Selective genotyping can increase power in quantitative trait association. One example of selective genotyping is two-tail extreme selection, but simple linear regression analysis gives a biased genetic effect estimate. Here, we present a simple correction for the bias. © The Author(s) 2011.published_or_final_versionSpringer Open Choice, 21 Feb 201

Kung, AWC

Kwan, JSH

Sham, PC

Behavior Genetics

HKU Scholars Hub

Title A simple bias correction in linear regression for quantitative traitassociation under two-tail extreme selectionAuthor(s) Kwan, JSH; Kung, AWC; Sham, PCCitation Behavior Genetics, 2011, v. 41 n. 5, p. 776-779Issued Date 2011URL http://hdl.handle.net/10722/145096Rights The Author(s)A Simple Bias Correction in Linear Regression for QuantitativeTrait Association Under Two-Tail Extreme SelectionJohnny S. H. Kwan • Annie W. C. Kung •Pak C. ShamReceived: 22 February 2011 / Accepted: 16 May 2011 / Published online: 29 May 2011 The Author(s) 2011. This article is published with open access at Springerlink.comAbstract Selective genotyping can increase power inquantitative trait association. One example of selectivegenotyping is two-tail extreme selection, but simple linearregression analysis gives a biased genetic effect estimate.Here, we present a simple correction for the bias.Keywords Bias correction  Linear regression  Selectivegenotyping  QTL association  Extreme selectionSelective genotyping can increase the power in the associationstudies of quantitative trait loci (QTL) (Chen et al. 2005;Huang and Lin 2007; Xiong et al. 2002; Kwan et al. 2009;Slatkin 1999; Van Gestel et al. 2000; Xing and Xing 2009). Bygenotyping only individuals with extreme phenotypes,genetic information is enriched compared to randomgenotyping of the same number of individuals. Examples ofselective genotyping include one-tail extreme selection, two-tail extreme selection and extreme-concordant and -discor-dant design (Abecasis et al. 2001). Tang (Tang 2010) provedthat the three score tests based on the prospective (Xiong et al.2002), retrospective (Wallace et al. 2006) and conditional(Huang and Lin 2007) likelihoods, were all equivalent in QTLassociation under selective genotyping, but Huang and Lin(Huang and Lin 2007) showed that the prospective test, whichis a linear regression of phenotype on the number of risk allelesat a QTL, gives a biased QTL effect estimate under two-tailextreme selection. Here, we present a simple bias correctionand validate the results through simulations.In a population sample, the direct regression of pheno-type on genotype can be written as,Y ¼ a1 þ b1X þ e1 ð1Þwhere Y and X are respectively the phenotype and QTLgenotype before selection. The regression estimator, b1^, isof our primary interest but is biased in a two-tail extremeselected sample (Huang and Lin 2007). Since the selection(S) on Y is conditionally independent of genotype (X) givenY, i.e., P(X|Y,S) = P(X|Y), the selection on Y should not, intheory, affect the reverse regression estimator, b2^, inX ¼ a2 þ b2Y þ e2 ð2Þandx ¼ a3 þ b2y þ e3 ð3Þwhere y and x are respectively, the phenotype and QTLgenotype after selection. DeMets and Halperin (DeMetsand Halperin 1977) showed that an unbiased estimator ofb1 of the same problem in a non-genetic (statistical)context can be given by,Edited by Sarah Medland.J. S. H. Kwan  P. C. Sham (&)Department of Psychiatry, LKS Faculty of Medicine,The University of Hong Kong, Pokfulam, Hong Kong, Chinae-mail: pcsham@hkucc.hku.hkA. W. C. KungDepartment of Medicine, The University of Hong Kong,Pokfulam, Hong Kong, ChinaA. W. C. KungResearch Centre for Heart, Brain, Hormone & Healthy Aging,The University of Hong Kong, Pokfulam, Hong Kong, ChinaP. C. ShamCentre for Reproduction, Development and Growth,The University of Hong Kong, Pokfulam, Hong Kong, ChinaP. C. ShamGenome Research Centre, The University of Hong Kong,Pokfulam, Hong Kong, China123Behav Genet (2011) 41:776–779DOI 10.1007/s10519-011-9475-0b1^¼ b2^VarðYÞVarðxÞ þ b22^VarðYÞ  VarðyÞ½ ð4ÞSince the reverse linear regression in Eq. 3 is valid inselected samples, instead of reusing the DeMets andHalperin’s derivation of the standard error (SE), we comeup with a simpler formula, which isSEðb1Þ^¼ b1^SEðb2Þ^ b2^ð5ÞTo validate our results, we simulated a population of 5,000individuals, containing a QTL under different scenarios:minor allele frequencies (MAF) of 10, 25 and 50%, andphenotype variance explained of none and 5%. Differentproportions of individuals were sampled (25 and 50%) atvarious ratios (1:1, 2:1 and 4:1) from the two tails of thetrait distribution. After 1,000 simulations, the average biasin b1^before and after correction, and average SE andempirical standard deviation (SD) of b1^after correction areshown in Table 1 and a plot of the beta distributions forone of the extreme cases is provided in Fig. 1. A bias wasseen in the raw b1^under the alternative, but this disap-peared after the adjustment. Also, the adjusted SE reflectedaccurately the true variation of the adjusted estimator.Table 1 The average bias, SE and empirical SD of the adjusted QTL effect estimate (b^) in linear regression for association studies of QTL undertwo-tail extreme selectionMAF(%)%Sampled.U/Lratioab = 0.00 b = 0.05Bias beforeadjustmentBias afteradjustmentAverageSEEmpiricalSDBias beforeadjustmentBias afteradjustmentAverageSEEmpiricalSD10 25 1:1 -0.001 0.000 0.039 0.040 0.094 0.000 0.037 0.0372:1 -0.007 -0.002 0.041 0.039 0.083 0.000 0.039 0.0404:1 0.002 0.001 0.046 0.047 0.056 0.000 0.044 0.04610 50 1:1 -0.002 -0.001 0.035 0.036 0.042 -0.001 0.033 0.0332:1 0.001 0.000 0.036 0.036 0.039 0.001 0.034 0.0344:1 -0.001 -0.001 0.039 0.039 0.025 0.001 0.037 0.03825 25 1:1 -0.001 0.000 0.027 0.028 0.094 0.000 0.026 0.0262:1 -0.002 -0.001 0.028 0.029 0.081 -0.001 0.027 0.0294:1 -0.003 -0.001 0.032 0.032 0.063 0.003 0.030 0.03125 50 1:1 0.001 0.000 0.024 0.023 0.044 0.001 0.023 0.0232:1 0.001 0.000 0.025 0.024 0.040 0.002 0.024 0.0244:1 0.001 0.001 0.027 0.027 0.026 0.002 0.026 0.02650 25 1:1 -0.001 0.000 0.024 0.023 0.094 0.000 0.022 0.0232:1 -0.001 0.000 0.025 0.024 0.084 0.001 0.023 0.0234:1 -0.004 -0.002 0.027 0.028 0.057 0.001 0.026 0.02750 50 1:1 0.000 0.000 0.021 0.020 0.042 0.000 0.020 0.0202:1 0.000 0.000 0.021 0.021 0.037 0.000 0.020 0.0204:1 -0.001 -0.001 0.023 0.024 0.023 0.000 0.022 0.022a Sample size ratio in the upper versus lower tail of the trait distributionFig. 1 Distributions of the estimated QTL effect size before (blue)and after (red) adjustment in linear regression for association studiesof QTL under two-tail extreme selection The case shown is when aquarter of the individuals were sampled at 1:1 ratio from the two tailsof the trait distribution from a population of 5,000 individuals, eachcontaining a QTL that had MAF of 10% and explained 5% of the traitvariation. The means (SEs) of the estimated QTL effect size (b^) fromlinear regression are, respectively, 0.144 (0.105) and 0.050 (0.036)before and after adjustment (Color figure online)Behav Genet (2011) 41:776–779 777123Next, to see whether the adjustment can be applied toa more complicated model, we repeated the above sim-ulation for two unlinked QTLs with or without epistasisand fitted the regression model below to test forepistasis:Y ¼ b1ðX1  X1Þ þ b2ðX2  X2Þ þ b3ðX1  X1ÞðX2  X2Þ þ e ð6Þwhere Y is the phenotype before selection, X1 and X2 arethe genotypes for the two QTLs. Epistasis is inferred whenTable 2 The average bias, SE and empirical SD of the adjusted epistatic effect estimate (b3^) in linear regression for association studies of QTLunder two-tail extreme selectionQTL1MAF(%)QTL2MAF(%)% Sampled. U/Lratioab3 = 0.00 b3 = 0.05B1 = b2 = 0.00 b1 = b2 = 0.05 b1 = b2 = 0.00 b1 = b2 = 0.05Bias SE (emp.SD)Bias SE (emp.SD)Bias SE (emp.SD)Bias SE (emp.SD)10 10 25 1:1 -0.001 0.093 (0.091) 0.001 0.084 (0.083) 0.001 0.089 (0.088) 0.003 0.079 (0.081)2:1 0.003 0.097 (0.098) -0.002 0.089 (0.092) 0.002 0.091 (0.088) 0.001 0.084 (0.089)4:1 0.000 0.109 (0.110) -0.006 0.101 (0.111) 0.001 0.102 (0.099) -0.001 0.095 (0.102)10 10 50 1:1 0.001 0.082 (0.082) 0.003 0.074 (0.073) -0.002 0.078 (0.078) 0.002 0.070 (0.071)2:1 -0.002 0.085 (0.086) 0.003 0.077 (0.080) 0.000 0.080 (0.079) 0.001 0.072 (0.074)4:1 -0.002 0.092 (0.093) -0.001 0.084 (0.092) -0.001 0.087 (0.087) 0.002 0.079 (0.082)10 25 25 1:1 0.001 0.064 (0.066) 0.001 0.058 (0.059) 0.001 0.061 (0.062) 0.000 0.055 (0.056)2:1 0.001 0.067 (0.067) -0.001 0.061(0.064) -0.001 0.063 (0.064) 0.001 0.058 (0.059)4:1 0.000 0.075 (0.075) -0.004 0.069 (0.074) 0.001 0.071 (0.069) -0.001 0.065 (0.070)10 25 50 1:1 0.001 0.057 (0.057) 0.002 0.051 (0.051) 0.000 0.054 (0.054) 0.000 0.048 (0.048)2:1 0.000 0.058 (0.058) -0.001 0.053 (0.053) 0.000 0.055 (0.055) -0.001 0.050 (0.051)4:1 0.001 0.064 (0.063) -0.003 0.058 (0.061) -0.001 0.060 (0.059) -0.001 0.055 (0.058)10 50 25 1:1 0.000 0.056 (0.056) 0.000 0.050 (0.050) 0.000 0.053 (0.053) -0.002 0.047 (0.047)2:1 -0.001 0.058 (0.058) -0.002 0.053 (0.055) -0.001 0.055 (0.053) -0.002 0.050 (0.052)4:1 0.001 0.065 (0.065) -0.005 0.060 (0.062) 0.001 0.062 (0.061) -0.003 0.056 (0.059)10 50 50 1:1 0.001 0.049 (0.048) 0.000 0.044 (0.044) -0.002 0.047 (0.047) 0.001 0.042 (0.042)2:1 -0.001 0.051 (0.051) 0.000 0.046 (0.046) -0.001 0.048 (0.048) -0.002 0.043 (0.044)4:1 0.000 0.055 (0.056) -0.003 0.050 (0.051) 0.001 0.052 (0.052) -0.001 0.047 (0.049)25 25 25 1:1 0.000 0.044 (0.045) 0.001 0.040 (0.039) 0.000 0.042 (0.042) -0.001 0.038 (0.037)2:1 0.001 0.046 (0.046) -0.002 0.042 (0.043) 0.001 0.044 (0.043) -0.003 0.040 (0.040)4:1 0.001 0.052 (0.051) -0.004 0.048 (0.051) 0.000 0.049 (0.049) -0.003 0.045 (0.046)25 25 50 1:1 0.001 0.039 (0.039) -0.001 0.035 (0.036) 0.000 0.037 (0.037) 0.000 0.033 (0.033)2:1 0.001 0.040 (0.040) -0.002 0.037 (0.038) 0.000 0.038 (0.039) -0.001 0.035 (0.035)4:1 0.000 0.044 (0.045) -0.004 0.040 (0.042) 0.000 0.042 (0.041) -0.001 0.038 (0.039)25 50 25 1:1 0.001 0.039 (0.038) 0.001 0.035 (0.034) 0.001 0.037 (0.036) 0.000 0.033 (0.033)2:1 -0.001 0.040 (0.041) -0.004 0.036 (0.037) -0.001 0.038 (0.038) -0.002 0.034 (0.035)4:1 0.000 0.045 (0.045) -0.005 0.041 (0.043) 0.000 0.043 (0.044) -0.003 0.039 (0.040)25 50 50 1:1 0.000 0.034 (0.034) 0.000 0.031 (0.031) -0.001 0.032 (0.033) 0.000 0.029 (0.029)2:1 -0.001 0.093 (0.091) -0.002 0.032 (0.032) 0.001 0.033 (0.033) -0.001 0.030 (0.030)4:1 0.003 0.097 (0.098) -0.003 0.035 (0.036) 0.000 0.036 (0.037) -0.003 0.033 (0.033)50 50 25 1:1 0.001 0.033 (0.034) -0.001 0.030 (0.031) -0.001 0.032 (0.031) -0.001 0.028 (0.028)2:1 0.000 0.035 (0.035) -0.002 0.031 (0.031) 0.000 0.033 (0.034) -0.003 0.030 (0.029)4:1 0.000 0.039 (0.039) -0.005 0.035 (0.035) 0.001 0.037 (0.037) -0.006 0.033 (0.034)50 50 50 1:1 0.000 0.029 (0.030) 0.000 0.026 (0.027) 0.000 0.028 (0.028) 0.000 0.025 (0.025)2:1 0.000 0.030 (0.030) -0.001 0.027 (0.027) -0.001 0.029 (0.029) -0.002 0.026 (0.026)4:1 -0.001 0.033 (0.033) -0.004 0.030 (0.030) -0.001 0.031 (0.031) -0.003 0.028 (0.028)a Sample size ratio in the upper versus lower tail of the trait distribution778 Behav Genet (2011) 41:776–779123b3 differs significantly from zero. Since mean-centering ofX1 and X2 alleviates collinearity between the main effectsand the epistatic term (Aiken et al. 1991; Jaccard et al.1990), we can model the regression as three independentregressions:Y ¼ b1ðX1  X1Þ þ e1 ð7ÞY ¼ b2ðX2  X2Þ þ e2 ð8ÞY ¼ b3ðX1  X1Þ ðX2  X2Þ þ e3 ð9Þand b3 in Eq. 9 was estimated as in Eq. 4. The results areshown in Table 2. In most cases, the adjustment workedwell. But caution must be taken when more genotyping arecarried out in one tail of the distribution than the otherbecause the adjustment might give an epistasis estimatorwith a small bias in the presence of main effects under thenull hypothesis.We showed that the bias in QTL effect estimate in linearregression for association under two-tail extreme selectioncan be corrected easily. Bearing this in mind, researchersmay use linear regression, which is simple and imple-mented in most statistical packages, in QTL associationunder selective genotyping.Acknowledgments This work was funded by Hong Kong ResearchGrants Council GRF HKU 774707, and The University of Hong KongStrategic Research Theme on Genomics, and the European Commu-nity’s Seventh Framework Programme under grant agreement No.HEALTH-F2-2010-241909 (Project EU-GEI).Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License which per-mits any noncommercial use, distribution, and reproduction in anymedium, provided the original author(s) and source are credited.ReferencesAbecasis GR, Cookson WOC, Cardon LR (2001) The power to detectlinkage disequilibrium with quantitative traits in selected sam-ples. Am J Hum Genet 68(6):1463–1474Aiken LS, West SG, Reno RR (1991) Multiple regression: testing andinterpreting interactions Inc. 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A simple bias correction in linear regression for quantitative trait association under two-tail extreme selection

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A simple bias correction in linear regression for quantitative trait association under two-tail extreme selection

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