Volume 5 Supplement 9

Genetic Analysis Workshop 17: Unraveling Human Exome Data

Open Access

Treating phenotype as given: a simple resampling method for genome-wide association studies

BMC Proceedings20115(Suppl 9):S60

DOI: 10.1186/1753-6561-5-S9-S60

Published: 29 November 2011

Abstract

Significance of genetic association to a marker has been traditionally evaluated through statistics that are standardized such that their null distributions conform to some known ones. Distributional assumptions are often required in this standardization procedure. Based on the observation that the phenotype remains the same regardless of the marker being investigated, we propose a simple statistic that does not need such standardization. We propose a resampling procedure to assess this statistic’s genome-wide significance. This method has been applied to replicate 2 of the Genetic Analysis Workshop 17 simulated data on unrelated individuals in an attempt to map phenotype Q2. However, none of the selected SNPs are in genes that are disease-causing. This may be due to the weak effect that each genetic factor has on Q2.

Background

The traditional approach to hypothesis testing is to construct a test statistic whose exact or approximate distribution under the null hypothesis is known. Alternative methods for assessing significance of a test statistic include resampling techniques, such as the bootstrap and permutation. These methods dominate modern genetic association studies.

Statistics used in these methods typically make use of a standardization factor, for instance, the standard error of the observed genetic effect size. Different model assumptions result in different standardization factors. For instance, in a case-control genetic association study, we are interested in testing whether the difference in the frequency of a reference allele between case subjects and control subjects is 0. The popular allelic test and the Cochran-Armitage test for trend are based on two different estimates of the variance of this difference. The estimate for the allelic test requires Hardy-Weinberg equilibrium in the combined case-control population, whereas the estimate for the Cochran-Armitage test does not. The properties of these two tests are rather different [1].

It is a brilliant idea to construct a test statistic through standardization. It makes it possible to assess the significance of the observed effect size with respect to a reference distribution. However, in genome-wide association studies there are typically tens of thousands of markers. Presumably most of these markers are under the null hypothesis (i.e., they are not associated). The presence of these null markers provides a natural reference distribution. Standardizing the effect size in this setting appears to be unnecessary and awkward.

We propose a novel resampling approach that aims to directly assess the significance of genetic effect size. The null distribution generated from this method is for a randomly selected single-nucleotide polymorphism (SNP). Hence it naturally addresses the genome-wide significance of genetic effect size at a SNP and obviates the issue of multiple testing. To define the genetic effect size, we observe that the phenotype remains the same in a genome scan. Only the testing locations are different. This observation leads to a simple representation of the genetic effect size. This approach is demonstrated by means of a genome-wide association study on phenotype Q2 in replicate 2 of the Genetic Analysis Workshop 17 (GAW17) simulated data on unrelated subjects.

Methods

Let n denote the number of individuals in a study. Let y i denote the measure of a quantitative phenotype y on individual i. The score of the genotype at a SNP is denoted by g. Assuming that the two alleles of the SNP are denoted by A and B, we set g = 0 for genotype AA, g = 1 for genotype AB, and g = 2 for genotype BB. The genotype score on individual i is denoted by g i .

To identify SNPs associated with phenotype y, we consider a regression in which the response variable is the genotype score g and the independent variable is the phenotype y. The (partial) regression sum of squares for g equals:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ1_HTML.gif
(1)
where https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq2_HTML.gif are the sample means of g i and y i , respectively. Because the denominator in expression (1) remains constant in a genome scan, a natural measure of association would be:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ2_HTML.gif
(2)
Note that in a case-control study https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq3_HTML.gif is proportional to the difference in the frequency of allele B between case subjects and control subjects. If there are p covariates x j , j = 1, …, p, then phenotype y i is replaced by the residual https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq4_HTML.gif from the following linear regression:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ3_HTML.gif
(3)

where x ij is the value of x j for individual i. Similarly, genotype g is replaced by the residual g* of the regression of g over x j , j = 1, …, p.

We use the statistic:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ4_HTML.gif
(4)

to measure the strength of association, where n* is the number of subjects whose genotype is nonmissing at the SNP being investigated. The sample mean of https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq5_HTML.gif is 0, so it is dropped from the definition of S. So is the sample mean of https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq6_HTML.gif . The purpose of using n* is to make the statistic S on the same scale because n* is expected to vary across the genome.

In comparison, the usual method for detecting association would consider the following regression:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ5_HTML.gif
(5)
The least-squares estimate of β1, denoted https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq7_HTML.gif , is:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ6_HTML.gif
(6)
The statistic for testing for association is:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ7_HTML.gif
(7)

where https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq8_HTML.gif is the standard error of https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq9_HTML.gif . This procedure is equivalent to testing whether the coefficient of g is 0 in a multiple linear regression that includes x i , …, x p as covariates and y as the response. Because the total sum of squares for https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq10_HTML.gif is fixed in a genome scan, there is a monotonic relationship between the regression sum of squares for Eq. (5) and the statistic T.

We propose the following resampling procedure to evaluate the genome-wide significance of S: (1) Compute residuals https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq11_HTML.gif and residuals https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq12_HTML.gif from regression (3). (2) Randomly select a SNP from a set of SNPs that are under the null hypothesis. This null set can be determined by using a histogram of p-value [2]. (3) Permute residuals https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq13_HTML.gif (or, equivalently, permute residuals https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq14_HTML.gif ). (4) Compute the statistic S for the randomly selected marker. (5) Repeat steps 2–4 the desired number of times, say K. (6) Let S( k ) denote the value of statistic S in the kth iteration. Suppose that the value of statistic S is s for the observed data. Its p-value is computed as:
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Equ8_HTML.gif
(8)

where I(·) is the indicator function satisfying I(True) = 1 and I(False) = 0.

Because the SNP selected in each iteration is random, the significance obtained in this way is genome-wide.

Results

We analyze phenotype Q2 in replicate 2. There are 24,487 SNPs and 697 individuals. Residuals https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq15_HTML.gif are computed from regression (3) with regressors Sex, Age, and Smoke. The residuals https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq16_HTML.gif are computed in the same way. Genome-wide p-values from the t-distribution with 1 degree of freedom for statistic T are presented in Figure 1. There are 19 SNPs with a p-value less than 5 × 10−4 (Table 1).
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Fig1_HTML.jpg
Figure 1

Genome-wide plot of p -values for the regular statistic T. The horizontal line corresponds to p = 5 × 10−4.

Table 1

Selected SNPs based on p-values of statistic T (p < 5 × 10−4)

     

p-value

Chromosome

Gene

SNP

Minor allele frequency

Base-pair position

Statistic T

Statistic S

1

NEK7

C1S8842

0.001435

196488886

3.55 × 10−4

1.53 × 10−1

2

R3HDM1

C2S3088

0.002152

136095600

1.32 × 10−4

1.18 × 10−1

6

BAT2

C6S2242

0.008608

31712334

4.78 × 10−4

5.59 × 10−2

7

MLXIPL

C7S1785

0.017217

72658237

3.43 × 10−4

1.59 × 10−2

8

NRG1

C8S1224

0.001435

32625157

4.28 × 10−4

1.56 × 10−1

 

RBM13

C8S1319

0.005022

33475616

1.67 × 10−4

4.63 × 10−2

 

ANK1

C8S1496

0.005022

41638505

2.92 × 10−5

6.35 × 10−2

 

CA3

C8S2634

0.029412

86539249

1.55 × 10−5

5.59 × 10−3

 

CNGB3

C8S2646

0.105452

87748419

6.52 × 10−5

5.72 × 10−4

 

NBN

C8S2710

0.022238

91064195

3.58 × 10−5

8.67 × 10−3

 

LPL

C8S442

0.015782

19849988

1.54 × 10−7

1.30 × 10−2

10

DUSP5

C10S5537

0.133429

112256813

4.77 × 10−4

7.31 × 10−4

11

PDGFD

C11S5292

0.008608

103413068

1.21 × 10−5

3.89 × 10−2

12

SLCO1A2

C12S1083

0.001435

21319511

4.28 × 10−4

1.56 × 10−1

 

LRRK2

C12S1884

0.023673

38978429

4.72 × 10−4

2.24 × 10−2

19

C19ORF6

C19S56

0.018651

963559

2.04 × 10−4

1.96 × 10−2

 

CAPNS1

C19S3468

0.012912

41325704

4.69 × 10−4

3.69 × 10−2

 

PSG9

C19S4603

0.030129

48465412

3.10 × 10−4

7.17 × 10−3

20

PRIC285

C20S2223

0.003587

61666107

2.59 × 10−5

6.27 × 10−2

The reference distribution for statistic S is obtained in the way described in the previous section. The value of K is set at 10,000,000. The histogram of p-values for parameter β1 in regression (5) shows no apparent deviation from uniform distribution (data not shown). Thus all SNPs are used to determine the reference distribution for statistic S. P-values obtained from the resampling procedure are presented in Figure 2. Eighteen SNPs have a p-value that is less than 5 × 10−4. None of these SNPs are close to the 19 SNPs that have p-values less than 5 × 10−4 for statistic T, except for C20S2223 on chromosome 20 (base-pair position 61666107; p = 2.59 × 10−5 for statistic T; resampling p = 6.27 × 10−2 for statistic S). C20S2223 is located in gene PRIC285 identified by statistic S.
https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_Fig2_HTML.jpg
Figure 2

Genome-wide plot of resampling p -values for the proposed statistic S. The horizontal line corresponds to p = 5 × 10−4.

Discussion and conclusions

Significance of genetic association at a SNP has been traditionally evaluated through a standardized test statistic such that its significance can be evaluated through a known distribution. For instance, in Eq. (7) the standard error of https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq17_HTML.gif serves as a standardization factor. Its p-value is assessed through a t-distribution or a standard normal distribution. This strategy works beautifully in general but is awkward in the context of genome-wide association studies. The presence of abundant unassociated SNPs provides a natural reference distribution that is pertinent to the data. We have proposed a simple statistic S and a resampling method for generating its reference distribution. Although a distributional assumption is necessary to compute the standard error of https://static-content.springer.com/image/art%3A10.1186%2F1753-6561-5-S9-S60/MediaObjects/12919_2011_Article_1111_IEq18_HTML.gif , it is not required in our method. More important, the resampling procedure has a built-in mechanism for handling genome-wide significance because the reference distribution is the distribution of an arbitrary SNP under the null hypothesis that there is no association. Application to the GAW17 simulated data on unrelated individuals (replicate 2) revealed several susceptible genes for phenotype Q2 that can be used as candidates for further investigation.

The essence of the proposed method is the existence of a large number of similar features—SNPs in the current context. The basic principle is applicable to other situations that involve a large amount of similar features, for instance, gene expression levels in gene expression studies.

We have focused on a continuous trait. As alluded to earlier, for a dichotomous trait, if there are no covariates, then the statistic S can be defined as the difference in frequency of the reference allele between the two phenotype groups. More generally, one can use a proportional odds model with genotype as the response and phenotype as the predictor. A proportional odds model is appealing for dealing with covariates and both continuous and dichotomous traits. We are actively investigating this possibility.

The proposed method is resampling based and requires intensive resampling. However, computation for each sample is fast. The overall computation time for K = 10,000,000 resampling iterations took less than two days on a computer running the Windows Vista operating system. The iteration coverage is about 408 per SNP (≈ K/24,487 SNPs). We have also used K = 100,000; the simulated p-value remains almost the same. The resampling procedure needs to be done only once, regardless of the number of SNPs.

We are investigating the utility of our approach in other association study settings, such as gene-gene interaction and population stratification.

We had no knowledge of the GAW17 disease-generating model when this analysis was conducted. It turns out that none of the genes in Table 1 and Table 2 are disease causing. This may be due to the weak effect that each genetic factor has on Q2.
Table 2

Selected SNPs based on resampling p-values of statistic S (p < 5 × 10−4)

     

p-value

Chromosome

Gene

SNP

Minor allele frequency

Base-pair position

Statistic T

Statistic S

1

LOC100133124

C1S2906

0.32

46294146

3.99 × 10−3

3.70 × 10−4

2

LY75

C2S4258

0.45

160419168

1.25 × 10−2

3.31 × 10−4

6

HLA-A

C6S1568

0.30

30020831

8.07 × 10−3

1.21 × 10−4

11

OR52H1

C11S662

0.31

5522627

1.31 × 10−3

7.35 × 10−5

  

C11S665

0.32

5522688

3.15 × 10−3

1.52 × 10−4

  

C11S675

0.32

5523065

1.59 × 10−3

8.52 × 10−5

 

GALNTL4

C11S1338

0.43

11310922

3.25 × 10−3

2.04 × 10−4

 

SYTL2

C11S4881

0.38

85113378

2.36 × 10−3

1.87 × 10−4

  

C11S4893

0.31

85114042

5.41 × 10−3

3.70 × 10−4

  

C11S4949

0.48

85134351

6.79 × 10−3

2.66 × 10−4

 

LOC100128794

C11S5677

0.30

107889417

8.72 × 10−3

4.65 × 10−4

12

OR6C76

C12S3054

0.26

54106388

4.77 × 10−3

4.37 × 10−4

17

RHOT1

C17S1743

0.35

27558083

6.77 × 10−3

2.81 × 10−4

 

SLFN13

C17S1964

0.42

30792467

9.33 × 10−3

2.33 × 10−4

 

PIP4K2B

C17S2376

0.40

34180257

7.62 × 10−3

3.55 × 10−4

 

CDC27

C17S3299

0.46

42589359

2.68 × 10−3

2.58 × 10−4

19

PSG3

C19S4193

0.34

47935058

1.73 × 10−2

4.00 × 10−4

20

PRIC285

C20S2305

0.22

61671020

5.23 × 10−3

3.72 × 10−4

Declarations

Acknowledgments

KW’s research is partly funded by the National Institute for Environmental Health Sciences (National Institutes of Health) (NIEHS/NIH) grant P30 ES05605 through the University of Iowa Environmental Health Sciences Research Center. JH’s work is partly supported by NIH grant R01 CA120988 and National Science Foundation (NSF) grant DMS 0805670. We would like to thank the two anonymous referees for their helpful comments and suggestions.

This article has been published as part of BMC Proceedings Volume 5 Supplement 9, 2011: Genetic Analysis Workshop 17. The full contents of the supplement are available online at http://​www.​biomedcentral.​com/​1753-6561/​5?​issue=​S9.

Authors’ Affiliations

(1)
Department of Biostatistics, University of Iowa
(2)
Department of Statistics and Actuarial Sciences, University of Iowa

References

  1. Sasieni PD: From genotypes to genes: doubling the sample size. Biometrics. 1997, 53: 1253-1261. 10.2307/2533494.View ArticlePubMed
  2. Storey JD, Tibshirani R: Statistical significance for genomewide studies. Proc Natl Acad Sci USA. 2003, 100: 9440-9445. 10.1073/pnas.1530509100.PubMed CentralView ArticlePubMed

Copyright

© Wang and Huang; licensee BioMed Central Ltd. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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