Biased Tests of Association: Comparisons of Allele Frequencies when Departing from Hardy-Weinberg Proportions

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1 American Journal of Epidemiology Copyright 1999 by The Johns Hopkins University School of Hygiene and Public Health All rights reserved vol. 149, No. 8 Printed in U.S.A Biased Tests of Association: Comparisons of Allele Frequencies when Departing from Hardy-Weinberg Proportions Daniel J. Schaid 1 ' 2 and Steven J. Jacobsen Association studies of genetic markers or candidate genes with disease are often conducted using the traditional case-control design. Cases and controls are sampled from genetically unrelated subjects, and allele frequencies compared between cases and controls using Pearson's chi-square statistic. An assumption of this analysis method is that the two alleles within each subject are statistically independent, at least when no association exists. This is equivalent to assuming that the frequencies of the genotypes in the general population comply with Hardy-Weinberg Equilibrium proportions, which may not always be the case. However, deviations from Hardy-Weinberg Equilibrium can inflate the chance of a false-positive association. These results demonstrate that when comparing the frequencies of two alleles between cases and controls, the chance of a false-positive association can be substantially increased if homozygotes for the putative high-risk allele are more common in the general population than predicted by Hardy-Weinberg Equilibrium. In contrast, Pearson's chisquare statistic can be conservative if the frequency of homozygotes for the high-risk allele is less than that predicted. A statistically valid method that corrects for deviations from Hardy-Weinberg Equilibrium is presented, so that the chance of a false-positive association is not greater than the acceptable level. Am J Epidemiol 1999;149: association; bias (epidemiology); case-control studies; chi-square statistic; genes; significance tests Association studies of candidate genes with disease have helped to decipher the genetic basis of many complex diseases. The case-control design provides an efficient method for assessing these associations. Unfortunately, many initial genetic associations found in case-control studies have been difficult to reproduce. This could be due to reporting bias (1), with the first publication representing an extreme outlier. Alternatively, bias in choice of cases or controls (2-4), choice of genetic markers or genotyping errors (5), unaccounted confounding factors (6), or improper analytic methods could explain the difficulty in replicating findings. An example of the difficulty in interpreting multiple case-control association studies is provided by the controversy of the association of alcoholism and the dopamine D 2 receptor. The first report (7) documented a large odds ratio of 8.7 {p < 0.001); a second study by the same group of investigators (8) reported a reduced odds ratio of 3.7 that was still statistically sig- Received for publication February 17,1998 and accepted for publication August 12, Abbreviation: HWE, Hardy-Weinberg Equilibrium. ' Department of Health Sciences Research Mayo Clinic/Mayo Foundation, Rochester, MN. 2 Department of Medical Genetics, Mayo Clinic/Mayo Foundation, Rochester, MN. Reprint requests to Dr. Daniel J. Schaid, Harwick 7, Mayo Clinic, 200 First Street S.W., Rochester, MN nificant. However, a number of other investigators have failed to reproduce these results (9). Metaanalyses on these studies (10, 11) concluded that the differences between cases and controls could be explained by variation in the genetic marker allele frequencies among the various ethnic groups, as well as sampling error. This demonstrates that the choice of appropriate controls can be difficult (2-4, 6), due to potential unmeasured confounding factors, such as ethnic background of cases and controls. Another potential source of error is the choice of analytic method. The distribution of genotypes is often compared between cases and controls by using Pearson's chi-square statistic for a 2 x G contingency table, where G is the number of observed genotypes. This method can have limited power when G is large, because of the large number of degrees-of-freedom. Alternatively, the frequencies of K alleles are often compared by cross-classifying both alleles of each person according to their case-control status, creating a 2 x K table. Frequencies are then compared with Pearson's chi-square statistic. The use of allele frequencies can be more appealing than comparing genotype frequencies because the sample size is twice as large (with each person contributing two alleles) and the degrees-of-freedom are fewer. However, the validity of Pearson's chi-square statis- 706

2 Biased Association when Departing from Hardy-Weinberg Proportions 707 tic requires the independence of alleles in the general population to maintain the correct false-positive (Type-I error) rate. When sampling nonrelated cases and controls, genotypes are independent among people, but alleles within genotypes may or may not be independent. Statistical independence of alleles is equivalent to the genotype frequencies complying with Hardy- Weinberg Equilibrium (HWE) proportions. For a simple case, consider two alleles, denoted A and B, where A is thought to be the high-risk allele, and B represents all other alleles. If p is the population frequency of allele A, then the HWE proportions for genotypes AA, AB, and BB are p 2, 2p(l - p), and (1 - p) 2. Independence of alleles can be tested by comparing the observed genotype proportions to those expected when there is HWE (12). Note that even if the general population is in HWE, the expected marker genotype proportions among diseased cases can deviate from HWE when a true association exists, and the amount of deviation depends on the genetic mechanism. For example, if a marker allele is associated with a disease because of a rare dominant disease susceptibility allele, then HWE is not expected to hold, yet, for association with a recessive disease susceptibility allele, HWE may hold among the cases, but with a marker allele frequency greater than in the general population (13, 14). Hence, testing for HWE should be performed among only controls, assuming that the disease is rare in the population. Deviations from HWE can be caused by multiple reasons, such as small population variation (random genetic drift), and the structure of the population. The latter may include inbreeding, assortative mating, stratification, or admixture of different ethnic groups. If a population is composed of a recent admixture of different ethnic groups that have different frequencies of marker alleles, then any trait more frequent in one of these ethnic groups will be positively associated with any marker allele that is more frequent in that group, even if the trait and marker locus are not genetically linked. This type of association is an example of confounding due to ethnic background. As a method to assess the potential for an admixed population, it has been proposed that testing for departures from HWE should be routinely performed for association studies (15). However, the impact of departures from HWE on the Type-I error rate has been ignored in many applications. When comparing allele frequencies between cases and controls, ignoring deviations from HWE can alter the Type-I error rate. Although this has been speculated to occur (16), the magnitude of the problem has not been explored. The purposes of this paper are to quantify the false-positive rate, and to offer guidelines and a correct analytic method to account for deviations from HWE when comparing allele frequencies in casecontrol studies. STATISTICAL METHODS Consider comparing the frequencies of two alleles between cases and controls. Although Pearson's chisquare statistic is often used to make this comparison, it is easier to present statistical properties by use of an equivalent statistic, based on the normal distribution. Let p d denote the estimated frequency of allele A among diseased cases, and let p c denote that among the non-diseased controls, where these estimates are obtained by simply counting alleles. A statistic to compare p d and p c is z = ~ Pc) here Vis the variance of (p d -p c ). If Vis correctly specified, z has an approximate standard normal distribution. When HWE exists, V can be written as (i) ^HWE ~~ ^77 I (2) where p is the underlying allele frequency common to both cases and controls under the null hypothesis of no association, and N d and N c are the number of cases and controls, respectively. The allele frequency p can be estimated by pooling cases and controls. When there are departures from HWE, V can be written (17) as NonHWE ~P) 2N d + Wj where P M is the frequency of AA homozygotes. This latter variance includes a measure of discrepancy between the frequency of AA homozygotes and that predicted by HWE; let 6 denote the discrepancy coefficient, where 6 = (P M - p 2 ). Under the null hypothesis of no association, the relative frequency P M can be estimated by pooling cases and controls, but it is not clear if this is the best approach when considering power. As an alternative to expression 3, one can estimate the variance of the allele frequency among cases, = W ~ (3)

3 708 Schaid and Jacobsen by using only cases to estimate p and P lar method for controls to estimate V. add these to compute AA' NooHWEj:' and a simiand then > = PI 'NonHWE = V, NonHWE,*/ + v,nonhwe,c- When HWE is falsely assumed to be true, and V uwc * HWb is used in expression 1, the Type-I error rate can be either inflated or deflated (i.e., conservative) relative to the assumed error rate. The Type-I error rate will be inflated when V iram is an underestimate of the true vanance, which occurs when 5 > 0, or, in other words, when the frequency of AA homozygotes exceeds that predicted by HWE. In contrast, the Type-I error rate is deflated when V HWE is greater than V NonHWE, which occurs when the frequency of AA homozygotes is less than that predicted by HWE. To examine the amount of deviation of the true Type-I error rate from that assumed, let z a be the quantile of a standard normal distribution that gives an assumed Type-I error rate for a two-sided test of a. Also, let z^^ and z NooHWE be the test statistics using Vjj^, and V NonHWE, respectively, in expression 1. Note that if HWE is false, then z^^, does not have a standard normal distribution, but z tioohwe does. Assuming HWE to be false, the Type-I error rate when using z lwfe can be evaluated by the following probability calculations: = PI - P ZNonHWE So, the square root of the variance ratio, ^NonHWE> which depends on the discrepancy coefficient 8, determines the true Type-I error rate. The effects of departure from HWE on the true Type-I error rate are considered separately in situations when there is an excess and a deficit of AA homozygotes. When there is an excess of AA homozygotes, 8 > 0. Because P M = p and p = P M + P^/2 imply that P^ = 2[p(l -p) - 8] and P w is bounded by 0 and 1, the maximum value of Ii is p{\ -p). At this maximum value, there are no AB heterozygotes, because P M =p and P^ = 0. An alternative way to express 8 is a fraction,/, of its maximal discrepancy value: 8 =fp(l - p)- Substituting this representation into the variance ratio results in Nominal Error Rate max = 0.17 ^ ^ I max = o.o i i i Fraction of Maximum Discrepancy from HWE FIGURE 1. True Type-I error rate as a function of fractional maximum discrepancy from Hardy-Weinberg Equilibrium (HWE) when there is an excess of AA homozygotes and the assumed Type-I error rate is either or 0.05.

4 Biased Association when Departing from Hardy-Weinberg Proportions 709 'HWE 'NonHWE 1+/' (5) which is independent of allele frequency. Substituting this variance ratio into expression 4 allows evaluation of the true Type-I error rate. When a deficiency of AA homozygotes exist, 8 < 0, and the maximum amount of negative discrepancy is -p 1. At this value, there are no AA homozygotes because P M = 0 and P^ = 2p. After expressing 5 as a fraction of its maximum negative value, the variance ratio can be written as 'HWE 'NonHWE (6) which depends on the allele frequency. In this simple case, only two alleles have been assumed. Thus, the comparison can focus on only one allele frequency. When K alleles are compared simultaneously by applying Pearson's chi-square statistic to &2xK contingency table, this approach can be extended to consider the dependence of alleles under the null hypothesis. RESULTS When the frequency of AA homozygotes exceeds that predicted by HWE, the Type-I error rate can be inflated, as illustrated in figure 1. Here, the true Type- I error rate is plotted as a function of the fractional maximum discrepancy, for an assumed Type-I error rate of either 5 percent or 1 percent. When discrepancy is at its maximum, the true Type-I error rate can be as high as 17 percent for an assumed rate of 5 percent, and as high as 7 percent for an assumed rate of 1 percent. When the frequency of AA homozygotes is less than that predicted by HWE, the Type-I error rate can be deflated, as illustrated in figure 2 for both a common allele (p = 0.25) and a rare allele (p = 0.05). For a common allele (p = 0.25), the Type-I error rate can be quite conservative, especially if the assumed error rate is 5 percent. The amount of conservatism is less when the assumed Type-I error rate is small, as for the assumed error rate of 1 percent in figure 2. As the allele frequency gets smaller, the amount of negative disequilibrium is also reduced, resulting in a less conservative Type-I error rate (e.g., when p = 0.05 in figure 2). The results in figures 1 and 2 are based on expression 4, which assumes that the sample size is large enough for the normal approximation to be adequate. To validate the adequacy of this approximation, simulations were performed. The genotypes for an equal number of cases and controls (N d = N c = 50 or 100) were sampled according to the probabilities P M - p 2 + l P /» where p = 0.10 and/= 0, 0.5, or 1.0. The maximum discrepancy, 5, was p{\ -p) for excess AA homozy ^ \. p =.O LU % Nominal Error Rate _p =.O5 p = I 1 1 I i i Fraction of Maximum Discrepancy from HWE FIGURE 2. True Type-I error rate as a function of fractional maximum discrepancy from Hardy-Weinberg Equilibrium (HWE) when there is a deficiency of AA homozygotes, the allele A is either common (p = 0.25) or rare (p = 0.05), and the assumed Type-I error rate is either or 0.05.

5 710 Schaid and Jacobsen TABLE 1. Type-I error rates for statistical methods with and without assumptions of Hardy-Welnberg Equilibrium (HWE) Frequency of AA homozygotes size (",= ") 0.0 Fraction of maximum discrepancy (l)t *««Excess Large-sample approximation Deficient Large-sample approximation Nq, number of cases; N c, number of controls. Type-I error rates for sample sizes of 50 and 100 are based on simulations; large sample approximation is based on expression 4 in the text, t f = 0 implies HWE, and f = 1 is the maximum departure from HWE. t Zywz, statistic assuming HWE; z tlaltttm, statistic with variance corrections for departure from HWE. gotes, and -p 2 for deficient AA homozygotes. For each sample size, 1,000 repetitions were sampled; for each sample, the frequency of allele A was compared between cases and controls using both z^^ (assuming HWE) and z NoaHWE (correcting the variance for deviations from HWE), with an assumed Type-I error rate of 5 percent. The simulated Type-I error rates are presented in table 1, along with those predicted by expression 4. For these sample sizes, the simulated Type-I error rates are close to those predicted, suggesting that expression 4 is a reliable indicator of the magnitude of false-positive results when HWE does not hold. Also, the simulations indicate that z SonHWE adequately corrects for deviations from HWE, achieving the assumed Type-I error rate of 5 percent. To illustrate the difference in statistical significance when considering departures from HWE, both z^^ and ^NMIHWE statistical tests were applied to data recently published on the association of a molecular variant of the angiotensinogen gene and coronary atherosclerosis. In the report by Ishigami et al. (18), the molecular variant of angiotensinogen that exists in exon 2, a thyminecystosine transition at nucleotide 704, was labeled a, and alleles which did not have this variant were labeled A. Among the 160 control subjects, 30 had genotype AA (18.8 percent), 51 had Aa (31.9 percent), and 79 had aa (49.4 percent). Among the 82 cases with coronary atherosclerosis, 6 were AA (7.3 percent), 22 were Aa (26.8 percent), and 54 were aa (65.9 percent). The frequencies of the a allele were 79 percent among cases and 65 percent among controls, but there was a significant excess of homozygotes among the controls (p = ). In pooled cases and controls, the amount of departure was 47 percent of the maximum departure. When ignoring this departure, z^^ = 3.17, giving a probability value of p = Taking this departure into account results in z NooHWE = 2.80 and/? = Thus, although both methods of analysis resulted in a statistically significant association for this example, the strength of significance was less after appropriately accounting for departures from HWE. DISCUSSION These results demonstrate that deviations from HWE can alter the assumed Type-I error rate, and that the true error rate occurs in a predictable manner. If AA homozygotes are more common in the general population than predicted by HWE, the chance of a falsepositive finding can be greater than assumed (11 percent if one-half of the maximum discrepancy, and up to a 17 percent chance, when the assumed chance is 5 percent). Population dynamics that can lead to an increased frequency of homozygotes are inbreeding, or a stratified population, also called Wahlund's principle in population genetics (19). If homozygotes occur less frequent than predicted by HWE, the true Type-I error rate will be less than assumed, leading to a conservative statistical comparison. For a common allele, the amount of conservatism can be substantial. However, the amount of conservatism depends on the allele frequency, and, as the candidate allele becomes more rare, the amount of conservatism becomes smaller. This finding may be important for associations of alleles that have a selective heterozygote advantage (20). In summary, these results demonstrate that the probability of a Type-I error can be underestimated when comparing frequencies of two alleles between cases and controls when there is a departure from HWE. Sasieni (16) recently speculated that the Type-I error rate would not be correct if HWE is falsely assumed.

6 Biased Association when Departing from Hardy-Weinberg Proportions 711 Sasieni suggested that the Pearson chi-square statistic should not be used to compare allele frequencies, but rather only genotype frequencies should be compared; trends in genetic relative risks could be assessed with Armitage's trend test (21). The results presented here quantify the biased Type-I error rate and demonstrate how to correctly account for dependencies of alleles where there are departures from HWE. ACKNOWLEDGMENTS This research was partially supported by grant no. GM51256 from the National Institutes of Health. REFERENCES 1. Begg CB, Berlin JA. Publication bias: a problem in interpreting medical data. J R Stat Soc [A] 1988;151: Wacholder S, McLaughlin JK, Silverman DT, et al. Selection of controls in case-control studies. I. Principles. Am J Epidemiol 1992;135: Wacholder S, Silverman DT, McLaughlin JK, et al. Selection of controls in case-control studies, n. Types of controls. Am J Epidemiol 1992;135: Wacholder S, Silverman DT, McLaughlin JK, et al. Selection of controls in case-control studies. HI. Design options. Am J Epidemiol 1992;135: Stefanski LA, Carroll RJ. Covariate measurement error in logistic regression. Ann Stat 1985;13: Falk CT, Rubinstein P. Haplotypc relative risks: an easy reliable way to construct a proper control sample for risk calculations. Ann Hum Genet 1987;51: Blum K, Nobel EP, Sheridan PJ, et al. Allelic association of human dopamine D2 receptor gene in alcoholism. JAMA 1990;263: Blum K, Noble EP, Sheridan PJ, et al. Association of the A1 allele of the D dopamine receptor gene with severe alcoholism. Alcohol 1991;8: Holden C. A cautionary genetic tale: the sobering story of D. Science 1994;264: Gelernter J, Goldman D, Risch N. The Al allele at the D 2 dopamine receptor gene and alcoholism. JAMA 1993;269: Pato CN, Macciardi F, Pato MT, et al. Review of the putatitve association of dopamine D receptor and alcoholism: a metaanalysis. Am J Med Genet 1993;48: Guo SW, Thompson EA. Performing the exact test of Hardy- Weinberg proportion for multiple alleles. Biometrics 1992;48: Risch N. A general model for disease-marker association. Ann Human Genet 1983;47: Thomson G. HLA disease associations: models for the study of complex human genetic disorders. Clin Rev Clin Lab Sci 1995;32: Tiret L, Cambien F. Departure from Hardy-Weinberg Equilibrium should be systematically tested in studies of association between genetic markers and disease. (Letter). Circulation 1995;92: Sasieni PD. From genotypes to genes: doubling the sample size. Biometrics 1997;53: Weir BS. Genetic data analysis. Sunderland, MA: Sinauer Associates, Inc, 1990: Ishigami T, Umemura S, Iwamoto T, et al. Molecular variant of angiotensinogen gene is associated with coronary atherosclerosis. Circulation 1995;91:951^ Li CC. First course in population genetics. Pacific Grove, CA: The Boxwood Press, 1976: Haiti DL, Clark AG. Principles of population genetics. 2nded. Sunderland, MA: Sinauer Associates, Inc, Armitage P. Tests for linear trends in proportions and frequencies. Biometrics 1955;ll: