Lee Responds to "Testing for Hardy-Weinberg Disequilibrium"

404 Am J Epidemiol 2003;158:404–405

American Journal of Epidemiology

Copyright © 2003 by the Johns Hopkins Bloomberg School of Public Health All rights reserved

Vol. 158, No. 5 Printed in U.S.A. DOI: 10.1093/aje/kwg152

Lee Responds to “Testing for Hardy-Weinberg Disequilibrium”

Wen-Chung Lee

From 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 work under the perspective of gene mapping in the postgenomic era. I share the same view with them that the method proposed in my paper amounts to a tree-shaking approach to harvesting the high-hanging fruit (a low-cost approach to generating 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 proposed method as a gene-localization tool; and 3) the false alarm due to unmeasured ethnicity.

To address the first issue, consider a marker, M, which is in 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 notapopula-tions changed to be consistent with my paper (2)):

, where f is the allele frequency of M in the source population, θ is the recombination fraction between M and A, t is the generation elapsed since the A gene was first introduced to the population, and q, R, Ψ1, and Ψ2 are defined the same as

in my paper (2). The equation shows that the Hardy-Weinberg disequilibrium coefficient of the M marker decays according to the function, (1 – θ)2t_{. However, the}

term still appears in the equation, meaning that the effect of the mode of inheritance of the A gene is largely preserved even though we are looking at the M marker. Weinberg and Morris’ assertion that “[s]uch a marker will display a

gene-dose relation to risk, even if the linked risk-related gene for which it serves as a surrogate works according to a recessive or a dominant model” (1, p. 401), is therefore incorrect.

A second consequence of the above equation is that the Hardy-Weinberg disequilibrium coefficient, D, decays more quickly than the linkage disequilibrium coefficient, δ = q(1 –

f) × (1 – θ)t_{, as the genomic distance between M and A}

increases (3). Thus, if a disease gene is not of too recent origin, a marker has to be closer to the gene to reach statis-tical significance using the HWT more than a marker has to be using the transmission/disequilibrium test. This implies that, in a Hardy-Weinberg population, a genome-wide HWT scan can fine map the putative disease-susceptibility gene(s), because in the very vicinity of the marker(s) with significant HWT, there may exist disease-susceptibility gene(s). This fine-mapping ability should be better for a HWT scan as compared 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. (Their method was proposed originally to correct the allelic chi-square statistic of a case-control design.) To be precise, a number of markers (e.g., 50 markers) are to be selected at random throughout the genome. It is unlikely that any such randomly 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 if the population is a Hardy-Weinberg population. (A chi-square distribution with 1 df has the expectation of one.) On the other hand, λ will tend to be greater than one if the popu-lation is stratified. By the principle of multiplicative scaling of chi-square distribution (4), one refers the adjusted statistic, HWT2_{/λ, to a 1-df chi-square distribution for each}

and every marker typed in the study. Such a correction procedure 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: wenchung@ha.mc.ntu.edu.tw).

D q 1( –f) R --- 2×(1–θ )2t×(Ψ₂–Ψ₁2) = Ψ2 Ψ1 2 –

Lee Responds to “Testing for Hardy-Weinberg Disequilibrium” 405

Am J Epidemiol 2003;158:404–405 REFERENCES

1. 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 2001;20:4–16.