estimatingcausaleffectsofgeneticriskvariantsforbreast ... · research article...

Research Article

Estimating Causal Effects of Genetic Risk Variants for BreastCancer Using Marker Data from Bilateral and Familial Cases

Frank Dudbridge1, Olivia Fletcher2, Kate Walker1, Nichola Johnson2, Nick Orr2, Isabel dos Santos Silva1, andJulian Peto1

AbstractBackground:Cases with a family history are enriched for genetic risk variants, and the power of association

studies can be improved by selecting cases with a family history of disease. However, in recent genome-wide

association scans utilizing familial sampling, the excess relative risk for familial cases is less than predicted

when compared with unselected cases. This can be explained by incomplete linkage disequilibrium between

the tested marker and the underlying causal variant.

Methods:Weshow that the allele frequency and effect size of the underlying causal variant can be estimated

bycombiningmarkerdata fromstudies that ascertain cases basedondifferent familyhistories. This allowsus to

learn about the genetic architecture of a complex trait, without having identified any causal variants. We

consider several validated common marker alleles for breast cancer, using our own study of high risk,

predominantly bilateral cases, cases preferentially selected to have at least two affected first- or second-degree

relatives, and published estimates of relative risk from standard case–control studies.

Results: To obtain realistic estimates and to accommodate some prior beliefs, we use Bayesian estimation to

infer that the causal variants are probably common, with minor allele frequency >5%, and have small effects,

with relative risk around 1.2.

Conclusion: These results strongly support the common disease common variant hypothesis for these

specific loci associated with breast cancer.

Impact: Our results agree with recent assertions that synthetic associations of rare variants are unlikely to

account for most associations seen in genome-wide studies. Cancer Epidemiol Biomarkers Prev; 21(2); 262–72.

�2011 AACR.

Introduction

Familial cases of disease are likely to segregate geneticrisk variants, and therefore offer increased efficiency oversporadic cases for detecting new variants (1–2). Largecollections of familial cases ascertained through geneticsclinics (3), and of cases with two primary cancers ascer-tained through cancer registries (4) have been developedspecifically for genetic studies of breast cancer. Theincreased efficiency of studies of these "genetically enrich-ed" cases results from the fact that, for example, the allelefrequency difference between cases with one affectedfirst-degree relative and population controls is about 1.5

times greater than in unselected cases, and there is anapproximately 2-fold higher difference for cases with 2affected first-degree relatives (1). In breast cancer, there isthe possibility of bilateral disease, among cases of whichthere is also a 2-fold higher difference. Recently, we (5)and others (6) have exploited the genetic enrichment ofbilateral and familial cases in genome-wide associationscans (GWAS) and thereby identified novel single nucleo-tide polymorphisms (SNP) associated with breast cancer.

For a given ascertainment scheme the excess risk, that isthe increase in effect size compared with a study ofsporadic cases and population controls, can be predictedfor variants having a direct causal effect on disease.However, the relationship for marker SNPs is less clearowing to incomplete linkagedisequilibrium (LD)betweenthemarker and the causal variant. Significant deviation ofthe excess risk from its expected value has been observedin familial cases of breast cancer (6), and here we showsimilar deviation in bilateral cases. The observed excessrisk is generally lower than that predicted for the causalvariant, implying that thegain in efficiency is less thanhadbeen anticipated.

Here, we give expressions for the relative risk formarker genotypes in bilateral and familial cases, in terms

Authors' Affiliations: 1Faculty of Epidemiology and Population Health,London School of Hygiene and Tropical Medicine; and 2BreakthroughBreast Cancer Research Centre, Institute of Cancer Research, London,United Kingdom

Corresponding Author: Frank Dudbridge, Faculty of Epidemiology andPopulation Health, London School of Hygiene and Tropical Medicine,Keppel Street, London WC1E 7HT, United Kingdom. Phone: 44-20-7927-2025; Fax: 44-20-7580-6897; E-mail: [email protected]

doi: 10.1158/1055-9965.EPI-11-0719

�2011 American Association for Cancer Research.

CancerEpidemiology,

Biomarkers& Prevention

Cancer Epidemiol Biomarkers Prev; 21(2) February 2012262

on April 26, 2021. © 2012 American Association for Cancer Research. cebp.aacrjournals.org Downloaded from

Published OnlineFirst October 25, 2011; DOI: 10.1158/1055-9965.EPI-11-0719

http://cebp.aacrjournals.org/

of the marker and causal genotype frequencies, LDbetween marker and causal variant, and the relative riskof the causal variant. We show that the attenuation of theexcess risk observed in recent studies can be explained byrealistic models of LD and causal relative risks.We then note that, given the marker relative risks in at

least 3 distinct sampling schemes, it is possible to infer thegenotype frequency and relative risk of the causal variant,even if the identity of that variant is unknown. This bearson recent debates over whether the numerous SNP asso-ciations emerging from GWAS are primarily driven byrare variants with larger effects than those estimated byGWAS, or by common variants with similar properties tothe tag SNPs used in GWAS. The former scenario wouldexplain more heritability than the latter (7), giving apartial account of the missing heritability problem (8).Recently, motivated by earlier observations that rare var-iation could explain common disease (9), Dickson andcolleagues argued that rare variations could stochasticallyoccur in coupling with a common SNP allele, creating"synthetic" association of the common variant (10). Butthis hypothesis has been challenged on theoretical andempirical grounds, with the evidence to date pointing to agreater role for common causal variation (11, 12). Thesearguments are based on simulations and whole-genomeaverages, and there has been little work on determiningwhether specific causal variants are rare or common, asthis would normally require complete resequencing ofassociated regions.Here, we address this issue for 10 specific loci that have

been consistently associated with breast cancer, by infer-ring the allele frequency and relative risk of the causalvariants from the marker relative risks estimated fromstudies of sporadic, bilateral, and familial cases. To allowfor sampling variation and to accommodate some priorbeliefs, we conduct a Bayesian estimation of causal effectsto show that the variants underlying these establishedassociations are probably common and have similar rel-ative risks to those observed for the marker SNPs. This isin agreement with recent simulation results (13) and, forthe first time, provides explicit support for the commondisease common variant (CDCV) hypothesis applied tobreast cancer. The approachwe develop can be applied tovarious familial sampling schemes, including those basedon concordant or discordant sib pairs, including twins,and is not limited to the studies of breast cancer describedhere.

Materials and Methods

SubjectsCases and controls were selected from the British

Breast Cancer Study (BBCS), details of which have beenpublished previously (4, 5). For this analysis, we includ-ed 1,695 cases, 1,564 of whom had 2 sequential orsimultaneous primary breast cancers, and 131 of whomhad at least 2 affected first-degree relatives. The excessrelative risk among cases with a second primary and

those with 2 affected first-degree relatives has beenshown to be equivalent (1) so the subsequent analysisassumes that all cases are bilateral. A total of 2001controls were ascertained as friends and nonblood rela-tives of the cases. All cases and controls were of self-reported white Caucasian ancestry and all controls werefree from breast cancer at enrolment in the study.Collection of blood samples and questionnaire informa-tion from case and control subjects was undertaken withinformed consent and in accordance with the tenets ofthe Declaration of Helsinki.

GenotypingDNAwas extracted from blood samples using conven-

tional methodologies and quantified using PicoGreen(Invitrogen). Genotype data from the BBCS was obtainedfor 10 SNPs that have been reproducibly associated withbreast cancer by GWAS (Table 1). Genotypes forrs13387042, rs4973768, and rs6504950 have been includedin previous publications (14, 15). Additional genotypingof rs1338740 and de novo genotyping of 3 SNPs(rs10941679, rs2046210, and rs11249433) was carried outusing Taqman nuclease assay, with reagents designed byAppliedBiosystems asAssays-by-Design andgenotypingdone using the ABI PRISM 7900HT Sequence DetectionSystem according to manufacturer’s instructions. For 4other SNPs (rs889312, rs2981582, rs3817198, andrs3803662) genotyping was done by KBioscience Ltd.,using their proprietary in house system (KASPar), acompetitive allele-specific PCR SNP genotyping systemthat uses FRET quencher cassette oligos.

Call rates for SNPs genotyped as part of this analysiswere 99.9% (rs1338740), 99.2% (rs10941679), 99.9%(rs2046210), 99.7% (rs11249433), 98.2% (rs889312), 97.8%(rs2981582), 98.9% (rs3817198), and 98.0% (rs3803662),and there was no evidence of deviation from HardyWeinberg equilibrium in controls, for any of the SNPs(all P > 0.05). Duplicate concordance based on a 3.5%random sample was 100% for all SNPs.

Published dataWe obtained summary OR estimates from published

studies using cases unselected for a family history (14–19). We also obtained summary estimates from a recentGWAS by Turnbull and colleagues that preferentiallyselected cases to have at least 2 affected first or second-degree relatives (6). The exact ascertainment criterionwas unspecified and we assumed that half the cases hadat least 2 affected first-degree relatives and the otherhalf at least 2 affected second-degree relatives. Thisapproximation was made following informal consulta-tion with those authors, but our conclusions turn outto be similar if we assume, say, that each case has at least1 affected first-degree and 1 affected second-degreerelative.

In what follows, we approximate the OR by the relativerisk, as this is appropriate for a rare disease and leads tosimplification in the analysis. Furthermore, several

Estimating Causal Effects of Genetic Variants Using Familial Cases

www.aacrjournals.org Cancer Epidemiol Biomarkers Prev; 21(2) February 2012 263




published studies used population-based controls, notselected to be disease free, for which the relative risk wasestimated directly.

Effect size in familial casesLet Y denote disease status, Y ¼ 1 for disease present

and Y ¼ 0 for disease absent. Let M be a diallelic markerwith genotypes {0, 1, 2} corresponding to the number ofminor alleles present. Let D be a diallelic variant withdirect causal effect, also with genotypes {0, 1, 2}.

The relative risk of causal genotype d, compared withbaseline 0, is

gDd¼ PrðY ¼ 1jD ¼ dÞ

PrðY ¼ 1jD ¼ 0Þ ð1:1Þ

and the relative risk of marker genotype m, compared with

baseline 0, is

gMm¼ PrðY ¼ 1jM ¼ mÞ

PrðY ¼ 1jM ¼ 0Þ

¼P

d2f0;1;2g PrðY ¼ 1jD ¼ dÞPrðD ¼ djM ¼ mÞPd2f0;1;2g PrðY ¼ 1jD ¼ dÞPrðD ¼ djM ¼ 0Þ

¼P

d2f0;1;2g gDdfDjMðdjmÞP

d2f0;1;2g gDdfDjMðdj0Þ ð1:2Þ

where fDjMðdjmÞ ¼ PrðD ¼ djM ¼ mÞ. The marker relative

risk thus depends only on the causal relative risks and the

conditional distribution of causal genotype given marker

genotype, which reflects the LD, or correlation, between the

2 genotypes.

Among bilateral cases of disease, the marker relativerisk is given by

gMm ;0 ¼P

d2f0;1;2g g2DdfDjMðdjmÞP

d2f0;1;2g g2DdfDjMðdj0Þ ð1:3Þ

where the subscript of g indicates that each case has an

affected 0th-degree relative. This expression also holds for

cases with an affected monozygotic twin. We assume that

controls are not selected for their disease status or family

history, and that cancers arise independently in each breast,

conditional on other individual-level risk factors.

For familial cases, themarker relative risk is obtainedbyconsidering the probability that affected relatives share acausal variant identical by descent (IBD). For first-degreerelatives the probability is 1/2 that a causal variant isshared IBD, and the probability that a subject with geno-type d is affected and has at least 1 affected first-degreerelative is given by

PrðY ¼ 1jD ¼ dÞX¥j¼1

PrðJ ¼ jÞ

� 1� 1� 12 PrðY ¼ 1jD ¼ dÞ þ PrðY ¼ 1Þ½ �� j� �

ð1:4Þ

where J denotes the number of first-degree relatives for the

subject. Assuming that disease risks are small, this is approx-

imated by

PrðY ¼ 1jD ¼ dÞ12 PrðY ¼ 1jD ¼ dÞ þ PrðY ¼ 1Þ½ �

�X¥j¼1

jPrðJ ¼ jÞ ð1:5Þ

so that the marker relative risk is

gMm;1 ¼P

d2f0;1;2g gDd

12 gDd

þ S� �

fDjMðdjmÞPd2f0;1;2g gDd

12 gDd

þ S� �

fDjMðdj0Þ ð1:6Þ

where S ¼ Pd2f0;1;2Þ gDd

fDðdÞ with fDðdÞ ¼ PrðD ¼ dÞ:In addition to the conditional distribution fDjMðdjmÞ, the

marker and causal relative risks are now also relatedthrough the causal genotype distribution fD(d). Similarly,for a case with at least 2 affected first-degree relatives, the

Table 1. Per-allele relative risks of associated SNPs considered in this study

SNP Band Gene MAF Unselected cases Familial cases Bilateral cases

rs11249433 1p11 0.39 1.16 (1.09–1.24)a 1.08 (1.02–1.15) 1.22 (1.09–1.35)rs13387042 2q35 0.49 1.12 (1.09–1.15)b 1.21 (1.14–1.29) 1.09 (0.97–1.21)rs4973768 3p22 SLC4A7 0.46 1.11 (1.08–1.13)c 1.16 (1.10–1.24) 1.11 (0.99–1.23)rs10941679 5p12 0.25 1.19 (1.11–1.28)d 1.11 (1.04–1.19)g 1.29 (1.15–1.43)rs889312 5q11 MAP3K1 0.38 1.12 (1.08–1.16)e 1.22 (1.14–1.30) 1.11 (1.01–1.21)rs2046210 6q25 0.34 1.15 (1.03–1.28)f 1.15 (1.08–1.22)g 1.12 (0.99–1.25)rs2981582 10q25 FGFR2 0.38 1.26 (1.22–1.29)e 1.43 (1.35–1.53) 1.39 (1.29–1.49)rs3817198 11p15 LSP1 0.3 1.07 (1.04–1.11)e 1.12 (1.05–1.19) 1.10 (1.00–1.20)rs3803662 16q12 TOX3 0.25 1.19 (1.15–1.23)e 1.30 (1.22–1.39) 1.41 (1.30–1.52)rs6504950 17q22 COX11 0.27 0.95 (0.92–0.97)c 0.92 (0.86–0.99)g 0.93 (0.79–1.07)

Estimates for unselected cases are taken from aref. (19), bref. (14), cref. (15), dref. (17), eref. (16), and fref. (18). Estimates for familial casesare taken from ref. (6). Estimates for bilateral cases are taken from this study (see Methods).gRelative risk for a proximal SNP in LD (r2 > 0.75) with the lead SNP.Abbeviation: MAF, minor allele frequency.

Dudbridge et al.

Cancer Epidemiol Biomarkers Prev; 21(2) February 2012 Cancer Epidemiology, Biomarkers & Prevention264




marker relative risk is approximately

gMm;1;1 ¼P

d2f0;1;2g gDd

14 gDd

þ S� �2


14 gDd

þ S� �2

fDjMðdj0Þð1:7Þ

Similar arguments give

gMm;2 ¼P

d2f0;1;2g gDd

14 gDd

þ 3S� �


14 gDd

þ 3S� �

fDjMðdj0Þ ð1:8Þ

for cases with at least 1 affected second-degree relative,

and

gMm;2;2 ¼P

d2f0;1;2g gDd

116 gDd

þ 3S� �2


116 gDd

þ 3S� �2

fDjMðdj0Þð1:9Þ

for cases with at least 2 affected second-degree relatives.

We use the geometric mean of equations (1.7) and (1.9) to

model the summary estimates of Turnbull and colleagues

(6), denoted by gM;fam.

For each case ascertainment scheme we define theexcess risk as the ratio of the log relative risk in selected,familial cases to the log relative risk in unselected, spo-radic cases. Thus for bilateral cases the excess risk for

genotype m islogðgMm ;0ÞlogðgMm Þ , which is 2 when disease and

marker genotypes are perfectly correlated.

Identification of causal effectsFrom equations (1.2) to (1.9), the marker and causal

relative risks are related through the conditional distri-bution of causal genotype given marker genotypefDjMðdjmÞ and the unconditional distribution of causal

genotypes fD(d). Given the marker relative risks from anumber of studydesigns, it is possible in principle to solvefor the causal effects. For diallelic marker and causalvariant, fDjMðdjmÞ is specified by 6 free parameters, fD(d)

by 2, and there are 2 causal relative risks gDd. All para-

meters could therefore be identified given the markerrelative risks from 10 different familial samplingschemes. This is more than is practical to obtain, butwith some mild assumptions we can reduce the para-meters to a practical number.First, we assume a multiplicative model of disease risk

at the causal variant, in which gD2¼ g2D1

. This assump-

tions fitsmost observedmarker associationswell, and canbe expected to extend to causal variants as well (20). Wefurther assume Hardy–Weinberg Equilibrium for bothmarker and causal variant, so that the genotype frequencydistributions can be parameterized by allele frequencies.Under these assumptions all the relative risks factorizeinto allelic terms and we may simply work withD andMtaking values in {0,1}. This leaves 4 free parameters to be

identified: fDjMð1j1Þ, fDjMð1j0Þ, fD(1), and gD1. Finally, not-

ing that

fDjMð1j0Þ ¼ 1� fDjMð1j1ÞfMð1ÞfDð1Þ

� �� fDð1ÞfMð0Þ ð1:10Þ

we assume that the marker allele frequencies fM(m) are

known, so that we have just 3 free parameters to identify.

Therefore, the marker relative risks from just 3 familial

ascertainment schemes are needed to solve for the effects of

the causal variant. Here, we use studies of unselected cases

[equation (1.2), from published data], cases with bilateral

disease [equation (1.3), from our data] and an equal mixture

of cases with at least 2 affected first- or second-degree rela-

tives [equations (1.7) and (1.9), from Turnbull and

colleagues].

Bayesian inference of causal effectsIn practice, we cannot solve for fDjMð1j1Þ, fD(1), and gD1

exactly because estimates of marker relative risks aresubject to sampling variation and do not conform toequations (1.2) to (1.9). Instead, we estimate the causalparameters using a likelihood calculated from the esti-mated marker effects, assuming that they are obtainedfrom independent samples. Letting b denote log relativerisk, b: ¼ logðg �Þ, the likelihood is

LðbD; fD; fDjM; b̂M; b̂M;0; b̂M;famÞ¼ �ðb̂M;bMðbD; fD; fDjMÞ; s2

MÞ� �ðb̂M;0;bM;0ðbD; fD; fDjMÞ; s2

M;0Þ� �ðb̂M;fam;bM;famðbD; fD; fDjMÞ; s2

M;famÞ ð1:11Þ

where �ð�;m; s2Þ is the normal density with mean m and

variance s2, the mean terms are obtained from equations

(1.2), (1.3), (1.7), and (1.9), and the variances are assumed

equal to their sample estimates.

Maximum likelihood estimation of the causal effectsis unsatisfactory for two reasons. First, in the datawe consider, the likelihood function can be multimodaland difficult to maximize numerically. Second, in anumber of instances the likelihood is maximized atparameter values that are unrealistic, such as a veryhigh causal relative risk but minimal correlationbetween causal and marker variant. This scenario isunreasonable because the SNPs studied were selectedas the most strongly associated within their regions,and so are likely to have the highest correlation with thecausal variant.

For these reasons,we conduct a Bayesian analysis usingthe likelihood in equation (1.11) and the following priordistributions. For the causal minor allele frequency(MAF), we follow the distribution of sequence variantsobserved in the ENCODE regions by the HapMap con-sortium (21). This strongly favors variants with frequency< 5%, with roughly equal probability given to frequencies> 20%. This distribution is closely approximated by a Beta






(0.345, 1.058) distribution scaled by 0.5, which we use asour prior for fD(1) (Table 2).

The causal relative risk is taken to be log-normal withmost of the distribution less than 3, the value abovewhichlinkage analysis is expected to be more powerful thanGWAS (11). We use a log-normal distribution with loca-tion 0 and scale dependent on the causal MAF accordingto the distribution proposed by Spencer and colleagues(ref. 13; Table 2). Together with our prior for the allele

frequency this represents 80% belief that 13 � gD1

� 3 and

90% belief that 15 � gD1

� 5, a strong but not overwhelm-

ing prior belief in moderate relative risks.The prior for fDjM ismore difficult to specify because the

range of values it can take depends upon the marker andcausal allele frequencies. The prior information in thiscase consists only of imprecise beliefs that the causal andmarker genotypes are in strong LD, but this notion is noteasy to parameterize and quantify. GWAS are designedon the premise that all common variants are correlatedwith a tag SNP at say r2 � 0.8, but we would like to allowthe possibility that causal variants are rare, as reflected inour prior for fD(1), which implies lower correlation withassociated tag SNPs.

Here, we quantify the LD between marker and causalvariants by the logOR (logOR) for association between thetwo minor alleles

� ¼ logfDjMð1j1Þ=fDjMð0j1ÞfDjMð1j0Þ=fDjMð0j0Þ ð1:12Þ

This is unbounded for all values of marker and causalallele frequencies, and the values of fDjM can be expressed

as functions of the allele frequencies and q. As the prior forq, we use a normal distribution with mean 6 and variance1.48, which corresponds to r2 � 0.8 when marker andcausal alleles have equal frequency of about 0.25, with

90%probability that 0.5� r2� 0.9 in that case. This reflectsgenerally held beliefs about tag SNPs for common causalvariants, while allowing strong statistical associationsbetween rare causal variants and common tag SNPs. Thisprior strongly favors a positive correlation between theminor alleles of themarker and the causal variant: where-as a negative correlation is possible, it would imply amuch lower r2 between the two alleles, which conflictswith the assumption that the most informative markerSNP has been chosen from the region.

It turns out that when this prior is used for the logOR,the posterior is virtually unchanged from the prior. Wenoticed a similar behavior for other priors that stronglyfavored a positive correlation between the two alleles,suggesting that the likelihood is almost independent ofq for high values of q. Therefore, the prior for the logORhas little effect on the posterior distributions for thecausal relative risk and allele frequency, which are ourmain parameters of interest. We return to this point in thediscussion.

We obtained posterior distributions for gD1, fD(1), and q

usingwinBUGSwith 100,000 samples, discarding the first10,000 and keeping all other settings at default values. Inaddition to posterior median, mode and 95% credibleintervals, we calculated the posterior probability that thecausal MAF is less than 5% or 1%.

Results

Marker relative risks in bilateral casesIn Table 1, we show the relative risks for 10 SNPs

estimated from our sample of bilateral cases along withrecent literature-based estimates for sporadic and familialcases. It is clear that there is an excess relative risk inbilaterals, confirming that bilateral sampling does offer again in efficiency. However, this excess is systematicallyless than 2, the value predicted for a causal variant, so thatthe gain in efficiency is less than had been anticipated.

These observations can be formally tested. Assumingnormality of the log relative risk estimates,

ðcb̂M � b̂M;0Þ2c2 s2

M þ s2M;0

x2ð1Þ ð2:1Þ

for each SNP, if the hypothesis holds that cbM ¼ bM;0. Sum-

ming equation (2.1) over all 10 SNPs gives a c2 variable on

10dfwhich can be regarded as a deviance for the parameter c.

The maximum likelihood estimate of c from Table 1 is 1.41

which is significantly different from both 1 (P ¼ 0.027) and 2

(P¼ 0.003). It is of note that no individual SNP had an excess

risk significantly less than 2, but the attenuation is significant

when considering all SNPs jointly.

Figure 1 illustrates the excess relative risk in bilateralcases for amarkerwith allele frequency 0.25, for a range ofcausal relative risks and allele frequency. The logORbetween marker and disease minor alleles is 6, although

Table 2. Prior distribution of MAF and SD of logrelative risk

MAF (%) ENCODE frequency (%) SD of log RR

0–5 46 1.215–10 13 0.8410–15 10 0.6115–20 6 0.4620–25 5 0.3625–30 5 0.330–35 4 0.2535–40 4 0.2340–45 4 0.2145–50 3 0.2

NOTE: "ENCODE frequency" gives the proportion of variantsin the ENCODE regions havingMAF in the stated range. "SDof log RR" gives the SD of the log relative risk for SNPsaccording to the model of Spencer and colleagues (13).Abbreviations: RR, relative risk; SD, standard deviation.

Dudbridge et al.





this value has very little effect on the figure. It is noticeablethat the excess risk exceeds 2 when the causal allele is lesscommon than the marker, but is less than 2 if the causalallele is more common. The attenuation is greater whenthe causal relative risk is higher. This pattern roughlyholds for all values of marker and causal parameters, andalso when the causal variant has dominant or recessiveaction (results not shown). Because the excess risk issystematically less than 2 for the markers shownin Table 1, this suggests that the causal variants mighthave risk allele frequency at least equal to that of themarker SNPs. However, this must be weighed against thegreater prior probability that the causal variant is rare,given the distribution of allele frequency in the genome.The following section addresses this question.

Inference of causal effectsWe applied Bayesian estimation of causal effects to the

estimated marker effects shown in Table 1. The markerMAFs are the most accurate currently available and theywere assumed to be known exactly. Tables 3 and 4 showsummaries of the posterior distributions of the causalallele frequency and relative risk of the 10 associated loci.Kernel density plots of these parameters are shown inFigs. 2 and 3.The posterior estimates of the causal allele frequency

have a wide range, roughly 0.1 to 0.4, whereas the esti-mates of the causal relative risks are all roughly between1.2 to 1.3. The evidence points strongly to the causalvariants being common. For only three loci, 11p15 (LSP1,rs3817198), 6q25 (rs2046210), and 17q22 (COX11,rs6504950), is there a reasonableprobability that the causalallele has frequency less than 5%, and for each of those theprobability that it is less than 1% is considerably lower.

Neither rs2046210 nor rs6504950 were directly typed inthe familial cases (6); instead a SNP in strongLDwasused,and this slight conflict of information may have kept theposteriors closer to theprior thanotherwise. Thegenerallyhigh probability of common causal variants is in spite ofthewide credible intervals on the allele frequencies,whicharise from the conflict between the prior that stronglyfavors rare variants, and the data which are more consis-tent with common variants. This result is consistent withrecent work suggesting that causal variants have similarproperties to tag SNPs (11–13, 22), and is in line with theCDCV hypothesis.

Sensitivity analysisAlthough our prior distributions reflect generally held

beliefs about causal variants, there are two importantquestions that can be asked of our approach. First, notingthat our posterior distributions consistently indicatedcommon causal variants, would our procedure identifya rare causal variant if one were present? Second, as wecould not solve for the causal effects exactly, how muchinformation was gained by combining the estimates fromthree study designs?

To address the first question, we considered a causalallele with frequency 0.1% and relative risk 3.35, andlogOR of 9 with a marker allele of frequency 5%. Thisrepresents a rare variant with effect size at 1 SD of itscorresponding distribution, and allele frequency fairlyclose to that of a common marker SNP. From equations(1.2) to (1.9) the predicted marker relative risk is 1.047 inunselected cases, 1.204 in bilateral cases and 1.222 in caseswith a mixture of family histories. The r2 between markerand causal alleles is only 0.019, but the excess risk isgreater than 2 in bilateral and familial cases, in linewith Fig. 1.

Figure 1. Excess relative risk inbilateral cases for amarkerwith allelefrequency 0.25. Excess risk is theratio of the log relative risk in bilateralcases to that in unselected cases.Each line corresponds to a frequencyof the minor allele at the causalvariant, which is associated to themarker minor allele with log OR of 6.

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

1.771.611.461.331.211.10

Exc

ess

risk

Causal RR

0.1

0.2

0.3

0.4

0.5

0.6






It is apparent that this configuration differs from themarker effects we observed for breast cancer, yet it repre-sents a rare causal variant that is consistent with our priorbeliefs and is strongly associated with the marker. Todetermine whether we could infer such a causal variantwith our approach, we used SE of 0.05, 0.06, and 0.03 for

b̂M1, b̂M1;0

, and b̂M1;fam, respectively, similar to the higher

values appearing in Table 1. We then sampled b̂M1, b̂M1;0

,

and b̂M1;famfrom the corresponding normal distributions

and inferred posterior distributions for the causal para-meters. This procedure was repeated 1,000 times.

On average, the posterior median for the causal allelefrequency was 0.10 and the mode was 0.0002. The meanprobability was 47% that the causal allele frequency wasless than 5%, and 36% that it was less than 1%, thecorresponding prior probabilities being 46% and 27%.The average posterior median of the causal relative riskwas 1.41 and the mode 1.19. Thus the presence of a rarevariant could be inferred reliably, although point estima-

tion of its allele frequency and relative risk seems to beheavily biased and the entire posterior distribution oughtto be considered when drawing inferences.

To address the second question of whether the familialsamples add information to the inference,we repeated theestimation of causal effects using only the marker relativerisks for unselected cases. We adjusted their SE to reflectthe total informationprovided from the 3 studies, using aninverse-variance formula

s ¼ �s�2M þ s�2

M;0 þ s�2M;fam

�12 ð2:2Þ

In this way, we can assess the information contributedspecifically by the study designs as distinct from theiradditional sample size.

The mean width of the 95% credible interval for thecausal allele frequency was 44%, compared with 43%when using 3 study designs. For the causal relative risk,the mean width was 2.04 compared with 0.78. The pos-terior median for the causal allele frequency was on

Table 3. Posterior distribution summaries for causal minor allele frequencies, estimated from 100,000MCMC samples

Band Gene Marker SNP Marker MAF Median Mode 95% CI Pr(<5%) Pr(<1%)

(Prior) 0.063 0.00 10�5–0.46 0.46 0.27

1p11 rs11249433 0.39 0.23 0.06 0.021–0.48 0.093 0.00602q35 rs13387042 0.49 0.28 0.25 0.069–0.49 0.0071 <10�5

3p22 SLC4A7 rs4973768 0.46 0.30 0.31 0.097–0.49 0.0017 <10�5

5p12 rs10941679 0.25 0.26 0.07 0.023–0.49 0.067 0.00345q11 MAP3K1 rs889312 0.38 0.25 0.21 0.045–0.49 0.032 <10�5

6q25 rs2046210 0.34 0.16 0.02 0.0055–0.48 0.22 0.05110q25 FGFR2 rs2981582 0.38 0.40 0.48 0.21–0.49 <10�5 <10�5

11p15 LSP1 rs3817198 0.3 0.16 0.03 0.0074–0.48 0.23 0.03816q12 TOX3 rs3803662 0.25 0.29 0.38 0.084–0.49 0.0018 <10�5

17q22 COX11 rs6504950 0.27 0.051 0.02 0.013–0.46 0.49 0.0033

NOTE: Pr[<5% (1%)], probability that causal allele frequency is less than 5% (1%).Abbreviation: CI, credible interval.

Table 4. Posterior distribution summaries for causal relative risks, estimated from100,000MCMCsamples

Band Gene Marker SNP Marker RR Median Mode 95% CI

(Prior) 1.00 1.00 0.14–7.241p11 rs11249433 1.16 1.13 1.09 1.06–1.882q35 rs13387042 1.12 1.19 1.14 1.11–1.673p22 SLC4A7 rs4973768 1.11 1.16 1.12 1.10–1.465p12 rs10941679 1.19 1.14 1.12 1.08–1.755q11 MAP3K1 rs889312 1.12 1.16 1.13 1.10–1.716q25 rs2046210 1.15 1.17 1.11 1.07–3.1210q25 FGFR2 rs2981582 1.26 1.30 1.29 1.25–1.4311p15 LSP1 rs3817198 1.07 1.13 1.08 1.06–2.4816q12 TOX3 rs3803662 1.19 1.24 1.22 1.18–1.5017q22 COX11 rs6504950 0.95 0.73 0.93 0.095–0.95

Dudbridge et al.





average 0.6 times thatwhen using 3 study designs,where-as the posterior median for the relative risk was 1.2 timeshigher. Although these results seem to give more supportfor a rare causal variant than the 3 studymodel, this is dueto the reduced information in the data (reflected in thewider credible intervals) tomove the estimates away fromtheir prior distributions. Moreover, the degree of supportis still weak. We see that the use of 3 study designs allowsstronger conclusions to be reached than a single study ofequivalent sample size, as a result of the increasednumberof parameters identifiable by including familial cases.

Discussion

One should be cautious about taking our estimates ofcausal effects too literally. They are dependent on priordistributions, and have wide credible intervals. Althoughwe have shown that our procedure could infer a rarevariant if one were present, point estimates of its allelefrequency and relative risk are heavily biased. Severalgroups are currently engaged in fine mapping and rese-quencing efforts in the regions studied, which will lead to

more direct estimates of causal effect sizes. Thus thequantitative estimates presented here will eventually beredundant, although it will be interesting to compare ourestimates with the actual causal effects when known.

Instead, we emphasize the qualitative nature of ourresults, which indicate that most, if not all, associationswith breast cancer so far identified by GWAS are likely tobe markers for common causal variants with modesteffects. This is consistent with the CDCV hypothesis thatoriginally motivated GWAS, but not with recent sugges-tions that many GWAS hits could be markers for rarecausal variants (10). In this respect, our results agree withother recent work in support of the CDCV hypothesis.Anderson and colleagues (11) argued that GWAS has lowpower to find a rare variant that had not already beendetected by linkage, and noted examples of resequencingprojects that had not identified rare variants underlying acommon GWAS hit. This includes currently unpublishedwork by theWellcome Trust Case–Control Consortium inwhich sequencing of 16 regions identified by GWAS didnot identify any underlying rare causal variant.Wray andcolleagues (12) show that the distribution of risk allele

Figure 2. Kernel density plots of prior and posterior distributions of the causal allele frequency at 10 breast cancer loci.






frequencies from currently known GWAS hits is consis-tent with the majority of these hits arising from commonvariants. Iles (22) showed that early findings of GWAShave been at loci forwhich the power is highest, which areindeed the common variants. Because we confined atten-tion to SNPs identified in the first wave of breast cancerGWAS, and have subsequently been replicated, weshould expect these loci to be enriched for commonvariants, and in this respect our results are unsurprising.But in contrast to these other studies, we are able toestimate causal effects for specific loci rather than averageproperties of all causal variants. Our results indicate thatthese loci are consistent with the general pattern of com-mon causal variation suggested by other work, and ourmethods can be applied to further markers that emergefrom GWAS.

Our approach cannot distinguish between the effect of asingle common variant and the average effect of a numberof variantswith a common total frequency.Although sucha scenario is theoretically possible for a complex disease(9),Wray andcolleagueshave arguedagainst this scenario

for the loci found to date (12). We cannot lend support toeither position here other than to note the fact that all 10SNPs indicated a common causal variant, suggesting thatif rare variants do underlie these associations then they doso either in large numbers or not at all.

Several authors (13, 20, 22) have used simulations toestimate the empirical conditional distribution of causalallele frequencies and relative risks, given that a markerwas identified by GWAS and subsequently replicated.Our approach to modeling the LD between markers andcausal variants is much simpler, but we found this modelhad little effect on the parameters of interest. We do notexplicitly model the process of marker discovery byGWAS, and in that respect our prior is more favorable torare variants, thus strengthening our conclusion that thecausal variants are common.

The use of familial cases in association studies is moti-vated by the excess relative risk in the ascertained samplecompared with a sample of unselected cases. We haveshown, however, that imperfect correlation betweenmar-kers and causal variants leads to an excess risk in familial

Figure 3. Kernel density plots of prior and posterior distributions of the causal relative risk at 10 breast cancer loci.

Dudbridge et al.





cases that differs from the predicted value. The differencecould be in either direction, and indeed when the causaland marker variants have similar frequency, the excessrisk is higher at the marker than at the causal variant, sothat the study design is evenmore efficient than predicted(Fig. 1). In our data, however, there was a systematicattenuation of the excess risk in bilateral cases, similar toobservations for familial cases in the study of Turnbulland colleagues (6), which is most consistent with causalvariants of higher frequency than the markers. The effi-ciency of bilateral sampling,while still greater than that ofunselected sampling, seems to be less than predicted, andthismay have implications for the design of future studiesof common genetic risk factors.Some other mechanisms can also lead to attenuation of

the excess risk. We assumed a multiplicative model inwhich each copy of the risk allele multiplies the diseaserisk to the same degree, but the true model could berecessive, dominant, or more general. We can rewriteequations (1.2) to (1.9) in terms of recessive or dominanteffects: it turns out that under a recessivemodel the excessrisk attenuates at higher causal frequencies thanunder themultiplicative model, whereas for a dominant model itattenuates at lower frequencies (results not shown). Dom-inant causal variants could therefore be more consistentwith rare variation than the multiplicative model consid-ered, but the relevantprobabilities remained lowwhenweassumed this model in our analyses, and for brevity wehave omitted these results.We have also assumed that effects act on the log-risk

scale, which is convenient as additional polygenic andenvironment effects cancel out of relative risk calculationsso we need not assume a model for them. If however theeffects act on say the logistic or probit scales, then theexcess relative riskwould be attenuated even at the causalvariant. We considered this possibility by allowing for anormally distributed polygenic random effect with meanzero and variance 2log2, consistent with a sibling rel-

ative recurrence of 2 (23). Acting on the logistic scale thiscould reduce the excess relative risk for the causal variantfrom 2 to 1.8 in bilateral cases, but this is less than thedegree of attenuation we observed in our data. Subgroupeffects, such as age or tumor subtype-specific risks, couldalso attenuate the marginal excess risk, but we did notobserve any such effects in our data.

We have shown that genetic markers of breast cancerhave lower excess risk in familial cases than had beenpredicted, leading to reduced improvements of efficien-cy in these study designs. However, this informationcan be usefully exploited to estimate the relative riskand allele frequency of the underlying causal variants.Despite using a prior distribution that favors rare var-iation, we showed that data from bilateral and familialcases strongly imply that the causal variants underlyingrecent GWAS findings are common with modest effects,in line with other recent work favoring the CDCVhypothesis. We look forward to the outcome of currentfine mapping projects to confirm the accuracy of thesepredictions.

Disclosure of Potential Conflicts of Interest

No potential conflicts of interest were disclosed.

Grant Support

This work was supported by the Medical Research Council (G1000718to F. Dudbridge), Cancer Research UK (C150/A5660 and C1178/A3947 toJ. Peto and Id.S. Silva), and Breakthrough Breast Cancer (O. Fletcher, N.Johnson, and N. Orr). We acknowledge NHS funding to the NIHR RoyalMarsden Biomedical Research Centre and the National Cancer ResearchNetwork (NCRN).

The costs of publication of this article were defrayed in part by thepayment of page charges. This article must therefore be hereby markedadvertisement in accordance with 18 U.S.C. Section 1734 solely to indicatethis fact.

Received July 26, 2011; revised September 14, 2011; acceptedOctober 17,2011; published OnlineFirst October 25, 2011.

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2012;21:262-272. Published OnlineFirst October 25, 2011.Cancer Epidemiol Biomarkers Prev Frank Dudbridge, Olivia Fletcher, Kate Walker, et al. Cancer Using Marker Data from Bilateral and Familial CasesEstimating Causal Effects of Genetic Risk Variants for Breast

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