ARTICLE Overcoming the Winner s Curse: Estimating Penetrance Parameters from Case-Control Data

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1 ARTICLE Overcoming the Winner Cure: Etimating Penetrance Parameter from Cae-Control Data Sebatian Zöllner and Jonathan K. Pritchard Genomewide aociation tudie are now a widely ued approach in the earch for loci that affect complex trait. After detection of ignificant aociation, etimate of penetrance and allele-frequency parameter for the aociated variant indicate the importance of that variant and facilitate the planning of replication tudie. However, when thee etimate are baed on the original data ued to detect the variant, the reult are affected by an acertainment bia known a the winner cure. The actual genetic effect i typically maller than it etimate. Thi overetimation of the genetic effect may caue replication tudie to fail becaue the neceary ample ize i underetimated. Here, we preent an approach that correct for the acertainment bia and generate an etimate of the frequency of a variant and it penetrance parameter. The method produce a point etimate and confidence region for the parameter etimate. We tudy the performance of thi method uing imulated data et and how that it i poible to greatly reduce the bia in the parameter etimate, even when the original aociation tudy had low power. The uncertainty of the etimate decreae with increaing ample ize, independent of the power of the original tet for aociation. Finally, we how that application of the method to cae-control data can improve the deign of replication tudie coniderably. Identification of the genetic variant that contribute to complex trait i an important current challenge in the field of human genetic. Although there i a teady tream of reported aociation, replication of finding i often inconitent, even for thoe aociation that do ultimately turn out to be genuine. 1, In large part, the difficultie of replication occur becaue mot genuine aociation have modet effect; hence, there i generally incomplete power to detect aociation in any given tudy. Thee challenge will undoubtedly continue a we move into the era of affordable whole-genome aociation tudie, through which it i poible to detect variant of mall effect anywhere in the genome. 3,4 When a tudy identifie a marker that how evidence of aociation with a dieae, it i common to etimate the impact of thi variant on the phenotype of interet. Thi impact i often expreed a an odd ratio that i, the ratio of the odd of manifeting the dieae in carrier of the rik allele to the odd of manifeting the dieae in noncarrier. A complete decription of the impact of a variant affecting a binary phenotype include two et of parameter: the frequencie of the genotype and the penetrance of the genotype. Thee parameter can be ued to ae the impact of the detected variant, a meaured, for example, by the attributable rik. Etimation of the trength of the effect of a genetic variant on the phenotype i alo helpful for planning ucceful replication tudie. Unfortunately, etimation of thee parameter with the ame data et that wa ued to identify the variant of interet i not traightforward, ince the data et doe not contitute a random population ample for two reaon. Firt, ample that are ued for aociation mapping are uually collected to overample affected individual relative to their frequency in the population (e.g., the ample might include equal number of cae and control). Second, and more eriouly, there i a major acertainment effect that occur when a variant i of interet pecifically becaue it wa ignificant for aociation. For a variant that i genuinely but weakly aociated with dieae, there may be only low or moderate power to detect aociation. Hence, when there i a ignificant reult, it may imply that the genotype count of cae and control are more different from each other than expected. Conequently, the etimate of effect ize are biaed upward. Thi effect, which i an example of the winner cure from economic, 5 depend trongly on the power of the initial tet for aociation. 6 If the power i high, mot random draw from the ditribution of genotype count will reult in a ignificant tet for aociation; thu, the acertainment effect i mall. On the other hand, if the power i low, conditioning on a ucceful aociation can will reult in a big acertainment effect. Thi problem i well appreciated in the field. It ha been oberved that the odd ratio of a dieae variant i uually overetimated in the tudy that firt decribe the variant. In a meta-analyi of 301 aociation tudie of 5 putative dieae loci, Lohmueller et al. concluded that, even though the replication tudie indicated that 11 of the loci were genuinely aociated with dieae, a triking 4 of the From the Department of Biotatitic and Pychiatry and Center for Statitical Genetic, Univerity of Michigan, Ann Arbor (S.Z.); and Department of Human Genetic, Univerity of Chicago, Chicago (J.K.P.) Received Augut 1, 006; accepted for publication January 8, 007; electronically publihed February 16, 007. Addre for correpondence and reprint: Dr. Sebatian Zöllner, Department of Biotatitic, Univerity of Michigan, 140 Wahington Height, Ann Arbor, MI zoellne@umich.edu Am. J. Hum. Genet. 007;80: by The American Society of Human Genetic. All right reerved /007/ $15.00 DOI: / The American Journal of Human Genetic Volume 80 April

2 5 loci were reported in the initial tudy to have an odd ratio higher than the etimate baed on ubequent replication tudie. One important implication of the winner cure i that it make it hard to deign appropriately powered replication tudie. If the ample ize of a replication tudy i choen on the bai of the odd ratio oberved in the initial tudy, then the replication will almot certainly be underpowered. 1 Göring et al. 6 were the firt to highlight the challenge of the winner cure for genome can. Studying the problem for variance-component linkage can of QTL, they howed that, for low-powered tudie, the naive etimate of the genetic effect at a ignificant locu i almot uncorrelated with the true underlying genetic effect; thi reult in a ignificant overetimation. They concluded that the only method for generating a ueful etimate i to collect a large, independent, population-baed ample of individual, without regard to phenotype, and then to phenotype and genotype each individual. Although uch a ample would provide unbiaed etimate of allele frequencie and penetrance parameter with properly calibrated confidence region, thi may be a prohibitively expenive olution to the problem. Thu, it i of interet to develop method that generate unbiaed etimate of the parameter of interet from the ame data et ued to detect the variant. Several author have ince decribed method that correct for the acertainment bia when etimating genetic effect after a whole-genome can for linkage. Allion et al. 7 noted that, for a pecified genetic model, the ditribution of the effect, contrained by acertainment bia, can be analyzed. Uing a method-of-moment approach, they calculated an etimate of the genetic effect. Siegmund 8 propoed lowering the confidence limit in the initial tet, thu accepting a large number of fale-poitive reult. A indicated above, thi lead to a high power for each individual tet and only a mall acertainment effect. Siegmund 8 then uggeted calculating CI and accounting for the high number of tet by increaing the tringency of the CI. To correct for acertainment, thi method alo require pecifying the genetic model. Sun and Bull 9 uggeted multiple method baed on randomly plitting the ample into a detection ample and an etimation ample. By comparing the etimate generated from the detection ample and the etimate from the etimation ample, they were able to calculate a correction factor for the acertainment effect. However, the reulting etimator i till omewhat biaed, and the SE of the corrected etimate i actually higher than the SE of the naive etimator. 9 In a omewhat analogou etting, one tudy of family-baed aociation tet propoed that the available information be plit into two orthogonal component. 10 Then, one component of the information i ued to validate promiing ignal from the other component. However, that tudy focued primarily on teting rather than etimation. Although ome of the method for correcting for the winner cure in linkage tudie could be extended to aociation tudie, aociation tudie differ from linkage tudie in everal repect. The ample collected for an aociation tudy i more imilar to a random population ample, which allow a more precie calculation of the ampling probabilitie. Furthermore, the power of an aociation tudy hould be much higher than the power of a linkage tudy of the ame trait, 11 o it i not clear how well the concluion of Göring et al. 6 apply to whole-genome can for complex dieae. The goal of the preent tudy wa to develop a method for generating corrected etimate of genetic-effect ize for a locu that wa identified in a ignificant tet for aociation. Intead of calculating an odd ratio or relative-rik parameter, we ue information about the population prevalence of the dieae to etimate directly the penetrance parameter of the variant of interet. Thi allow u to perform the etimation for any pecific genetic model (e.g., additive or dominant) or for a completely general genetic model. We decribe an algorithm for calculating the approximate maximum-likelihood etimate (MLE) of the frequencie and the penetrance parameter of the genotype and aociated confidence region. We find that, for a variety of genetic model, our etimator correct the acertainment effect and provide reaonably accurate etimate and well-calibrated CI while lightly underetimating the genetic effect. We how that thee corrected etimate provide a far better bai for deigning replication tudie than do the naive uncorrected etimate. Lat, we how an application of the method to the aociation of the Pro1Ala polymorphim in PPARg with type diabete. 1 Method Model We developed a fairly general model for calculating the likelihood of a et of penetrance parameter and genotype frequencie conditional on having oberved a ignificant ignal for aociation at a certain biallelic marker. It i aumed that ignificance i determined according to a prepecified tet and type 1 error rate (a). In a data et of n a affected individual and n u unaffected individual, we conider the three genotype g 1, g, and g 3 indicating, repectively, the minor-allele homozygote, the heterozygote, and the major-allele homozygote. Let the data D p (a 1,,a 3,u 1,u 3) be the count of thee genotype in affected and unaffected individual that contitute the ignificant ignal for aociation. Furthermore, let f p (f 1,,f 3) be the population frequencie of the genotype, let v p (v 1,,v 3) be the penetrance, and let F be the population prevalence of the dieae phenotype. F, which i aumed to be known from independent data, will be ued to contrain the ample pace for the other parameter a follow: 3 i F p vf. (1) We plit the acertainment into two part. Let B indicate that, a required, the marker of interet how ignificant aociation at level a, and let S be the experimental deign of ampling n a af- i 606 The American Journal of Human Genetic Volume 80 April 007

3 fected individual and n u unaffected individual, regardle of the prevalence F. We uepr S (7) a a horthand for Pr(7FS). We calculate the likelihood L(v,f) and obtain an MLE for v and f, uing the equation L(v,f) p Pr (DFB,v,f) Pr (BFD,v,f) # Pr (DFv,f) Pr (DFv,f) p p. Pr (BFv,f) Pr (BFv,f) Thi reult i obtained uing the fact that the data D contitute, by definition, a ignificant reult, o D implie B; hence, Pr (BFD,v,f) p 1. The numerator on the right ide of equation () i the likelihood of the oberved genotype count, and the denominator i the power of the tet ued in the initial genome can. The numerator i maximized at the naive penetrance etimate. Meanwhile, the denominator (power) i made maller a the penetrance value move cloer together. Thi ha the effect of tilting the maximum likelihood toward maller difference among the penetrance. Notice that equation () i undefined when the power i 0 (e.g., if all the fi p 0); however, power p 0 implie that oberving a ignificant reult i impoible. Since we condition our etimation on oberving a ignificant reult, thi cae can be ignored. Under the aumption that the ample of affected and unaffected individual are only a mall proportion of the affected and unaffected individual in the population, Pr (GFv,f) i the product of two multinomial ditribution () 3 3 n a! n u! ai ui Pr S(DFv,f) p 3 Pr (gifa) 3 Pr(giFU), (3) a i! u i! where A indicate the affected phenotype and U the unaffected phenotype. Pr (gifa) i the probability that a randomly elected affected individual carrie genotype g i and can be calculated and Pr (AFg i)pr(g i) v i# fi Pr (gifa) p p Pr (A) F (1 v i )f Pr (g FU) p i i. 1 F A general expreion for the denominator of equation () i Pr (BFv,f) p Pr (D Fv,f), (4) S S i Di ignificant where the D i repreent all ignificant realization of the data vector D. For many deign of tet for aociation, it i poible to calculate the power of the initial tet exactly (ee appendix A). To apply thi algorithm to tet that do not have a imple method of power calculation, equation (4) can be evaluated by ampling Di conditional on v and f and approximating Pr (BFv,f) by Monte Carlo integration. Thee calculation aume that control are elected to not how the phenotype of interet. If random (unphenotyped) control are ued, equation (3) i modified by replacing Pr (gifu) with Pr(g i). Note that, although, in the initial can for aociation, everal tet can be performed at each of many marker, the multiple teting affect the etimate only indirectly through the choice of the level of ignificance, a. The equation can be extended to etimate gene-gene interaction parameter. To ae the interaction of m loci in the genome, all poible combination of genotype have to be conidered, o there are k p 3 m tate. Enumerating thee tate 1,,k, the et of genotype count can then be expreed a D p {u 1,,u k,a 1,,a k}, and the goal i to etimate the vector of population genotype frequencie f p (f 1,,f k) and the vector of penetrance v p (v,,v ) by extending equation (1) (4). The Algorithm 1 k We deigned a two-tage algorithm to etimate the population frequencie of the underlying genotype and their penetrance parameter conditional on a known dieae prevalence F, by maximizing L(v,f). In the firt tep, we generated an approximate likelihood urface by ampling m p 30,000 independent et of parameter conditional on F (i.e., that atify eq. [1]). The three genotype were aumed to be in Hardy-Weinberg proportion in the overall population. We then calculated the likelihood L(DFf 1,,f 3, v 1,,v 3) for each parameter et and elected the et with the highet likelihood a a firt approximation of the point etimate of the parameter. In the econd tep, we improved thi etimate by perturbing each parameter value by a mall value e and accepting the new parameter value if the likelihood i higher than the likelihood of the old maximum. By repeating thi procedure 3,000 time, reducing e with every repetition, we generated highly table etimate. We aeed the fidelity of thi algorithm by analyzing a et of data et multiple time, and we oberved that the parameter etimate differed by a magnitude of only To generate etimate for a known genetic model, we repeated the analyi in parameter pace that are contrained accordingly. We generated 95% confidence region by comparing the likelihood of all initial m parameter point with the likelihood of the point etimate. We included all point for which twice the difference of log-likelihood wa!95th percentile of a x ditribution with 3 df. (The model contain three free parameter: an allele frequency and three penetrance that are jointly contrained to produce the overall dieae prevalence F.) For impler genetic model, we ued fewer degree of freedom a appropriate. Analyi of the Algorithm We teted our algorithm by imulating aociation can data and then etimating the true underlying parameter according to the following cheme: 1. Simulate genotype count D on the bai of minor-allele frequency f and penetrance v 1, v, and v 3 for a ample of n cae and n control, according to equation (3), under the aumption of Hardy-Weinberg equilibrium in the general population.. Perform x tet of ignificance for the imulated data, comparing allele count in cae and control at ignificance level a p If ignificant, move to tep 3; otherwie, return to tep Generate three etimate ( ˆfv ˆ ˆ ˆ 1,v,v3) on the bai of D: a. without correcting for the acertainment effect (naive etimator), The American Journal of Human Genetic Volume 80 April

4 b. with ue of the correction for acertainment decribed here, and c. imulating a econd, independent ample and generating the etimate on the bai of the econd ample. Uing thi procedure, we performed two imulation tudie: (1) a imple example to explore the baic propertie of the winner cure and the propoed correction and () a et of tudie imulated under a large number of penetrance model, to ae the effect of genetic model, ample ize, and power of the initial tudy. Simulation tudy 1. We et the minor-allele frequency f p 0. and conidered three additive combination for the penetrance parameter (v 1,v,v 3) : (0.18, 0.14, 0.1), (0.5, 0.175, 0.1), and (0.3, 6 0.1, 0.1). At a ample ize of n p 400 and a p 10, thee three parameter et yielded power of 0.03, 0.54, and 0.97, repectively. We aumed that the underlying model wa known to be additive, and we contrained the parameter pace accordingly. For each etimate, we calculated the etimated additive component (v ˆ 1 ˆv ) and compared the reulting ditribution. Simulation tudy. We ampled random data et of 1,000 parameter combination for each n {100,00,400,800}, generating the minor-allele frequency f from a uniform ditribution on (0.05,0.5), drawing v 1 and v 3 independently from a uniform ditribution on (0,1), and etting 1 vp (v1 v 3) (additive model). We ampled from uch data et to have approximately uniformly ditributed power in the initial tet for aociation, by evaluating 100 parameter et with a power between and 0.1, 100 et with a power between 0.1 and 0., and o on; the lat category wa 100 parameter et with a power between 0.9 and All other parameter et were rejected. For each parameter et, we generated one et of genotype count D. On the bai of D, we obtained one etimate under the aumption that the underlying genetic model wa known to be additive, contraining the parameter pace accordingly, and a econd etimate under the aumption that the model wa unknown. We ummarized the difference between the etimated penetrance vˆ, ˆ, and ˆ 1 v v3 and the true value v 1, v, and v 3 a the um of quared deviation (q) 3 ˆ q p (vi v i). To evaluate the bia of the different etimation method, we calculated the true additive effect v v1 and compared it with the etimate of the ame effect, calculating (v v ) (vˆ v ˆ 1 1) D p v v1 for corrected etimate and uncorrected etimate. Our reult indicate that the winner cure ha only a mall effect on etimate of the allele frequency f (data not hown). Furthermore, we evaluated whether the true underlying parameter lie within the confidence region by comparing the likelihood L(v,f ) and L(v,f ˆ ˆ ). We calculated the relative ize of the confidence region, dividing the number of ampled parameter point that were in the confidence region by 30,000, the total number of parameter point ampled during tep 1 of the etimation procedure. We repeated the analyi by imulating data under four genetic model: dominant ( v1 p v), receive ( v p v3), multiplicative ( v p v 1# v3), and a general three-parameter model ( v ampled independently). When we performed the ame analyi, we found no qualitative difference among model (data not hown). To ae the effect of mipecifying the dieae prevalence, we imulated 1,000 aociation tudie of 400 cae and 400 control in which the pecified prevalence wa 0%, 40%, or 60% of the true prevalence parameter. We oberved that the etimate of the penetrance parameter were affected by thi change, in the ene that the mean quare error increaed with increaing mipecification. However, the median odd ratio of the minor allele compared with that of the major allele wa unaffected. Furthermore, when ample ize were calculated for a replication tudy with an allele-baed tet, ample ize predicted under a mipecified prevalence were a precie a ample ize predicted under the correct prevalence. To etimate the ample ize neceary for a replication tudy, we calculated the minimum ample ize ( ŝ) neceary to achieve 6 a power of 0.80 at a ignificance level of 10 under the aumption of a tet that compared allele frequencie on the bai of ˆv and ˆf generated with the uncontrained model (ee appendix A). We performed the ame computation for each point in the confidence region, conidering the maximum of the reult a the upper bound of ample ize neceary to replicate the aociation ( ŝ m ). We repeated thi calculation for three additional tet of aociation, comparing (1) the number of homozygote for the minor allele, () the number of homozygote of the major allele, and (3) the number of heterozygote in a x tet. We elected the tet that required the mallet of thoe four ample ize a the bet tet for a replication tudy and recorded the ample ize required under thi bet tet. Reult For a qualitative aement of the algorithm, we elected three parameter et with additive penetrance repreenting a low-powered tudy (0.03), a moderately powered tudy (0.54), and a high-powered tudy (0.97). For each parameter et, we imulated 100 ignificant aociation tudie of 400 affected and 400 unaffected individual and obtained three etimate of the penetrance parameter: (1) without correcting for the acertainment effect, () with application of the algorithm decribed earlier to correct for acertainment, and (3) with an independent ample that wa not conditioned to how a ignificant aociation reult. For the low-powered parameter et, the uncorrected etimate for each of the 100 imulated tudie i coniderably higher than the true underlying value, clearly demontrating the effect of the winner cure (fig. 1). In moderately powered tudie, the uncorrected etimate i till clearly biaed toward overetimation of the genetic effect: in 93 of the 100 tudie, the etimated effect of the variant i higher than the true underlying effect ize. Only in the high-powered tudy are the etimate centered on the true underlying value. In contrat, the etimate that i corrected for acertain- 608 The American Journal of Human Genetic Volume 80 April 007

5 Figure 1. Etimate of the genetic effect generated for three parameter et with low, intermediate, and high power. For each parameter et, 100 ignificant aociation tudie were generated, and the underlying parameter were etimated by three different method: (1) not correcting for acertainment (diamond), () correcting for acertainment (circle), and (3) collecting a econd unacertained ample and baing the etimate on that ample (quare). The vertical axi how the etimated genetic effect; the horizontal axi group the etimate by the power of the initial tudy. The horizontal line in each power category indicate the true underlying genetic effect, and the hort horizontal bar indicate the average of each ditribution of 100 data et. uncorrected etimate for low- and moderately powered tudie are more tightly clutered around the biaed average, wherea etimate generated with the corrected method are more widely dipered. Thee reult illutrate the problem of the winner cure in low-powered aociation tudie and how that our algorithm generate nearly unbiaed etimate of the penetrance parameter. To ae the bia of our method in a more ytematic fahion, we calculated the relative difference between the etimated and true underlying genetic effect, D, for each data et generated in imulation tudy (ee the Simulation tudy ection) that wa imulated with an additive model. Without correction for acertainment, the genetic effect i overetimated by 0%, on average, over all parameter et, independent of ample ize. After applying the correction for acertainment, we underetimate the additive effect of the cauative allele by 1% of the true additive effect. To ae the effect of the power of the initial tet for aociation on thee biae, we tratified the imulated aociation can by the power of each can and averaged the bia over all parameter et in power interval of 0.10, tarting in a bin of power. Figure how the average bia for cae and control ample ize between 100 and 800. Figure indicate that the bia of the uncorrected etimate i trongly dependent on the power of the underlying tet for aociation (olid line). If the power of the initial can i low (!10%), then ment yield a ditribution of etimate that i centered on the true value in all three cae. The average of the etimated effect i cloe to the true value, regardle of the underlying power, although the etimate are biaed lightly downward. However, in every et of imulated data et analyzed with acertainment correction, we oberve a cluter of imulated data et in which the etimated genetic effect i far below the true effect. Thee point repreent cae-control tudie in which the tet tatitic i only lightly larger than the critical value C. The probability of oberving uch a tet tatitic near C conditional on oberving a tet tatitic 1C i large for genetic model with mall effect ize, but it i not very large for model with moderate or large effect ize. In the model we conidered, whenever the tet tatitic wa cloe to C, a model of low genetic effect had the highet likelihood. Furthermore, under mot genetic model, ome data et generated a tet tatitic near C. In thee data et, the effect ize wa underetimated after correction for acertainment. Thu, the corrected etimate ha a bia toward underetimating the effect ize. We can alo oberve difference in the variance of the etimate. In general, figure 1 reveal that etimate of genetic effect can vary widely, even if no acertainment bia i introduced. In comparion with the ditribution of etimate generated from the unacertained ample, the Figure. Bia of the uncorrected and corrected etimate of the additive genetic effect. For each of the ample ize, the data et ha been tratified into 10 categorie of power indicated on the horizontal axi. The vertical axi indicate the average relative bia oberved in each power category. We performed imulation with four ample ize, a indicated by the legend. The olid line how the bia of etimate of penetrance parameter that were generated without correction for acertainment, wherea the dahed line how the bia of etimate generated while correcting for acertainment. The American Journal of Human Genetic Volume 80 April

6 Figure 3. Accuracy of point etimate for penetrance parameter dependent on ample ize (horizontal axi). All etimate were corrected for acertainment. The height of each bar indicate the average difference between the true and inferred parameter meaured a q (ee the Method ection), and the black portion of the bar diplay the median q tatitic. The dark gray bar how the q error of etimate generated without conditioning on a genetic model. The white bar how the q error of an etimate generated from an unacertained ample of the ame ize without knowing the underlying model. the additive effect i overetimated by 70%, on average, when the acertainment effect i not taken into account. In contrat, if the power of the initial tet i 10.90, then the uncorrected etimate ha almot no bia. By comparion, figure illutrate that the corrected etimate i biaed toward underetimating the genetic effect but that the power of the initial tudy ha only limited effect on the bia of the corrected etimate (dahed line). The average bia for parameter et with power!0.10 ( 17%) i only lightly more pronounced than the average bia of data et for which the power i between 0.60 and 0.70 ( 11%). Only if the power of the initial can i i the bia of the corrected etimate negligible. We analyzed the impact of ample ize on the preciion of the corrected etimate, uing data et generated with imulation cheme (ee the Simulation tudy ection). Since the winner cure effect i dependent on the power of the initial tet for aociation, it can be expected that, for a given et of parameter, the preciion of an etimate will increae with ample ize, ince the power rie with ample ize. To determine whether ample ize would alo have a direct effect on the preciion of the etimate beyond the effect of increaed power, we conditioned each data et of 1,000 imulated can to have a uniform ditribution of power in the initial tet for aociation. Here, we preent data imulated with the additive model. Generating data with other genetic model yielded imilar reult. Figure 3 diplay the difference between the etimated and the true parameter a a function of ample ize. The dark gray bar how the average q error for etimate generated with our method, under the aumption that the true underlying model i unknown. The reult indicate that the accuracy of the etimated penetrance trongly depend on ample ize. Analyi of aociation tudie with 800 cae and control reulted, on average, in roughly a quarter of the q error of the ame analyi of tudie with 100 cae and control. Figure 3 alo how that the median of the error core (black part of each bar) wa coniderably lower than the average, indicating that the ditribution of the relative error and q over 1,000 imulated data et conit of motly etimate with a mall error and a few etimate with a high error. Thee tatitic can be appreciated by comparing them with an etimate generated from an independent ample of equal ize with the aumption of unknown penetrance parameter (fig. 3, white bar). Unurpriingly, the unacertained ample generated more-precie etimate (maller q value) than did the acertained ample. Neverthele, comparion of the value between the ample ize revealed that the median q of an independent ample wa roughly the ame a the median q of an acertained ample with twice the ample ize. We can quantify the uncertainty of the point etimate a a four-dimenional confidence region correponding to the three penetrance parameter and the allele frequency. Aeing whether the confidence region were well calibrated, we oberved the true value falling in the contructed 95% confidence region in 94.7% of all imulated data et. Thu, our tatitic yield appropriate confidence region. The volume of thi confidence region ummarize the extent of uncertainty a a ingle tatitic. We etimated thi volume for each imulated data et relative to the ize of the underlying parameter pace and oberved a trong effect of ample ize on the ize of the confidence region. For each doubling of the ample ize, the average volume of the confidence region wa approximately halved. Furthermore, the reult indicate that etimate generated from a econd independent ample have only lightly maller confidence region than do the bia-corrected etimate obtained from the original aociation ample. Replication Studie Uing the MLE of minor-allele frequency and penetrance parameter, we calculated the ample ize ( ŝ) required to achieve power of 0.80 at a ignificance level of a p 10 6 in a replication tudy with an allele-baed tet for aociation. Firt, we applied thi analyi to etimate from 100 imulated aociation tudie, hown in the Low power ection of figure 1, generated in imulation tudy 1. For each etimated et of parameter, we calculated the replication ample ize ŝ and the upper bound (ee the Method ection). We compared thee etiŝ m 610 The American Journal of Human Genetic Volume 80 April 007

7 mated ample ize with the true ample ize required to replicate the aociation with power of 0.80 and oberved that none of the ample ize baed on uncorrected etimate wa ufficient (fig. 4); none generated a power For 7% of all imulated data et, even the upper bound of thi ample ize ŝ m i too mall to generate ufficient power. In contrat, the ample ize calculated uing the corrected etimate are centered on the true required ample ize. Neverthele, the ditribution of ample ize i quite broad. For many imulated data et, the calculated replication ample ize i quite different from the truth. Since the effect ize i low, penetrance etimate that differ by 0.0 reult in very different replication ample ize. The uncertainty in etimate of mall effect ize i reflected in the large upper bound of each etimate; only 4 of the 100 data et reulted in an ŝ m!10,000. A imilar level of un- certainty can be oberved in etimate generated from unacertained ample; here, we oberve that only 35 unacertained ample reulted in ŝ m!10,000. However, thi analyi aume that the tet for the replication tudy i choen independent of the etimated penetrance parameter. Intead, we can elect the tet that i mot powerful for the etimated parameter. To evaluate thi approach, we compared two trategie for planning replication tudie: (1) elect a tet for aociation independent of the etimated parameter and then etimate the neceary ample ize and () elect the tet tatitic that i mot powerful for the etimated penetrance parameter and etimate the ample ize required for a replication tudy with that tet. We applied both trategie to data et imulated with imulation tudy, uing threeparameter model, receive model, and multiplicative model for variou different ample ize (table 1). All data were analyzed uing the three-parameter model; thu, no knowledge about the underlying genetic model wa aumed. Firt trategy. For each data et, we calculated the ample ize ŝ required to replicate the aociation with an allele-baed tet. The median ratio between ŝ and the true ample ize i lightly 11.0 (table 1, column ). The median deviation between the true and the etimated neceary ample ize relative to the true neceary ample ize i 0.4, indicating that, in 50% of all imulated cae, the predicted neceary ample ize i over- or underetimated by 40%. Both the median ratio and the median deviation eem to be unaffected by the underlying genetic model or the ample ize. Second trategy. To deign the replication tudy on the bai of the etimated penetrance parameter, we calculated the replication ample ize ŝ for four different tet of aociation, electing the one that required the mallet ˆ (i.e., ˆ min ). The four tet aeed the difference between affected and unaffected individual in (1) the homozygote for the minor allele, () the homozygote for the major allele, (3) heterozygote, and (4) allele count. We then calculated the true ample ize neceary for the elected optimal tet and the ratio between the etimated ample ize and the neceary true ample ize; ee appendix A for detail about thee calculation. The reult for thee comparion are een in column 3 and 5 of table 1, which diplay the ame tatitic a decribed above, wherea the lat column how the ratio of ŝ min generated with trategy to ŝ generated with trategy 1. Other than for the firt trategy, the ummary tatitic depend on ample ize and on the underlying genetic model. If the optimal tet i the allele-baed tet employed in the firt trategy, we expect no improvement from applying the econd trategy. Thi can be oberved for data generated with the multiplicative model in which the allele-baed tet wa almot alway the mot powerful tet. In contrat, for parameter et generated with the threeparameter model, the allele-baed tet i only ometime optimal. Thu, the required ample ize for replication tudie deigned with the econd trategy i about half (38% 59%) of the ample ize required under the firt trategy. We alo oberve that the median ratio ŝ : i cloe to 1, and the median normalized difference between the etimated value and the true value i 0. ( ), indicating that ample ize calculated here are centered on the true required ample ize and are reaonably accurate. Finally, for data imulated with the receive model, the Table 1. Deign of Replication Studie Baed on the Etimated Penetrance Parameter Sample Size and Genetic Model Allele Baed ŝ Replication Tet Fˆ F ŝ min Bet Model Fˆ min F Ratio 100/100: Multiplicative Three parameter Receive /00: Multiplicative Three parameter Receive /400: Multiplicative Three parameter Receive /800: Multiplicative Three parameter Receive NOTE. Deigning ample of the indicated ize (number of cae/number of control) were imulated auming a multiplicative model, a three-parameter model, or a receive model. For each replicate imulation, the ample ize ŝ to obtain 80% power in a replication tudy wa etimated. Here, ŝ i compared with the true required ample ize for two tet trategie: with ue of the allelebaed tet or with choice of the bet of everal tet tatitic. Column and 4 how the median ratio, and column 3 and 5 diplay the median relative difference. The final column how the ratio between the etimated ample ize for the firt and the econd trategy. The American Journal of Human Genetic Volume 80 April

8 Figure 4. Etimated ample ize for a replication tudy. We ued the 100 parameter etimate for the low-powered tudy decribed in figure 1 to calculate the ample ize required to achieve power for a p 10 in a replication tudy. The vertical axi indicate the calculated ample ize, and the horizontal axi how the method ued to etimate the ample ize; the quare repreent reult baed on point etimate; the diamond how reult baed on upper 95% bound. The horizontal line how the actual required ample ize (1,61), and the hort horizontal bar diplay the average of each et of point etimate. allele-baed tet i never the bet tet to ue. Thu, ue of the econd trategy generate replication ample ize that are lower than the ample ize generated with the firt trategy; in our analyi, we oberved that the econd trategy required 40% of the ample required under the firt trategy. However, for penetrance etimate baed on mall initial ample ize, the ample ize required in the replication tudy i omewhat underetimated. For ample ize that are more appropriate for whole-genome aociation tudie, thi ratio i cloe to 1. The median relative difference between the neceary ample ize and the etimated ample ize i 0.4. Thu, the reult ugget that ue of the data to inform the deign of the replication tudy i an advantageou trategy. It provide a maller relative error and a bigger proportion of tet for which we can provide an upper bound for the neceary ample ize. Mot importantly, it reduce the required number of ample that need to be typed for the replication tudy while retaining the targeted power. Data Example A well-known example of the winner cure in practice i the Pro1Ala ubtitution in PPARg and it aociation with type diabete. 1 The firt publihed report of thi aociation wa baed on a comparion of the frequency of the PPARg proline homozygote in 91 Japanee American affected with type diabete and 54 healthy control. The tudy oberved a ignificantly maller proportion of proline homozygote in unaffected individual (45) than in affected individual (87). From thee data, the author etimated a 4.35-fold rik that carrier of the Pro/Pro genotype will develop type diabete. However, among five follow-up tudie with ample ize of 450 1,00 proband, only one tudy found a ignificant replication of the reult A larger follow-up tudy combined with a meta-analyi of all the previou tudie, excluding the initial report, included 13,000 individual. 19 The tudy did find a ignificant aociation of the Pro/Ala polymorphim with type diabete. However, the meta-analyi etimated the odd ratio of the Pro allele to be 1.5, much lower than the original etimate of We applied the method decribed in the preent tudy to the initial PPARg data, 1 to etimate the allele frequency and penetrance parameter and to calculate from thoe value the odd ratio for the Pro/Pro genotype. We aumed that the aociation had been found with a tet comparing the number of oberved Pro/Pro genotype at a ignificance level of.05. Furthermore, we et the prevalence of type diabete in Japanee American to the worldwide prevalence of 0.08 (ee the work of Wild et al. 0 ). Variation of the dieae prevalence between and 0.04 changed the etimated penetrance parameter without affecting the odd ratio. Thu, moderate change in the aumed prevalence do not affect our concluion. Without contraining the genetic model, we generated an MLE of for the allele frequency and penetrance of 0.0 for Ala/Ala, for Ala/Pro, and 0.08 for Pro/ Pro. From thee etimate, we calculated the odd ratio of the Pro/Pro genotype and the odd ratio of the Pro allele (table ). Furthermore, we calculated the odd ratio for every point in the 95% confidence region to generate CI. The reulting CI clearly exclude the naive etimate of 4.35 but are conitent with the odd ratio etimated by Althuler et al. 19 Becaue of the low count of heterozygote and Ala/Ala homozygote in the ample, the penetrance etimate for thoe two genotype were quite variable, and underdominant model, in which the penetrance of Ala/ Pro i maller than the penetrance of Ala/Ala, were included in the confidence region. Thu, the 95% CI of the odd ratio include value!1. Contraining the penetrance parameter to Pro-receive model did not affect our concluion. On the bai of thee etimate, we calculated that a minimum ample ize of,100 i required to replicate the aociation, with the aumption of a population frequency of 0.15 of the Ala allele. Thi i coniderably larger than the ample ize ued by the early and inconitent replication attempt but maller than the ample ued in the eventual tudy and meta-analyi that ultimately provided trong upport for the aociation. Thu, thi analyi would have predicted that the initial replication attempt were underpowered. 61 The American Journal of Human Genetic Volume 80 April 007

9 Table. Etimated Odd Ratio of the Pro1Ala Mutation in the PPARg Gene Data Source Pro/Pro Odd Ratio (CI) Pro Odd Ratio (CI) Deeb et al., 1 uncorrected etimate 4.35 ( ) 4.54 ( ) Deeb et al., 1 corrected etimate 1.09 (.47.63) 1.1 (.39.09) Althuler et al ( ) NOTE. The cae-control data collected by Deeb et al. 1 howed a ignificant aociation between the Pro/Pro genotype and type diabete. From their data, we etimated the odd ratio for both the Pro/Pro genotype and for the proline allele. The firt row how the reult without correction for the acertainment effect, and the econd row how the reult obtained for thee data with ue of our method. The third row provide an etimate of the odd ratio of the proline allele generated from a large meta-analyi. 19 Dicuion Etimate of the genetic effect of a newly detected variant are confounded by acertainment. It ha been argued that the only way to obtain an unbiaed etimate i to generate a econd independent ample. 6 However, thi view i overly peimitic. A we how here, we can, in fact, obtain corrected etimate that are far le biaed than the naive etimate. We have performed extenive teting of the algorithm, conditioning on a uniform ditribution of tet power to allow u to draw inference about the impact of ample ize without being confounded by the power of the initial tudy. The reult indicate that our algorithm generate an accurate etimate of the penetrance parameter of a dieae allele. An independent ample of at leat half the ize of the initial cae-control ample i required to obtain an etimate that i more accurate than the corrected etimate baed on the initial ample. The algorithm alo provide well-calibrated confidence region for the penetrance parameter, thu quantifying the uncertainty of each etimate. We have hown that the ample ize of the aociation can trongly influence the preciion of the etimate and the ize of the confidence region, independent of the underlying genetic model. Neverthele, we oberved a mall bia toward underetimation of the genetic effect that i independent of the ample ize. The power of the initial aociation can alo ha a major impact on the ize of the confidence region but ha only little influence on the preciion of the point etimate or on the extent of the negative bia. Thu, we concluded that the ample ize ued in whole-genome aociation tudie will provide ufficient data to generate ueful etimate of the penetrance parameter of an allele. For implicity, we aumed that the initial tet for aociation i a imple comparion of allele frequencie between cae and control. Application of the algorithm to any other teting method i generally traightforward, requiring only an efficient way to calculate the power of the initial can for aociation conditional on the penetrance and frequency parameter. Although thee calculation are uually imple for ingle-point method, they may be more complicated for multipoint method, neceitating the etimation of power by computer imulation. Furthermore, the etimation procedure alo require a good undertanding of the ampling cheme that wa ued to collect cae and control. In the reult we preented, it wa aumed that cae and control were randomly drawn from affected and unaffected people in a population. In real aociation tudie, thi may not be the cae; for example, ample are often drawn from patient at one or more medical facilitie. However, a model of random ampling erve a a good approximation of thi opportunitic acertainment. 1 Extending the algorithm to other ampling deign i poible, although ome care hould be taken in interpreting the reult, a illutrated by the following example. In a commonly ued tudy deign, particularly for dieae with late onet age, random member of the population, rather than unaffected individual, are ampled a the control group. For thi type of ample, modification of equation (3) will take into account the fact that the control group provide no information about the penetrance parameter; the ret of the algorithm i unchanged. In general, the algorithm in thi work can be applied to any ampling cheme that can be decribed in a imple tatitical model. Problem will arie if the acertainment trategy i not clearly documented. Epecially in meta-analye, thi i likely to be the cae. For uch data, it may be more fruitful to ae the error in the etimation procedure by imulating different acertainment trategie or to adopt boottrap algorithm imilar to the one preented by Sun and Bull. 9 A more general problem for etimating genetic rik of aociation data i that, in practice, the manifetation of a complex dieae not only will be governed by the allele at one locu but will alo depend on environmental and genetic covariate. If thee covariate are unknown, they cannot be accounted for by the etimation procedure. Thu, we are etimating marginal penetrance, ummarizing the effect of other genomic loci and environmental factor on the dieae a the penetrance parameter of one allele. If tudie are performed in different population, the frequency of genetic covariate might vary ignificantly acro ample, reulting in very different marginal penetrance. However, thi problem i not pecific to the decribed algorithm; a long a we are aware of only a ubet The American Journal of Human Genetic Volume 80 April

10 of all covariate, we are confined to calculating marginal rik of thee known covariate. Thi problem can be omewhat alleviated by extending the cheme preented here to etimation of genotype-environmental interaction or the effect of multiple loci at once. By etimating the penetrance of combination of genotype, it i poible to calculate an MLE for model of gene-gene or gene-environment interaction. Thu, it i poible to compare the reulting likelihood with the likelihood generated under a model of no interaction. Since the two model are neted, the upport for an interaction parameter can be teted by a likelihood-ratio tet. Thu, the interaction between an allele and a covariate can be identified and quantified, and the covariate can be controlled for in further tudie. We have alo hown that the corrected etimate can be ued to calculate the ample ize required to replicate a new finding. Although the etimate are biaed toward weaker genetic effect, thi reult in overetimating the required ample ize by only 10%. Furthermore, it i poible to quantify the uncertainty in that etimate of the ample ize, allowing the reearcher to plan the tudy accordingly. Moreover, we have demontrated that etimating the et of penetrance parameter provide another major advantage for planning replication tudie: baed on the three parameter, it i poible to predict what kind of tet would be mot powerful in the replication tudy and to deign the tudy accordingly. We have demontrated that even a rough comparion between imple tet wa ufficient to reduce the required ample ize ignificantly. In ummary, we have provided an algorithm that generate etimate of the frequency and penetrance of a variant while correcting for the winner cure. Providing thee etimate in future publication of aociation-mapping reult hould facilitate evaluation of the genetic effect of a variant, the planning of replication tudie, and comparion of the reult of multiple ignal oberved in the ame region. Acknowledgment We thank Graham Coop, Michael Boehnke, Laura Scott, Richard Spielman, and two anonymou reviewer for helpful comment. Thi work wa upported in part by National Intitute of Health grant HG0077 and HL The C program ued for the method and imulation tudie i available from S.Z. (S.Z. Web ite). Appendix A Power Calculation Multiple method exit to calculate the power of a tet for aociation. The mot commonly ued algorithm ue a normal approximation of the x tet tatitic (e.g., the work of Pritchard and Przeworki 13 ). Thi approximation can be imprecie, epecially if the reulting power i mall (!0.01). Since calculating the likelihood of each parameter point require a precie calculation of the power, we intead ue equation (4), umming over all realization of the # table that reult in a ignificant tet for aociation. To peed up computation, we calculate in advance all critical value cv 1(c a) and cv (c a) of allele-count of allele 0 in the unaffected ample (c u ) conditional on the allele count in the affected ample (c a ) that reult in a ignificant Pearon x tet, by olving the quadratic equation K na Kca K n u c u( ) c u(k c a) (Kca ca c a )! 0 N n N N n u for c u, where n a i the affected ample ize, n u i the unaffected ample ize, N p na nu, and K i the critical value of a x 1 ditribution at the appropriate ignificance level. Uing thoe critical value, we can then calculate the power of the aociation can with na c v1(i) n u S S a u S a u ip0 jp0 jpc v(i) [ ] Pr (BFv,f) p Pr (c p i,c p jfv,f) Pr (c p i,c p jfv,f), a where the ummed probabilitie are calculated uing equation (3). To calculate the ample ize for replication tudie, we ued the normal approximation mentioned in the work of Pritchard and Przeworki. 13 Under the null hypothei, the difference ( ) c c z u a N 0,, n n n u a with z p f(1 ˆ f) ˆ if n p n p n and fˆ i the frequency of allele 0 in the ample. With q p Pr (0FU) and q p Pr (0FA) a u u a 614 The American Journal of Human Genetic Volume 80 April 007