency causal variant scenario , we found that the meta - analysis of single - point association results had greatest power to detect at least one of the causal alleles , with the p - value meta - analysis having 100% power , followed by the fixed effects of or - based meta - analysis with 98% power ( table 2 ) .
however , these meta - analyses also gave rise to a very high false - positive rate , with the p - value - based and fixed effects or - based approaches identifying at least one false positive association in 72% and 51.8% of the replicates , respectively .
only 4 out of 1163 replicates correctly detected association at all five causal variants through the p - value - based meta - analysis , whereas none of the replicates achieved this in the or - based meta - analysis approaches ( table 2 ) .
the locus - wide collapsing tests examining low - frequency and rare variants ( maf < 0.05 ) in aggregate demonstrated lower power to detect association . here , the p - value - based meta - analysis method performed worst with 15.7% power , whereas the or - based fixed effects approach performed best with 22.7% power .
the allele - matching method demonstrated clear power advantages , with 74.7% power to detect association at the locus level , despite the underlying allelic heterogeneity ( figure 3 ) .
our findings demonstrate the low sensitivity of current approaches to detect common causal variants in the presence of allelic heterogeneity when meta - analyzing across genetically heterogeneous populations .
the field of statistical genetics is active in developing and testing new approaches to overcome these issues and to enable powerful trans - ethnic mapping .
we found the choice of methodological approach to detecting low - frequency variants in the era of whole - exome and whole - genome sequencing to greatly affect association study power .
however , the power of individual locus - wide methods critically depends on the allelic architecture of disease .
for example , in these simulations there were only one or two low frequency causal variants in each population , and the collapsing method has been designed to perform optimally in the presence of an accumulation of multiple rare variants .
single - point approaches to the meta - analysis of low - frequency variants performed poorly with respect to false - positive rate and , within the power constraints of our study , almost never detected all causal alleles .
it appears that given a large enough sample size , allele - matching methods provide good power to detect association at the locus level in the presence of allelic heterogeneity . as the currently sparse set of next - generation sequencing empirical data for complex traits begins to grow , the need for powerful meta - analysis methods that account for heterogeneity increases .
the development of such approaches will allow the next set of large - scale meta - analyses to take place , and will therefore pave the way for successful novel locus discovery .
meta - analysis has proven a useful tool in genetic association studies . 
 allelic heterogeneity can arise from ethnic background differences across populations being meta - analyzed ( for example , in search of common frequency variants through genome - wide association studies ) , and through the presence of multiple low frequency and rare associated variants in the same functional unit of interest ( for example , within a gene or a regulatory region ) 
 . the latter challenge will be increasingly relevant in whole - genome and whole - exome sequencing studies investigating association with complex traits . here 
 , we evaluate the performance of different approaches to meta - analysis in the presence of allelic heterogeneity . 
 we simulate allelic heterogeneity scenarios in three populations and examine the performance of current approaches to the analysis of these data . 
 we show that current approaches can detect only a small fraction of common frequency causal variants . 
 we also find that for low - frequency variants with large effects ( odds ratios 23 ) , single - point tests have high power , but also high false - positive rates . 
 p - value based meta - analysis of summary results from allele - matching locus - wide tests outperforms collapsing approaches . 
 we conclude that current strategies for the combination of genetic association data in the presence of allelic heterogeneity are insufficiently powered .