404 Am J Epidemiol 2003;158:404–405American Journal of EpidemiologyCopyright © 2003 by the Johns Hopkins Bloomberg School of Public HealthAll rights reservedVol. 158, No. 5Printed in U.S.A.DOI: 10.1093/aje/kwg152Lee Responds to “Testing for Hardy-Weinberg Disequilibrium” Wen-Chung LeeFrom the Graduate Institute of Epidemiology, College of Public Health, National Taiwan University, Taipei, Taiwan.Received for publication April 28, 2003; accepted for publication May 8, 2003.Abbreviation: HWT, Hardy-Weinberg disequilibrium test.I appreciate Weinberg and Morris’ thoughtful commen-tary (1) on my paper (2). In their article, they put my workunder the perspective of gene mapping in the postgenomicera. I share the same view with them that the methodproposed in my paper amounts to a tree-shaking approach toharvesting the high-hanging fruit (a low-cost approach togenerating hypotheses aimed at localizing disease-susceptibility genes for complex human diseases).However, some issues raised by Weinberg and Morris (1)deserve scrutiny. These are 1) the power of the Hardy-Weinberg disequilibrium test (HWT) when a single-nucle-otide polymorphism is a “marker” but is not a disease-susceptibility “gene” itself; 2) the utility of the proposedmethod as a gene-localization tool; and 3) the false alarmdue to unmeasured ethnicity.To address the first issue, consider a marker, M, which isin linkage disequilibrium with a disease-susceptibility gene,A. Jiang et al. (3) showed that, for the M marker, the Hardy-Weinberg disequilibrium coefficient in the affected popula-tion is (with the notations changed to be consistent with mypaper (2)):,where f is the allele frequency of M in the source population,θ is the recombination fraction between M and A, t is thegeneration elapsed since the A gene was first introduced tothe population, and q, R, Ψ1, and Ψ2 are defined the same asin my paper (2). The equation shows that the Hardy-Weinberg disequilibrium coefficient of the M marker decaysaccording to the function, (1 – θ)2t. However, the term still appears in the equation, meaning that the effect ofthe mode of inheritance of the A gene is largely preservedeven though we are looking at the M marker. Weinberg andMorris’ assertion that “[s]uch a marker will display a gene-dose relation to risk, even if the linked risk-related gene forwhich it serves as a surrogate works according to a recessiveor a dominant model” (1, p. 401), is therefore incorrect.A second consequence of the above equation is that theHardy-Weinberg disequilibrium coefficient, D, decays morequickly than the linkage disequilibrium coefficient, δ = q(1 –f) × (1 – θ)t, as the genomic distance between M and Aincreases (3). Thus, if a disease gene is not of too recentorigin, a marker has to be closer to the gene to reach statis-tical significance using the HWT more than a marker has tobe using the transmission/disequilibrium test. This impliesthat, in a Hardy-Weinberg population, a genome-wide HWTscan can fine map the putative disease-susceptibility gene(s),because in the very vicinity of the marker(s) with significantHWT, there may exist disease-susceptibility gene(s). Thisfine-mapping ability should be better for a HWT scan ascompared with a transmission/disequilibrium test scan.As for the problem of unmeasured ethnicity (hidden strati-fication), the “genomic control” method of Reich and Gold-stein (4) can be used for a correction of the HWT. (Theirmethod was proposed originally to correct the allelic chi-square statistic of a case-control design.) To be precise, anumber of markers (e.g., 50 markers) are to be selected atrandom throughout the genome. It is unlikely that any suchrandomly selected marker will be tightly linked to a disease-susceptibility gene. Therefore, the mean square HWT(denoted as λ) of these “null markers” will be close to one ifthe population is a Hardy-Weinberg population. (A chi-square distribution with 1 df has the expectation of one.) Onthe other hand, λ will tend to be greater than one if the popu-lation is stratified. By the principle of multiplicative scalingof chi-square distribution (4), one refers the adjustedstatistic, HWT2/λ, to a 1-df chi-square distribution for eachand every marker typed in the study. Such a correctionprocedure should reduce the number of false positive results.Reprint requests to Dr. Wen-Chung Lee, Graduate Institute of Epidemiology, National Taiwan University, No. 1, Jen-Ai Road, Section 1, Taipei, Taiwan (e-mail: [email protected]).Dq 1 f )–(R-------------------21 θ)2t–(×Ψ2(×Ψ12)–=Ψ2Ψ12–Lee Responds to “Testing for Hardy-Weinberg Disequilibrium” 405 Am J Epidemiol 2003;158:404–405REFERENCES1. Weinberg CR, Morris RW. Invited commentary: testing for Hardy-Weinberg disequilibirum using a genome single-nucle-otide polymorphism scan based on cases only. Am J Epidemiol 2003;158:401–3.2. Lee W-C. Searching for disease-susceptibility loci by testing for Hardy-Weinberg disequilibirum in a gene bank of affected individuals. Am J Epidemiol 2003;158:397–400.3. Jiang R, Dong J, Wang D, et al. Fine-scale mapping using Hardy-Weinberg disequilibrium. Ann Hum Genet 2001;65:207–19.4. Reich DE, Goldstein DB. Detecting association in a case-con-trol study while correcting for population stratification. Genet Epidemiol