Genetic Epidemiology 35 : 133–138 (2011)Quantifying and Correcting for the Winner’s Cursein Quantitative-Trait Association StudiesRui Xiao1and Michael Boehnke21Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine, Philadelphia, Pennsylvania2Department of Biostatistics and Center for Statistical Genetics, University of Michigan, Ann Arbor, MichiganQuantitative traits (QT) are an important focus of human genetic studies both because of interest in the traits themselvesand because of their role as risk factors for many human diseases. For large-scale QT association studies including genome-wide association studies, investigators usually focus on genetic loci showing significant evidence for SNP-QT association,and genetic effect size tends to be overestimated as a consequence of the winner’s curse. In this paper, we study the impactof the winner’s curse on QT association studies in which the genetic effect size is parameterized as the slope in a linearregression model. We demonstrate by analytical calculation that the overestimation in the regression slope estimatedecreases as power increases. To reduce the ascertainment bias, we propose a three-parameter maximum likelihood methodand then simplify this to a one-parameter method by excluding nuisance parameters. We show that both methods reducethe bias when power to detect association is low or moderate, and that the one-parameter model generally results in smallervariance in the estimate. Genet. Epidemiol. 35:133–138, 2011.r 2011 Wiley-Liss, Inc.Key words: quantitative trait; winner’s curse; ascertainment bias; genome-wide association study; linear regression;maximum likelihoodContract grant sponsor: National Institutes of Health (NIH); Contract grant numbers: HG000376; DK062370.Correspondence to: Rui Xiao, Department of Biostatistics and Epidemiology, University of Pennsylvania School of Medicine,Philadelphia, PA 19104-6021. E-mail: [email protected] 28 May 2010; Revised 5 October 2010; Accepted 28 October 2010Published online 31 January 2011 in Wiley Online Library (wileyonlinelibrary.com/journal/gepi).DOI: 10.1002/gepi.20551INTRODUCTIONFor complex disease genetics research in humans,remarkable progress has been made recently with anumber of genome-wide case-control association studiespublished. In parallel, there have been increasing efforts toinvestigate the association between genotype and disease-related quantitative trait (QT) at population level(www.genome.gov/gwastudies). One rationale behindQT studies is that, because the traits examined are inmany cases risk factors for disease, identified quantitativetrait loci (QTL) may also be disease predisposing loci.A commonly used method to detect the SNP-QTassociation is to regress the observed trait values on ascore based on the individual’s SNP genotype. The slopeof the linear regression is a measure of the strength of thegenetic effect. As in disease case-control associationstudies, for QT association studies, investigators usuallyfocus on genetic loci showing significant evidence forSNP-QT association. As a consequence of the winner’scurse [Lohmueller et al., 2003], the effect size estimatortends to overestimate the true genetic effect size. Severalinvestigators have studied the winner’s curse effect in thecontext of QT linkage analysis [Go¨ring et al., 2001;Siegmund, 2002; Allison et al., 2002; Sun and Bull, 2005;Wu et al., 2006] or disease association analysis [Zo¨llner andPritchard, 2007; Garner, 2007; Yu et al., 2007; Zhong andPrentice, 2008; Ghosh et al., 2008; Xiao and Boehnke, 2009].In this paper, we study the winner’s curse effect in thecontext of QT association studies. We quantify analyticallythe impact of the winner’s curse on the estimate of thegenetic effect size parameterized as the linear regressionslope as a function of sample size, allele frequency,and statistical significance level. We then describe anascertainment-corrected maximum likelihood methodsimilar to that we and others derived for case-controldisease association studies [Zo¨llner and Pritchard, 2007;Xiao and Boehnke, 2009] to correct for this bias. Wedescribe both a fully parameterized model in whichwe estimate the intercept, slope, and error of the linearregression model, and a simplified model that focusesonly on the regression equation slope parameter resultingin a one-parameter model. We also consider a meansquare error (MSE) weighted estimator [Zhong andPrentice, 2008] calculated as the weighted average of theuncorrected and corrected estimators using MSE as theweight. We compare the performance (bias, standard error,and MSE) of these ascertainment-corrected maximumlikelihood estimators (MLEs) and that of the naı¨ve,uncorrected estimators.As for case-control studies [Zo¨llner and Pritchard, 2007;Xiao and Boehnke, 2009], we find that (1) the factors thatresult in overestimation of the regression slope can besummarized by study power alone, independentof sample size and allele frequency, and that overestima-tion decreases as power increases; (2) compared to theuncorrected estimator of the regression slope, ther 2011 Wiley-Liss, Inc.ascertainment-corrected estimators based on the one- andthree-parameter models result in reduced absolute biaswhen study power is low or moderate, and havecomparable absolute bias when power is high; (3) theMSE of the ascertainment-corrected MLE of the regressionslope based on the one-parameter model is generallysmaller than that for the three-parameter model; and isalso smaller than the uncorrected estimator when power islow or moderate; and (4) the MSE-weighted estimatorgenerally improves the ascertainment correction comparedto the three- and one-parameter model-based ascertain-ment-corrected MLEs. We recommend the use of theMSE-weighted version of the one-parameter-basedascertainment-corrected maximum likelihood method forestimation of genetic effect size in large-scale quantitativetrait association studies.METHODSMODELS AND ASSUMPTIONSWe assume N independent samples genotyped at anautosomal QTL with alleles A and a. Let p be the fre-quency for the minor allele a. For individual i, let yibethe trait value and Xibe the genotype score, dependingon the genetic model we assume. For example, Xi5 0for AA, 1 for Aa, and 2 for aa, if we assume an additivemodel or Xi5 0 for AA or Aa, and 1 for aa for a recessiveallele a.To test for SNP-QT association, we assume the linearregression