A hierarchical and modular approach to the discovery of robust associations in genome-wide association studies from pooled DNA samples

1471-2156-9-6 1471-2156 Methodology article A hierarchical and modular approach to the discovery of robust associations in genome-wide association studies from pooled DNA samples Sebastiani Paola sebas@bu.edu Zhao Zhenming zmzhao@bu.edu Abad-Grau M Maria mabad@ugr.es Riva Alberto ariva@ufl.edu Hartley W Stephen shartley@bu.edu Sedgewick E Amanda asedge@bu.edu Doria Alessandro Alessandro.Doria@joslin.harvard.edu Montano Monty mmontano@bu.edu Melista Efthymia emelista@bu.edu Terry Dellara laterry@bu.edu Perls T Thomas thperls@bu.edu Steinberg H Martin mhsteinb@bu.edu Baldwin T Clinton cbaldwin@bu.edu

Department of Biostatistics, Boston University School of Public Health, Boston 02118 MA, USA

Department of Software Engineering, University of Granada, Granada 18071, Spain

Department of Molecular Genetics, University of Florida at Gainesville, Gainesville 32611 FL, USA

Bioinformatics Program, Boston University School of Engineering, Boston 02116 MA, USA

Joslin Diabetes Center, Harvard Medical School, Boston 02215 MA, USA

Department of Medicine, Boston University School of Medicine, Boston 02118 MA, USA

Geriatric Section, Boston Medical Center, Boston 02118 MA, USA

BMC Genetics 1471-2156 2008 9 1 6 http://www.biomedcentral.com/1471-2156/9/6 18194558 10.1186/1471-2156-9-6 06 9 2007 14 1 2008 14 1 2008 2008 Sebastiani et al; licensee 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.

Abstract

Background

One of the challenges of the analysis of pooling-based genome wide association studies is to identify authentic associations among potentially thousands of false positive associations.

Results

We present a hierarchical and modular approach to the analysis of genome wide genotype data that incorporates quality control, linkage disequilibrium, physical distance and gene ontology to identify authentic associations among those found by statistical association tests. The method is developed for the allelic association analysis of pooled DNA samples, but it can be easily generalized to the analysis of individually genotyped samples. We evaluate the approach using data sets from diverse genome wide association studies including fetal hemoglobin levels in sickle cell anemia and a sample of centenarians and show that the approach is highly reproducible and allows for discovery at different levels of synthesis.

Conclusion

Results from the integration of Bayesian tests and other machine learning techniques with linkage disequilibrium data suggest that we do not need to use too stringent thresholds to reduce the number of false positive associations. This method yields increased power even with relatively small samples. In fact, our evaluation shows that the method can reach almost 70% sensitivity with samples of only 100 subjects.

Background

The availability of genotyping assays for hundreds of thousands of single nucleotide polymorphisms (SNP)s is making genome wide association (GWA) studies more accessible to a broad range of genotype-phenotype investigations. The promise of this technology is that it will accelerate gene discovery for polygenic diseases and complex phenotypes of Mendelian disorders because data for all genes can be obtained simultaneously 12. At the same time, the large number of significance tests performed is expected to result in a large number of false positive association signals. In fact, the number of signals observed by chance may well be greater than those that are authentic 3. Thus, the development of analytic methods and strategies to distinguish authentic signals from those due to chance will contribute significantly to disease-gene association studies.

Here we describe a modular procedure to analyze data from pooling-based GWA studies that use the Illumina SNP microarray technology 4. Rather than genotyping individual samples, the pooling-based technology types a carefully constructed pool of DNA samples that can be used to infer allele frequencies and is an affordable alternative to GWA studies that are still a financial burden for many investigators. Several studies have shown the usefulness of pooling-based GWA studies to discover SNPs associated with disease 56789 using well calibrated methods 7101112, and a variety of methods to estimate allele frequencies from pooled-based studies that use the Affymetrix microarray technology have been proposed 1314. Our objective is twofolds: (i) we wish to assess reproducibility and accuracy of the algorithm proposed by Illumina to detect chromosomal aberrations when used to estimate allele frequencies from pooled DNA samples 15; and (ii) we propose a modular approach to the analysis of pooling-based GWA studies that limits the loss of power due to both the use of pools of DNA samples and the issue of multiple comparisons.

Several studies apply stringent thresholds on the significance level that is required to determine significant SNP-phenotype associations 161718. Contrary to this approach, our method integrates Bayesian tests for general associations 19 with decision rules based on the structure of linkage disequilibrium (LD) discovered through the International HapMap project 20, and other machine learning techniques to reduce the number of false positive associations. We also describe a hierarchical procedure to summarize the findings in terms of genes that can be further synthesized into gene sets using Gene Ontology annotations 21, pathways 2223, or chromosomal bands. We evaluate this method using data from the sixty unrelated CEPH parents used for the International HapMap project 20 and two independent datasets. The first is a study of fetal hemoglobin (HbF) levels in African American subjects with sickle cell anemia and the objective is to discover genetic modulators of HbF. The second dataset is a study of exceptional longevity in a cohort of centenarians. In both datasets, using our novel analytic approach, we identified association signals in genes previously known to affect these phenotypes. The method is implemented in the R package and can be integrated with other R packages for genetic analysis, or GWA studies 2425. We develop the method for the analysis of pooled DNA samples 2627, but the approach can be easily extended to the analysis of samples that are individually genotyped.

Results

We ran three sets of experiments to assess the reproducibility and accuracy of the estimates of the allele frequencies derived from pooled DNA samples, as well as the sensitivity and specificity of our modular procedure.

Experiment 1: accuracy and reproducibility

We obtained DNA samples from the 60 unrelated parents used in the HapMap CEU panel and created 2 duplicated pools of 30, 45 and 60 samples each (Table 1 provides a summary). The pooled DNA samples were analyzed in duplicates with the Illumina Sentrix HumanHap300 Genotyping BeadChip (v.1) and b-allele frequencies were estimated using the Illumina LOH and Copy Number analysis tool. The reproducibility was assessed by the agreement between allele frequency estimates in the two replicate samples for each pool (Table 1). Shown in Figure 1 is the scatter plot of two independent replicates of allele frequency estimates for the 22842 SNPs tagging chromosome 1 (top), and the 5452 SNPs tagging chromosome 22 (bottom) obtained with pools of 60 samples. The plots show a high degree of agreement that is confirmed for different sample sizes as shown by the results summarized in Table 1. Plots for other chromosomes are in the supplementary material 28.

Table 1

Summary of the results of Experiment 1.

Number of pools

Sample size

Average difference

Standard deviation

Correlation

Average error

Standard deviation

Correlation

0.0043

0.0303

0.9940

0.0304

0.0565

0.9860

0.0011

0.0295

0.9956

0.0331

0.0573

0.9890

0.0164

0.0498

0.9873

0.0451

0.0668

0.9890

Column 1: pool description; Column 2: number of samples per pool; Column 3: average difference between estimates of allele frequencies in repeated pools; Column 4: standard deviation of differences; Column 5: correlation between repeated allele frequency estimation. Column 6: average difference between estimates of allele frequencies from pooled DNA samples and individually genotyped samples. Column 7: standard deviation of the differences; Column 8: Correlation between estimates of allele frequencies from pooled DNA samples and individually genotyped samples.

Figure 1

The reproducibility of the allele frequency estimates is shown by the scatter plot of repeated estimates of allele frequency inferred from pooled DNA samples (left)

The reproducibility of the allele frequency estimates is shown by the scatter plot of repeated estimates of allele frequency inferred from pooled DNA samples (left). The labels "run 1" and "run 2" in the x- and y-axis specify each replication. The accuracy of the allele frequency estimates is shown by the scatter plot of the estimates of allele frequency inferred from pooled DNA samples (y-axis in the right plots) and those computed from individually genotyped samples (x-axis). The analysis of the other chromosomes shows similar results.

We assessed the accuracy of the allele frequency estimates from pooled DNA samples by comparing the average estimates over the replicated pools with the allele frequencies computed using individually genotyped DNA samples that are available from the web site of the HapMap project 29. A scatter plot of part of the results is displayed in Figure 1 for pools of 60 samples. The error analysis summarized in Table 1 suggests that, on average, the allele frequency based on the analysis of replicated pooled DNA samples differ from those based on individually genotyped data by approximately ± 0.04 but the error can be as large as ~0.12 = 0.04+2×0.06/√2 thus making differences in allele frequencies smaller than 0.24 difficult to detect because of technical errors. However, our analysis shows that less than 5% of the estimates based on pools of DNA samples differ from those based on individually genotyped samples by more than 0.12, and less than 10% differ by more than 0.08. This suggests reducing the minimum detectable allele frequency difference to 0.15 with a 10% chance of error. Furthermore, we have observed that amplifying DNA does not appear to affect either the reproducibility or the accuracy of the analysis.

To infer the effective sample size to be used in the analysis, we also looked at the distribution of the ratio between the two types of allele frequency estimates: say p(Si) = ni/n and q(Si) where ni is the frequency of the minor allele of the SNP Si computed from the samples that were typed individually, n is the overall sample size, and q(Si) is the frequency of the same minor allele computed from the analysis of the pooled DNA samples, in the different sets. The analysis demonstrated that log(q(Si)/p(Si)) has approximately a normal distribution with 0 mean and standard deviation 0.35. From this data, we deduced that about 95% of allele frequency estimates derived from the pooled DNA samples can be assumed to be within the interval p(Si) exp(± 1.95 × 0.35) from which we derive the empirical relation between p(Si) and q(Si): 0.51 p(Si) < q(Si) < 1.98 p(Si) with a range of uncertainty of 1.47 ni/n. The inequality suggests that when we infer allele frequency from pooled DNA samples, we have a loss of precision approximately equivalent to using 2/3 (= 1/1.47) of the original DNA sample size. We call this the "effective sample size" used in the calculation of the Bayesian test of association.

Experiment 2: specificity

To estimate the false positive rate (FPR) we used real data from pools of DNA samples to create artificial sets of pools. The original pools are described in Table 2 and were generated in duplicates to discover genetic variants associated with exceptional longevity 30, and fetal hemoglobin expression in subjects with sickle cell anemia 31. The Illumina Sentrix HumanHap300 Genotyping BeadChip was used for all the experiments. We created the artificial sets of pools by mixing replicates of different pool sets. For example, we generated a set of two pools by taking one replicate of the pooled DNA samples from the female centenarians and one replicate of the pooled DNA samples from the younger female controls, and we constructed a second set of two pools by taking the remaining replicates from the two sets (See Figure 2 for an example). Because the two artificial sets of pools are homogeneous relative to the phenotype, the differences in allele frequencies between the two sets can be attributed to chance, and the SNPs with significant differences in allele distribution are false positives. We repeated this analysis by mixing different types of pools of DNA samples and using a BF>3, together with the LD and regional filters, we observed a false positive rate ranging between 0.001 and 4×10-4 with a mean of 0.001, and an average of 300 SNPs selected by chance. Note that this number is substantially smaller than the number of false positive associations that we would expect by chance using a BF>3. This threshold is equivalent to accepting an association when the posterior probability of the association is greater than 0.75, so that we expect 1 in 4 associations to be false. Also the specificity of the selected and significant genes was very high: The number of genes that by chance were selected in two unrelated analyses was 9 and this number was further reduced to 7 when we limited attention to significant genes. These numbers should provide a reference when we examine the reproducibility of findings in different studies, because we expect that, by chance alone, we would have an agreement in about 0.1% of findings. We note that long genes that are tagged by a larger number of SNPs are more likely to be selected by chance in different studies. Figure 3 displays the log10 BF in the 1,114 false positive associations generated in approximately the 106 association tests. The plot shows an exponential decay of the BF so that the chance to observe a very large BF has an exponential decay, and the probability of observing a BF greater than 10 by chance is 6 × 10-4, whereas the probability of observing a BF greater than 100 is 3 × 10-4, and greater than 1000 is 2 × 10-4. This analysis however shows that trying to reduce the false positive rate by imposing a stringent threshold on the BF would likely reduce the power of relatively small association studies and require unrealistically large sample sizes.

Table 2

Summary of the pools of DNA samples that were used for the validation of the analytical method. Each pool was done in duplicates.

Phenotype

Sample size

Exceptional longevity

130 male centenarians

130 male controls

130 female centenarians

100 female controls

Fetal hemoglobin expression

55 sickle cell anemia subjects with fetal hemoglobin below 3% of the total hemoglobin

54 sickle cell anemia subjects with fetal hemoglobin above 6.5% of the total hemoglobin

Figure 2

Example of the artificial pool sets that we created to assess the specificity of the procedure

Example of the artificial pool sets that we created to assess the specificity of the procedure. As an example, the top four pools were generated to compare the genome of centenarians (pools 1 and 2) with that of younger controls (pools 3 and 4). The two artificial pool sets are obtained by mixing pools of centenarians DNA with those of controls.

Figure 3

Distribution of the log₁₀Bayes factor in 1,114 false positive associations generated in approximately 10⁶association tests with an estimated false positive rate 5 10^-6

Distribution of the log₁₀Bayes factor in 1,114 false positive associations generated in approximately 10⁶association tests with an estimated false positive rate 5 10^-6. The analysis shows that the chance to observe a very large Bayes factor has an exponential decay, and the probability of observing a Bayes factor greater than 10 by chance is 6 10^-4, the probability of observing a Bayes factor greater than 100 is 3 10^-4, greater than 1000 is 2 10^-4.

We also run experiments to assess the effect of LD and regional filters on the specificity. Using the same simulated sets, we run the analysis by using only a BF>3 to select the significant SNPs, and also examined the effect of adding either the LD filter or the regional filter or both on the false positive rate. Our results suggest that the LD filter reduces the false positive rate by 43%, while the regional filter alone increases the false positive rate by approximately 25%, and both filters decrease the false positive rate by 20%. These results are consistent with the intuition that the regional filter increases the power by finding clusters of SNPs that individually have small effects and would be disregarded by a one-SNP-at-a-time analysis. However, the effect is to slightly increase the false positive rate. This conjecture is confirmed in the next experiments that we conducted to assess the sensitivity.

Experiment 3: sensitivity

In related work we are analyzing pools of DNA samples as a screening tool to discover genetic variants associated with exceptional longevity 30, and fetal hemoglobin (HbF) expression in subjects with sickle cell anemia 31. As an indication of the sensitivity of technology and analytic method, we searched for SNPs in the Illumina Sentrix HumanHap300 Genotyping BeadChip that have been reported associated with either trait in independent studies, and verified whether an association was found based on the pooled DNA samples.

HbF experiment

We created two pools using DNA samples from 55 patients in the top and 54 patients in the bottom quartile of HbF concentrations. These patients were part of a clinical trial described in 32. The pools were run in duplicates, and the data analyzed using the method proposed here. We searched the literature and found 36 SNPs with rs numbers that were reported associated with different levels of HbF 31333435. Thirteen of these SNPs are in the Illumina array, and only 3 of these were found associated in our analysis with a BF greater than 1, and 2 with a BF greater than 3. The moderate effect of the other 10 SNPs (odds ratios between 0.55 and 1.76) is consistent with the weak associations reported by other investigators and would not be detectable with our sample size of about 60 subjects per group. In fact, a sample size of 60 subjects would give at most 30% power to detect an odds ratio of 1.75 when the MAF in one group is 0.5. We also found 23 SNPs in the Illumina array that are within 150 kb of the other 23 reported SNPs, and are associated with HbF levels with a BF greater than 1, and 13 of these had a BF greater than 3. Fifteen of these SNPs were typed as part of the HapMap project and ten of these are in strong LD (Bayes D' > 0.8)36. Thus the analysis based on pooled DNA samples discovered association of 26 SNPs, and for 13 of them the association was strong. This analysis suggests a sensitivity of 72% and if we limit attention to associations supported by a BF of at least 3 the sensitivity is 36%. The details of the associations are in Table 3.

Table 3

List of SNPs that are known to be associated with different levels of HbF and results of the analysis based on pooled DNA samples.

Number

SNP

Band

Gene

Validated

Distance

P_H

P_L

rs1143637

2q13

IL1B

rs12469600

16976

1.39

0.13

0.04

3.08

rs31481

5q23.3

IL3

rs40401

724

1.00

2.32

0.48

0.65

0.48

rs271158

6q23.2

rs271156

641

1.00

22.86

0.53

0.77

0.35

rs454877

6q23.2

EYA4

rs211433

475

2.43

0.49

0.32

2.09

rs212770

6q23.2

EYA4

rs11154727

4274

1.25

0.12

0.04

3.03

rs2295199

6q23.2

rs2295199

0.51

0.83

0.89

0.58

rs210948

6q23.2

MYB

rs210798

17242

2.17

0.78

0.90

0.38

rs509342

6q23.2

PDE7B

rs560713

23645

0.95

1.47

1.00

0.95

17.80

rs2076192

6q23.3

MAP7

rs2076193

44438

0.97

1.30

0.52

0.37

1.88

rs997139

6q23.3

MAP7

rs3799419

2100

1.00

1.35

0.06

0.14

0.36

rs3778314

6q23.3

MAP7

rs2181096

25746

0.89

57.04

0.61

0.85

0.28

rs2237262

6q23.3

MAP3K5

rs3799472

18405

0.98

1.03

0.62

0.76

0.53

rs2012700

6q23.3

PEX7

rs2012700

1.50

0.34

0.50

0.52

rs717088

6q23.3

PEX7

rs717088

0.47

0.37

0.47

0.65

rs3799476

6q23.3

PEX7

rs3799479

44021

1.98

0.12

0.25

0.42

rs1342645

6q23.3

PEX7

rs1342645

0.67

0.77

0.86

0.55

rs1342641

6q23.3

rs1342642

22583

0.06

6.12

0.96

0.84

4.98

rs1322393

6q23.3

IL20RA

rs1322394

661

1.00

3.00

0.23

0.40

0.44

rs44450

6q23.3

rs276568

3229

1.00

47.32

0.37

0.64

0.34

rs1349115

8q12.1

TOX

rs1349115

0.69

0.29

0.41

0.59

rs10504269

8q12.1

TOX

rs10504269

0.61

0.72

0.82

0.58

rs6997859

8q12.1

TOX

rs6997859

0.37

0.23

0.17

1.45

rs12155519

8q12.1

TOX

rs12155519

0.94

0.58

0.44

1.77

rs1947178

8q12.1

TOX

rs1947178

21.86

0.10

0.29

0.28

rs746867

8q12.1

TOX

rs746867

0.29

0.35

0.76

rs389349

8q12.1

TOX

rs389349

0.42

0.95

0.93

1.43

rs851800

8q12.1

TOX

rs396720

23622

0.15

9.20

0.28

0.11

3.24

rs380620

8q12.1

TOX

rs2561145

300069

0.58

10.67

0.08

0.24

0.26

rs2043190

9q34.11

ASS

rs540140

609

19.56

0.99

0.88

24.90

rs7482144

11p15.4

HBG2

rs3813727

20257

2.52

0.17

0.32

0.43

rs723623

15q13.3

C15orf16

rs6493688

8825

0.51

1.31

0.93

0.84

2.66

rs1867380

15q22.31

AQP9

rs1867380

0.28

0.82

0.84

0.85

rs4489951

15q22.31

MAP2K1

rs4489951

17.66

0.36

0.60

0.38

rs1440372

15q22.31

SMAD6

rs2469141

65753

0.03

2181.00

0.68

0.33

4.24

rs8038623

15q22.31

SMAD3

rs6494633

16831

87.23

0.88

0.65

4.09

rs2227319

17q21.1

CSF3

rs2071369

1460

1.00

3.42

0.90

0.77

2.88

Column 1: row number; Column 2: SNP ID; Column 3: Cytogenic band; Column 4: Genes tagged by the SNP; Column 5: SNP in the Illumina array that was used to compare the association. If the SNP to be validated was not in the array, we searched for the closest SNP within 100 kb from that to be validated with a positive Bayes Factor; Column 6: distance between the two SNPs; Column 7: Bayes D' between the two SNPs, an NA means that the SNPs originally reported as associated with HbF is not in the HapMap data. Column 8: Bayes Factor; Column 9–10: estimates of allele frequencies in the pools of DNA from patients with high HbF and low HbF; Column 11: Odds ratio. The SNPs 6, 13, 14, 16, 20–26, 32 and 33 are in the array and SNPs 13, 24 and 33 were found associated with different levels of HbF. Highlighted in bold are the associations confirmed by our analysis.

Longevity experiment

We created pools of DNA samples from unrelated centenarians and younger controls. Because there is evidence of gender effect 37 --- 85% of centenarians are female--- we created distinct pools for males and females as summarized in Table 2. We searched the literature and found 36 SNPs with rs numbers reported as associated with longevity 383940. Seven of these 36 SNPs are in the Illumina array, and five of these seven were found associated in our analysis. For 21 of the remaining 29, we found SNPs within 100 kb that were associated with longevity in either the males and female comparisons, or both. These SNPs are reported in Table 4. The analysis suggests approximately 67% sensitivity, and this is consistent with the sensitivity estimated with the HbF experiment. We also noted that the regional filter helped identify some of the associations that would be lost with a tight threshold on the BF. As an example, the SNP rs2227956 on HSPA1A was found associated with longevity in males only when the regional filter is used, and the two SNPs in WRN -a well known longevity gene in mice 41 – were selected by the regional filter. A similar form of sensitivity analysis is to see whether the GSEA analysis can lead to discover sets of functionally similar genes that are known to be associated with longevity. GSEA analysis of the centenarian cohort revealed several enriched GO biological categories (Table 4 and manuscript in preparation). Among the significantly enriched categories were genes associated with immune response (e.g., CSF3) and DNA repair (e.g., XRCC4), see Table 4. Intriguingly, CSF3 (also known as GCSF) is reported to influence migration of stem cells between the bone marrow and blood 4243 and appears to promote regeneration of myocardial tissue 444546, which has clear relevance to longevity. The gene XRCC4 has a well established role in DNA repair 47, and unrepaired DNA has been reported to accelerate ageing, possibly through dysregulating the IGF/growth axis 48. Therefore, a comprehensive analysis of these and other genes present with the enriched gene sets that we detected will be essential to fully appreciate pathways engaged that contribute to the longevity phenotype.

Table 4

List of SNPs that were found associated with exceptional longevity in different studies, and results based on the analysis of pooled DNA samples.

Males

Females

Index

SNP

Band

Gene

Validated

Distance

P_L

P_C

Validated

Distance

P_L

P_C

rs1870377

4q12

KDR

rs2305945

1128

1.00

1.5

0.84

0.75

1.71

rs2305945

1128

1.00

1.1

0.77

0.68

1.62

rs2866164

4q23

MTP

rs7693203

9187

3.4

0.82

0.72

1.83

rs1057613

14042

8.6

0.64

0.49

1.90

rs750032

4q24

PPP3CA

rs2850971

3819

41.4

0.87

0.74

2.39

rs2850971

3819

10.8

0.82

0.69

2.12

rs951085

4q24

rs9999238

29363

0.98

0.2

0.81

0.83

0.85

rs9999238

29363

0.98

10.9

0.81

0.67

2.09

rs28360135

5q14.2

XRCC4

rs1382367

40462

295053.4

0.32

0.59

0.33

rs1382367

40462

2.6

0.50

0.63

0.58

rs1799945

6p22.2

HFE

rs1572982

3188

0.99

28228.6

0.67

0.42

2.78

rs1572982

3188

0.99

11.2

0.44

0.61

0.51

rs9380254

6p21.33

MICA

rs1131896

780

1.00

5744.1

0.67

0.87

0.32

rs1131896

780

1.00

0.8

0.83

0.84

0.96

rs2227956

6p21.33

HSPA1L

rs2227956

12.6

0.10

0.20

0.42

rs2227956

0.3

0.18

0.14

1.31

rs1800797

7p15.3

IL6

rs2056576

5019

0.64

0.2

0.70

0.72

0.91

rs2056576

5019

0.99

1.7

0.75

0.64

1.70

rs662

7q21.3

PON1

rs662

0.2

0.33

0.29

1.21

rs662

2.8

0.32

0.21

1.77

rs1799983

7q36.1

NOS3

rs2373929

18701

0.16

6.1

0.50

0.64

0.57

rs2373929

18701

0.07

0.6

0.63

0.54

1.45

rs2251621

8p12

PURG

rs13269094

8189

0.66

1936.2

0.07

0.22

0.26

rs1362911

23018

0.47

>10^6

0.69

0.96

0.10

rs2725362

8p12

WRN

rs3024239

158

4942.7

0.34

0.57

0.39

rs2725362

27.2

0.48

0.66

0.48

rs1346044

8p12

WRN

rs1346044

257.9

0.36

0.19

2.44

rs1346044

0.4

0.28

0.21

1.46

rs5744256

11q23.1

IL18

rs243908

16909

0.61

>10^6

0.63

0.29

4.02

rs243908

16909

0.96

0.2

0.28

1.01

rs675

11q23.2

APOA4

rs581015

5559

0.28

1.3

0.61

0.51

1.55

rs10502189

371

1.00

4.8

0.05

0.00

56.22

rs1467558

11p33

CD44

rs1467558

2.0

0.91

0.83

1.97

rs1467558

0.2

0.84

0.86

0.87

rs9536314

5.1

0.24

0..36

0.55

rs9536314

99.0

0.36

0.18

2.56

rs861539

XRCC3

rs861539

0.15

0.75

0.74

1.05

rs861539

68.7

0.74

0.55

2.27

rs8052394

16q12.2

MT1A

rs7189840

9798

0.14

0.59

0.57

1.09

rs7189840

9798

54.1

0.56

0.37

2.18

rs1800776

16q13

CETP

rs3764261

1910

1.1

0.59

0.69

0.65

rs3764261

1910

0.15

0.66

0.67

0.97

rs5882

16q13

CETP

rs5882

0.19

0.39

0.35

1.19

rs5882

0.17

0.37

0.40

0.88

rs704

17q11.2

VTN

rs2027993

12085

0.99

17.3

0.45

0.61

0.53

rs2027993

12085

0.99

1.1

0.58

0.47

1.57

rs4344

17q23.3

ACE

rs4343

693

1.00

1.4

0.69

0.78

0.61

rs4343

693

1.00

6.4

0.66

0.51

1.87

rs2252673

19p13.2

INSR

rs2059807

15691

0.23

1.5

0.63

0.73

0.63

rs2059807

15691

0.31

0.2

0.75

0.72

1.16

rs1799782

XRCC1

rs9394611128

11001

1.4

0.05

0.10

0.47

rs939461

11001

12.6

0.09

0.02

5.88

Column 1: row number; Column 2: SNP ID; Column 3: Cytogenic band; Column 4: Genes tagged by the SNP; Column 5: SNP in the Illumina array that was used to compare the association as described in the caption of Table 3; Column 6: distance between the two SNPs; Column 7: Bayes D' between the two SNPs; Column 8: Bayes Factor; Column 9–10: estimates of allele frequencies in the pools of DNA from male centenarians and younger controls; Column 11: Odds ratio. Columns 12–18: as columns 5–11 but for the pools comparing female centenarians to younger controls. The SNPs 8, 10, 14, 16–19, 22 are in the array and SNPs 8, 14, 17–19 were found associated with longevity in females and/or males. Highlighted in bold are the associations confirmed by our analysis.

Discussion and Conclusion

We have developed a hierarchical and modular approach to the analysis of genome wide genotype data based on pooled DNA samples. The method incorporates quality control data, information about linkage disequilibrium, Bayesian association tests, physical distance and gene ontology to identify associations warranting further investigation. Our evaluation using real data has shown the accuracy, reproducibility, sensitivity and specificity of the method.

Compared to other approaches, the integration of Bayesian tests with information about linkage disequilibrium and other machine learning techniques implies that we do not need to use too stringent thresholds to reduce the number of false positive associations. The implication of this fact is an increased power even with relatively small samples. In fact, our estimate of the sensitivity shows that the method can reach almost 70% sensitivity with samples of only 100 subjects.

Although we developed the approach to analyze pooled DNA samples, the method can also be used for the analysis of individually genotyped samples.

Methods

Genotyping

For the HbF study, DNA was obtained from the 60 subjects with HbF levels below the first quartile of the distribution, and the 60 subjects with HbF levels above the third quartile who were enrolled in the Multicenter Study of Hydroxyurea (MSH) study in Sickle Cell Anemia 49. DNA samples from 260 centenarians and a control group of 230 subjects were obtained from the New England Centenarian Study: a cross-sectional study of individuals aged 97 and older conducted at the Boston Medical Center. CEPH DNA samples of the sixty unrelated parents used for the International HapMap project 20 were obtained from the Coriell Institute, Camden, NY and used to compare the accuracy and reproducibility of the estimates of allele frequency in pooled DNA samples compared to individually genotyped samples. For DNA pool construction and to ensure that each individual contributed equally to the pool, we first measured DNA stock solutions using a fluorimetric method (RNAseP) against a standard curve constructed from known concentrations of human genomic DNA. We then diluted the stock solutions to 10 ng/ul and measured the concentrations of these working solutions by means of PicoGreen. In the case of samples for which the CV of the three measurements was greater than 10%, quantification was repeated in triplicate until the CV was smaller than 10. Measurements were highly reproducible, with a correlation coefficient of 0.97 between the third measurement and the average of the first two. Based on these concentrations, 50 ng of DNA were added to the pool for each individual. The pools of DNA were analyzed on the Sentrix HumanHap300 bead chip (Illumina) according to the manufacture's protocol. The data used in the HbF and longevity studies will be released with companion publications. We make available the data derived from pools of CEPH DNA samples from the supplementary web site 28. The HbF and longevity studies were approved by the Institutional Review Boards of Boston University.

Association test

The overall analytic strategy is shown in Figure 4. The first module is a statistical procedure to test the allelic association between each individual SNP and the phenotype. The input data are the allele relative frequencies estimated with the "b-allele frequency" value provided by the Illumina Beadstudio genotype module. This value represents the relative proportion of each allele in the DNA sample and is used by the Illumina loss of heterozygosity (LOH) and Copy Number analysis tool 15 to detect chromosomal aberrations and copy numbers by comparing the normalized intensity of the test sample (the pooled DNA samples) to a reference sample. We use the estimate of the allele frequencies θ_ijin test and control pools to reconstruct the expected allele frequencies as nij=ni∗θij MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOBa42aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpcqWGUbGBdaqhaaWcbaGaemyAaKgabaGaey4fIOcaaGGacOGae8hUde3aaSbaaSqaaiabdMgaPjabdQgaQbqabaaaaa@39B3@ where ni∗ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemOBa42aa0baaSqaaiabdMgaPbqaaiabgEHiQaaaaaa@2FAF@ is the effective sample size in pool i, the index i = 1 for cases and i = 2 for controls, and the index j = A,B denotes the A or B allele. We then use a Bayesian test of association to compare the distributions of allele frequency in the two different pools. The test is described in 1950 and assumes that prior probabilities are available for the model of allelic association and the model of no association – say p(M_a), p(M_i) – and then uses the data to update these prior probabilities p(M_a), p(M_i) into the posterior probabilities by using Bayes' theorem. The decision rule is then to select the model of association if its posterior probability is at least 3 times larger than the posterior probability of the model of no association (as suggested in reference 51). Formally, the ratio of the posterior probabilities is

B a y e s t e s t : p ( M a | n i j ) p ( M i | n i j ) = p ( n i j | M a ) p ( n i j | M i ) × p ( M a ) p ( M i ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegeKCPfgBaGabaiaa=jeacaWFHbGaa8xEaiaa=vgacaWFZbGaa8hiaiaa=rhacaWFLbGaa83Caiaa=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@7EB6@

and the ratio of the marginal likelihood functions p(n_ij| M_a)/p(n_ij| M_i) is known as the Bayes factor (BF). When the prior probabilities of the two models are equal, the BF is equivalent to the posterior odds. Assuming the conjugate Beta distribution for the allele frequencies, the BF can be calculated in closed form and the formula for this calculation is reported in the appendix. To take into account the issue of multiple testing, we can use prior information about the number of SNPs that we expected to be associated to make the selection stronger. For example, if we expect 1,500 SNPs associated with the phenotype, then the prior odds for the alternative hypothesis of association are 0.005/0.995 when we test 300,000 SNPs, and the decision rule becomes to accept that a SNP is associated with the phenotype if the posterior odds for the association are at least 3 × 0.995/0.005 = 597. Initial experiments described in the Evaluation section suggest that a robust choice for an effective sample size is 2/3 of the original pool, and this is consistent with a larger sample size needed with the analysis of pooled DNA samples 52. We note here that one advantage of this modular procedure is that the Bayesian test can be replaced by a standard χ²test for allelic association.

Figure 4

Schematic summary of the modular approach to the analysis of GWA data

Schematic summary of the modular approach to the analysis of GWA data.

Filtering out false positives

Although we can take into account the issue of multiple comparisons by choosing appropriate prior odds for an association, the consequence of this approach is to reduce power and to require large sample sizes to detect associations with a small effect. This consequence can be problematic in studies where cases are relatively rare such as the study of exceptional longevity in which cases are subjects who lives 100 years and older. To fully exploit the power of small scale studies we developed a series of data filters that remove unreliable or suspicious associations (see Figure 4). The first filter is specific for allele association analysis using pooled DNA samples and accounts for the lack of precision of the technology. The other two filters take into account redundancy as well as reciprocal information of SNPs based on the LD structure of the human genome. So, rather than using "SNP pruning" as in PLINK to remove SNPs that are in LD 53, we leverage on dependencies determined by LD to improve the detection of false positive while reducing the false negative rate.

Quality control (QC) filter

The function of this filter is based on an extensive evaluation of the accuracy and reproducibility of the allele frequency estimates that are computed with the Illumina software. Allele frequencies obtained from genotyping of pooled DNA samples were compared with those derived from genotyping of individual samples (detailed in the below Evaluation section). The results suggest that alleles with minor allele frequency MAF <0.15 as well as differences in allele frequency of less than 0.15 are not reliable. We therefore filter out all SNPs with these characteristics, as well as those SNPs for which repeated estimates of allele frequencies in replications of the same pool differ by more than 0.15.

Linkage disequilibrium (LD) filter

SNPs in LD with each other would be expected to show similar patterns of association if the signal is authentic whereas a single SNP in a LD block showing association is more likely to represent a spurious association. Therefore, our procedure automatically checks this condition and disregards the associations for SNPs that are not supported by positive associations with other SNPs in the same LD block. To this end, we used genotype data collected within the HapMap project to compute pairwise measures of LD for all consecutive pairs of SNPs in the HumanHap300 platform. The estimation of LD was based on a novel Bayesian version of D' that we introduced in 36. As the traditional D', our Bayesian estimator is defined in the interval [0;1] regardless of the allele frequency so that it is easier to interpret than other measures of correlation like r2 but it is much less biased toward disequilibrium. We use a Bayesian D' > 0.7 between pairs of consecutive SNPs as suggestive of strong LD and we filter out all the associations of the SNPs whose adjacent SNPs that are in strong LD are not associated with the phenotype. The value 0.7 was chosen based on experiments reported in 36 showing that the Bayesian D' rarely exceeds 0.7 under no LD. The Bayesian D' values for each pair of consecutive SNPs were built for Caucasians using the DNA samples from unrelated parents of thirty trios of the CEPH (Utah residents with ancestry from northern and western Europe, also known as CEU) and similarly for Africans, using Yoruba in Ibadan Nigeria. These data are available from the supplementary material web site 28.

Regional association filter

The rationale of this filter is that a region or gene showing authentic association would be expected to show a greater number of SNPs associated than would be expected by chance. In this filter, we analyze the data using a sliding window of 20 SNPs, and summarize the global measure of association within the window as the product of the posterior probabilities of associations of the 20 SNPs. Here, we assume that the association tests are independent, so that the product of the posterior probability of association of the individual SNPs becomes a measure of the global association of the region tagged by the 20 SNPs. Windows with a global measure of association exceeding 0.5 [or product of BFs >1] are then selected for further inspection. Although LD between SNPs in a window may introduce dependencies, the global measure of association does not seem to be affected by this approximation. Figure 5 shows some examples.

Figure 5

Relation between the pattern of LD (x-axis) and the global measure of association (y-axis) in the regional filter

Relation between the pattern of LD (x-axis) and the global measure of association (y-axis) in the regional filter. The pattern of LD is measured by the average of the Bayes D' between consecutive SNPs in the region, and the global measure of association is the joint probability of association in the region. The two figures in the top half show the relation using data from the study of fetal hemoglobin in the sickle cell anemia subjects. The two figure in the bottom half show the relation using data from the longevity study. The different extent of LD reflect the fact that sickle cell anemia subjects are all African American while centenarians in the longevity study are all Caucasians The correlations in the four sets are 0.03, 0.18, 0.018, -0.10.

Hierarchical summary

The list of SNPs that are selected by the association test and the filters are labeled as "significant SNPs". The list is annotated by the SNP physical position, the position relative to known genes, the allele frequencies estimated in different populations, and the cytogenetic band. This information is collected through SNPPer 54 that integrates information from the UCSC human genome browser and dbSNP 55. As a further level of summary we use those SNPs that are linked to genes to create a set of selected genes and a set of significant genes. The first set consists of genes that are tagged by at least one significant SNP. The set of significant genes is a subset of the selected genes and consists of genes in which the global measure of association given by the product of the posterior probability of association of the gene tagging SNPs is greater than 0.5 [Or equivalently, the product of BFs exceeds 1].

Ranking

We rank the significant genes by the global measure of association. To rank the selected genes we score them by two further measures that weigh the likelihood of selecting a gene by chance. In fact, there are genes that are tagged by a large number of SNPs: for example CSMD1 in chromosome 8 is tagged by 614 SNPs in the HumanHap300 array, and assuming a 5% false positive rate, we would expect about 30 SNPs to be selected from this gene by chance in any analysis. To take this issue into account, each selected gene is assigned 3 scores: the global measure of association, the ratio of selected SNPs relative to the number of tagging SNPs, and the probability of selecting this number by chance using the hyper-geometric distribution. Each score determines a ranking and then the sum of the ranks is defined as a final ranking of selected genes.

Gene set enrichment analysis

To evaluate selected and/or significant genes for enrichment of biological categories associated with a variable phenotype, we implement a stand-alone version of the EASE statistical software 56. This program computes a modified Fisher's exact probability score for observing the frequency of a biological category associated with a phenotype (e.g., dementia, sickle phenotype, infection), compared with the likelihood of identifying that category by chance given the total number of genes in the data set. An adjusted score is then reported representing the upper bound of the distribution of Jackknife Fisher exact probabilities for observing an enriched biological category. Enriched categories are then inspected for biological trends and overlapping or related categories, based on significance scores or categories with a p-value << 0.05. For more detail, see Hosack et al. 56.

Authors' contributions

PS developed and implemented the analytic method, conceived the evaluation and drafted the manuscript. ZZ contributed to the implementation and evaluation. MAG, AR, SH, AS, developed the support material for the various filters and hierarchical summary. AD helped to design and conduct the experiments with the CEPH samples. MM contributed to the development of the hierarchical method and the interpretation of the analysis results. EM and CB carried out the pooling-based GW genotyping. DT, TTP, MS contributed to the development and evaluation of the method using the longevity study and the HbF study. All authors read and approved the final manuscript.

Appendix

Derivation of the Bayes Factor

This Bayesian association test assumes that the allele frequencies follow a binomial distribution with probabilities θ_ijthe index i = 1 for cases and i = 2 for controls, and the index j = A,B denotes the A or B allele [See Table 5 for an example]. Under the hypothesis of general association, the parameters θ_ijdescribing the allele distributions in cases and controls follow different probability distributions, while the parameters θ_ijfollow the same probability distribution under the hypothesis of no association. Therefore, the likelihood function under the hypothesis of general association M_ais

Table 5

parameters and allele frequencies from pooled DNA samples

Allele A

Allele B

Cases (one pool)

n_1A= p(A|cases)*(2n₁)

n_1B= p(B|cases)*(2n₁)

2n₁

Controls (one pool)

n_2A= p(A|controls)*(2n₂)

n_2B= p(B|controls)*(2n₂)

2n₂

n_A

n_B

M a : p ( θ i j | n i j ) ∝ θ 1 A n 1 A ( 1 − θ 1 A ) n 1 B θ 2 A n 2 A ( 1 − θ 2 A ) n 2 B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyta00aaSbaaSqaaiabdggaHbqabaGccqGG6aGocqWGWbaCcqGGOaakiiGacqWF4oqCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcYha8jabd6gaUnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiykaKIaeyyhIuRae8hUde3aa0baaSqaaiabigdaXiabdgeabbqaaiabd6gaUnaaBaaameaacqaIXaqmcqWGbbqqaeqaaaaakiabcIcaOiabigdaXiabgkHiTiab=H7aXnaaBaaaleaacqaIXaqmcqWGbbqqaeqaaOGaeiykaKYaaWbaaSqabeaacqWGUbGBdaWgaaadbaGaeGymaeJaemOqaieabeaaaaGccqaH4oqCdaqhaaWcbaGaeGOmaiJaemyqaeeabaGaemOBa42aaSbaaWqaaiabikdaYiabdgeabbqabaaaaOGaeiikaGIaeGymaeJaeyOeI0Iae8hUde3aaSbaaSqaaiabikdaYiabdgeabbqabaGccqGGPaqkdaahaaWcbeqaaiabd6gaUnaaBaaameaacqaIYaGmcqWGcbGqaeqaaaaaaaa@642B@

While the likelihood function under the hypothesis of no association M_iis:

M i : p ( θ i j | n i j ) ∝ θ A n A ( 1 − θ A ) n B MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyta00aaSbaaSqaaiabdMgaPbqabaGccqGG6aGocqWGWbaCcqGGOaakiiGacqWF4oqCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcYha8jabd6gaUnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiykaKIaeyyhIuRae8hUde3aa0baaSqaaiabdgeabbqaaiabd6gaUnaaBaaameaacqWGbbqqaeqaaaaakiabcIcaOiabigdaXiabgkHiTiab=H7aXnaaBaaaleaacqWGbbqqaeqaaOGaeiykaKYaaWbaaSqabeaacqWGUbGBdaWgaaadbaGaemOqaieabeaaaaaaaa@4DC7@

We assume independent Beta distributions to model the prior distributions of the parameters that are defined as

M a : p ( θ i j ) ∝ θ 1 A α 1 A − 1 ( 1 − θ 1 A ) β 1 B − 1 θ 2 A α 2 A − 1 ( 1 − θ 2 A ) β 2 B − 1 and M i : p ( θ i j ) ∝ θ A α A − 1 ( 1 − θ A ) β B − 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqbaeaabiqaaaqaauaabeqabiaaaeaacqWGnbqtdaWgaaWcbaGaemyyaegabeaakiabcQda6iabdchaWjabcIcaOGGaciab=H7aXnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiykaKIaeyyhIuRae8hUde3aa0baaSqaaiabigdaXiabdgeabbqaaiab=f7aHnaaBaaameaacqaIXaqmcqWGbbqqaeqaaSGaeyOeI0IaeGymaedaaOGaeiikaGIaeGymaeJaeyOeI0Iae8hUde3aaSbaaSqaaiabigdaXiabdgeabbqabaGccqGGPaqkdaahaaWcbeqaaiab=j7aInaaBaaameaacqaIXaqmcqWGcbGqaeqaaSGaeyOeI0IaeGymaedaaOGae8hUde3aa0baaSqaaiabikdaYiabdgeabbqaaiab=f7aHnaaBaaameaacqaIYaGmcqWGbbqqaeqaaSGaeyOeI0IaeGymaedaaOGaeiikaGIaeGymaeJaeyOeI0Iae8hUde3aaSbaaSqaaiabikdaYiabdgeabbqabaGccqGGPaqkdaahaaWcbeqaaiab=j7aInaaBaaameaacqaIYaGmcqWGcbGqaeqaaSGaeyOeI0IaeGymaedaaaGcbaGaeeyyaeMaeeOBa4Maeeizaqgaaaqaaiabd2eannaaBaaaleaacqWGPbqAaeqaaOGaeiOoaOJaemiCaaNaeiikaGIae8hUde3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGPaqkcqGHDisTcqWF4oqCdaqhaaWcbaGaemyqaeeabaGae8xSde2aaSbaaWqaaiabdgeabbqabaWccqGHsislcqaIXaqmaaGccqGGOaakcqaIXaqmcqGHsislcqWF4oqCdaWgaaWcbaGaemyqaeeabeaakiabcMcaPmaaCaaaleqabaGae8NSdi2aaSbaaWqaaiabdkeacbqabaWccqGHsislcqaIXaqmaaaaaaaa@8AEC@

Where the hyper-parameters are chosen as α_1A= α_2A= α_A/2 = α/4 and

β_1B= β_2B= β_B/2 = α/4. The parameter α is the overall prior precision and can be set based on prior information. The likelihood function and the prior distribution of the parameters are used to compute the marginal likelihood as the expected likelihood function, where the expectation is taken over the parameter distribution. Formally

p(M | n_ij) = ∫p(θ_ij| n_ij)p(θ_ij)dθ_ij

Compared to the maximum likelihood that returns the likelihood function evaluated in the estimate of the parameters, the marginal likelihood incorporates the uncertainty about the parameters by averaging the likelihood functions for different parameter values. This conceptual difference is fundamental to understand the different approach to model selection: in the classical framework, model selection is based on the maximized likelihood and its sampling distribution to take into account sampling variability, for fixed parameter values. In the Bayesian framework, model selection is based on the marginal likelihood which takes into account the parameter variability, for fixed sample values. Therefore, no significance testing is performed when using this approach to model selection. Our experience with the Bayesian procedure to model selection is that it is usually more robust to false associations. The calculation of the marginal likelihood can be done in closed form and it is easy to show that

p ( M a | n i j ) = Γ ( α 1 A + β 1 B ) Γ ( α 2 A + β 2 B ) Γ ( α 1 A + β 1 B + n 1 ) Γ ( α 2 A + β 2 B + n 2 ) × Γ ( α 1 A + n 1 A ) Γ ( β 1 B + n 1 B ) Γ ( α 2 A + n 2 A ) Γ ( β 2 B + n 2 B ) Γ ( α 1 A ) Γ ( α 2 A ) Γ ( β 1 B ) Γ ( β 2 B ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiCaaNaeiikaGIaemyta00aaSbaaSqaaiabdggaHbqabaGccqGG8baFcqWGUbGBdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcMcaPiabg2da9KqbaoaalaaabaGaeu4KdCKaeiikaGccciGae8xSde2aaSbaaeaacqaIXaqmcqWGbbqqaeqaaiabgUcaRiab=j7aInaaBaaabaGaeGymaeJaemOqaieabeaacqGGPaqkcqqHtoWrcqGGOaakcqWFXoqydaWgaaqaaiabikdaYiabdgeabbqabaGaey4kaSIae8NSdi2aaSbaaeaacqaIYaGmcqWGcbGqaeqaaiabcMcaPaqaaiabfo5ahjabcIcaOiab=f7aHnaaBaaabaGaeGymaeJaemyqaeeabeaacqGHRaWkcqWFYoGydaWgaaqaaiabigdaXiabdkeacbqabaGaey4kaSIaemOBa42aaSbaaeaacqaIXaqmaeqaaiabcMcaPiabfo5ahjabcIcaOiab=f7aHnaaBaaabaGaeGOmaiJaemyqaeeabeaacqGHRaWkcqWFYoGydaWgaaqaaiabikdaYiabdkeacbqabaGaey4kaSIaemOBa42aaSbaaeaacqaIYaGmaeqaaiabcMcaPaaakiabgEna0MqbaoaalaaabaGaeu4KdCKaeiikaGIae8xSde2aaSbaaeaacqaIXaqmcqWGbbqqaeqaaiabgUcaRiabd6gaUnaaBaaabaGaeGymaeJaemyqaeeabeaacqGGPaqkcqqHtoWrcqGGOaakcqWFYoGydaWgaaqaaiabigdaXiabdkeacbqabaGaey4kaSIaemOBa42aaSbaaeaacqaIXaqmcqWGcbGqaeqaaiabcMcaPiabfo5ahjabcIcaOiab=f7aHnaaBaaabaGaeGOmaiJaemyqaeeabeaacqGHRaWkcqWGUbGBdaWgaaqaaiabikdaYiabdgeabbqabaGaeiykaKIaeu4KdCKaeiikaGIae8NSdi2aaSbaaeaacqaIYaGmcqWGcbGqaeqaaiabgUcaRiabd6gaUnaaBaaabaGaeGOmaiJaemOqaieabeaacqGGPaqkaeaacqqHtoWrcqGGOaakcqWFXoqydaWgaaqaaiabigdaXiabdgeabbqabaGaeiykaKIaeu4KdCKaeiikaGIae8xSde2aaSbaaeaacqaIYaGmcqWGbbqqaeqaaiabcMcaPiabfo5ahjabcIcaOiab=j7aInaaBaaabaGaeGymaeJaemOqaieabeaacqGGPaqkcqqHtoWrcqGGOaakcqWFYoGydaWgaaqaaiabikdaYiabdkeacbqabaGaeiykaKcaaaaa@B8B8@

p ( M i | n i j ) = Γ ( α ) Γ ( α + n ) Γ ( α 1 A + α 2 A + n A ) Γ ( β 1 B + β 2 B + n B ) Γ ( α 1 A + α 2 A ) Γ ( β 1 B + β 2 B ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemiCaaNaeiikaGIaemyta00aaSbaaSqaaiabdMgaPbqabaGccqGG8baFcqWGUbGBdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcMcaPiabg2da9KqbaoaalaaabaGaeu4KdCKaeiikaGccciGae8xSdeMaeiykaKcabaGaeu4KdCKaeiikaGIae8xSdeMaey4kaSIaemOBa4MaeiykaKcaamaalaaabaGaeu4KdCKaeiikaGIae8xSde2aaSbaaeaacqaIXaqmcqWGbbqqaeqaaiabgUcaRiab=f7aHnaaBaaabaGaeGOmaiJaemyqaeeabeaacqGHRaWkcqWGUbGBdaWgaaqaaiabdgeabbqabaGaeiykaKIaeu4KdCKaeiikaGIae8NSdi2aaSbaaeaacqaIXaqmcqWGcbGqaeqaaiabgUcaRiab=j7aInaaBaaabaGaeGOmaiJaemOqaieabeaacqGHRaWkcqWGUbGBdaWgaaqaaiabdkeacbqabaGaeiykaKcabaGaeu4KdCKaeiikaGIae8xSde2aaSbaaeaacqaIXaqmcqWGbbqqaeqaaiabgUcaRiab=f7aHnaaBaaabaGaeGOmaiJaemyqaeeabeaacqGGPaqkcqqHtoWrcqGGOaakcqWFYoGydaWgaaqaaiabigdaXiabdkeacbqabaGaey4kaSIae8NSdi2aaSbaaeaacqaIYaGmcqWGcbGqaeqaaiabcMcaPaaaaaa@79D2@

Where G is the gamma function and the ratio produces the Bayes factor.

Acknowledgements

Supported by NHLBI grants R21 HL080463 (PS); R01 HL68970 (MHS); K-24, AG025727 (TP); K23 AG026754 (D.T.). We thank the anonymous reviewers and editors for their helpful suggestions.

Genome-wide association study of 14,000 cases of seven common diseases and 3,000 shared controls Nature 2007 447 7145 661 678 10.1038/nature05911 17554300 What genome-wide association studies can do for medicine Christensen K Murray JC N Engl J Med 2007 356 11 1094 1097 10.1056/NEJMp068126 17360987 Efficiency and power in genetic association studies de Bakker PI Yelensky R Pe'er I Gabriel SB Daly MJ Altshuler D Nat Genet 2005 37 11 1217 1223 10.1038/ng1669 16244653 Highly parallel genomic assays Fan JB Chee MS Gunderson KL Nat Rev Genet 2006 7 8 632 644 10.1038/nrg1901 16847463 Identification of disease causing loci using an array-based genotyping approach on pooled DNA Craig DW Huentelman MJ Hu-Lince D Zismann VL Kruer MC Lee AM Puffenberger EG Pearson JM Stephan DA BMC Genomics 2005 6 138 1262713 16197552 10.1186/1471-2164-6-138 Identification of a novel risk locus for progressive supranuclear palsy by a pooled genomewide scan of 500,288 single-nucleotide polymorphisms Melquist S Craig DW Huentelman MJ Crook R Pearson JV Baker M Zismann VL Gass J Adamson J Szelinger S Corneveaux J Cannon A Coon KD Lincoln S Adler C Tuite P Calne DB Bigio EH Uitti RJ Wszolek ZK Golbe LI Caselli RJ Graff-Radford N Litvan I Farrer MJ Dickson DW Hutton M Stephan DA Am J Hum Genet 2007 80 4 769 778 1852701 17357082 10.1086/513320 Identification of PVT1 as a candidate gene for end-stage renal disease in type 2 diabetes using a pooling-based genome-wide single nucleotide polymorphism association study Hanson RL Craig DW Millis MP Yeatts KA Kobes S Pearson JV Lee AM Knowler WC Nelson RG Wolford JK Diabetes 2007 56 4 975 983 10.2337/db06-1072 17395743 Genomic DNA pooling for whole-genome association scans in complex disease: empirical demonstration of efficacy in rheumatoid arthritis Steer S Abkevich V Gutin A Cordell HJ Gendall KL Merriman ME Rodger RA Rowley KA Chapman P Gow P Harrison AA Highton J Jones PB O'Donnell J Stamp L Fitzgerald L Iliev D Kouzmine A Tran T Skolnick MH Timms KM Lanchbury JS Merriman TR Genes Immun 2007 8 1 57 68 10.1038/sj.gene.6364359 17159887 Quantitative trait locus association scan of early reading disability and ability using pooled DNA and 100K SNP microarrays in a sample of 5760 children Meaburn EL Harlaar N Craig IW Schalkwyk LC Plomin R Mol Psychiatry 2007 17684495 Single nucleotide polymorphism (SNP) allele frequency estimation in DNA pools using Pyrosequencing Lavebratt C Sengul S Nat Protoc 2006 1 6 2573 2582 10.1038/nprot.2006.442 17406511 Allelotyping of pooled DNA with 250 K SNP microarrays Wilkening S Chen B Wirtenberger M Burwinkel B Forsti A Hemminki K Canzian F BMC Genomics 2007 8 77 1839100 17367522 10.1186/1471-2164-8-77 Identification of the sources of error in allele frequency estimations from pooled DNA indicates an optimal experimental design Barratt BJ Payne F Rance HE Nutland S Todd JA Clayton DG Ann Hum Genet 2002 66 Pt 5-6 393 405 10.1046/j.1469-1809.2002.00125.x 12485472 Applicability of DNA pools on 500 K SNP microarrays for cost-effective initial screens in genomewide association studies Docherty SJ Butcher LM Schalkwyk LC Plomin R BMC Genomics 2007 8 214 1925094 17610740 10.1186/1471-2164-8-214 Genotyping pooled DNA using 100K SNP microarrays: a step towards genomewide association scans Meaburn E Butcher LM Schalkwyk LC Plomin R Nucleic Acids Res 2006 34 4 e27 1368655 16478714 10.1093/nar/gnj027 Reliable high-throughput genotyping and loss-of-heterozygosity detection in formalin-fixed, paraffin-embedded tumors using single nucleotide polymorphism arrays Lips EH Dierssen JW van Eijk R Oosting J Eilers PH Tollenaar RA de Graaf EJ van't Slot R Wijmenga C Morreau H van Wezel T Cancer Res 2005 65 22 10188 10191 10.1158/0008-5472.CAN-05-2486 16288005 Controlling the false discovery rate - a practical and powerful approach to multiple testing Benjamini Y Hochberg Y J Roy Stat Soc B Met 1995 57 1 289 300 Significance analysis of microarrays applied to the ionizing radiation response Tusher VG Tibshirani R Chu G Proc Natl Acad Sci U S A 2001 98 9 5116 5121 33173 11309499 10.1073/pnas.091062498 Using linkage genome scans to improve power of association in genome scans Roeder K Bacanu SA Wasserman L Devlin B Am J Hum Genet 2006 78 2 243 252 1380233 16400608 10.1086/500026 A tutorial on statistical methods for population association studies Balding DJ Nat Rev Genet 2006 7 10 781 791 10.1038/nrg1916 16983374 A haplotype map of the human genome International HapMap Consortium Nature 2005 437 7063 1299 1320 1880871 16255080 10.1038/nature04226 The Gene Ontology (GO) project in 2006 Nucleic Acids Res 2006 34 Database issue D322 6 1347384 16381878 KEGG: kyoto encyclopedia of genes and genomes Kanehisa M Goto S Nucleic Acids Res 2000 28 1 27 30 102409 10592173 10.1093/nar/28.1.27 From genomics to chemical genomics: new developments in KEGG Kanehisa M Goto S Hattori M Aoki-Kinoshita KF Itoh M Kawashima S Katayama T Araki M Hirakawa M Nucleic Acids Res 2006 34 Database issue D354 7 1347464 16381885 10.1093/nar/gkj102 GenABEL: an R library for genome-wide association analysis Aulchenko YS Ripke S Isaacs A van Duijn CM Bioinformatics 2007 23 10 1294 1296 10.1093/bioinformatics/btm108 17384015 An R package for analysis of whole-genome association studies Clayton D Leung HT Hum Hered 2007 64 1 45 51 10.1159/000101422 17483596 Multistage Sampling for Genetic Studies Elston RC Lin DY Zheng G Annu Rev Genomics Hum Genet 2007 17506660 DNA Pooling: a tool for large-scale association studies Sham P Bader JS Craig I O'Donovan M Owen M Nat Rev Genet 2002 3 11 862 871 10.1038/nrg930 12415316 Supplementary material http://www.bu.edu/sicklecell/downloads/Projects/web_supplement_pooling/index.html The International HapMap Project Gibbs RA Belmont JW Hardenbol P Willis TD Yu F Yang H Ch'ang LY Huang W Liu B Shen Y Tam PK Tsui LC Waye MM Wong JT Zeng C Zhang Q Chee MS Galver LM Kruglyak S Murray SS Oliphant AR Montpetit A Hudson TJ Chagnon F Ferretti V Leboeuf M Phillips MS Verner A Kwok PY Duan S Lind DL Miller RD Rice JP Saccone NL Taillon-Miller P Xiao M Nakamura Y Sekine A Sorimachi K Tanaka T Tanaka Y Tsunoda T Yoshino E Bentley DR Deloukas P Hunt S Powell D Altshuler D Gabriel SB Zhang H Matsuda I Fukushima Y Macer DR Suda E Rotimi CN Adebamowo CA Aniagwu T Marshall PA Matthew O Nkwodimmah C Royal CD Leppert MF Dixon M Stein LD Cunningham F Kanani A Thorisson GA Chakravarti A Chen PE Cutler DJ Kashuk CS Donnelly P Marchini J McVean GA Myers SR Cardon LR Abecasis GR Morris A Weir BS Mullikin JC Sherry ST Feolo M Daly MJ Schaffner SF Qiu R Kent A Dunston GM Kato K Niikawa N Knoppers BM Foster MW Clayton EW Wang VO Watkin J Sodergren E Weinstock GM Wilson RK Fulton LL Rogers J Birren BW Han H Wang H Godbout M Wallenburg JC L'Archeveque P Bellemare G Todani K Fujita T Tanaka S Holden AL Lai EH Collins FS Brooks LD McEwen JE Guyer MS Jordan E Peterson JL Spiegel J Sung LM Zacharia LF Kennedy K Dunn MG Seabrook R Shillito M Skene B Stewart JG Valle DL Jorde LB Cho MK Duster T Jasperse M Licinio J Long JC Ossorio PN Spallone P Terry SF Lander ES Nickerson DA Boehnke M Douglas JA Hudson RR Kruglyak L Nussbaum RL Nature 2003 426 6968 789 796 10.1038/nature02168 14685227 Understanding the determinants of exceptional longevity Perls T Terry D Ann Intern Med 2003 139 5 Pt 2 445 449 12965974 Predicting clinical severity in sickle cell anaemia Steinberg MH Br J Haematol 2005 129 4 465 481 10.1111/j.1365-2141.2005.05411.x 15877729 Effect of hydroxyurea on mortality and morbidity in adult sickle cell anemia: risks and benefits up to 9 years of treatment Steinberg MH Barton F Castro O Pegelow CH Ballas SK Kutlar A Orringer E Bellevue R Olivieri N Eckman J Varma M Ramirez G Adler B Smith W Carlos T Ataga K DeCastro L Bigelow C Saunthararajah Y Telfer M Vichinsky E Claster S Shurin S Bridges K Waclawiw M Bonds D Terrin M Jama 2003 289 13 1645 1651 10.1001/jama.289.13.1645 12672732 Fetal hemoglobin in sickle cell anemia: association with single nucleotide polymorphisms in TOX (8q12). Ma Q Baldwin CT Safaya S Kutlar A Farrer LA Steinberg MH Hum Mol Genet 2007 Polymorphisms near a chromosome 6q QTL area are associated with modulation of fetal hemoglobin levels in sickle cell anemia Wyszynski DF Baldwin CT Cleves MA Amirault Y Nolan VG Farrell JJ Bisbee A Kutlar A Farrer LA Steinberg MH Cell Mol Biol (Noisy-le-grand) 2004 50 1 23 33 15040424 A network model to predict the risk of death in sickle cell disease Sebastiani P Nolan VG Baldwin CT Abad-Grau MM Wang L Adewoye AH McMahon LC Farrer LA Taylor JG Kato GJ Gladwin MT Steinberg MH Blood 2007 110 7 2727 2735 10.1182/blood-2007-04-084921 17600133 Bayesian estimates of linkage disequilibrium Sebastiani P Abad-Grau MM BMC Genet 2007 8 1 36 1924864 17592642 10.1186/1471-2156-8-36 Anti-aging teleconference: what is anti-aging medicine? Arking R Butler B Chiko B Fossel M Gavrilov LA Morley JE Olshansky SJ Perls T Walker RF J Anti Aging Med 2003 6 2 91 106 10.1089/109454503769684775 14614799 Unique lipoprotein phenotype and genotype associated with exceptional longevity Barzilai N Atzmon G Schechter C Schaefer EJ Cupples AL Lipton R Cheng S Shuldiner AR Jama 2003 290 15 2030 2040 10.1001/jama.290.15.2030 14559957 The quest for genetic determinants of human longevity: challenges and insights Christensen K Johnson TE Vaupel JW Nat Rev Genet 2006 7 6 436 448 10.1038/nrg1871 16708071 Human aging genomic resources http://genomics.senescence.info/genes/longevity.html Mutations in the WRN gene in mice accelerate mortality in a p53-null background Lombard DB Beard C Johnson B Marciniak RA Dausman J Bronson R Buhlmann JE Lipman R Curry R Sharpe A Jaenisch R Guarente L Mol Cell Biol 2000 20 9 3286 3291 85622 10757812 10.1128/MCB.20.9.3286-3291.2000 G-CSF induces stem cell mobilization by decreasing bone marrow SDF-1 and up-regulating CXCR4 Petit I Szyper-Kravitz M Nagler A Lahav M Peled A Habler L Ponomaryov T Taichman RS Arenzana-Seisdedos F Fujii N Sandbank J Zipori D Lapidot T Nat Immunol 2002 3 7 687 694 10.1038/ni813 12068293 Disruption of the CXCR4/CXCL12 chemotactic interaction during hematopoietic stem cell mobilization induced by GCSF or cyclophosphamide Levesque JP Hendy J Takamatsu Y Simmons PJ Bendall LJ J Clin Invest 2003 111 2 187 196 151860 12531874 10.1172/JCI200315994 G-CSF prevents cardiac remodeling after myocardial infarction by activating the Jak-Stat pathway in cardiomyocytes Harada M Qin Y Takano H Minamino T Zou Y Toko H Ohtsuka M Matsuura K Sano M Nishi J Iwanaga K Akazawa H Kunieda T Zhu W Hasegawa H Kunisada K Nagai T Nakaya H Yamauchi-Takihara K Komuro I Nat Med 2005 11 3 305 311 10.1038/nm1199 15723072 Endogenous G-CSF and CD34+ cell mobilization after acute myocardial infarction Leone AM Rutella S Bonanno G Contemi AM de Ritis DG Giannico MB Rebuzzi AG Leone G Crea F Int J Cardiol 2006 111 2 202 208 10.1016/j.ijcard.2005.06.043 16051386 G-CSF and AMD3100 mobilize monocytes into the blood that stimulate angiogenesis in vivo through a paracrine mechanism Capoccia BJ Shepherd RM Link DC Blood 2006 108 7 2438 2445 1895560 16735597 10.1182/blood-2006-04-013755 Interplay of p53 and DNA-repair protein XRCC4 in tumorigenesis, genomic stability and development Gao Y Ferguson DO Xie W Manis JP Sekiguchi J Frank KM Chaudhuri J Horner J DePinho RA Alt FW Nature 2000 404 6780 897 900 10.1038/35009138 10786799 A new progeroid syndrome reveals that genotoxic stress suppresses the somatotroph axis Niedernhofer LJ Garinis GA Raams A Lalai AS Robinson AR Appeldoorn E Odijk H Oostendorp R Ahmad A van Leeuwen W Theil AF Vermeulen W van der Horst GT Meinecke P Kleijer WJ Vijg J Jaspers NG Hoeijmakers JH Nature 2006 444 7122 1038 1043 10.1038/nature05456 17183314 Fetal hemoglobin in sickle cell anemia: determinants of response to hydroxyurea. Multicenter Study of Hydroxyurea Steinberg MH Lu ZH Barton FB Terrin ML Charache S Dover GJ Blood 1997 89 3 1078 1088 9028341 Genetic dissection and prognostic modeling of overt stroke in sickle cell anemia Sebastiani P Ramoni MF Nolan V Baldwin CT Steinberg MH Nat Genet 2005 37 4 435 440 10.1038/ng1533 15778708 Bayes factor Kass RE Raftery AE J Am Statist Assoc 1995 90 773 795 10.2307/2291091 The impacts of errors in individual genotyping and DNA pooling on association studies Zou G Zhao H Genet Epidemiol 2004 26 1 1 10 10.1002/gepi.10277 14691952 PLINK: A Tool Set for Whole-Genome Association and Population-Based Linkage Analyses Purcell S Neale B Todd-Brown K Thomas L Ferreira MA Bender D Maller J Sklar P de Bakker PI Daly MJ Sham PC Am J Hum Genet 2007 81 3 559 575 10.1086/519795 17701901 SNPPer http://snpper.chip.org/ A SNP-centric database for the investigation of the human genome Riva A Kohane IS BMC Bioinformatics 2004 5 33 395999 15046636 10.1186/1471-2105-5-33 Identifying biological themes within lists of genes with EASE Hosack DA Dennis G Jr. Sherman BT Lane HC Lempicki RA Genome Biol 2003 4 10 R70 328459 14519205 10.1186/gb-2003-4-10-r70