detecting association using epistatic information

Genetic Epidemiology 31 : 894909 (2007)

Detecting Association Using Epistatic Information

Juliet Chapman1 and David Clayton2

1London School of Hygiene and Tropical Medicine, London, United Kingdom2Diabetes and Inflammation Laboratory, Cambridge Institute for Medical Research, University of Cambridge, Cambridge, United Kingdom

Genetic association studies have been less successful than expected in detecting causal genetic variants, with frequent non-replication when such variants are claimed. Numerous possible reasons have been postulated, including inadequate samplesize and possible unobserved stratification. Another possibility, and the focus of this paper, is that of epistasis, or gene-geneinteraction. Although unlikely that we may glean information about disease mechanism, based purely upon the data, it maybe possible to increase our power to detect an effect by allowing for epistasis within our test statistic. This paper derives anappropriate omnibus test for detecting causal loci whist allowing for numerous possible interactions and compares thepower of such a test with that of the usual main effects test. This approach differs from that commonly used, for example byMarchini et al. [2005], in that it tests simultaneously for main effects and interactions, rather than interactions alone. Thealternative hypothesis being tested by the omnibus test is whether a particular locus of interest has an effect on diseasestatus, either marginally or epistatically and is therefore directly comparable to the main effects test at that locus. The paperbegins by considering the direct case, in which the putative causal variants are observed and then extends these ideas to theindirect case in which the causal variants are unobserved and we have a set of tag single nucleotide polymorphisms (tagSNPs) representing the regions of interest. In passing, the derivation of the indirect omnibus test statistic leads to a novelindirect case-only test for interaction. Genet. Epidemiol. 31:894909, 2007. r 2007 Wiley-Liss, Inc.

Key words: tag SNPs; epistasis; association studies; multiple tests; power

The Supplemental materials described in this article can be found at http://www.interscience.wiley.com/jpages/0741-0395/suppmatContract grant sponsor: Wellcome Trust and Juvenile Diabetes Research Foundation.Correspondence to: Juliet Chapman, London School of Hygiene and Tropical Medicine, Keppel Street, London, UK.Received 9 February 2007; Revised 27 April 2007; Accepted 21 May 2007E-mail: [email protected] online 24 July 2007 in Wiley InterScience (www.interscience.wiley.com).DOI: 10.1002/gepi.20250

INTRODUCTION

In the search for disease susceptibility (DS) lociassociation studies have focused upon the detectionof the marginal effects of a single locus [Agresti,1990] or single set of tag single nucleotide poly-morphisms (SNPs) [Xiong et al., 2002; Chapmanet al., 2003; Fan and Knapp, 2003; Clayton et al.,2004]. Such analyses have lead to an arguablydisappointing number of positive findings in com-plex diseases, for which many different reasons havebeen postulated [Pritchard and Rosenberg, 1999;Fanous and Kendle, 2005]. This lack of findings isperhaps unsurprising when considering the natureof such diseases, since we expect these diseases to bethe result of a complicated network of numeroussusceptibility loci, each of which is likely to haveonly a small effect when considered in isolation.Related to these ideas is the possibility that suscept-ibility loci act together to cause disease and thereforethe effect of these loci is obscured when searchingfor simple main effects. Such interaction is generallyreferred to as epistasis although this term is

fraught with confusion [Cordell et al., 2001; Cordell,2002]. Further complications occur due to the factthat statistical interactions do not necessarily equateto underlying biological interactions and thereforemany have questioned whether there is any use insearching statistically for epistatic effects [Cordell,2002; Coredell et al., 2001]. In the light of the recentlack of findings, however, interest in searching forepistatic effects has resurfaced, where the focus ofsuch analyses is no longer upon finding biologicalinteractions but in the hope that we might be able touse the extra information to detect susceptibility locimore easily. Within this paper we will consider thedetection of a particular locus of interest whileallowing for possible pairwise interactions withnumerous other loci, which we will refer to assecondary or modifying loci.In the next section we introduce notation and a

simple pairwise model for the direct case, in whichthe genotype at the putative causal locus isobserved. Based on this model, we derive a pseudo-score test statistic that tests simultaneously for a maineffect at the locus of interest and an interaction

r 2007 Wiley-Liss, Inc.

between this locus and a secondary locus. Asmentioned above, there are likely to be a largenumber of secondary loci that we think may interactwith our locus of interest and therefore we need toadjust for the multiple tests carried out. We proposethe use of the maximum of these pseudo-scoretests, across all secondary loci, as an overall test ofeffect at the primary locus. This test we shall refer toas the omnibus test. Such an approach differs fromthat usually used, for example by Marchini et al.[2005] and Evans et al. [2006], since these approachestest for (all pairwise) interactions only. The alter-native hypotheses of these interaction only ap-proaches are testing for epistatic effects of two ormore loci and as such differ from the alternative ofthe main effects tests, each of which tests for an effectat just a single locus. This means that is difficult tomake a fair comparison between the power of theseusual epistatic approaches to that of the single locusapproach. For example, considering just a single pairof loci, do we compare the power of the singlepairwise interaction only test with the power of themain effects approach to detect (a) both of the lociinvolved in the interaction or (b) either of these loci?Our proposed test, side steps this problem since thealternative being tested is simply any effect at thelocus of interest, either marginally or epistatically,and as such is directly comparable to the main effectstest. If we obtain a significant finding, this can onlytell us about the likelihood of an effect at the locusof interest and tells us nothing definitive about anyeffect at the secondary locus.We then generalise this model to the indirect case,

in which the putative causal loci are not observedand we instead base our analysis upon markers intight linkage disequilibrium (LD) with these causalloci. Using this latent model we are able to derive anindirect version of the maximum pseudo-score test.As a by-product of this derivation we also necessa-rily define a novel indirect case-only test forinteraction, which searches (just) for an interactionbetween two sets of multivariate data. Finally wereport the results of simulation studies comparingthe power of the omnibus test with that of the maineffects tests, in both the direct and indirect case.

METHODS

DIRECT MODEL

We begin by considering the situation in which wehave a primary locus of interest and a singlesecondary locus that modifies the behaviour of thisprimary locus. We will start by assuming that thelocation of both of these loci are known and that wemay observe the genotypes at each locus. We denotethe genotype of the primary locus as Z1 and that ofthe modifying locus as W1. If we represent the

diseased and normal alleles by 1 and 0, respectively,then the genotype at a single locus will be denotedby the sum of the pair of alleles within an individual,taking the values 0, 1 or 2. Defining the diseasestatus of an individual by Y, we choose to model theway in which the primary and secondary loci affectdisease status using the following logistic regressionmodel [McCullagh and Nelder, 1989]

logitY a b1Z b2W gZW:Note that this model is a simplification of the fullmodel that would include dominance effects at bothloci. However, the more complicated model is likelyto include many more parameters than is necessary(or desirable), having two degrees of freedom (df)for each main effect and four df for the epistaticeffect. The above model, on the other hand allowsfor a large amount of flexibility while retainingrelative simplicity and fewer df. The model above isbased upon the prospective likelihood (i.e. thelikelihood of Y given fixed Z1 and W1). Thecorresponding joint model can be viewed as model-ling the probabilities of cells within a three-waycontingency table (Y, Z1, W1). If we let py;z;wrepresent the probability of an individual havingdisease status y, genotype z1 at the primary locusand genotype w1 at the secondary locus, then thesecell probabilities can be modelled in terms of thefollowing log-linear model

logpy;z;w m e1y e2z e3w e12yz e13yw e23zw e123yzw:

Agresti [1990] demonstrates the correspondencebetween this log-linear model and the originallogistic model such that the log-linear parametere12 equates to the logistic main effects parameter ofthe primary locus, b1, (i.e. its main effect on thedisease risk) and similarly that e13 equates to thelogistic main effects parameter of the secondarylocus, b2. The third parameter, e123, then equates tothe logistic "interaction" parameter between theprimary and secondary locus, g. That is, it equatesto the interactive effect of the primary and secondarylocus upon the risk of disease [McCullagh andNelder, 1989]. The importance of this correspondencewill become clear in the following section, where wewish to make assumptions about the distribution ofthe loci, which is only possible in models for whichthe loci are treated as random variables.

DIRECT PSEUDO-SCORE TEST

Since our interest lies in testing whether theprimary locus is causal, we would like to test forboth a main effect and an interaction and wetherefore wish to define a test statistic that testssimultaneously whether b15 0 and g5 0, irrelevant

895Epistasis and tag SNPs

Genet. Epidemiol. DOI 10.1002/gepi

of whether there exists a main effect at themodifying locus. For example, we may alreadyknow that the secondary locus has a strongeffect and we are not interested in simply detectingan effect at this locus again. Normally we mightderive the appropriate score for testing a particularset of parameters by differentiating the likelihoodwith respect to these parameters of interest [Cox andHinkley, 1974]. However, interaction terms arereputedly difficult to detect and we therefore wishto make use of the assumption that the primaryand modifying loci are independent of one anotherin the population, since this will allow us to increasethe power of our test statistic (so long as theassumption holds) [Weinberg and Umbach, 2000].Since we wish to make assumptions about theindependence of loci we need to consider theproblem from the joint point of view, in which theloci are regarded as random variables. In this casethe relevant likelihood is complex and no closedform for the score is known. Therefore we suggestthe use of a pseudo-score test, in which we coupletogether a score for testing the main effect (b1) and ascore that tests for an interaction (g). Both are theneasily obtained. We denote this pseudo-score vec-tor as uT5 (u1, u12), where u1 is the score for the maineffects test and u12 is the score for the chosen test forinteraction.The first component, u1, we define to be the

score for the usual main effects score test statistic,which is defined by

u1 XNi1

Yi YZi ;

where the subscript i denotes the ith individual andN denotes the number of individuals within thestudy. Even when the modifying locus is causal, thiscomponent still has zero expectation under the null(of no effect at the primary locus) due to theassumption that the primary locus is independentof the secondary locus and therefore gives a validmain effects test.Since we are assuming that the primary locus is

independent of the secondary locus, the case-onlytest for interaction is appropriate for testing theinteraction term, g [Weinberg and Umbach, 2000].This is desirable since it is both of simple form andhas increased power. This test is derived based uponthe estimate of the e123 parameter from the jointlikelihood, which, as mentioned above, is equivalentto the g parameter from the prospective likelihood.Weinberg and Umbach [2000] show that when thetwo loci are independent the e123 parameter isequivalent to the odds ratio between the two lociin case individuals only; hence when the loci areindependent we may test for interaction simply bytesting for association between the two loci in cases

only. Therefore we could model this interaction interms of a regression for the effect of the secondarylocus upon the primary locus in case individuals only(i.e. when Yi5 1). The appropriate score for testingthis association in cases only (and therefore interac-tion) is thus defined by

u12 XNi1

YiZi ZY1Wi ;

where ZY1 is the mean Zi across case individuals

only. When the primary and modifying loci areindependent in the population, this score componenthas zero expectation under the null (of no effect atthe primary locus) and is therefore a valid teststatistic for interaction. The increased power occurssince fewer cells of the contingency table are beingused to estimate the interaction parameter/scoreand therefore this estimate/score has smaller var-iance and greater power, compared to the usualestimate/score which is based upon all cells ofthe table.The second component of the pseudo-score we

therefore choose to be the score for this case-onlytest, u12. To complete the definition of the pseudo-score test we let v define the variance of the vectoru under the null hypothesis, such that

v v1 c

c v12

;

where v1 and v12 are the null variances of u1 and u12,

respectively, and c their null covariance. Since ourdata was retrospectively sampled (Z and W given Y)we can now treat Y as being fixed. Within thetechnical report we redefine u1 and u12 conditionalupon fixed Y and calculate their null variances to bedefined by

v1 N N1N1

NVar Z and

v12 N1 1 Var ZjY 1 Var WjY 1;where N1 is the number of cases within the sample,Var (Z1) is the variance of Z1 within the wholepopulation and Var (Z1|Y5 1) and Var (W1|Y5 1)are the variances of Z1 and W1 in case individualsonly. These variances are unknown but may beconsistently estimated by

v1 N N1N1NN 1

XNi1

Zi Z2 and

v12 1

N1 1XN1i1

Zi ZY12XN1i1

Wi WY12;

where we assume that the first N1 individuals arethe cases and the last (NN1) are the controls andZY1 and W

Y1 define the mean of Z

1 and W1 incase individuals only and Z the mean of Z1 across

896 Chapman and Clayton


all individuals within the sample. Under the nullhypothesis and the assumption that the primaryand secondary loci are independent, the twoscore components are uncorrelated [see technicalreport; Chapman and Clayton, 2007a] and thereforec is equal to zero. We therefore estimate the nullvariance matrix, v by

v v1 00 v12

;

and the pseudo-score test is then defined by

t uTv1u u21

v1 u

212

v12 t1 t12;

where t1 is the usual main effects score test and t12 isthe case-only score test for interaction.

MULTIPLE MODIFYING LOCI

Since we are unlikely to be able to choose a singlemodifying locus, a priori, we will need to carry outnumerous pairwise tests between the primary locusand different secondary loci. Suppose that we wishto consider a set ofMmodifying loci, then we will becarrying out M different tests of the null hypothesis,that the primary locus is not causal and the rejectionof just one of these tests will lead to the rejection ofour null hypothesis. In other words, the null isrejected if the maximum of these M tests is rejected.In this way the set of these M tests form a single testof the null hypothesis, by way of their maximumvalue. The appropriate omnibus test statistic, foran effect at the primary locus, is then equal to themaximum of these M tests.To illustrate this, we will now define the genotype

at the mth modifying locus as Wm , for m5 1,y,M.The mth test statistic is then testing simultaneouslyfor a main effect at the primary locus, Z, and anepistatic effect between the primary locus and themth modifying locus and is of the form t(m)5 t11t1m, as shown above.The omnibus test statistic is then simply equal to

the maximum value of this test statistic over all Mmodifiers,

tomni maxm1; ... ;M

tm maxm1; ... ;M

t1 t1m t1 max

m1; ... ;Mt1m:

In order to find an appropriate P-value for thisomnibus test, we need to calculate its distributionunder the null hypothesis of no effect at the primarylocus (either marginally or epistatically). Unfortu-nately there is no known form for this specific teststatistic. However, when all primary and secondaryloci are mutually independent of one another, we areable to use Monte Carlo integration [Press et al.,1992] to simulate from the required null distribution

(see technical report). When this assumption doesnot hold we can use a reasonably simple permuta-tion argument, under which we randomly permutethe genotypes at the primary locus (see technicalreport).

INDIRECT MODEL

In this section, we consider the indirect case, inwhich the functional variants are unobserved, andinstead we have one region of primary interest andM secondary regions, each assumed to contain asingle putative causal variant. Each of these regionsis assumed to be tagged by a set of tag SNPs and thesecondary sets of tag SNPs are assumed to beindependent of those tag SNPs in the primaryregion. We let XZ denote the genotype vectorcorresponding to the set of tag SNPs in the primaryregion, which is assumed to contain the primarylocus of interest, Z. This vector is of length nZ, whichis equal to the number of tag SNPs within theprimary region and its jth component then recordsthe number of 1 alleles present at the jth tag SNP,with the assignment of the 1 label arbitrary. Thegenotype vector corresponding to the set of tag SNPsfor each of the M modifying regions is then denotedas XWm and is of length nWm, for m5 1,y,M. Ourmodel assumes that region m contains the putativemodifying locus Wm. As in Chapman et al. [2003],we assume that the causal loci may be estimated interms of the tag SNP genotype vectors via a simplelinear model, such that

Z dTZXZ and Wm dTmXWm; for m 1; . . . ; M:Considering the model of the primary regionand just the mth modifying region, we followthe arguments of Chapman et al. [2003] and derivethe indirect pseudo-score using the following twosteps:

1. take the expectation of the direct pseudo-score,given the observed tag SNPs, and

2. maximise the resulting expected score withrespect to a particular constraint upon thevariance of the score (Lagrange multiplierapproach).

Using Step (1), the expectation of the indirectpseudo-score components, corresponding to b1 andg, are therefore defined by

U01 EZu1jXZ

XNi1

Yi YE Zi jXZi

dTZXNi1

Yi YXZi;



and

U01m XNi1

YiE Zi ZWmijXZi;XWmi

XNi1

YiE Zi ZjXZi

E WmijXWmi

dTZXNi1

YiXZi XZ XTWmidm

respectively, where the second step above makes useof the assumption of independence between regions.Since both of these score components contain

unknown parameter values, namely dZ and dm,which are not estimable from the observed data,we follow Chapman et al. [2003] and use theLagrange multiplier argument of step (2), above,whereby we choose those parameter values thatmaximise the score component conditional uponsome constraint on the variance. For simplicity, wemaximise each component separately. Chapmanet al. [2003] have already done this for the maineffects component. They define U01 dTZU1 andV01 VarU01 dTZVarU1dZ dTZV1dZ, where V1 isthe null variance of U1, which is defined by

U1 XNi1

Yi YXZi:

Chapman et al. [2003] maximise U01 with respectto the constraint that V01 is equal to 1 and find thatthe appropriate value for dTZ is V

11 U1. Consequently,

the appropriate main effects test component isdefined by

T1 U0T1 V011U01 UT1 dZ dTZV1dZ

1dTZU1

UT1V11 U1 UT1V11 V1V11 U1 1

UT1V11 U1

UT1V11 U1:Thus, T1 is equivalent to the test statistic for scoreU1,with null variance V1. In other words, the dZ can bedropped from the score component altogether andthe first component of the pseudo-score test becomessimply U1, as defined above, and then the nullvariance of U1 can be estimated by

V1 N N1N1NN 1

XNi1

XZi XZ XZi XZ T:

Under the null, of no effect of the primary locus, T1has an asymptotic w2 distribution of nZ df.The derivation of the appropriate indirect compo-

nent for the interaction term can be calculatedsimilarly (see Appendix). (Note that since we aremaximising the terms separately, we reestimate dZ

within this term). We find that dZ may be estimatedby the first eigenvector of the matrix n1m C

Tn1Z C, inwhich nZ and nm are the estimated variances of theprimary and mth secondary region and C is theirestimated covariance matrix, all evaluated in caseindividuals only. Similarly dm may be estimated by thefirst eigenvector of the matrix n1Z Cn

1m C

T. The scorecomponent itself, turns out to be the square root ofthe corresponding maximum eigenvalue of both ofthe matrices above (the eigenvalues are equal). Thiseigenvalue is sometimes known as the maximumcanonical correlation and we will denote this byCCZ,Wm [Kendall and Stuart, 1979]. The square rootof this canonical correlation no longer has zeroexpectation under the null. To obtain a valid teststatistic, we need to take away its null expectation,EH0, so that it has zero expectation under the nullhypothesis. Therefore the second component of ourpseudo-score test is defined by

U1m ccZ;Wmp EH0:The null expectation, EH0, is estimated by apermutation argument described in the appendix.Using the same permutation approach we can alsoestimate the null variance, V1m (see appendix). Weassume, as is true in the direct case and confirmedby simulation in the indirect case, that the scorecomponents, Um and U1m are independent of eachother. The indirect pseudo-score test between theprimary region and the mth secondary region cantherefore be defined as

Tm UT1V11 U1 U21mV1m

T1 T1m:Unlike the first component, the null distribution ofthe second component above, T1m, does not have asimple distribution of closed form, however, we notethat U1m may be written in the form d

TZCdm. Since we

estimate dZ and dm from the data the conditions forU1m being asymptotically normally distributed (andT1m having a w

2 distribution on 1df) are not formallymet, however the quantile-quantile plot (qqplot)within the appendix (see Fig. 9) suggests that thenull distribution of T1m is well approximated by aw2 1 distribution and we can use this fact to calculateappropriate Pvalues and estimate power (see tech-nical report). This result is somewhat counter-intuitive since the interaction term has the potentialof having higher dimension than the main effectterm, however the interaction term can be viewed asbeing analogous to Tukeys one degree of freedomfor non-additivity in the analysis of variance[Turkey, 1949].Note, also, that we could use a version of

the canonical correlation test to test for the maineffect component (i.e. b15 0) by substituting the



multidimensional secondary region for the univari-ate disease status. In this case the n1m C

Tn1Z C matrixbecomes VarY1CovY;XZVarXZ1CovXZ;Y,which is a scalar value and simply equal to themultivariate R2 between disease and the nZ tagSNPs. We will denote this R2 value by R2Y to drawattention to the fact that it differs from the usual R2

often quoted, since it measures correlation betweendisease and tag SNPs rather than between aparticular locus and tag SNPs. The squareroot ofthis first eigenvalue is therefore simply equal to RY.In fact the form of the indirect main effect score testthat we propose earlier in the paper, T1, can bewritten in the form N 1R2Y. In this particular casethe asymptotic distribution of this test is known tobe w2 on nZ df and there is no need to use a bruteforce approximation to a w2 1 distribution.An alternative, less convenient, test statistic

may be defined by estimating dZ and dm bycorresponding coefficients of a logistic regressionmodel of disease status upon these regions (as isdone when calculating U1). In this alternativeapproach dZ would be defined, as above, by d

TZ

V11 U1 and dm would be defined by dTm V1WmUWm,

where UWm is the main effects score if we weretesting for a main effect at the mth secondary locusand VWm is its estimated variance under the null.Although not of key interest to this paper, we alsobriefly consider tests based upon these logisticregression coefficients within our simulations, laterin the paper, and we refer to the corresponding teststatistic as Tlog; the logistic regression case-only testfor interaction.

INDIRECT OMNIBUS TEST

The indirect omnibus test statistic is then definedby

Tomni maxm1; ... ;M

Tm maxm1; ... ;M

T1 T1m T1 max

m1; ... ;MT1m:

As in the direct case, if we are willing to assumemutual independence between all primary andsecondary loci we can use Monte Carlo integrationto simulate the null distribution of this test statistic(see technical report). When this assumption doesnot hold, we may again make use of a permutationargument in which the allocation of the primaryregion is permuted across all individuals (detailswithin technical report).

SIMULATION STUDY

We now examine the properties of our approachby simulation. We consider simulations under twoalternative hypotheses; one with epistasis andanother without.

ALTERNATIVE HYPOTHESES

Epistatic alternative hypothesis. There are nu-merous alternative hypotheses that are consistentwith an epistatic effect. We have chosen to focusupon a particularly simple alternative but one underwhich we may expect the usual marginal test to havelow power. This alternative model assumes that theprimary locus is causal only through its modificationby a single secondary locus within the set of Mpossible modifying loci. Without loss of generality,we assume this true modifying locus to be the firstlocus, i.e. m5 1. This model is defined by

logitY a gZW1 ;which may be more clearly illustrated by consider-ing the relative risks of the joint distribution of theprimary and modifying locus as shown in Table I.Note that we assume no main effects.It is clear that the usual marginal test would have

difficulty in detecting such an effect as there is littleinformation retained in the margins. Despite thisfact, it is unclear whether the omnibus test statisticwould have greater power, since many more testsare carried out and the adjustment for so manymultiple tests may diminish any increase in power.Main effects alternative hypothesis. This paper

is also interested in how much power we may loseby allowing for epistasis, when the true underlyingmodel is simply a multiplicative main effect modelat the primary locus, i.e.

logitY a b1Z:This means that compared to those individuals with0 mutant alleles, those people with 1 and 2 copies ofthe mutant allele have relative risks of expb1 andexpb12, respectively.

THE DIRECT CASE

For simplicity, our simulations assume that allprimary and secondary loci are mutually indepen-dent of one another. In this case we need not carryout a full simulation of datasets but we are ableto use asymptotic distributions to simulate thedistribution of the omnibus test statistic under theappropriate alternative (see technical report). Using

TABLE I. Relative risks consistent with our epistaticalternative

Locus W1

0 1 2

0 1 1 1Locus Z 1 1 exp (g) (exp (g))2

2 1 (exp (g))2 (exp (g))4



these simulated alternative distributions we can usestandard techniques to calculate the power of theomnibus test statistic, under both the epistatic andmain effects alternatives.The graphs in Figures 1 and 2 compare the

power between the single marginal test of theprimary locus (t1), with that of the omnibus epistatictest statistic (tomni) under the epistatic alternativeabove. To generate these graphs we allowed theepistatic relative risk (expg as defined in Table I), torange between 1.2 and 2 and assumed that both locihad a susceptibility allele frequency of 0.1(Fig. 1) and 0.2 (Fig. 2). We assumed a sample of2,000 cases and 2,000 controls and the powers shownare based upon Pvalues of 0.01 (a and b) and 0.0001(c and d). We consider both M5 100 (a and c) andM5 1,000 (b and d). For each calculation in theomnibus test we used P5 1,000,000 simulationsunder the Monte Carlo integration approach toestimate the critical value of the test and P5 10,000samples estimate the power of the test. Fewer

simulations are required to calculate the power sinceunder the alternative we have moved away from thetails of the distribution.The graphs demonstrate that we have more power

to detect the primary locus using multiple tests thatallow for epistatic effects rather than the usualmarginal test. Only for small relative risk effects (1.1-1.2) is the marginal test ever slightly more powerful.Surprisingly, an increase in the number of testscarried out in the omnibus test (100 to 1,000) onlydecreases the power of this test statistic by arelatively small amount. For example consideringgraphs (c) and (d) in Fig. 2, we see that the power ofthe omnibus test, when the relative risk is equal to1.3, is about 50% when 100 other loci are includedand about 35% when 1,000 are included. It alsoappears that as the level of type 1 error, a, becomesmore strict the omnibus test becomes increasinglymore powerful compared to the marginal test. It isalso clear that for smaller DS allele frequencies thegain in power is greater than for more common DS

a.

1.2 1.4 1.6 1.8 2.0 2.2

0.0

0.2

0.4

0.6

0.8

1.0

epistatic relative risk

powe

r

Omninbus Test (M=100)

Main Effects Test

b.

0.0

0.2

0.4

0.6

0.8

1.0

powe

r


Main Effects Test

c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


Main Effects Test

d.

0.0

0.2

0.4

0.6

0.8

1.0

powe

r



Main Effects Test

1.2 1.4 1.6 1.8 2.0 2.2epistatic relative risk

1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6 1.8 2.0 2.2

Fig. 1. Comparison of power between the usual marginal test and the omnibus test that allows for numerous epistatic effects, when thedisease susceptibility allele frequencies are equal to 0.1, our samples consist of 2,000 cases and 2,000 controls and the true model is an

epistaticmodel. Graphs (a) and (c) assumeM5 100 modifying loci within the omnibus test and (b) and (d) assumeM5 1,000. Powers in(a) and (b) are based upon an a level of 0.01 and (c) and (d) an a of 0.0001.



loci. This is likely to be due to the fact that for rarervariants the marginal effect practically disappearsbut that the epistatic effect is still detectable.Although these findings are surprising they arecorroborated by the findings of Marchini et al.[2005].The graphs in Figures 3 and 4, again compare the

power of the omnibus test with the marginal test, forloci with allele frequencies 0.1 and 0.2, respectively.However, in this case the true underlying model is asimple main effects model at the primary locus. Weagain let M equal both 100 and 1,000 and assumesamples of 2,000 cases and 2,000 controls. We alsoconsider Pvalues of 0.01 and 0.0001. In this case weallow the marginal relative risk, (expb1), to rangebetween 1.2 and 1.7.These graphs show that when only a main effect

occurs, little power is lost when using the omnibustest as opposed to the main effects test. This mayseem surprising, however, the main effect is presentin all M tests and they are therefore highly

correlated, meaning that the extra penalty formultiple testing using the omnibus test is relativelyslight. In the omnibus test there is practically nodifference between the power of carrying outM5 100 tests and M5 1,000 tests, which may beexpected considering the above observations. Whenthe allele frequencies are lower, the difference inpower between the marginal test and omnibus testincreases slightly. However, the allele frequency onlyhas a small effect upon the power of both tests undera true main effects model. In other words, when thecausal process is a main effect model both teststatistics are less susceptible to changes in the allelefrequency compared with the effect under anepistatic causal model.

MULTIPLE TAG SNPS

When the regions of interest contain more thanone tag SNPs, it is no longer possible to define acontrol population based upon allele frequencies

a.

1.2 1.4 1.6 1.8 2.0 2.2

0.0

0.2

0.4

0.6

0.8

1.0


powe

r

Omninbus Test (M=100)Main Effects Test

b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


1.2 1.4 1.6 1.8 2.0 2.2

1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6 1.8 2.0 2.2

Fig. 2. Comparison of power between the usual marginal test and the omnibus test that allows for numerous epistatic effects, when thedisease susceptibility allele frequencies are equal to 0.2, our samples consist of 2,000 cases and 2,000 controls and the true model is an

epistaticmodel. Graphs (a) and (c) assumeM5 100 modifying loci within the omnibus test and (b) and (d) assumeM5 1,000. Powers in(a) and (b) are based upon an a level of 0.01 and (c) and (d) an a of 0.0001.



and R2Y alone and we need to define the structureof each region (i.e. the sets of tag SNPs and theR2 values between these loci and between them andthe causal locus). For simplicity we will assume thatthe primary region and each of the M secondaryregions all have the same structure and we will carryout a number of different simulation studies, eachone based upon real data from different regions ofthe genome, with slightly different structures. Thegenotype populations we use are based upon realgenotype data, from which haplotype populationsare estimated via the expectation-maximisationalgorithm and then appropriate genotype popula-tions formed under the assumption of Hardy-Weinberg equilibrium. For each region, we haveselected a set of tag SNPs based upon the method ofChapman et al. [2003] and have then randomlyselected a single remaining locus from the region torepresent the causal locus. Note that these SNPs areequal to those chosen in Chapman and Clayton[2007b]. Table II gives some details of the seven

regions that we consider. Again, since we assume allregions are independent, we are able to simulate theapproximate null distribution of the omnibus teststatistic using Monte Carlo integration. However,due to the complex structure of the regions andthe complicated test statistics we need to samplewhole case-control datasets from the alternativepopulation and calculate the omnibus test statisticfor each sample, in order to generate the distributionof the omnibus test statistic under the alternativehypothesis.We first compare the power of the indirect

omnibus test with the indirect marginal test fromChapman et al. [2003] (which is in fact equal to T1),when the epistatic alternative is true. In the case ofthe omnibus test we consider both the test formedusing the canonical correlation score, Tomni, and thatderived using fitted values of the logistic regressionof disease upon tag SNPs, Tlog. We consider variousdifferent regions with different numbers of tag SNPsand different structures that lead to different levels

a.

1.2 1.4 1.6 1.8 2.0

0.0

0.2

0.4

0.6

0.8

1.0

allelic relative risk

powe

r

Omnibus Test (M=100)Main Effect test

b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


1.2 1.4 1.6 1.8 2.0

1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0

Fig. 3. Comparison of power between the usual marginal test and the omnibus test that allows for numerous epistatic effects, when the

disease susceptibility allele frequencies are equal to 0.1, our samples consist of 2,000 cases and 2,000 controls and the true model is amain effects model. Graphs (a) and (c) assume M5 100 modifying loci within the omnibus test and (b) and (d) assume M5 1,000.Powers in (a) and (b) are based upon an a level of 0.01 and (c) and (d) an a of 0.0001.



of ability to predict the causal locus within eachregion. We have selected just one untagged locus inthe region as being the causal locus. The details ofthese regions and their chosen DS loci are shown inTable II. Within each region we assume that we havea sample of 1,000 cases and 1,000 controls andassume a type 1 error of size 0.001. To calculate thecritical value we use P5 1,000,000 simulations andto estimate the power we use only P5 10,000simulations. As above we consider the cases ofM5 100 secondary loci (Fig. 5) and also M5 1,000secondary loci (Fig. 6).

These graphs, again, demonstrate that it is possibleto gain power by using a multiple testing omnibusstrategy as opposed to the usual marginal test. Theserious lack of power with the INS4 (c) andTRANCE (d) regions are caused by the rare causalallele frequencies; 0.0802 and 0.0676, respectively.Note however that for larger effects (i.e. expg 2)there is considerably more power using the omnibustest compared to the marginal test and also that thecanonical correlation test has more power than thefitted logistic regression test. However in regionswith larger allele frequencies (IL21 (a) and INS1 (b))

a.

1.2 1.4 1.6 1.8 2.0

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


1.2 1.4 1.6 1.8 2.0

1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0

Fig. 4. Comparison of power between the usual marginal test and the omnibus test that allows for numerous epistatic effects, when the

disease susceptibility allele frequencies are equal to 0.2, our samples consist of 2,000 cases and 2,000 controls and the true model is a

main effects model. Graphs (a) and (c) assume M5 100 modifying loci within the omnibus test and (b) and (d) assume M5 1,000.Powers in (a) and (b) are based upon an a level of 0.01 and (c) and (d) an a of 0.0001.

TABLE II. Details of DS loci chosen within each region

Region Kb SNPs Common SNPs DS allele frequency Locus R2 Haplotype R2 No. of tag SNPs

IL21 8 15 10 0.2684 0.9748 1 4INS1 10 12 12 0.218656 0.8039 0.9798 3INS4 13 12 0.0802 0.8008 0.8787 3TRANCE 33 26 14 0.0676 0.8566 0.9234 3

SNPs, single nucleotide polymorphism; DS, disease susceptibility.



these two omnibus tests have very similar power.This may be explained by the fact that when theallele frequency is low the approximation of thecausal locus using the disease status becomesincreasingly inaccurate, where as the canonicalcorrelation is able to use all possible informationcontained in the tag SNPs. We notice that, in regionsin which the causal allele has a reasonable allelefrequency, we may still gain considerably greaterpower by using the omnibus test as opposed to themarginal test. We also notice that the difference inpower between using M5 100 tests and M5 1,000tests is greater in the indirect case, compared to thedirect case. In general we note that indirect tests aremore susceptible to changes in factors such as thecausal allele frequency and number of tests carriedout. In general the greater the value of R2, the biggerthe increase in power of the omnibus test statistic.The graphs also show that large R2 alone is notsufficient to increase power and that if the causallocus is very rare then neither test will havesufficient power. We also see that the number oftag SNPs representing a region make little differenceto the increase in power, this is to be expected sincethe epistatic tests remain on effectively 1 df,

although more tag SNPs may result in a greatervariance of the canonical correlation.We now assume that the underlying model is a

simple multiplicative model at the primary locus.We wish to calculate the amount of power we lose bymisspecifying the model and using the indirectomnibus test as opposed to using the appropriateindirect marginal test statistic. The scenario consid-ered is exactly the same as that above except for theunderlying model. In this case we allow the allelicrisk to range between 1.2 and 1.7, we assume 1,000cases and 1,000 controls and a Pvalue of 0.001. Weexplore the power within the same four regions asabove (a) IL21, (b) INS1, (c) INS4 and (d) TRANCEand assume the same DS locus. We again considerM5 100 (Fig. 7) and M5 1,000 (Fig. 8) tests withinthe omnibus test. Note that in the case of a true maineffect model, both test components corresponding tothe canonical correlation and the logistic regressioncase only test follow their null distribution, whichare asymptotically equal. This means that theomnibus tests will have the same power and wetherefore only need to plot the power of one of them.These graphs, again, show that little power is

lost if the main effects model is misspecified and

a.

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.0

0.2

0.4

0.6

0.8

1.0


powe

r

Main Effects TestOmnibus Test (M=100)Log. Reg Test, (M=100)

b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.


0.0

0.2

0.4

0.6

0.8

1.0

powe

rMain Effects TestOmnibus Test (M=100)Log. Reg Test, (M=100)

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Fig. 5. Comparison of power between the usual indirect marginal test and the omnibus indirect test that allows for M5 100 possibleepistatic effects, assumes samples of 1,000 case and 1,000 controls and that the model is an epistatic model.



we carry out the omnibus test, rather than themarginal test.

DISCUSSION

Within this paper we have proposed an "omnibus"test statistic to test for an effect at a locus of interest,allowing for possible epistatic effects with numerousother loci. We have also extended the case-only testfor interaction to the indirect case and used this toform an indirect version of the omnibus test statistic,above. Under both the direct and indirect scenariowe compared the power of the appropriate pseudo-score test with the appropriate main effects testunder both a true epistatic model and a true maineffects model. We demonstrated that in both thedirect and indirect case, testing for numerousinteractions may increase our power to detect DSloci when epistasis occurs and also that relativelylittle power is lost when the true model is a simplemain effects model. In general we showed thatmultiple testing may not be such a problem as hasbeen feared previously, so long as we can correct forit appropriately, and it appears that testing fornumerous interactions may well be worthwhile,despite the increased number of multiple tests.

Our simulations, however, only consider a parti-cular model of epistasis, generating our data basedupon the same model under which we derived theappropriate test statistic. It would be interesting tosee how these results transfer when the true modelof epistasis differs from the model of epistasissimulated. Note that since our model includes (andwe test for) main effects as well as interactions it isreasonably flexible and can include interactions ofnumerous different forms. As a more extremeexample, suppose a variant at the primary locusincreases the risk of disease in the absence of avariant at the secondary locus but the variant at thesecondary locus counter-acts the effect at theprimary locus so that the primary locus appears tohave no effect in those hetrozygous for the second-ary variant and appears to be protective in thosehomozygous for the secondary variant. This situa-tion can easily be modelled within our proposedmodel by setting b2 to 0 and b1 to be equal to 1/g.Table III shows the relative risks when we assumethis model for b15 1.5. Many other exotic models ofepistasis are definable through use of the modeldefined. In these cases we would expect theomnibus test to again increase power compared tothe main effects test, since the model underlying the

a.

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.

0.0

0.2

0.4

0.6

0.8

1.0


powe

rMain Effects TestOmnibus Test (M=1000)Log. Reg Test, (M=1000)

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.01.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Fig. 6. Comparison of power between the usual indirect marginal test and the omnibus indirect test that allows for M5 1,000 possible

epistatic effects, assumes samples of 1,000 case and 1,000 controls and that the model is an epistatic model.



omnibus test can reflect the true situation (testing forboth the main effect and the interaction) and givenour results, we would not expect the correction formultiple testing to diminish this property. The hopeis that many models of interaction that do notexactly fit within this framework may be wellenough represented by the model and the scorecomponents well enough disturbed from their nulldistributions that our power to detect such scenarioswill remain high. Since little is known about true orlikely forms of interation, the question remainswhether such statistical models can realisticallyrepresent real genetic data.Another interesting consideration would be to

investigate different forms of alternative hypothesisacross the set of secondary loci. For example, whenmore than just one of these secondary loci interact

with the locus of interest, or perhaps when the trueunderlying model is a higher order interactionbetween a subset of the secondary loci. Note thatour combination of test statistics into the maximum(omnibus) test is motivated by the need to adjust theset of tests for multiple testing. In fact, combining thetest statistics in this way makes an implicit assump-tion that all pairwise interactions act independentlyof one another, for example that they do not stemfrom a single higher order interaction.Since our test statistic makes use of the case-only

test for interaction, we must be wary that theomnibus test statistic is only valid when all modify-ing loci are independent of the primary locus. Whenthis assumption is violated the test statistic is nolonger valid, therefore care must be taken to ensurethat the primary locus is independent of all themodifying locus. One could consider only those loci,say, 100 kb from the primary locus as possiblemodifiers. In the indirect case, in which we aredealing with regions rather than loci, possiblecorrelation between primary and secondary regionsis likely to be less of a problem since edges of regionsoften coincide with edges of LD blocks and thereforedefined regions are likely to be independent ofone another. When we are unwilling to make thisassumption our options are more limited. In the

a.

1.2 1.3 1.4 1.5 1.6 1.7

0.0

0.2

0.4

0.6

0.8

1.0


powe

r

Main Effects TestOmnibus Test (M=100)

b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.

powe

r0.

00.

20.

40.

60.

81.

0



1.2 1.3 1.4 1.5 1.6 1.7

1.2 1.3 1.4 1.5 1.6 1.7 1.2 1.3 1.4 1.5 1.6 1.7

Fig. 7. Comparison of power between the usual indirect marginal test and the omnibus indirect test that allows for M5 100 possibleepistatic effects, assumes samples of 1,000 case and 1,000 controls and that the model is a main effects model.

TABLE III. Table of relative risks for more complexmodel of epistasis

Z2

0 1 2

0 1 1.5 2.25Z1 1 1 1 1

2 1 0.67 0.44



direct case we could simply substitute the case-onlyscore for the usual score for the interaction term,however the resulting omnibus test statisticbecomes more complicated since the pseudo-scorecomponents are now no longer independent of oneanother and we are unable to make use of the MonteCarlo integration approach. In the indirect approachthe situation is far more difficult and no simple testcan be derived using the approach above, howeverChatterjee et al. [2006] suggest an appropriate teststatistic for this situation. Note that, although oursimulations make the assumption that the secondaryloci (regions) are mutually independent of oneanother, the omnibus test statistic remains validwhen this is not the case. In fact in this case ofdependence between secondary loci (but not be-tween primary and secondary loci), findings arelikely to be even more favourable to the omnibus testas the amount of multiple testing is effectivelyreduced. However in this case the need for thepermutation argument to calculate appropriatePvalues means that this situation is computerintensive.Despite these considerations, the results shown are

encouraging evidence that tests for epistasis may beof use in the search for loci involved in complexdisease. This is welcome news in the face of multiple

testing concerns at the dawn of an era in whichgenotyping of huge numbers of markers is fastbecoming a reality.

ACKNOWLEDGMENTS

We are grateful to our colleagues in the Diabetesand Inflammation Laboratory for sharing the SNPsequence data used to obtain the results shown inTable II and Figures 58. We are also grateful to JohnWhittaker for his helpful suggestions and ananonimous reviewer for insightful comments.

REFERENCESAgresti A. 1990. Categorical Data Analysis. New York: Wiley.Chapman JM, Clayton DG. 2007a. Detecting association

using epistatic information: technical addendum. Available athttp://homepages.lshtm.ac.uk/encdjcha/

Chapman JM, Clayton DG. 2007b. One degree of freedom fordominance in indirect association studies. Genet Epidemiol31:261271.

Chapman JM, Cooper JD, Todd JA, Clayton DG. 2003. Detectingdisease associations due to linkage disequilibrium usinghaplotype tags: a class of tests and the determinants ofstatistical power. Hum Hered 56:1831.

Chatterjee N, Kalaylioglu Z, Moslehi R, Peters U, Wacholder S.2006. Poweful multilocus tests of genetic association in the

a.

1.2 1.3 1.4 1.5 1.6 1.7

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


b.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


c.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


d.

0.0

0.2

0.4

0.6

0.8

1.0


powe

r


1.2 1.3 1.4 1.5 1.6 1.7

1.2 1.3 1.4 1.5 1.6 1.7 1.2 1.3 1.4 1.5 1.6 1.7

Fig. 8. Comparison of power between the usual indirect marginal test and the omnibus indirect test that allows for M5 1,000 possibleepistatic effects, assumes samples of 1,000 case and 1,000 controls and that the model is a main effects model.



presence of gene-gene and gene-environment interactions. AmJ Hum Genet 79:10021016.

Clayton D, Chapman J, Cooper J. 2004. The use of unphasedmultilocus genotype data in indirect association studies. GenetEpidemiol 27:415428.

Cordell H. 2002. Epistasis: what it means, what it doesnt mean,and statistical methods to detect it in humans. HumMol Genet11:24632468.

Cordell H, Todd J, Hill N, Lord C, Lyons P, Peterson L, Wicker L,Clayton D. 2001. Statistical modelling of interlocus interactionsin a complex disease: rejection of the multiplicative model ofepistasis in type 1 diabetes. Genetics 158:357367.

Cox D, Hinkley D. 1974. Theoretical Statistics. London: Chapmanand Hall.

Evans DM, Marchini J, Morris AP, Cardon LR. 2006. Two-stagetwo-locus models in genome-wide association. PLOS Genet2:14241432.

Fan R, Knapp M. 2003. Genome association studies of complexdiseases by case-control designs. Am J Hum Genet 72:850868.

Fanous AH, Kendle KS. 2005. Genetic heterogeneity, modifiergenes and quantitative phenotypes in psychiatric illness:searching for a framework. Mol Psychiatry 10:613.

Kendall M, Stuart A. 1979. The Advanced Theory of Statistics, VolI, 4th edition. New York: MacMillan.

Maier LM, Chapman JM, Howson JMM, Clayton DG, et al. 2005.No evidence of association of, or interaction between, the il-4ralpha, il4 and il-13 genes in type 1 diabetes. Am J Hum Genet76:517521.

Marchini J, Donnelly P, Cardon L. 2005. Genome-wide strategiesfor detecting multiple loci that influence complex diseases. NatGenet 37:413417.

McCullagh P, Nelder JA. 1989. Generalized Linear Models, 2ndedition. Monographs on Statistics and Applied Probability.London: Chapman and Hall.

Press W, Teukolsky S, Vetterling W, Flannery B. 1992. Numericalrecipes in C. 2nd edition. Cambridge: Cambridge University Press.

Pritchard J, Rosenberg N. 1999. Use of unlinked genetic markers todetect population stratification in association studies. AmJ Hum Genet 65:220228.

Tukey J. 1949. One degree of freedom for non-additivity.Biometrics 5:232242.

Weinberg C, Umbach D. 2000. Choosing a retrospective design toassess joint genetic and environmental contributions to risk.Am J Epidemiol 152:197203.

Xiong M, Zhao J, Boerwinkle E. 2002. Generalized t2 test forgenome association studies. Am J Hum Genet 70:12571268.

APPENDIX: INDIRECT CASE-ONLYTEST FOR INTERACTION

We will derive an appropriate form for the indirectepistatic score component. Taking the expectation ofthe direct epistatic score component, under the null,the indirect epistatic score component between theprimary and mth secondary locus can be written inthe form

U#1m dTZXN1i1

YiXZi XZ XTWmidm

N1 1dTZCdm;

where C is the sample covariance between XZ andXWm in the set of cases only. Note that this matrix isnot neccessarily symmetrical, having dimension nZby nWm. Since (N11) is a constant and since we willstandardise U1m

# to form our test statistic, we candrop (N11) and from now on will define theappropriate score component simply as

U1m dTZCdm:We again consider the Lagrange multiplier techni-que and now choose the parameters, dZ and dm, sothat this score component is maximised under theconstraint that the pair of underlying causal locihave unit variance,

dTZnZdZ 1 and dTmnmdm 1;where nZ is the variance of XZ and nm is the varianceof XWm in cases only. Therefore we want to maximisethe function

U1m l1constraint 1 l2constraint 2 dTZCdm l11 dTZnZdZ l21 dTmnmdm;

with respect to the parameters dZ, dm, l1, l2.

qqdZ

Cdm 2l1nZdZ 0qqdm

CTdZ 2l2nmdm 0qql1

1 dTZnZdZ 0qql2

1 dTmnmdm 0

Substituting the third and fourth equations, into thefirst and second equations, respectively, we find that

U1m dTZCdm 2l1 2l2 2l;where we now replace l1 and l2 by the single value l.Rearranging the first equation in the form,

dZ n1Z Cdm2l

we may substitute this into the second equation andrearrange, so that

n1m CTn1Z C 4l2I

dm 0:

This shows that 4l2 may be chosen to be themaximum eigenvalue of the matrix Mat1 n1m C

Tn1Z C, where dm is the corresponding eigenvec-tor. A similar argument shows that dZ is theeigenvector corresponding to the maximum eigen-value of the matrix Mat2 n1Z Cn1m CT, where thiseigenvalue is equal to that of the first matrix, Mat1.We noted earlier that the appropriate score, U1m, isdefined as 2l, and this means that the appropriatescore is equal to the square root of the maximum



eigen value of both matrices Mat1 and Mat1. For easeand speed we choose the smallest of these matricesto calculate the score. This eigen value is referred toas the maximum canonical correlation between XZand XWm and we will denote it as ccZ,Wm, so that theepistatic score component may be defined byU1ma ccZ;Wmp . Note that C, nm and nZ are allcalculated in the set of case individuals only and thatthe score component above corresponds to anindirect form of the case only test for interactionon 1df.Under the null hypothesis, the expectation of U1 is

zero, however the expectation of U1m is nonzerosince the nuisance parameters are chosen to max-imise the correlation between Z1 andWm . Therefore,in its present form U1m(a) will not form a valid teststatistic and we need to take away the nullexpectation of

ccZ;Wm

p. This we may estimate from

the data by the permutation argument describedbelow. The appropriate score is then of the form

U1m ccZ;Wmp EH0;where EH0 is the expectation of

ccZ;Wm

punder the

permutation argument. We can form a score likeindirect test for interaction using the statistic

T1m U21mV1m

;

where V1m is the null variance ofccZ;Wm

pwhich can

again be estimated by the same permutation argu-ment that gave us EH0.This permutation argument depends upon the

exact hypothesis being tested; are we searching forinteraction alone or are we looking for any effectat the primary locus (as is the aim of this paper). Inboth cases we are aiming to generate a number (NP)of null samples and for each sample we thencalculate the appropriate square root of the canonicalcorrelation (cps, for s 1; . . . ;NP). The expectedvalue of

ccZ;Wm

punder the null can then be

estimated by the mean value across all null samples,EH0

PNPs1 cps=NP. The null variance of

ccZ;Wm

pis then simply estimated by V1m PNPs1cps EH02=NP.Considering the score test form of this test

statistic we might expect that it follows a central w2

distribution on 1 df under the null hypothesis. Weinvestigated this by simulations and found this to bethe case. Figure 9 shows the related qqplot. The topline of circles shows the qqplot between the w2

1 distribution and the indirect case-only test statistic(T1m). Although it appears to differ quite a bit withinthe tails, in fact only 63 out of the 10,000 simulationshave quantiles above 8 (0.0063%) and this is to beexpected due to the simulated nature of thequantiles. The qqplot of the alternative, but closelyrelated, logistic regression case-only test statistic(Tlog) compared to the w

2 1 distribution are shown bythe bottom line of triangles in Fig. 9. This logisticregression test statistic is also a good fit to the w2

1 distribution (only 43/10,000 simulated test statis-tics are below 8), however the divergence appears tobe in a different direction the canonical correlationtest statistic.Note that if testing only for a single epistatic effect

we can use the canonical correlation as the simpletest statistic and generate an appropriate P valueusing the same permutation argument and simplycounting the proportion of permuted data sets thathave a test statistic more extreme then the observedvalue. This is the test statistic we used to test forepistasis in Maier et al. [2005].

20151050

05

1015

2025

QQplot of null distribution of both test statistics against the chisquare 1 distribution

chisquare quantiles

test

sta

tistic

qua

ntile

s

Fig. 9. Quantile-quantile plot between the v2 (1) distributionand the null distribution of the T212a and T

212b test statistics.



detecting association using epistatic information

Documents