Detecting Marker-Disease Association by Testing for Hardy-Weinberg Disequilibrium at a Marker Locus

Transcription

1 Am. J. Hum. Genet. 63: , 1999 Detecting Make-Disease Association by Testing o Hady-Weinbeg Disequilibium at a Make Locus Dahlia M. Nielsen, 1, M. G. Ehm, 1 and B. S. Wei 1 Bioinomatics Depatment, Glaxo Wellcome, Inc., Reseach Tiangle Pak, NC; and Pogam in Statistical Genetics, Depatment o Statistics, Noth Caolina State Univesity, Raleigh Summay We eview and extend a ecent suggestion that ine-scale localization o a disease-susceptibility locus o a complex disease be done on the basis o deviations om Hady-Weinbeg equilibium among aected individuals. This deviation is diven by linkage disequilibium between disease and make loci in the whole population and equies a heteogeneous genetic basis o the disease. A inding o make-locus Hady-Weinbeg disequilibium theeoe implies disease heteogeneity and make-disease linkage disequilibium. Although a lack o depatue o Hady-Weinbeg disequilibium at make loci implies that disease susceptibility weighted linkage disequilibia ae zeo, given disease heteogeneity, it does not ollow that the usual measues o linkage disequilibium ae zeo. Fo disease-susceptibility loci with moe than two alleles, theeoe, cae is needed in the dawing o ineences om make Hady-Weinbeg disequilibia. Intoduction We will ee to ine mapping as attempting to naow what may be a 10-cM egion indicated by linkage analysis to an X1-cM egion containing the disease-susceptibility locus. Fine-mapping methods o qualitative and quantitative phenotypic taits have been unde constant development in ecent yeas. Simple Mendelian taits with high penetance ae oten ine mapped by ecombinant mapping: typing makes evey 1 cm, detemining haplotypes by use o extended amily inomation, and identiying ecombination events on eithe side o the supposed disease-susceptibility locus Received June 6, 1998; accepted o publication Septembe 9, 1998; electonically published Octobe 6, Addess o coespondence and epints: D. M. G. Ehm, Glaxo Wellcome, Inc., Bioinomatics Depatment, 5 Mooe Dive, Reseach Tiangle Pak, NC mge3716@glaxowellcome.com The Ameican Society o Human Genetics. All ights eseved /98/6305/0031$0.00 (Boehnke 1994). Glase et al. (1995) illustate this appoach in a seach o the gene esponsible o amilia hypeinsulinism. In the absence o high penetance o suicient numbes o patients, linkage-disequilibium methods in isolated populations have been used. Hästbacka et al. (199) used linkage disequilibium to map diastophic dysplasia (DTD) in Finland and indicated that the DTD gene should lie!0.06 cm om the CSF1R gene, which was late conimed (Hästbacka et al. 1994). Although the identiication o disease-susceptibility loci o complex taits has been slow, seveal ine-mapping methods have been employed. Identiication o IDDM (the insulin gene) was accomplished by use o linkage disequilibium (Bennett et al. 1995). Association studies played a key ole in implicating the apolipopotein E gene in late-onset Alzheime disease and heat disease (Code et al. 1993). Although methods o mapping simple Mendelian diseases use extended amilies collected o the genomic scan to eine disease-gene locations, ine-mapping techniques o complex diseases use samples with vaying chaacteistics. Seveal methods have been poposed. Association methods use unelated cases and contols. Taditional tansmission/disequilibium tests equie aected childen and thei paents (Spielman et al. 1993; Kaplan et al. 1997). Seveal tests using data sets that ae simple to collect and ae moe simila to those used in a genomic scan have been poposed. Tégouët et al. (1997) have poposed making use o estimating equations to estimate association paametes in samples o nuclea amilies o vaying sizes and in mixtues o elated and unelated individuals. Matin et al. (1997) have poposed two test statistics o association that use data om all aected childen (and thei paents) in a nuclea amily. Spielman and Ewens (1998) poposed a test statistic that tests o linkage disequilibium by use o aected and unaected siblings. Fede et al. (1996) have suggested that ine localization o a disease-susceptibility locus could be accomplished by use o deviation om Hady-Weinbeg (HW) equilibium among aected individuals. Fede et al. (1996) studied heeditay hemochomatosis (HH), a common autosomal ecessive disode o 1531

2 153 Am. J. Hum. Genet. 63: , 1999 ion metabolism. As descibed in thei pape, pevious localization o the HH gene placed it nea the majo histocompatibility complex on chomosome 6p and!1 cm om the HLA-A gene, although many epots have been contadictoy. Linkage-disequilibium studies conimed the existence o a ounde eect. To poceed with the localization o a gene involved in this disease, Fede et al. (1996) developed 45 shot tandem-epeat polymophism and single nucleotide polymophism (SNP) makes lying within an 8-cM egion suspected to contain the gene. All 45 makes wee typed in 101 HH patients and 64 contols. To estimate the position o the gene elative to these closely spaced makes, Fede et al. (1996) used the measue p excess (Bengtsson and Thomson 1981; Lehesjoki et al. 1993). This is a measue o linkage disequilibium that, in the pesence o linkage, is expected to be maximized at the make neaest the gene. Fede et al. (1996) plotted p excess o each make along the make map. This plot had a peak epesenting the maximum p excess in the egion; howeve, the peak was not vey shap, causing concen about the accuacy o these esults. In examining the data used in this study, Fede et al. (1996) noted that, among the aected individuals, thee appeaed to be an excess o homozygosity at the make loci. They consideed seveal explanations o this, the most likely o which poved to povide the basis o a new measue o linkage disequilibium. They noted that, o heteogeneous ecessive taits such as those which they wee studying, not only will an excess o homozygosity exist among aected individuals, but also this excess homozygosity should decease with deceased linkage disequilibium between the make loci and the disease-susceptibility locus. A disequilibium measue based on this obsevation has the advantage that only aected individuals need to be collected and genotyped, as opposed to the case-contol type studies necessay o most measues o association, such as p excess. In thei pa- pe, Fede et al. (1996) plotted a measue o HW disequilibium within thei HH-aected individuals, o each make along the make map. This plot had a maximum at appoximately the same point in the map as did p excess, but the peak was much shape. Fom this, Fede et al. (1996) concluded that thei initial esults om the p excess measue had been conimed and that the egion in which the gene lies had been moe accuately deined. We wee impessed with these esults and wee inteested in exploing the popeties o this measue. We have extended the model and have examined some geneal esults o this and othe measues. We also compae a test o HW disequilibium, to a diect test o linkage disequilibium. Methods Recessive Disease Model Fede et al. (1996) examined a heteogeneous ecessive model in which a subset o the disease cases ae due to a mutation in the egion o inteest and in which othe disease cases ae due to eithe unelated genetic loci o nongenetic actos. I A is used to denote the disease allele and Ā is used to denote all othe alleles at the disease-susceptibility locus, this model can be summaized as P (AectedFAA) 1, P (AectedFAA) w, and P (AectedFAA) w, whee w is the pobability that an individual will exhibit the disease because o causes othe than this locus. With the assumption o andom mating in the population, genotype and allele pobabilities at the disease-susceptibility locus among aected individuals can be calculated and include P (AAFAected) PAAFAected p A /, P (AFAected) pafaected p A(pA wp A )/, whee is the pevalence o the disease in the population. We have used pafaected and PAAFAected to di- eentiate between equencies among aected individuals and the whole-population equencies, denoted as pa and PAA, espectively. Fo this model, pa w(1 p A ). Depatue om HW equilibium at the disease-susceptibility locus can be measued by the disequilibium coeicient DAA PAA pa (Wei 1996). Among aected individuals, this coeicient becomes DAAFAected PAAFAected pafaected w(1 w)p (1 p )/. Fede et al. (1996) quantiied depatue om HW equilibium at the disease-susceptibility locus by use o the measue F A, deined as (Ho H e)/(1 H e), whee Ho and He ae the obseved and the expected homozygosities, espectively. Although they did not give an explicit expession o this quantity, it appeas to us that they used the omulation P P p p F AAFAected AAFAected AFAected AFAected A 1 p p A A D AAFAected/(pAp) A w(1 w)p p/. A A Association between the disease-susceptibility allele A and a make allele M can be expessed by use o the A A

3 Nielsen et al.: Detecting Make-Disease Association 1533 linkage-disequilibium measue DAM PAM pq A M, whee q M is the equency o make allele M. This quan- tity compaes the equency ( P AM ) o haplotypes caying both alleles A and M with the poduct o the sepaate equencies o the two alleles. D AM is positive when make allele M is moe likely to be associated with disease-susceptibility allele A than would be expected by chance. Fede et al. (1996) also discussed HW disequilibium at a biallelic make locus. With the assumption o andom mating in the whole population, pobabilities o the make alleles and make genotypes conditioned on having the disease include PMMFAected [(1 w)(pq A M D AM) wq M]/, q [wq (1 w)p (pq D )]/. MFAected M A A M AM The HW disequilibium coeicient at the make locus among aected individuals is DMMFAected w(1 w)d AM/. (1) This is non-0 only i w is neithe 1 no 0, implying that the disease must be heteogeneous, and that, i thee is linkage disequilibium, D AM ( 0. The HW-depatue measue o Fede et al. (1996) o the make locus is P P q q F MMFAected MMFAected MFAected MFAected M 1 qm qm D MMFAected/(qMq M ) w(1 w)d /( q q ). AM M M As stated by Fede et al. (1996), FM DAMF A, () whee DAM D AM/pA pq A MqM. Equations (1) and () captue the essential point that HW disequilibium at a make locus among aected individuals depends on the whole-population linkage disequilibium between the make locus and the disease locus. Although it is the latte quantity that is o inteest, it is easie to test o the ome. A test o HW disequilibium at the make locus can seve as a test o linkage disequilibium. It should be noted, howeve, that the measue F M poposed by Fede et al. (1996) depends on the values qm and qm, which ae whole-population paametes and cannot be estimated by use o aected individuals alone. A common diect measue o linkage disequilibium is the quantity p excess (Bengtsson and Thomson 1981; Lehesjoki et al. 1993). This measue compaes the equency o a make allele M among aected individuals ( q MFAected ) vesus the equency among unaected in- dividuals ( ). It is deined as q MFUnaected qmfaected qmfunaected pexcess. 1 q MFUnaected Fo the model o Fede et al. (1996), q [wq (1 w)p (pq D )]/, MFAected M A A M AM q (1 w)[q p (pq D )]/(1 ), MFUnaected M A A M AM so that (1 w)pd A AM p excess. (3) (1 ) [ q (1 w)pd /(1 ) ] M A AM Theeoe pexcess is popotional to DAM, and it eaches its maximum at the make with the geatest disequilibium with the disease. Note that w must be!1. HW disequilibium is popotional to the squae o disequilibium, so that F M is expected to be a moe sensitive indicato o linkage in the pesence o linkage disequilibium (eq. []). This appeas to have been the case in the analyses epoted by Fede et al. (1996). Geneal Disease Model We wished to know whethe equation (1) might be genealized to othe disease models, and so we consideed a moe geneal model with disease susceptibility aected by a locus with an abitay numbe o alleles, denoted by A. Unde this model, the conditional pobability that an individual has the disease, given that the individual has genotype AA s at the disease-susceptibility locus, is s. We will ee to these values as pen- etances, although we ecognize that, o some o the AA s genotypes, the s values should popely be called phenocopy ates. These values could equivalently be called pevalences : they epesent the pevalence o the disease within a genotypic class. This elates the notation s to the use o, the whole-population pevalence (the unconditional pobability that an individual has the disease). This value is SS t s spp s, whee p is the population equency o allele A at the disease-susceptibility locus and whee HW equilibium is assumed. In addition to the genotypic penetances s, we ind it convenient to deine an allelic penetance, : Ssp s s, which is the conditional pobability that an individual will have the disease, given that the individual has allele A (the othe allele being a andom allele om the population). Note that p. We conside a make locus with alleles M i occuing at equencies q i. Fo such a make, we ae likely to concentate on those alleles that show positive associ-

4 1534 Am. J. Hum. Genet. 63: , 1999 ations with the disease, meaning that they have highe equencies among aected than among unaected individuals. I P i is the population equency o haplotypes caying disease-susceptibility allele A and make allele Mi, then the population linkage disequilibium Di between these alleles is deined as Di Pi pq i. These coeicients sum to 0 ove all the alleles at eithe locus, so that SDi SiDi 0. We also wish to descibe the linkage disequilibium between make allele M i and the disease locus as a whole, and we do so by weighting the D i tems by the allelic penetances. This measue is witten as di S t Di SSp t s s sdi and sums to 0 ove i. The quantity d i is 0 i all disease susceptibility locus alleles have the same penetances. Fo a disease-susceptibility locus with two alleles, A1 and A, di is a multiple o D1i and so is popotional to the usual linkage-disequilibium coeicient and will maximize at the same point as does linkage disequilibium. In this two-allele case, i the penetances ae not the same, a 0 value o d i implies that thee is no linkage disequilibium between disease susceptibility and make loci. Fo a disease-susceptibility locus with moe than two alleles, howeve, it is possible o d i to be nea o equal to 0 even when thee is linkage disequilibium, since the values o D i do not all have the same sign and may have a (penetance-weighted) sum close to 0. The penetance-weighted linkage-disequilibium coeicient allows simple expessions o make-allele equencies among aecteds: qi d Aected qi (d/), as is shown in Appendix A. This equation shows that makeallele equencies among aected individuals deviate om the oveall population equencies by an amount that depends on the stength o association between the make allele and the disease-susceptibility alleles, weighted by the penetances o those alleles. A simila expession holds o the make-allele equency among unaected individuals, qi d Unaected qi [d i/(1 )], so that, in the whole population, qi qifaected (1 )q ifunaected. As a genealization o equation (3), the quantity p o make allele M becomes excess i p excess i. (1 ) [(1 q ) d /1 ( ) ] I M i is a make allele showing a positive association with the disease, then pexcess x 0, so that di x 0, and i these two quantities ae maximized togethe. Howeve, it is not necessay that each individual linkage-disequilibium coeicient D i be positive. Fo the geneal disease model, discussion o makelocus HW disequilibium equies an additional summay measue o linkage disequilibium. This quantity, dij, is deined o pais o make alleles, Mi and Mj, instead o o single make alleles: d SS DD. d i i i ij t s s i sj We tem it genotypic disequilibium, as opposed to the allelic disequilibium di. Note that idij jdij 0. Among aected individuals, the make-lo- cus homozygote HW disequilibium coeicients can now be witten as d d ii i DiiFAected PiiFAected qifaected, and the heteozygote disequilibia (Wei 1996) ae D P q q ijfaected ijfaected ifaected sfaected (dij dd) i j. Fo this moe geneal model, it is not clea that the HW disequilibia, DijFAected and DiiFAected, ae maximized when the linkage disequilibia, d i, ae maximized. It is clea, howeve, that some pattens o non-0 linkage disequilibium will esult in 0 depatue om HW equilibium at a make locus. Convesely, a depatue om HW equilibium at a make locus povides evidence both o linkage disequilibium between make and disease-susceptibility loci and o heteogeneity o disease susceptibility. Test Statistics We have discussed two measues that can be used to chaacteize make/disease associations. One is p excess, which is diectly popotional to linkage disequilibium measued in unelated aected and unaected individuals, and the othe is the HW-disequilibium coeicient measued among aected individuals. To compae these two appoaches, we conside the statistical powe o coesponding test statistics. A widely used statistical test o association based on unelated aected and unaected individuals that is, a case-contol design is the (m 1) -d x test based on the statistic x CC when the make locus has m alleles. When the make alleles have sample equencies p and ifaected pifunaected among n aecteds and n unaecteds, (pifaected p ifunaected) xcc 4n. (4) p p i ifaected ifunaected When altenatives to the null hypothesis o no disequilibium ae o the Pitman type (i.e., depatues tend towad 0 with sample size), the noncentality paamete o this statistic is (Meng and Chapman 1966)

5 Nielsen et al.: Detecting Make-Disease Association 1535 l CC ( qifaected q ) 4n q q ifunaected i ifaected ifunaected d i i i i 4n. (1 ) { q (1 )d / [ (1 ) ]} Elsewhee (Kaplan et al. 1997), we have witten the sum in this expession as I. To test o HW disequilibium at the make locus among the same total numbe o individuals, n aect- eds, the test statistic is (Wei 1996) x n x HW P ( iifaected qifaected) HW i q ifaected ( P q q ijfaected ifaected jfaected) n.! q q i j ifaected jfaected This has m(m 1)/ d and a noncentality paamete o (dij dd) i j lhw n. (q d )(q d ) i j i i j j When thee ae just two make alleles, m, the powe o the two x tests can be compaed diectly by compaing lcc with lhw. Fo this case, HW disequilib- ium decays at a ate popotional to the squae o linkage disequilibium (Appendix B). This indicates that the measue o HW disequilibium should be a moe sensitive indicato o position, decaying moe quickly than the measue o linkage disequilibium as the distance between the make locus and the disease-susceptibility locus inceases. Simulations To illustate ou theoetical esults, we peomed simulations o evolving populations segegating o a biallelic disease-susceptibility locus and seveal biallelic makes. We peomed these simulations unde ou dieent disease models, epesenting special cases o the geneal model. Analytical esults o these special cases can be ound in Appendix B. Fo the ou models, we peomed x tests o linkage disequilibium and o HW disequilibium and compaed the estimated powe o the esults. Fo all ou models, we consideed a make allele M, at equency qm.0, that had a positive association with the disease allele. Ou ist simulated model was the heteogeneous ecessive model o Fede et al. (1996), (5) with pa.10 and w.05. Fo these paametes, the maximum linkage disequilibium expected is.08. The second model was also o the Fede et al. (1996) type, but with dieent paamete values. Fo this model, we chose the paametes pa.05 and w.05. Since pa is smalle in the second model, less linkage disequilibium is possible, eaching a maximum expected value o.04, one-hal o what was expected in the ist model. The thid model was an additive model o penetance. We set the eect o the disease-causing allele (A) at.50 and set the eect o the nondisease allele ( Ā) at 0. This yields AA 1.0, AA.5, and AA.0. The equency o the disease allele in the population, p A, was.10. Fo the additive model, HW disequilibium is expected to be negative (Appendix B) and will incease in absolute value with inceasing linkage disequilibium. A multiplicative model o penetance was assumed o the outh set o simulations. We set the eect o the disease-causing allele (A) at.9 and set the eect o the nondisease allele ( Ā) at.05. This leads to AA.8100, AA.0450, and AA.005. The equency o the dis- ease allele in the population, p A, was.10. We did not expect to see any HW disequilibium among the aected individuals (Appendix B). A summay o the paamete values used in these ou models can be ound in table 1. Fo ou simulated populations, we consideed make loci positioned at distances o 0 cm om the diseasesusceptibility locus, consideing one make evey 0.5 cm. The populations stated at geneation G 0 with complete association between the disease allele and one allele at each make locus, then evolved o 50 geneations o andom mating. Fo each model, we etained the ist 100 populations, which, ate 50 geneations, had not expeienced substantial genetic dit at the disease locus. Fo a population to be accepted, the equency o the disease allele at the end o the evolution could not deviate om the oiginal equency by We made no adjustments o genetic dit at the make locus. Results Powe To detemine the powe to detect HW and linkage disequilibia, we peomed the x tests x and x (eqs. Table 1 Paametes o the Simulated Disease Models Model (Type) AA AA a AA p A p B D max 1 (heteogeneous ecessive) (heteogeneous ecessive) 3 (additive) (multiplicative) a Values ae maximum expected disequilibium. CC HW

6 1536 Am. J. Hum. Genet. 63: , 1999 [4] and [5], espectively) on samples taken om each population. Fo the case-contol test, we sampled 50 aected and 50 unaected individuals om each population. Fo the test o HW disequilibium, we sampled 100 aected individuals. We epeated both tests 5,000 times o each population, ecoding the pecentage o times that we ejected the hypothesis o no disequilibium. This ejection pecentage gave us an estimate o the powe o the espective tests. The compaisons o these esults can be seen in igue 1. The symbols in this igue ae box plots o the esults; the bottom and top edges o the box ae located at the sample 5th and 75th pecentiles, the point joined by the connecting line is the median, and the whiskes extend the ange o the esults. This igue shows that, o the Fede et al. type models (igs. 1A and 1B), the powe to detect HW disequilibium is geate in geneal than the powe to detect linkage disequilibium. This is paticulaly notewothy in the case o the second Fede et al. type model (ig. 1B), in which the powe to detect linkage disequilibium by use o xcc is not vey dieent om the a.05 nominal level. The additive model (ig. 1C) shows high powe o both tests. Fo the multiplicative model (ig. 1D), we expected to ind no HW disequilibium among aected individuals. These expeiments showed that, although the powe to detect HW disequilibium was vey low o this model, it was vey equently above the a.05 nominal level. This appeas to be caused by inceased vaiance o HW-disequilibium values, which is ceated by sampling o aected individuals. Figue 1 eveals the vaiability o the powe o the two x tests. Fo seveal o these expeiments, the powe o vaied om the nominal.05 level to values close x HW x CC to 1.0. was less vaiable in its powe. In these expeiments, we geneated linkage disequilibium in the pesence o physical linkage. Thus, both tests showed educed powe at geate distances between the loci. As expected, in the thee models in which we expected to ind HW disequilibium, the powe to detect HW disequilibium decayed moe quickly than did the powe to detect linkage disequilibium. Size To detemine the size o ou tests, we simulated a second set o populations unde the same ou disease models but segegating o a biallelic make located at 50% ecombination om the disease locus. We peomed the same sampling and testing expeiments as descibed above. The esults om these expeiments ae displayed in igue. These esults showed that the casecontol test, x CC, was consevative; in most cases ex- amined, it ejected the tue null hypothesis at a ate less than the a.05 nominal level. The ejection ate o the test o HW disequilibium,, appeaed to be cen- x HW teed aound the nominal a.05 level o the ist thee models but was highe o model 4, the multiplicative model. Fo model 4, the vaiance o the ejection ate appeaed to be quite lage. It was, howeve, vey simila to that seen in the populations geneated with linked makes, as shown in igue 1. Discussion We have examined depatues om HW equilibium that ae ceated by sampling o individuals on the basis o the pesence o a disease phenotype. These depatues om equilibium ae ceated because the selection citeion is based on disease-susceptibility genotypes, athe than on independently selected alleles. Alleles within genotypes that cone geate susceptibilities ae epesented in the sample at dispopotionally high ates. Disequilibium is expected to be geatest at the diseasesusceptibility locus itsel, since this is the acto that detemines the selection citeion. Loci that ae phenotypically neutal but ae somehow associated with the disease-susceptibility locus, such as genetic makes in linkage disequilibium with the disease-susceptibility locus, also expeience dispopotionate genotype selection. As the degee o association between disease susceptibility and make loci deceases, HW disequilibium at the make locus is also expected to decease. We have examined measues that captue this elationship, potentially oeing ine-mapping techniques that can be peomed on samples o aected individuals when an appopiate contol sample is not available. Fo a geneal disease model that consides an abitay numbe o alleles at the disease-susceptibility locus, we have poposed the measues di and dij. These ae summay measues, useul in quantiying the linkage disequilibia between make allele M i and the disease-susceptibility alleles. Since these measues allow o a simple expession o the make-allele equencies within aected and unaected individuals, they can be eadily incopoated into many established measues o association. This allows o simple intepetation o these measues. Unde cetain disease models, and when physical linkage and linkage disequilibium exist between the makes and a disease-susceptibility locus, conventional tests o HW disequilibium at make loci can be used to ine map disease-susceptibility loci. Fo biallelic locus models (in which both the disease-susceptibility locus and the make loci have only two alleles pe locus), HW disequilibium is popotional to the squae o linkage disequilibium (Appendix B). This indicates that measues o HW disequilibium ae expected to decay moe apidly than diect measues o linkage disequilibium as linkage disequilibium diminishes. The esults o Fede et al. (1996) illustate this: thei cuve plotting the make map vesus HW equilibium is shape than

7 Figue 1 Powe esults o the x tests o linkage disequilibium (gay-shaded boxes) and HW disequilibium (blackened boxes). A and B, ist and second heteogeneous ecessive models o Fede et al. (1996); C, additive model; and D, multiplicative model. The symbols epesent the ange o the popotions o times the hypothesis o no disequilibium was ejected o the 100 populations. The bottom and top edges o the box epesent the sample 5th and 75th pecentiles, the point joined by the connecting line is the median, and the whiskes extend the ange o the esults.

8 1538 Am. J. Hum. Genet. 63: , 1999 Figue Size o the x tests o linkage disequilibium (gayshaded boxes) and HW disequilibium (blackened boxes). Models 1 and wee the ist and second heteogeneous ecessive models o Fede et al. (1996), model 3 was the additive model, and the multiplicative model was model 4. These symbols epesent the popotion o times a tue null hypothesis was ejected. thei cuve plotting the make map vesus p excess, a mea- sue o linkage disequilibium. Fo a geneal disease model, allowing o two o moe alleles at the make and disease-susceptibility loci, the elationship between linkage disequilibium and HW disequilibium becomes less clea. Howeve, depatue om HW equilibium at a make locus povides evidence both o linkage disequilibium between make and susceptiblity loci and o heteogeneity o disease susceptibility, so tests o HW disequilibium could still be useul. Thee ae some caveats that should be consideed when a geneal disease model is examined. Fo simple biallelic-locus models, the intepetation o disequilibium measues is staightowad. Howeve, when moe than two alleles exist at the make and/o disease-susceptibility locus, complications aise. Fo these models, summay measues may be used to quantiy association between loci; howeve, although disequilibia between speciic alleles may exist, these disequilibia may cancel out when combined into the summay measue. This poses a challenge in the mapping o loci involved in complex taits, since it is doubtul that many o the taits o inteest ae biallelic. With the use o SNPs as genetic makes, some o the poblems with multiple alleles disappea. In this case, d d1. Howeve, i thee ae moe than two alleles at the disease-susceptibility locus, then the poblem o the disequilibia between the diseasesusceptibility alleles and the make allele canceling to within d 1 is still a concen. Tests o HW disequilibium will be the most poweul when lage amounts o disequilibium within a sample o aected individuals ae expected. The amount o HW disequilibium expected depends on both the degee to which the disease-susceptibility locus aects disease status and the manne in which the alleles within a genotype inteact. In the sampling o aected individuals, genotypes will be sampled popotionally to the ate o disease susceptibility that they cone. By deinition, HW disequilibium is the dieence between genotype popotions and the poduct o the popotions o the composite alleles. I the alleles within a genotype act in a multiplicative manne to cause inceased levels o disease susceptibility, the genotypes ae expected to be selected popotionally to the poduct o the allele equencies. Thus, with disease models in which the alleles act in a multiplicative manne, HW disequilibium is not expected to be ceated in the sample. The moe the eects o alleles deviate om multiplicative inteactions, the geate the amount o HW disequilibium that is expected. This is seen in the theoetical esults o Appendix B and is illustated in the esults o the simulations that we have peomed. We note that, i the penetances s ae egaded as genotypic values, much o the theoy in this pape can be applied to the study o quantitative taits. Acknowledgments This wok was suppoted in pat by National Institutes o Health gant GM45344 to Noth Caolina State Univesity. Paticula thanks ae due to D. Michael Wagne, Glaxo Wellcome, Inc., o suggesting that we investigate moe caeully the methods discussed. Appendix A Fo disease susceptibility locus homozygotes, the two-locus genotypes and thei equencies ae AAMM i i (pq i D i), AAMM (pq D )(pq D ), i ( j; i j i i j j those o disease susceptibility locus heteozygotes ae

9 Nielsen et al.: Detecting Make-Disease Association 1539 AAMM (pq D )(pq D ), ( s, s i i i i s i si AAMM (pq D )(pq D ) s i j i i s j sj (pq D )(pq D ), ( s, i ( j. j j s i si Among aected people, theeoe, the make genotype equencies ae 1 P (pq D )(pq D ) iifaected s i i s i si s qidi d ii qi, 1 P [(pq D )(pq D ) ijfaected s i i s j sj s (pq D )(pq D )] j j s i si (qidj qjd i) dij qq i j, i ( j, whee di SSp s s sdi S t Di and dij SS s sdd i sj. Adding ove genotypes povides the make allele equencies: Appendix B j(i [ j (qidj qjd i) dij] 1 q P P ifaected iifaected ijfaected qq i j d i q i. Heteogeneous Recessive Model Fo disease susceptibility locus alleles A and A, make alleles M and M, 1, and w, AA AA AA Geneal Biallelic Model Fo disease susceptibility locus alleles A and A make alleles M and M, dm [p A( AA AA ) (1 p )( )]D, A AA AA AM dmm ( AA AA AA )D AM, (AAAA AA )DAM DMMFAected. Additive Susceptibilities I s a as, then a p, d a D, i i dij 0, adi sasdsj DijFAected ( )( ) X 0, i s. ap sasps Multiplicative Susceptibilities I s aa s, then ( ) a p, ( )( ) d a p a D, i i d a D a D, D 0. ijfaected ( )( ) ij i s sj s and pa w(1 p A), dm (1 w)pd A AM, d (1 w)d, MM AM w(1 w)d AM D MMFAected x 0. Reeences Bengtsson BO, Thomson G (1981) Measuing the stength o associations between HLA antigens and diseases. Tissue Antigens 18: Bennett ST, Lucassen AM, Gough SCL, Powell EE, Undlien DE, Pitchad LE, Meiman ME, et al (1995) Susceptibility to human type 1 diabetes at IDDM is detemined by tan-

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