Abstract
In single nucleotide polymorphism (SNP) data analysis, the allelic odds ratio and its confidence interval (CI) are usually used to evaluate the association between disease and alleles at each SNP. The usual formula for calculating the CI of the allelic odds ratio based on the Hardy–Weinberg equilibrium (HWE) may, however, lead to errors beyond the control assured by the nominal confidence level if HWE is not true. We therefore present a generalized formula for CI that does not assume HWE. CIs calculated by this generalized formula are likely to be wider than those by the usual method if the Hardy–Weinberg disequilibrium (HWD) is toward a relative deficiency of the heterozygotes (fixation index greater than 0), whereas they are likely to be narrower if HWD is toward a relative excess of the heterozygotes (fixation index less than 0). A simulation experiment to examine the influence of the generalization was performed for the case where 2% of SNPs had a fixation index greater than 0. The result revealed that the generalized method slightly decreased the mean number of falsely detected SNPs.
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Acknowledgments
We thank Professor Toshiya Sato at Kyoto University and Dr. Takashi Sozu at Tokyo University of Science for their valuable advice in improving this paper. We are grateful to the anonymous reviewers for their useful comments, which greatly improved this paper. This study was supported by the Program for Promotion of Fundamental Studies in Health Sciences of the National Institute of Biomedical Innovation of Japan.
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Appendix: Mathematical details
Appendix: Mathematical details
Let the population probabilities of genotypes “XX”, “Xx”, and “xx” be π1, π2 and π3 (π1 + π2 + π3=1), respectively, then those of alleles “X” and “x” for a SNP are given by Eq. 10.
When we use the fixation index (Li et al. 1953) defined by
(πi; i=1, 2, 3) are expressed as Eq. 12.
Therefore, (π1, π2, π3) is equivalent to (P 1, P 2, F).
When the Hardy–Weinberg equilibrium (HWE) holds, F=0 and Eq. 12 reduces to Eq. 13; that is, the second term in the right side of Eq. 12 represents the degree of disequilibrium.
For a random sample of size n, the observed frequency (n 1, n 2, n 3); (n 1 + n 2 + n 3=n) of genotypes (XX, Xx, xx) is distributed as trinomial distribution Tn(n; π1, π2, π3) and, therefore, the maximum likelihood estimator of πi (i=1, 2, 3) is p i=n i/n (i=1, 2, 3). Likewise, the maximum likelihood estimators of P 1 and allele odds \({P_{1} } \mathord{\left/ {\vphantom {{P_{1} } {{\left({1 - P_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left({1 - P_{1} } \right)}}\;\hbox{are}\; \hat{P}_{1} = p_{1} + {p_{2}} \mathord{\left/ {\vphantom {{p_{2} } 2}} \right. \kern-\nulldelimiterspace} 2 = {{\left({2n_{1} + n_{2} } \right)}} \mathord{\left/ {\vphantom {{{\left({2n_{1} + n_{2} } \right)}} {{\left({2n} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left({2n} \right)}},\;\hbox{and}\;{\hat{P}_{1} } \mathord{\left/ {\vphantom {{\hat{P}_{1} } {{\left({1 - \hat{P}_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left({1 - \hat{P}_{1} } \right)}},\) respectively.
Since the means, variances, and covariance of p 1 and p 2 are given (Bishop et al. 1975; Agresti 2001) by
the mean and variance of \(\hat{P}_{1}\) is, after a simple but tedious algebra, derived as Eqs. 15 and 16.
When F=0, the last term is the well-known formula for binomial proportion for the size 2n and probability P 1. It reflects that the distribution of the frequency of allele X under HWE is the same as that of allele X randomly chosen from 2n alleles with P 1 as the proportion of X.
Since \(\hat{P}_{1}\) tends to P 1 in probability when n tends to infinity, the logarithm of estimated allelic odds, \(\log {\left({{\hat{P}_{1}} \mathord{\left/ {\vphantom {{\hat{P}_{1} } {{\left({1 - \hat{P}_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left({1 - \hat{P}_{1} } \right)}}} \right)},\) can be approximated by the first order Taylor expansion as Eq. 17.
Consequently, the mean and variance of \(\log {\left({{\hat{P}_{1} } \mathord{\left/ {\vphantom {{\hat{P}_{1} } {{\left({1 - \hat{P}_{1} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left({1 - \hat{P}_{1} } \right)}}} \right)}\) are asymptotically approximated by Eqs. 18 and 19:
When we consider the populations of cases and controls of a disease, the association between allele and disease is conventionally represented by the allele odds ratio ψ defined by Eq. 20, where the case and the control are differentiated with the second subscript 1 (case) and 2 (control).
Consider we have random samples of size n .1 and n .2 from cases and controls, respectively. Then the maximum likelihood estimator \(\hat{\psi}\) of ψ is given by Eq. 21, where \(\hat{P}_{11}\;\hbox{and}\;\hat{P}_{12}\) are the maximum likelihood estimators based on samples of case and control, respectively.
Since the sample of case and that of control can be assumed independent, we obtain Eqs. 22 and 23.
where F 1 and F 2 are fixation indices of case and control, respectively.
When we construct an asymptotic confidence interval of log (ψ) with confidence level 1−α, we should replace \(V{\left\{ {\log {\left({\hat{\psi }} \right)}} \right\}}\) with its estimator given by Eq. 24.
where \(\hat{F}_{1}, \hat{F}_{2}\) are as follows:
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Sato, Y., Suganami, H., Hamada, C. et al. The confidence interval of allelic odds ratios under the Hardy–Weinberg disequilibrium. J Hum Genet 51, 772–780 (2006). https://doi.org/10.1007/s10038-006-0020-6
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DOI: https://doi.org/10.1007/s10038-006-0020-6
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