Abstract
Background
Linkage Disequilibrium (LD) is a powerful approach for the identification and characterization of morphological shape, which usually involves multiple genetic markers. However, multiple testing corrections substantially reduce the power of the associated tests. In addition, the principle component analysis (PCA), used to quantify the shape variations into several principal phenotypes, further increases the number of tests. As a result, a powerful multiple testing correction for simultaneous largescale geneshape association tests is an essential part of determining statistical significance. Bonferroni adjustments and permutation tests are the most popular approaches to correcting for multiple tests within LD based Quantitative Trait Loci (QTL) models. However, permutations are extremely computationally expensive and may mislead in the presence of family structure. The Bonferroni correction, though simple and fast, is conservative and has low power for largescale testing.
Results
We propose a new multiple testing approach, constructed by combining an Intersection Union Test (IUT) with the Holm correction, which strongly controls the familywise error rate (FWER) without any additional assumptions on the joint distribution of the test statistics or dependence structure of the markers. The power improvement for the Holm correction, as compared to the standard Bonferroni correction, is examined through a simulation study. A consistent and moderate increase in power is found under the majority of simulated circumstances, including various sample sizes, Heritabilities, and numbers of markers. The power gains are further demonstrated on real leaf shape data from a natural population of poplar, Populus szechuanica var tietica, where more significant QTL associated with morphological shape are detected than under the previously applied Bonferroni adjustment.
Conclusion
The Holm correction is a valid and powerful method for assessing geneshape association involving multiple markers, which not only controls the FWER in the strong sense but also improves statistical power.
Keywords:
Bonferroni; Holm; QTL mapping; LD; Multiple correctionBackground
Linkage Disequilibrium (LD)based Quantitative Trait Loci (QTL) studies now involve largescale numbers of genetic markers and play a significant role in identifying underlying genetic variants for complex quantitative traits such as morphological shape or human disease [15]. A major issue for LD based QTL mapping is in determining significance levels for the testing of multiple individual markers. Three reasons add to the complexity of this multiple testing correction. First, new genotyping techniques make it common to measure tens of thousands of markers. The more statistical tests that we perform for identifying significant genetrait associations, the more likely we are to reject the null hypothesis when it is true. This problem is also called the inflation of the type I error [6,7]. Second, high dimensional shape traits, often quantified by multiple principal components, dramatically increase the number of multiple tests by as much as three or more times [5,8,9]. Third, independence of test statistics is not guaranteed because correlations between markers lead to highly complicated and unknown dependency structures.
The Bonferroni correction, as one of the most popular multiple correction approaches, is known to be conservative and have low power for largescale tests [10]. Permutations, although the current gold standard for assessing significance levels in genetic mapping studies with multiple markers, is extremely time consuming due to its computational burden, and may not work well if the population has family structure [6,1113]. Therefore, it is necessary to seek alternative approaches that can improve the power for largescale simultaneous individual marker tests while preserving control of the familywise error rate (FWER) under nominal significance thresholds (e.g. α = 0.05) [14,15].
In this article, we propose a uniformly more powerful sequentially rejective multiple testing approach that strongly controls the FWER for the LD based shape mapping model, by merging Holm's procedure [16] with the Intersection Union Test (IUT) [17]. The new procedure makes no assumptions on the joint distribution of the test statistics. The advantage of the Holm correction over the standard Bonferroni correction has been known to statisticians for over 35 years [16,1820] but has not yet gained traction in LD based QTL mapping.
A critical challenge in largescale LD association tests is the increase in the false positive rate if selected markers are not in complete LD with each other. In this case, the power is likely reduced (or false negative rate inflates) if the correction for multiple comparisons is overly conservative or if independence is assumed for markers with strong LD associations with each other. Despite the fact that the false discovery rate (FDR) is very popular and has been extensively used in multiple hypothesis testing [21], the FWER, the probability of making at least one type I error, exerts a more stringent control over the FDR. Since FDR is controlled only for all selected markers and provides no promise of control for an arbitrarily selected subset of the significant markers, researchers may detect more spurious QTLs using the FDR in place of the FWER as they often consider only a subset of the significant results [22]. Therefore, we recommend controlling the FWER rather than FDR whenever only the most promising results are valued, such as in LD based QTL mapping.
Detecting a significant shape QTL requires two hypothesis tests [5], the first testing for the association between QTL and shape, and the second testing for the LD between the observable marker and underlying QTL. Currently, Bonferroni corrections are applied separately to two families of hypotheses, one family consisting of the first hypothesis test for all markers, and the other family consisting of the second hypothesis test for all markers. Only those markers showing significance within both families, after correcting for multiple tests, are identified as linked to a QTL. This amounts to performing an IUT with the two test statistics for each marker and applying a Bonferroni correction for multiple markers. Although the LD based QTL model has been successful in locating significant QTLs [5,23,24], two improvements can be made within the multiple hypothesis testing aspect. First, several geneshape association tests were made separately for each principal component (PC). Since these PCs quantify the original high dimensional shape variations from different directions, the multiple testing correction should account for these separate PCs as well as for multiple markers. Currently, multiple PCs are not accounted for in the multiplicity correction. Second, we introduce the uniformly more powerful Holm adjustment on the pvalues resulting from the IUT, which shows greater power than the Bonferroni approach.
The significance of the power advantage of the Holm method over the Bonferroni method is demonstrated through both simulations and a real leaf shape data of a natural population of poplar, Populus szechuanica var tietica. We detect more significant QTL than were previously detected in the literature while still ensuring strong control of the FWER. Since sample size, Heritability, and number of markers all determine the power, we illustrate the power differences for Heritabilities of 0.1 and 0.4, sample size small (100), medium (300), and large (500), and number of markers changing from 1, 10, 50, 100, 500 to 1,000.
Results
Power simulation
We investigated a simulation study to quantify the power advantage of the Holm adjustment over the standard Bonferroni adjustment within the LD based QTL mapping model of [5]. The QTL, phenotype, and markers were generated under the assumptions of the alternative hypotheses in (3) and (4). The QTL was generated using an assigned probability of q = 0.7 for the major allele. For each individual i, Qi = l with l ∈ {1, 2, 3} was used to code the QTL genotypes of aa, Aa, and AA, respectively. The normally distributed phenotype dependent on the value of the QTL is generated as Y_{i}(Q_{i }= l) ~ N (µl, σ). The means for the phenotype Y corresponding to the values of the QTL were set at µ_{1 }= 8, µ_{2 }= 10 and µ_{3 }= 12. markers were then generated using the conditional probability of the marker genotype given the value of the QTL genotype for each individual. In general, for an LD based QTL mapping model, researchers genotype the marker first and then use the marker to generate a QTL based on the conditional probability of QTL genotype given marker genotype. However, for our purposes, we are interested in extending from single marker mapping to multiple marker mapping. Therefore, we derive the conditional probability of marker genotype given QTL genotype (see Table 1) from the Bayes Rule in Equation (1) and Table 2.
Table 1. The theoretical conditional probabilities of marker genotype (columns) given QTL genotype (rows).
Table 2. The theoretical joint distribution probabilities of marker and QTL haplotypes.
Sample sizes of n = 100, 300, and 500 were used to represent small, medium, and large sample sizes, respectively. The number of markers per simulation was set at m = 1, 10, 50, 100, 500, and 1,000 to show the initial power under the single marker scenario and the corresponding decreasing power trend as the number of markers increases. Finally, the heritability was set at two values, H^{2 }= 0.1 and 0.4, corresponding to high and low error variance [25]. The model error variance σ^{2 }was computed using the heritability and genetic variance of the QTL. Power estimates were averaged over 1,000 simulations.
The simulation results, depicted in Figure 1 and shown in Additional file 1, demonstrate the power comparison of the Holm adjustment with the traditional Bonferroni adjustment. These results provide an experimental reference for researchers about how power varies among different sample size n, the number of markers m, and the degree of heritability (H^{2}). As expected, the power under high heritability (B: H^{2 }= 0.4) is much higher than that of the low heritability (A: H^{2 }= 0.1) and the power under large sample size (n = 500, blue curves) is much higher than that of the small sample size (n = 100, green curves). Under high heritability (H^{2 }= 0.4) and a larger sample size (n = 500), the power of the Holm multiplicity adjustment remains high, at least 99%, as the number of markers vary from 1 to 1,000. However, in practice it is often expensive to collect so many sample measurements, so these results are useful in deciding the opportunity costs in power for smaller sample sizes. It is worth noting that for moderate numbers of markers, the power increase of the Holm over the Bonferroni adjustment allows for maintaining the same power level of the Bonferroni adjustment while decreasing the sample size of the study or increasing the number of markers, a great advantage for researchers. For example, with a medium sample size of 300, the Holm correction maintains a power of 95% for 100 markers. Even when number of markers increase to 1, 000, the power achieved by the Holm correction is still as high as 80%.
Additional file 1. Additional file 1 includes a single table showing the results of the power simulation as depicted in Figure 1.
Format: PDF Size: 55KB Download file
This file can be viewed with: Adobe Acrobat Reader
Figure 1. Power comparison between the Holm adjustment and standard Bonferroni adjustment under different sample size, number of markers, and heritability (A: H^{2 }= 0.1, B: H^{2 }= 0.4).
Although the power increase of the Holm adjustment improves moderately over the standard Bonferroni adjustment for the case of high heritability (H^{2 }= 0.4) when the sample size is small (n = 100), these findings are comparable to seminal results found by previous multiplicity improvements over their competitors [16,21]. For the worst case when the data has extremely large variance (H^{2 }= 0.4) and relatively much smaller sample size (n = 100), the improvement of Holm over Bonferroni is not obvious. However, it is not an issue of multiple hypothesis testing but an issue of the least sample size necessary to guarantee a decent level of power.
All in all, the Holm method generally shows a valuable increase in power over the Bonferroni adjustment under the majority of simulated circumstances, including different combinations of sample size, numbers of markers, and Heritability. As long as the sample size is reasonably large in comparison to the variance to guarantee decent power, the improvement of the Holm correction over the Bonferroni is consistently meaningful.
Poplar leaf shape QTL mapping project
To show how the power advantage of the Holm approach leads to increased scientific discovery over the Bonferroni adjustment, we apply it to a real poplar leaf shape QTL mapping study [5]. The study design used a representative leaf from each of 106 poplar trees (i.e., Populus szechuanica var. tibetica belonging to the Tacamahaca section) that was randomly selected and photographed for shape QTL analysis. The trees were also genotyped for a panel of 29 microsatellite markers (16 of them were considered). A Radius Centroid Contour (RCC) approach was used to represent the leaf shape (phenotype) with a high dimensional curve. The first three principal components (PCs) were selected to capture the majority variation of leaf shape from different directions to quantify the original high dimensional shape curves respectively. Significant QTLs affecting the shape variability (i.e., affecting the most important PCs) were mapped through the statistical LD based QTL mapping model [5]. Previously, the standard Bonferroni adjustment was used to control the FWER for the multiple markers [5, Table 1]. However, the researchers did not consider the multiple testing correction issue introduced by multiple PCs and their reported results treated as a family of hypotheses only the multiple markers within each PC. After including the multiple PCs within the family of interest, we found slightly different results, even under the previously applied Bonferroni correction.
After applying our proposed Holm approach to provide a comprehensive multiple correction,
not only including the multiple PCs in the correction but also including multiple
markers within each PC, the Holm correction successfully detects all significant microsatellites
that were detected by the Bonferroni correction. Further, the Holm correction detects
one more marker, marker 10, that was not detected previously. Figure 2 demonstrates the bivariate plot of the two test statistics
Figure 2. Bivariate plot of the two test statistics
Figure 3 illustrates the genotypic shape effects according to the different genotypes (AA, Aa, aa) of the QTL identified by marker 10 on PC 3. It is evident that the effect of aa produces shorter leaf tips and more degrees of deltoidness at leaf base compare to the other genotypes. The effect of AA and Aa are very similar, which indicates a dominance effect. Although significant genotype differences can be observed visually, it is nevertheless not detected under the Bonferroni correction. This confirms the practical relevence of the increased sensitivity of the Holm correction over the Bonferroni. Figure 4 illustrates the RCC curves of leaf shape as a function of radial angle θ explained by the different genotypes (AA, Aa, aa) of the QTL identified by marker 10 on PC 3. It is evident that the RCC curve of aa has a higher peak when θ is close to π/2 but a lower dynamic pattern when θ is close to 3π/2, which matches the interpretations of above Figure 3 visualized from the leaf image domain. We believe that the advantages of our proposed approach will be more remarkable for a larger number of markers.
Figure 3. The genotypic shape effects according to the different genotypes (AA, Aa, aa) of the QTL identified by marker 10 on PC 3. This shows the increased sensitivity of the Holm approach as the effect of the aa QTL genotype is noticeably different from that of genotypes AA and Aa, but nevertheless, corresponds to information which was not detected under the Bonferroni correction.
Figure 4. RCC curves of leaf shape as a function of radial angle θ explained by the different genotypes (AA, Aa, aa) of the QTL identified by marker 10 on PC 3. This shows the increased sensitivity of the Holm approach as the effect of the aa QTL genotype is noticeably different from that of genotypes AA and Aa, but nevertheless, corresponds to information which was not detected under the Bonferroni correction.
Conclusion
Detecting significant genes that affect complex traits such as shape or disease through LD based QTL mapping has been popular in many disciplines [15,2633]. The new genotyping techniques make it possible to simultaneously consider tens of thousands of markers, bringing substantial challenges for multiple testing. In addition, high dimensional shape traits, often involving multiple PC components, have been widely used and add yet another demand for a powerful and computationally efficient approach to adjust for multiple tests [5,8,9].
These multiple tests require an adjustment on the resulting pvalues in order to preserve control of the familywise error rate (FWER) at a prespecified level α. Making the alpha level more stringent will create less errors, but it may lower the chance of detecting more real effects [7]. The FDR has been widely used as the error rate of interest. Typically however, a subset of the significant results are directly reported and therefore the FWER is the more desirable form of error rate to control [22]. The current standard approach in LD based QTL mapping is to apply a Bonferroni adjustment to correct for multiplicity and preserve control of the FWER. As is well known, the Bonferroni correction is overly conservative for large numbers of tests, but the advantages of simplicity without independence assumptions on the corresponding family of tests continue to make it popular. Permutation, although has been the gold standard for assessing significance levels in studies with multiple markers, is extremely time consuming, computationally intensive, and may not work well if the population has family structure [6,1113].
In this article, we propose an uniformly more powerful multiple correction approach by integrating Holm [16] with the IUT test, which is assured strong control of the FWER under arbitrary dependencies among the test statistics. The advantage of Holm over Bonferroni actually has been recognized to statisticians for over 35 years [16,1820] but is new to LD based QTL mapping. The significance of the power advantage of the Holm correction over the Bonferroni, has been established theoretically [16]. This work demonstrates the power advantage in LD based QTL mapping empirically through both simulation study and real data. As long as the sample size is reasonablly large in comparison to the variance to guarantee a decent power, the improvement of the Holm correction over the Bonferroni is consistent and meaningful.
Methods
LD based QTL model
To map the rough location of the QTL regulating shape, we apply the mixture model of [5]. Under this model, QTL mapping is accomplished by statistically modeling the genotypic variation through not only the association between phenotype and the putative QTL, but also the LD between the putative QTL and marker. Since the marker genotype is observable, the probabilities of a putative QTL genotype can be inferred by the conditional probability of QTL genotype (A) given the marker genotype (M), as long as there exists LD between the marker and putative QTL [5].
The mixture model of [5] assumes each individual's phenotype Yi, i = 1, . . . , n, is a random variate from density f_{l}(Y_{i}θ_{l}), where l ∈ {1, 2, 3} denote three distinct QTL genotypes. Each QTL genotype is assumed to induce a separate distribution of phenotypes. Typically, normal distributions are assumed for each f_{l}(Y_{i}θ_{l}) with θ_{l }= (µ_{l}, σ). From these assumptions, the corresponding likelihood is expressed as [5]
where ωli is the conditional probability of individual I having QTL genotype l given their marker genotypes, µl is the phenotypic mean for QTL genotype l, σ is the common variance for all genotypes, and fl(Yiµl, σ) is the probability density of observations for individual I at QTL genotype l [5,25,34].
The probability of the marker's major allele (M) is denoted by p, and correspondingly 1 − p for the minor allele (m). Similarly, the probability of the QTL's major allele (A) is denoted by q, and correspondingly 1−q for the minor allele (a). Together, the marker and QTL form four haplotypes (MA, Ma, mA, and ma) with corresponding frequencies p_{11 }= pq+D, p_{10 }= p(1−q)−D, p_{01 }= (1−p)q −D, and p_{00 }= (1−p)(1−q)+D, respectively. Here, D is the linkage disequilibrium between marker and QTL. The conditional probabilities ω_{li }of the QTL's various genotypes (AA, Aa, and aa) can be calculated upon the observed marker genotypes (MM, Mm, and mm) from the joint probabilities in Table 2[25,5]. The EM algorithm is then applied to the likelihood in (2) to obtain maximum likelihood estimates for all parameters [5,25].
Two hypothesis tests
Through the likelihood in (2), the hypotheses
On the other hand, linkage disequilibrium, denoted by D, between the marker and QTL can be tested by means of the hypotheses
The test statistic used to judge whether or not the QTL is significantly associated with marker is [5,35]:
Here,
In general, the IntersectionUnion test (IUT) is defined as [39]
Here Γ is a finite or infinite set containing index of tests, θ is the unknown parameters under testing, and Θ_{γ }specifies the statement of null hypothesis test for each index γ. Suppose that for each γ ∈ Γ, {x : T_{γ }(x) ∈ R_{γ}} is the rejection region for each test
In the context of LD based QTL mapping, the tests of the above hypotheses (3) and
(4) must be performed simultaneously to make the final conclusion, i.e., a significant
QTL regulating shape is not detected unless both null hypotheses in (3) and (4) are
rejected.
The Holm correction
The Holm multiplicity correction [16] applies a "sequentially rejective Bonferroni test" to all currently nonrejected hypothesis in a stepdown manner. The first step of our proposed approach is to use an IUT to obtain m pvalues. Then, we order the tests from the one with the smallest pvalue to the one with the largest pvalue as p_{(1)}, . . . , p_{(m) }according to the usual order statistics notation. The smallest pvalue, p_{(1)}, corresponding to the ordered hypothesis H_{(1)}, is then tested with the usual Bonferroni correction of α_{1 }= α/m. If H_{(1) }is declared significant, then the method continues by testing p_{(2) }against α_{2 }= α/(m − 1). So long as rejections continue to occur, p_{(i) }is compared to α_{i }= α/(m − i + 1) until finally p_{(m) }is compared to α_{m }= α. If for any i ∈ {1, 2, . . . , m} the hypothesis H_{(i) }is not rejected, then the method stops and H_{(i)}, . . . , H_{(m) }are retained, i.e., not rejected. Therefore, the procedure stops when the first nonsignificant test is obtained or when all the tests have been performed.
To be exact, let p_{(1)}, . . . , p_{(m)}, denote the ordered pvalues corresponding to the ordered m hypotheses obtained from IUTs, H_{(1)}, . . . , H_{(m)}. The ordered pvalues for the test of H(j) are then compared to the thresholds α_{j }where
and all testings will stop at the jth test for which the first nonrejection occurs, i.e., the j for which p_{(j) }> α_{j}. Because the denominators are m − j + 1 instead of m, Holm's procedure never rejects fewer hypotheses than the Bonferroni procedure does.
A multiple test procedure for testing hypotheses H_{1}, . . . , H_{m }is said to have multiple level of significance α (for free combinations) if for any nonempty index set I ⊆ {1, 2, 3, . . . , m} the supremum of the probability
Importantly, as proved in [16], strong control of the FWER is ensured under the Holm adjustment, no matter the dependency structure of the corresponding test statistics. A simple, but elegant proof of the strong FWER control of this approach, under arbitrary dependence structures of the test statistics, is given in [16]. The idea behind the proof is as follows. Let I denote the set of indices corresponding to the true null hypotheses, and let k denote the number of hypotheses in I, so that k ≤ m. The well known Boole's Inequality shows that the FWER (the probability of at least one type I error) is controlled by the Bonferroni method so long as at most α/k is applied to the testing of all k true nulls. Specifically,
Given the nature of the sequential testing of the Holm adjustment, at any stage j of testing, the number of true nulls remaining to be tested (k) will always be less than or equal to the number of hypotheses remaining to be tested (m−j+1). Hence, any true null will always be tested by at least α/(m − j + 1) ≤ α/k, ensuring strong control of the FWER no matter how many or which nulls are true.
Just as with the Bonferroni method, Holm's method is a distribution free approach to the multiple hypothesis testing issue. More importantly it is uniformly more powerful than the Bonferroni method as it will compare P_{(2)}, . . . , P_{(m) }to larger thresholds α_{i}, i = 2, . . . , m than will the Bonferroni method. Therefore, it is clear that the Holm method should always be preferred over the Bonferroni method from a theoretical perspective. In the following sections, we will illustrate the benefit of Holm over Bonferroni from an application perspective.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GF initiated the project, supervised the main ideas, closely guided several details, provided the estimation programming, wrote and revised the manuscript. GS participated in the development of the Holm method, made programs for multiple testing, performed data analysis, and drafted the manuscript. JS involved in idea development discussions, checked the validation of the method, and revised the manuscript.
Acknowledgements
This work was supported by a Utah State University VPR Research Catalyst Grant. Publication costs for this article were funded by the corresponding author's institution.
This article has been published as part of BMC Genetics Volume 15 Supplement 1, 2014: Selected articles from the International Symposium on Quantitative Genetics and Genomics of Woody Plants. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcgenet/supplements/15/S1.
References

Sachidanandam R, Weissman D, Schmidt SC, Kakol JM, Stein LD, Marth G, Sherry S, Mullikin JC, Mortimore BJ, Willey DL, Hunt SE, Col CG, Coggill PC, Rice CM, Ning Z, Rogers J, Bentley DR, Kwok PY, Mardis ER, Yeh RT, Schultz B, Cook L, Davenport R, Dante M, Fulton L, Hillier L, Waterston RH, McPherson JD, Gilman B, Schaffner S, Van Etten WJ, Reich D, Higgins J, Daly MJ, Blumenstiel B, Baldwin J, StangeThomann N, Zody MC, Linton L, Lander ES, Altshuler D: A map of human genome sequence variation containing 1.42 million single nucleotide polymorphisms.
Nature 2001, 409:928933. PubMed Abstract  Publisher Full Text

Ardlie KG, Kruglyak L, Seielstad M: Patterns of linkage disequilibrium in the human genome.
Nat Rev Genet 2002, 3:299309. PubMed Abstract  Publisher Full Text

Weiss KM, Clark AG: Linkage disequilibrium and the mapping of complex human traits.
Trends Genet 2002, 18:1924. PubMed Abstract  Publisher Full Text

Wang L, Luzynski K, Pool JE, Janoušek V, Dufkov´a P, Vyskočilov´a MM, Teeter KC, Nachman MW, Munclinger P, Machol´an M, Pi´alek J, Tucker PK: Measures of linkage disequilibrium among neighbouring SNPs indicate asymmetries across the house mouse hybrid zone.

Fu G, Bo W, Pang X, Wang Z, Chen L, et al.: Mapping shape QTLs using a radiuscentroidcontour model.
Heredity 2013, 110:511519. PubMed Abstract  Publisher Full Text

Nicodemus KK, Liu W, Chase GA, Tsai YY, Fallin MD: Comparison of type I error for multiple test corrections in large singlenucleotide polymorphism studies using principal components versus haplotype blocking algorithms.
BMC genetics 2005, 6(Suppl 1):S78. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Langlade NB, Feng XZ, Dransfield T, Copsey L, Hanna AI, Thebaud C, Bangham A, Hudson A, Coen E: Evolution through genetically controlled allometry space.
P Natl Acad Sci USA 2005, 102:1022110226. Publisher Full Text

Drake AG, Klingenberg CP: Largescale diversification of skull shape in domestic dogs: Disparity and modularity.
Am Nat 2010, 175:289301. PubMed Abstract  Publisher Full Text

Nakagawa S: A farewell to bonferroni: the problems of low statistical power and publication bias.
Behav Ecol 2004, 15:10441045. Publisher Full Text

Gao X, Starmer J, Martin ER: A multiple testing correction method for genetic association studies using correlated single nucleotide polymorphisms.
Genet Epidemiol 2008, 32:361369. PubMed Abstract  Publisher Full Text

Han B, Kang HM, Eskin E: Rapid and accurate multiple testing correction and power estimation for millions of correlated snps.
PLoS Genet 2009, 5(4):e1000456.
doi:10.1371/journal.pgen.1000456
PubMed Abstract  Publisher Full Text  PubMed Central Full Text 
Doerge RW: Mapping and analysis of quantitative trait loci in experimental populations.
Nat Rev Genet 2002, 3:4352. PubMed Abstract  Publisher Full Text

Nyholt DR: A simple correction for multiple testing for singlenucleotide polymorphisms in linkage disequilibrium with each other.
Am J Hum Genet 2004, 74:765769. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Johnson R, Nelson GW, Troyer JL, Lautenberger JA, Kessing BD, Winkler CA, O'Brien SJ: Accounting for multiple comparisons in a genomewide association study (gwas).
BMC Genomics 2010, 11:724. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Holm S: A simple sequentially rejective multiple test procedure.

Berger RL: Mltiparameter hypothesis testing and acceptance sampling.
Technometrics 1982, 24:295300. Publisher Full Text

Goeman JJ, Mansmann U: Multiple testing on the directed acyclic graph of gene ontology.
Bioinformatics 2008, 24:537544. PubMed Abstract  Publisher Full Text

Aickin M, Gensler H: Adjusting for multiple testing when reporting research results: the Bonferroni vs Holm methods.
American Journal of Public Health 1996, 86(5):726728. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Simes RJ: An improved Bonferroni procedure for multiple tests of significance.
Biometrika 1986, 73(3):751754. Publisher Full Text

Benjamini Y, Hochberg Y: Controlling the false discovery rate: a practical and powerful approach to multiple testing.

Goeman JJ, Solari A: Tutorial in biostatistics: multiple hypothesis testing in genomics.
Statist Med 2014.
DOI: 10.1002/sim.6082

Lou X, Casella G, Littell RC, Yang MCK, Johnson JA, Wu R: A haplotypebased algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis.

Das K, Wu R: A statistical model for the identification of genes governing the incidence of cancer with age.

Wang Z, Wu R: A statistical model for highresolution mapping of quantitative trait loci determining HIV dynamics.
Statist Med 2004, 23:30333051. Publisher Full Text

Terwilliger JD, Weiss KM: Linkage disequilibrium mapping of complex disease: Fantasy or reality? Curr.
Opin Biotechnol 1998, 9:578594. Publisher Full Text

Service SK, Lang TDW, Freimer NB, Sandkuijl LA: Linkagedisequilibrium mapping of disease genes by reconstruction of ancestral haplotypes in founder populations.
Am J Hum Genet 1999, 64:17281738. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Martin ER, Lai EH, Gilbert JR, Rogala AR, Afshari AJ, Riley J, Finch KL, Stevens JF, Livak KJ, Slotterbeck BD, Slifer SH, Warren LL, Conneally PM, Schmechel DE, Purvis I, PericakVance MA, Roses AD, Vance JM: SNPing away at complex diseases: Analysis of singlenucleotide polymorphisms around APOE in alzheimer disease.
Am J Hum Genet 2000, 67:383394. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Goldstein D, Weale M: Linkage disequilibrium holds the key.
Curr Biol 2001, 11:576579. Publisher Full Text

Reich D, Cargill M, Bolk S, Ireland J, Sabeti PC, Richter DJ, Lavery T, Kouyoumjian R, Farhadian SF, Ward R, Lander ES: Linkage disequilibrium in the human genome.
Nature 2001, 411:199204. PubMed Abstract  Publisher Full Text

Dawson E, Abecasis GR, Bumpstead S, Chen Y, Hunt S, Beare DM, Pabial J, Dibling T, Tinsley E, Kirby S, Carter D, Papaspyridonos M, Livingstone S, Ganske R, Lõhmussaar E, Zernant J, Tõnisson N, Remm M, Magi R, Puurand T, Vilo J, Kurg A, Rice K, Deloukas P, Mott R, Metspalu A, Bentley DR, Cardon LR, Dunham I: A firstgeneration linkage disequilibrium map of human chromosome 22.
Nature 2002, 418:544548. PubMed Abstract  Publisher Full Text

Morris RW, Kaplan NL: On the advantage of haplotype analysis in the presence of multiple disease susceptibility alleles.
Genet Epidemiol 2002, 23:221233. PubMed Abstract  Publisher Full Text

Skipper L, Wilkes K, Toft M, Baker M, Lincoln S, Hulihan M, Ross OA, Hutton M, Aasly J, Farrer M: Linkage disequilibrium and association of mapt h1 in parkinson disease.
Am J Hum Genet 2004, 75:669677. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Fu G, Berg A, Das K, Li J, Li R, et al.: A statistical model for mapping morphological shape.

Brown AHD: Sample size required to detect linkage disequilibrium between two or three loci.
Theor Popul Biol 1975, 8:184201. PubMed Abstract  Publisher Full Text

Hedrick PW: Gametic disequilibrium measures: Proceed with caution.

Pritchard JK, Przeworski M: Linkage disequilibrium in humans: Models and data.
Am J Hum Genet 2001, 69:114. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Mueller JC: Linkage disequilibrium for different scales and applications.
Brief Bioinform 2004, 5:355364. PubMed Abstract  Publisher Full Text

Berger RL: Likelihood ratio tests and intersectionunion tests.
Advances in statistical decision theory and applications 1997, 225237.