a central goal of genome wide association studies ( gwas ) is to identify genetic risk factors for complex disorders . in order to find the disease genetic risk factors in a population , gwas measures dna sequence variations across human genome ( bush and moore , 2012 ) . practitioners in medical sciences and bioinformatics use gwas to investigate the relations in different disorders ; gwas of different cancers ( easton and eeles , 2008 ) , gwas of pancreatic cancer ( amundadottir et al . , 2009 ) . the idea of genetic variations with alleles that are common in the population may explain much of the heritability of common diseases , see ( reich and lander , 2001 ) and ( schork et al . , 2009 ) . review of gwas can be found in several texts and papers , see ( moore et al . , 2010 ) among others . in the simplest form of association mapping , a set of markers are genotyped in both sample of cases and sample of unrelated controls and then using different association tests , allele frequency differences or genotype frequency differences at each marker will be studied ( pritchard and donnelly , 2001 ) . the main idea behind gwas studies relies on the fact that if a mutation has positive correlation with susceptibility of a disease , then that mutation is expected to be more frequent among affected individuals than those unaffected individuals ( pritchard and donnelly , 2001 ) . hence , considering the existence of linkage disequilibrium ( ld ) between the marker locus and the susceptibility mutation , the marker close to the disease mutation may also present a frequency difference between case and control group of study ( pritchard and donnelly , 2001 ) . case - control traits can be analysed using either logistic regression or contingency table techniques ( bush and moore , 2012 ) . contingency table methods examine the deviation from independence that is expected under the null hypothesis of observing no association between the disease under study and the measured allelic / genotyping frequency differences ( bush and moore , 2012 ) . pearson chi - squared test and the related fisher 's exact test are the most widely used tests for independence of the rows and columns of the contingency table ( bush and moore , 2012 ) . it should be noted that the association tests are performed separately for each individual marker and depending on the aim of study , the data for each marker with minor allele a and major allele a can be represented either as genotype count ( e.g. , a / a , a / a and a / a ) or allele count ( e.g. , a and a ) ( clarke et al . , 2011 ) . it is widely believed that the allelic association test with 1 degrees of freedom ( df ) is more reliable than the genotypic test with 2 df . however , it is imperative to note that this superior performance can only be considered for the case of having the penetrance of the heterozygote genotype between the penetrance of the two homozygote genotypes ( clarke et al . , 2011 ) . when the distribution of genotypes in the population deviates from hardy - weinberg proportions ( hwe ) , of which additive , dominant and recessive models are all examples ( clarke et al . , 2011 ) , the frequency of genotypes rather than alleles should be compared by the cochran - armitage test for trend ( sasieni , 1997 ) . for more information on different models thus , the advantage of the cochran - armitage trend test in comparison to pearson 's chi - square test is that it possesses the superior conservation and is not dependent on the hwe assumption ( sasieni , 1997 ) . therefore a number of authors have recommended to use the cochran - armitage trend test as the genotype - based test for association ( sasieni , 1997 , corcoran et al . , 2000 , li , 2008 , risch and merikangas , 1996 , risch , 2000 ) . it should also be noted that the allelic and trend statistic are equivalent when the combined sample is in hwe ( sasieni , 1997 ) . however , a major drawback of model based methods is that the statistical properties depend on the choice of weights . thus , the model miss - specifications minimize the power of the test ( sasieni , 1997 , corcoran et al . , 2000 , li , 2008 , risch and merikangas , 1996 , risch , 2000 ) . furthermore , escott - price et al . ( 2013 ) showed that , although in most scenarios the cochran - armitage trend test is more powerful than the chi - squared test of genotype counts , the advantage is not substantial . even , when the disease locus is extremely biased from the additive model , the chi - squared test of genotype counts can be more powerful than the cochran - armitage trend test due to the choice of scores for each genotype in the trend test ( escott - price et al . although , there are considerable studies about the advantages and disadvantages of cochran - armitage trend test , to the best of our knowledge , there is a small number of researches which evaluated the distribution of p - values for this association test . in this paper the distribution of the p - values derived by the cochran - armitage trend test has been studied and it has been shown that unlike the considered presumption those p - values obtained by this test are not uniformly distributed . to overcome this issue , we introduce a new method , based on the bootstrap technique , for computing the p - value of the cochran - armitage trend test . the bootstrap method has become a standard tool in statistical analysis and is an indispensable tool for testing statistical hypotheses . using resampling , bootstrap approximates the sampling distribution of a statistic under the null ( or the alternative ) hypothesis . bootstrap provides a practical complement to asymptotic parametric inference , hence have attracted many attentions in the applied . the efficiency of the nonparametric bootstrap method has also been shown by amiri and von rosen ( 2011 ) in which for example in the case of the pearson chi - squared statistic with a yates ' correction and fisher 's exact test , remarkable improvement has been achieved . the pearson chi - squared statistic with a yates ' correction and fisher 's exact test , are quite conservative and fail to reject the null hypothesis and can not be recommended to test independence with small sample sizes . section 3 studies the alternative approach to draw the inference including the bootstrap version of cochran - armitage trend test . section 4 investigates the proposed method using the monte carlo simulation , which show they are the accurate tests in terms of the significant level and statistical power . section 4 also demonstrates the improvements in goodness - of - fitness achieved by the introduced bootstrap approach . the cochran - armitage 's trend test is a widely used test for trend among binomial proportions which uses the genotype contingency table ( table 1 ) in a different manner than pearson 's test . power is very often improved as long as the probability of having disease increases with the number of disease - associated alleles . in genetic association studies in which the underlying genetic model is unknown , the additive version of this test is most commonly used . in order to measure the effect of genotype i and to detect particular types of association , we introduce a weight wi the special choice ( w0 ,  w1 ,  w2 )  =  ( 0,1,2 ) , represents the additive effect of allele a. ( see table 2 . ) let us consider a single - marker locus with two possible alleles which are commonly denoted by a and a. thus , each individual has three possible genotypes aa  ,  aa , and aa . in the following we denote the two alleles by 0 and 1 instead of a and a and the genotypes by 0  ,  1  ,  2 , the sum of the two allele indices involved . the case - control data can then be summarized according to genotypes as shown in table 1 . here , ( n0  ,  n1  ,  n2 ) are counts of the genotypes in cases and ( m0  ,  m1  ,  m2 ) are counts of the genotypes in controls , and ( n0  ,  n1  ,  n2 ) are counts of the genotypes in case - control samples . let n and m be the total number of cases and controls , respectively , and the total sample size , n  =  n  +  m. as cases and controls are independently sampled the genotype counts for cases and controls follow independent multinomial distributions with parameters ( p0  ,  p1  ,  p2 ) , and ( p0  ,  p1  ,  p2 ) , respectively , where pi and pi , i  =  0,1,2 , are the genotype probabilities in cases and controls.n0n1n2:multinp0p1p2,m0m1m2:multimp0p1p2. under the null hypothesis of no association , h0  :  pi  =  pi for i  =  0,1,2 . the cochran - armitage 's trend test statistic for the data in table 1 is given by(1)t = nnn1 + 2n2nn1 + 2n22nnnnn1 + 4n2n1 + 2n22 . the statistic in eq . ( 1 ) follows the chi - square distribution with one degree of freedom ( df ) , see ( armitage , 1955 ) . let us denote the cochran - armitage trend test as ca in the rest of work . agresti ( 2007 ) states ca in terms of the pearson chi - squared statistic . consider a contingency table 2    j with ordered column , see table 1 . let nj  ~  bin(nj ,  pj ) , j  =  0  , ,  j    1 , it is of interest to test the following null hypothesis(2)h0:p0=p1=  =pj1,h1:pipj,ij . one can use the ordinary least square approach for testing . let w=njwj / n , pj = nj / nj and p^=n / n . the prediction equation isp^j = p^+^wiw,where^=njpjp^wjwnjwjw2 . using the pearson chi - squared statistics(4)x2=jnjpjp^2p^1p^=z2+x2l~j12 , wherez2=2^p^1p^jnjwjw2,x2l=1p^1p^jnjp~jp^j2,under linear probability model x(l )  ~  j    2 that using the application of cochran 's theorem , z  ~  1 . it can be used to test h0  :    =  0 for the linear trend , the test of independence using z is called the cochran - armitage ( ca ) trend test . the bootstrap method has brought a vast new body of statistics in the form of nonparametric approaches to model uncertainty , in which not only the individual parameters of the probability distribution , but also the entire distribution are sought ( amiri , 2013 ) . this has led to a versatile tool for data analysis , in particular in the field of statistical hypothesis tests . two monographs on the bootstrap method written by efron and tibshirani ( 1994 ) and davison and hinkley ( 1997 ) are very useful in this regard in that they focus more on applications than on the theoretical approach . the idea of the bootstrap method is to approximate the sampling distribution of the proposed statistic , and this technique is based on resampling , which provides a practical complement to asymptotic parametric methods . the flexibility and robustness of this technique , especially in situations where the violation of assumptions is being dealt with , can be counted as two advantages of the technique ( good , 2013 ) . amiri and von rosen ( 2011 ) use the bootstrap to carry out the test of the contingency table . in order to test the association using the bootstrap method , the resampling should be performed on eij , where it is held for the expected value in the ( i ,  j)th cell . the principle of the bootstrap test is the performance of bootstrap resampling under the null hypothesis , which is explained in ( efron and tibshirani , 1994 ) and ( davison and hinkley , 1997 ) . the null hypothesis of the lack of the association in the contingency table is h0  :  pij  =  pi.p.j , it leads to h0  :  eij  =  oi.o.j / o .. , and therefore resampling under the null hypothesis is resampling on eij rather than oij , where oij is held for the observed value . let x be the resampled that is done under null hypothesis and has j    1 , since x  =  z  +  x(l ) , z has 1 and can be used to test h0  :  p0  =  p1  =    =  pj    1 . the test can be done using the following steps,1.calculate t or z.2.resample data under eij , and obtain the contingency table n  =  { n0 ,    ,  nj    1 ,  m0 ,    ,  mj    1 } , where n*~multine00n  e2j1n.3.repeat the second step b times , and calculate tb or zb , b  =  1  ,    ,  b.4.estimate p - value usingpvalue=#tb*>tb . resample data under eij , and obtain the contingency table n  =  { n0 ,    ,  nj    1 ,  m0 , repeat the second step b times , and calculate tb or zb , b  =  1  ,    ,  b. estimate p - value usingpvalue=#tb*>tb . another approach is to consider a parametric bootstrap . to this end , consider each allele or column are produced from the independent pdf . under the null hypothesis(5)fpn=j=0j1njnjpnj1pnjnj , that is actually a product of four binomial pdf . in order to estimate p , the maximum likelihood estimation ( mle ) of it can be used i.e. , p^=nn . the procedure of the test is the same as below , just in the step 3 , the resampled contingency tables are generated using fp^n=j=0j1njnjp^nj1p^njnj . the test can be done using the following steps,1.calculate t or z.2.resample data under eij , and obtain the contingency table n  =  { n0 ,    ,  nj    1 ,  m0 ,    ,  mj    1 } , where n*~multine00n  e2j1n.3.repeat the second step b times , and calculate tb or zb , b  =  1  ,    ,  b.4.estimate p - value usingpvalue=#tb*>tb . resample data under eij , and obtain the contingency table n  =  { n0 ,    ,  nj    1 ,  m0 , repeat the second step b times , and calculate tb or zb , b  =  1  ,    ,  b. estimate p - value usingpvalue=#tb*>tb . another approach is to consider a parametric bootstrap . to this end , consider each allele or column are produced from the independent pdf . under the null hypothesis(5)fpn=j=0j1njnjpnj1pnjnj , that is actually a product of four binomial pdf . in order to estimate p , the maximum likelihood estimation ( mle ) of it can be used i.e. , p^=nn . the procedure of the test is the same as below , just in the step 3 , the resampled contingency tables are generated using fp^n=j=0j1njnjp^nj1p^njnj . this section demonstrates the validity of the proposed methods for the inference of cochran - armitage trend test . in order to study the finite sample properties of the proposed approaches , the proposed methods are simultaneously based on the same simulated data in order to provide a meaningful comparison of various algorithms . in total 5000 simulations were performed . in order to make a comparative evaluation of the procedures , we seek the certain desirable features such as the actual significance level . in order to produce the simulation , the data is generated using ( n0 ,  n1 ,  n2 )  :  multi(n ,  ( 0.2,0.4,0.2 ) ) and ( m0 ,  m1 ,  m2 )  :  multi(m ,  ( 0.2,0.4,0.2 ) ) . the q - q plot of the p - value of the proposed tests are given in fig . 1 , which shows that the p - value using the bootstrap tests fit better to the uniform distribution , that admits the bootstrap can be nominated to draw the inference . racine and mackinnon ( 2007 ) suggest(6)pvalue=#tb*>tb+1+ub+1 , where u  :  unif(0 ,  1 ) . under null hypothesis , p(p    value  <   )  =   for any finite b , specially if the number of bootstrap is not large . the simulated data are generated using ( n0 ,  n1 ,  n2 )  :  multi(n ,  ( 0.2,0.4,0.2 ) ) and ( m0 ,  m1 ,  m2 )  :  multi(m ,  ( 0.4,0.2,0.2 ) ) . 2 . the violin plot is a combination of a box plot and a kernel density plot ; it starts with a box plot , and then adds a rotated kernel density plot to each side of box plot that provides a better indication of the shape of distribution and summary of data . 3 illustrates the q - q plot of generated random number of size 100,000 from 1 . the results confirm that the statistic with distribution 1 suffer from the lack of goodness - of - fitness in the right tail . this fact is also quite evident for cochran - armiatage trend test . in order to study the efficiency the proposed approaches , we generate 2000 tables with ( n0 ,  n1 ,  n2 )  :  multi(n ,  ( 0.2,0.4,0.2 ) ) and ( m0 ,  m1 ,  m2 )  :  multi(m ,  ( 0.2,0.4,0.2 ) ) , where n  =  m  =  20 . the q - q plot from 1 and the bootstrap approach is given in fig . this article explores the genetic association study for the case - control design that draw the inference of the equality of the genotype frequencies . gwas represent important challenges and opportunities in bioinformatics as they enable modeling of complex genotype - phenotype relationships using the mathematical and statistical approaches . such models aids us to understand and interpret genetic association studies and promotes the development of powerful algorithms to examine genotype - phenotype relationships . in this paper it was shown that the proposed bootstrap can be used to test the genetic association . the results confirm that the p - value of the proposed approaches fits better to the uniform distribution , specially on the right side . another advantage of the proposed tests require less assumption in comparison with the conventional method . the results also support that the proposed approaches can be successfully employed to test the genetic association . extending the proposed idea in this paper to obtain a better test that is more robust under choosing weights for the cochran - armitage trend test is our future research plan .