population admixture ( cryptic heterogeneity ) represents a potentially serious problem in case - control association studies . allele frequencies tend to differ between countries and even between different regions in a single country . one of the first methods to deal with the deleterious effects of heterogeneity is genomic control ( gc ) , which assesses the extent of inflation of  in terms of the gc factor ,  , and then divides each  by . additional methods have since been introduced , notably principal components analysis   and logistic regression with components of multidimensional scaling ( mds ) as covariates . here , we propose a novel approach based on deleting individuals that appear as outliers . this approach specifically addresses the situation of a relatively small number of individuals that do not belong to the main portion of the study sample . it is intuitive that one way to deal with heterogeneity is to remove individuals not belonging to a sample . punishing all individuals by rolling back all test statistics as it is done in the gc method . however , removing outliers has to be carried out in a statistically satisfactory manner . to decide how many and which individuals to remove , we proceed as follows : based on the commonly used identity by state ( ibs ) metric , similarity between two individuals is defined as the ibs between two individuals , averaged over all snps . in the coordinate system of the c largest mds components ( here we use c = 4 throughout ) , each individual is at some distance from the center . that individual with the largest distance from the center is considered a potential outlier . initially , the pearson  is computed in a 2  2 contingency table for each snp , where the two rows correspond to cases and controls , and the two columns represent the snp alleles . after retaining the p value , p0 , for the largest  over all the snps , the first potential outlier is removed and another largest  is computed ( at whatever snp in the genome it occurs ) leading to p1 , and so on . we proceed until a predefined maximum number , m , of individuals has been removed . the sequence of p values ( p0 , p1 ,  , pm ) initially either decreases ( p0 > p1 ) or increases ( p0 < p1 ) . in the first case , assume that the smallest p value , pmin , among the m + 1 values occurs at step k , that is , after k outliers have been removed . we then take t = pmin as our overall test statistic . in the latter case , we search for the first ( local ) minimum p value , t , or , if none occurs , we retain t = pm , with t again being our test statistic . in each of a sufficiently large set of randomization samples ( labels case and control are randomly permuted ) , the whole approach is repeated , and we obtain the significance level associated with t as the proportion of randomization samples with t values at least as small as the observed t. note that there may be a different snp with largest  in different steps of outlier removal . the technique of finding the smallest p value among several model assumptions and obtaining the ( genome - wide ) significance level associated with this smallest p value is not new . we previously applied this principle in comparing disease association of sets of snps , where each set contains different numbers of snps . this has led to our set association method , which is more powerful than snp by snp analysis   and has successfully been applied in various studies . by design , our approach always removes at least one individual . in this sense , it furnishes trimmed results . trimming is well known in classical statistics as a procedure for eliminating outliers . here we apply this principle to case - control association studies . for a simple power comparison , we assume a total of 1,000 independent snps , with the last snp conferring disease susceptibility . we further assume a total sample size of 200 individuals , of which 10 are outliers . the 190 non - outliers are equally divided into cases and controls while we consider 3 scenarios for the 10 outliers : ( 1 ) 5 cases and 5 controls , ( 2 ) 2 cases and 8 controls , and ( 3 ) no cases and 10 controls , where the latter scenario represents the ( perhaps common ) situation that controls tend to be chosen from a different population segment than that furnishing cases . for the 999 non - disease snps with alleles a and b , allele frequencies p(a ) are randomly picked between 0.10 and 0.50 for non - outliers , and between 0.10 and 0.90 for outliers ( for details , see online suppl . material , see www.karger.com/doi/10.1159/000320422 ) . the disease ( functional ) snp has alleles d and d , with the former conferring disease susceptibility . its allele frequency , p(d ) , is set to 0.30 in non - outliers and is chosen randomly from 0.10 through 0.90 in outliers . we consider dominant and recessive inheritance , with h denoting the penetrance for non - susceptibility genotypes , while the penetrance for disease conferring genotypes is given by rh . power to detect the disease snp is computed as a function of the penetrance ratio , r = rh / h , where r = 1 represents the null hypothesis of no genetic effect . the maximum number of outliers to be removed is set at m = 20 ( 10% of the sample size of 200 ) . we compare the following 4 test procedures , where each is applied to the disease snp . pearson  test with gc correction , that is , all  are divided by the gc parameter ,  , where  = observed median  for all snps divided by the median of the  distribution with 1 d.f . logistic : logistic regression analysis for each snp in turn , additive allele test ( 1 d.f . ) . logistic - mds : logistic regression analysis ( 1 d.f . ) with the largest 4 mds components as covariates . outliers : our approach for removing individuals that extremely deviate from the center in an mds coordinate space . at r = 1 , for each of the 4 methods , 5,000 datasets are generated under dominant and recessive inheritance , and critical thresholds for the test statistics are chosen such that the resulting significance level ( proportion of significant results ) is exactly equal to 0.05 . for a simple power comparison , we assume a total of 1,000 independent snps , with the last snp conferring disease susceptibility . we further assume a total sample size of 200 individuals , of which 10 are outliers . the 190 non - outliers are equally divided into cases and controls while we consider 3 scenarios for the 10 outliers : ( 1 ) 5 cases and 5 controls , ( 2 ) 2 cases and 8 controls , and ( 3 ) no cases and 10 controls , where the latter scenario represents the ( perhaps common ) situation that controls tend to be chosen from a different population segment than that furnishing cases . for the 999 non - disease snps with alleles a and b , allele frequencies p(a ) are randomly picked between 0.10 and 0.50 for non - outliers , and between 0.10 and 0.90 for outliers ( for details , see online suppl . the disease ( functional ) snp has alleles d and d , with the former conferring disease susceptibility . its allele frequency , p(d ) , is set to 0.30 in non - outliers and is chosen randomly from 0.10 through 0.90 in outliers . we consider dominant and recessive inheritance , with h denoting the penetrance for non - susceptibility genotypes , while the penetrance for disease conferring genotypes is given by rh . power to detect the disease snp is computed as a function of the penetrance ratio , r = rh / h , where r = 1 represents the null hypothesis of no genetic effect . the maximum number of outliers to be removed is set at m = 20 ( 10% of the sample size of 200 ) . we compare the following 4 test procedures , where each is applied to the disease snp . pearson  test with gc correction , that is , all  are divided by the gc parameter ,  , where  = observed median  for all snps divided by the median of the  distribution with 1 d.f . logistic : logistic regression analysis for each snp in turn , additive allele test ( 1 d.f . ) . logistic - mds : logistic regression analysis ( 1 d.f . ) with the largest 4 mds components as covariates . outliers : our approach for removing individuals that extremely deviate from the center in an mds coordinate space . at r = 1 , for each of the 4 methods , 5,000 datasets are generated under dominant and recessive inheritance , and critical thresholds for the test statistics are chosen such that the resulting significance level ( proportion of significant results ) is exactly equal to 0.05 . power of the different analysis methods was somewhat dependent on model assumptions , but the logistic - mds method overall did best , followed by our outliers method . table 1 shows results for dominant inheritance and outliers consisting of 2 cases and 8 controls ( all results of power simulations are given in online suppl . we combined results for each value of r and 6 model assumptions ( dominant / recessive , 3 splits of cases versus controls in outliers ) and computed average power over these 36 conditions ( online suppl . as the last row of table 1 shows , this ranking makes the logistic - mds method the winner , closely followed by our outliers method . this power simulation is rather simple and is mainly designed to demonstrate that our outliers method is competitive . in particular , additional power simulations are provided in the online supplementary material , for example , for a trait influenced by two susceptibility loci and for different population structures . the pearson - gc method presumably suffers from the potentially severe protection from false - positive results . in fact , computing p values from  tables for the pearson - gc method leads to type i errors much smaller than 0.05 ( details not shown ) but , as mentioned , in our simulations the type i error was constant for all methods . to demonstrate our outliers method , we applied it and the 3 other approaches discussed here to a published dataset on parkinson disease with approximately 540 case and control individuals and approximately 408,000 snps genome wide . to make results comparable and allow for genome - wide correction for multiple testing , p values were estimated in permutation samples . in this analysis as table 2 shows , the outliers method furnished the smallest p value of 0.076 , which is not formally significant , although nearly so . the smallest nominal p value in the outliers method was obtained after 3 individuals had been removed as outliers ( fig . the significance level associated with this smallest p value is estimated to be 0.076 . without removing outliers , the p value of the largest test statistic (  ) is equal to 0.120 . thus , the outliers method resulted in a considerable improvement , although it did not furnish a significant result . if , for argument 's sake , we transform p values into  with 2 d.f . , we find  of c1 = 5.15 for p = 0.076 , and c2 = 4.24 for p = 0.120 . as  is proportional to sample size , the ratio , c1/c2 = 1.22 , reflects a virtual gain of 22% in sample size obtained by our method . of course , this argument is artificial since we do not know whether these p values reflect true or false positives . to demonstrate our outliers method , we applied it and the 3 other approaches discussed here to a published dataset on parkinson disease with approximately 540 case and control individuals and approximately 408,000 snps genome wide . to make results comparable and allow for genome - wide correction for multiple testing , p values were estimated in permutation samples . in this analysis as table 2 shows , the outliers method furnished the smallest p value of 0.076 , which is not formally significant , although nearly so . the smallest nominal p value in the outliers method was obtained after 3 individuals had been removed as outliers ( fig . the significance level associated with this smallest p value is estimated to be 0.076 . without removing outliers , the p value of the largest test statistic (  ) is equal to 0.120 . thus , the outliers method resulted in a considerable improvement , although it did not furnish a significant result . if , for argument 's sake , we transform p values into  with 2 d.f . , we find  of c1 = 5.15 for p = 0.076 , and c2 = 4.24 for p = 0.120 . as  is proportional to sample size , the ratio , c1/c2 = 1.22 , reflects a virtual gain of 22% in sample size obtained by our method . of course , this argument is artificial since we do not know whether these p values reflect true or false positives . so - called  obvious outliers are often removed in an ad - hoc manner , and there may not be good statistical justifications for doing so . in particular , if outliers are removed by trial and error , that is , if they are removed only when this leads to a reduction in p value , then such a procedure clearly tends to increase the false - positive rate of results . here , we developed a statistically rigorous procedure for removing outliers while maintaining correct type i error . we carried out additional power simulations under various conditions and also analyzed one more real dataset . these simulations confirm our conclusions based on results shown in table 1 ; they also show that the outliers method often does best with recessive modes of inheritance . in addition , at least in the two real datasets analyzed here , for the best snps , the outliers method yields the smallest p values . as is well known , an alternative to removing outliers is to allow for them in the analysis , which may be done by including principal components as covariates in logistic regression analysis . the two approaches may do equally well in practice , although our power calculations have given the logistic regression approach ( with mds components ) a slight advantage .
background / aimsin human case - control association studies , population heterogeneity is often present and can lead to increased false - positive results . 
 various methods have been proposed and are in current use to remedy this situation.methodswe assume that heterogeneity is due to a relatively small number of individuals whose allele frequencies differ from those of the remainder of the sample . 
 for this situation , we propose a new method of handling heterogeneity by removing outliers in a controlled manner . in a coordinate system of the c largest principal components in multidimensional scaling ( mds ) , 
 we systematically remove one after another of the most extreme outlying individuals and each time recompute the largest association test statistic . 
 the smallest p value obtained within m removals serves as our test statistic whose significance level is assessed in randomization samples.resultsin power simulations of our method and three methods in current use , averaged over several different scenarios , the best method turned out to be logistic regression analysis ( based on all individuals ) with mds components as covariates.conclusionour proposed method ranked closely behind logistic regression analysis with mds components but ahead of other commonly used approaches . in analyses of real datasets 
 our method performed best .