Suppose there is a diversity of haplotypes constructed by two SNPs each with two alleles designated as 1 and 0. Let p and q be the 1-allele frequencies for the first and second SNP, respectively. Thus, the 0-allele frequencies at these two different SNPs will be 1 - p and 1 - q. The two SNPs that are segregating in a natural human population form four haplotypes, [11], [10], [01], and [00], whose frequencies are constructed by allele frequencies and linkage disequilibrium (D) between the two SNPs, i.e.,

p 11 = p q + D p 10 = p ( 1 − q ) − D p 01 = ( 1 − p ) q − D p 00 = ( 1 − p ) ( 1 − q ) + D MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5FB7@

The parameters contained in equation (1) can be used to describe some important aspects of the genetic structure and diversity of a natural population. Thus, differences in genetic architecture between the two different sexes can be characterized by these sex-specific parameters. Because it is easy to derive the closed forms for estimating haplotype frequencies 14, we will estimate the linkage disequilibrium from the estimated haplotype frequencies.

Let ΘMp=(pM11,pM10,pM01,pM00) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGae8hMde1aaSbaaSqaaiabb2eannaaBaaameaacqWGWbaCaeqaaaWcbeaakiabg2da9iabcIcaOiabdchaWnaaBaaaleaacqqGnbqtdaWgaaadbaGaeGymaeJaeGymaedabeaaaSqabaGccqGGSaalcqWGWbaCdaWgaaWcbaGaeeyta00aaSbaaWqaaiabigdaXiabicdaWaqabaaaleqaaOGaeiilaWIaemiCaa3aaSbaaSqaaiabb2eannaaBaaameaacqaIWaamcqaIXaqmaeqaaaWcbeaakiabcYcaSiabdchaWnaaBaaaleaacqqGnbqtdaWgaaadbaGaeGimaaJaeGimaadabeaaaSqabaGccqGGPaqkaaa@48F9@ and ΘFp=(pF11,pF10,pF01,pF00) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabg2da9iabcIcaOiabdchaWnaaBaaaleaacqqGgbGrdaWgaaadbaGaeGymaeJaeGymaedabeaaaSqabaGccqGGSaalcqWGWbaCdaWgaaWcbaGaeeOray0aaSbaaWqaaiabigdaXiabicdaWaqabaaaleqaaOGaeiilaWIaemiCaa3aaSbaaSqaaiabbAeagnaaBaaameaacqaIWaamcqaIXaqmaeqaaaWcbeaakiabcYcaSiabdchaWnaaBaaaleaacqqGgbGrdaWgaaadbaGaeGimaaJaeGimaadabeaaaSqabaGccqGGPaqkaaa@48B3@ be the vectors of haplotype frequencies among males and females, respectively. All the genotypes for the two SNPs are consistent with diplotypes, except for the double heterozygote, 10/10, that belongs to a diplotype of either [11] [00] or [10] [01] (Table 1). Assuming that the population is at Hardy-Weinberg equilibrium, the frequency of a diplotype is expressed as the product of the frequencies of the two haplotypes that construct it. Table 1 characterizes the differences in diplotype frequencies between the males and females.

If haplotypes triggers an effect on a quantitative trait, this means that at least one haplotype performs differently from the rest of the haplotypes. Without loss of generality, let haplotype 11 be such a distinct haplotype, called risk haplotype, designated as A. All the other non-risk haplotypes, [10], [01] and [00], are collectively expressed as A¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebaaaa@2CF8@. The risk and non-risk haplotypes form three composite diplotypes AA (symbolized as 2), AA¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqaeKafmyqaeKbaebaaaa@2E03@ (symbolized as 1) and A¯A¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebacuWGbbqqgaqeaaaa@2E1B@ (symbolized as 0). The genotypic values of the three composite diplotypes may be different between the two sexes, arrayed in (μM2,μM1,μM0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaeqiVd02aaSbaaSqaaiabb2eannaaBaaameaacqaIYaGmaeqaaaWcbeaakiabcYcaSiabeY7aTnaaBaaaleaacqqGnbqtdaWgaaadbaGaeGymaedabeaaaSqabaGccqGGSaalcqaH8oqBdaWgaaWcbaGaeeyta00aaSbaaWqaaiabicdaWaqabaaaleqaaOGaeiykaKcaaa@3BE6@ for the males and (μF2,μF1,μF0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaeqiVd02aaSbaaSqaaiabbAeagnaaBaaameaacqaIYaGmaeqaaaWcbeaakiabcYcaSiabeY7aTnaaBaaaleaacqqGgbGrdaWgaaadbaGaeGymaedabeaaaSqabaGccqGGSaalcqaH8oqBdaWgaaWcbaGaeeOray0aaSbaaWqaaiabicdaWaqabaaaleqaaOGaeiykaKcaaa@3BBC@ for the females, respectively. Let (a_M, d_M) and (a_F, d_F) be the additive and dominance genetic effects due to the risk and non-risk haplotypes in males and females, respectively.

Assume that a total of n subjects (including n_M males and n_F females) sampled from the population are phenotyped for a quantitative trait. In each sex, there are nine possible genotypes for the two SNPs, each genotype with an observed number generally expressed as nr1r′1/r2r′2M MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOBa42aa0baaSqaaiabdkhaYnaaBaaameaacqaIXaqmaeqaaSGafmOCaiNbauaadaWgaaadbaGaeGymaedabeaaliabc+caViabdkhaYnaaBaaameaacqaIYaGmaeqaaSGafmOCaiNbauaadaWgaaadbaGaeGOmaidabeaaaSqaaiabb2eanbaaaaa@39DE@ for the males and nr1r′1/r2r′2F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOBa42aa0baaSqaaiabdkhaYnaaBaaameaacqaIXaqmaeqaaSGafmOCaiNbauaadaWgaaadbaGaeGymaedabeaaliabc+caViabdkhaYnaaBaaameaacqaIYaGmaeqaaSGafmOCaiNbauaadaWgaaadbaGaeGOmaidabeaaaSqaaiabbAeagbaaaaa@39D0@ for the females (r₁ ≥ r'₁, r₂ ≥ r'₂, r₃ ≥ r'₃ = 1,0).

Many physiological traits scale with body weight (W) according to a power function with a certain allometric exponent. Thus, we implement this allometric scaling law to describe the phenotypic value of a trait for subject i within male or female subpopulations in terms of the haplotypes considered as

y M i = α M W M i β M + x M i a M + z M i d M + e M i , y F i = α F W F i β F + x F i a F + z F i d F + e F i , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7649@

where (α_M, β_M) or (α_F, β_F) are body weight-related allometric coefficients, (xMi,zMi) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaemiEaG3aaSbaaSqaaiabb2eannaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcYcaSiabdQha6naaBaaaleaacqqGnbqtdaWgaaadbaGaemyAaKgabeaaaSqabaGccqGGPaqkaaa@3731@ or (xFi,zFi) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaemiEaG3aaSbaaSqaaiabbAeagnaaBaaameaacqWGPbqAaeqaaaWcbeaakiabcYcaSiabdQha6naaBaaaleaacqqGgbGrdaWgaaadbaGaemyAaKgabeaaaSqabaGccqGGPaqkaaa@3715@ are the indicator variables associated with the additive and dominance effects, respectively, and eMi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyzau2aaSbaaSqaaiabb2eannaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@3008@ or eFi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyzau2aaSbaaSqaaiabbAeagnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@2FFA@ is the residual error, normally distributed as N(0,σM2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta4KaeiikaGIaeGimaaJaeiilaWIaeq4Wdm3aa0baaSqaaiabb2eanbqaaiabikdaYaaakiabcMcaPaaa@3487@ or N(0,σF2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOta4KaeiikaGIaeGimaaJaeiilaWIaeq4Wdm3aa0baaSqaaiabbAeagbqaaiabikdaYaaakiabcMcaPaaa@3479@. The genotypic values of composite diplotypes and variance are arrayed by a quantitative genetic parameter vector ΘMq=(αM,βM,aM,dM,σM2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGae8hMde1aaSbaaSqaaiabb2eannaaBaaameaacqWGXbqCaeqaaaWcbeaakiabg2da9iabcIcaOiabeg7aHnaaBaaaleaacqqGnbqtaeqaaOGaeiilaWIaeqOSdi2aaSbaaSqaaiabb2eanbqabaGccqGGSaalcqWGHbqydaWgaaWcbaGaeeyta0eabeaakiabcYcaSiabdsgaKnaaBaaaleaacqqGnbqtaeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabb2eanbqaaiabikdaYaaakiabcMcaPaaa@45C8@ for the males and ΘFq=(αF,βF,aF,dF,σF2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGXbqCaeqaaaWcbeaakiabg2da9iabcIcaOiabeg7aHnaaBaaaleaacqqGgbGraeqaaOGaeiilaWIaeqOSdi2aaSbaaSqaaiabbAeagbqabaGccqGGSaalcqWGHbqydaWgaaWcbaGaeeOrayeabeaakiabcYcaSiabdsgaKnaaBaaaleaacqqGgbGraeqaaOGaeiilaWIaeq4Wdm3aa0baaSqaaiabbAeagbqaaiabikdaYaaakiabcMcaPaaa@4574@ for the females, respectively.

The log-likelihood of haplotype frequencies, genotypic values of composite diplotypes and residual variances given sex-specific phenotypic (y_M, y_F) and SNP data (S_M, S_F) is factorized into two parts, expressed as

log ⁡ L ( Θ M p , Θ M q ; Θ F p , Θ F q | y M , S M ; y F , S F ) = log ⁡ L ( Θ M p , Θ F p | S M , S F ) + log ⁡ L ( Θ M q , Θ F q | y M , S M , Θ M p ; y F , S F , Θ F p ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabiGaaaqaaaqaaiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOGGabiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemiCaahabeaaaSqabaGccqGGSaalcqWFyoqudaWgaaWcbaGaeeyta00aaSbaaWqaaiabdghaXbqabaaaleqaaOGaei4oaSJae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYcaSiab=H5arnaaBaaaleaacqqGgbGrdaWgaaadbaGaemyCaehabeaaaSqabaGccqGG8baFieqacqGF5bqEdaWgaaWcbaGaeeyta0eabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGnbqtaeqaaGqaaOGae03oaSJae4xEaK3aaSbaaSqaaiabbAeagbqabaGccqGGSaalcqGFtbWudaWgaaWcbaGaeeOrayeabeaakiabcMcaPaqaaiabg2da9aqaaiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemiCaahabeaaaSqabaGccqGGSaalcqWFyoqudaWgaaWcbaGaeeOray0aaSbaaWqaaiabdchaWbqabaaaleqaaOGaeiiFaWNae43uam1aaSbaaSqaaiabb2eanbqabaGccqGGSaalcqGFtbWudaWgaaWcbaGaeeOrayeabeaakiabcMcaPiabgUcaRiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemyCaehabeaaaSqabaGccqGGSaalcqWFyoqudaWgaaWcbaGaeeOray0aaSbaaWqaaiabdghaXbqabaaaleqaaOGaeiiFaWNae4xEaK3aaSbaaSqaaiabb2eanbqabaGccqGGSaalcqGFtbWudaWgaaWcbaGaeeyta0eabeaakiabcYcaSiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemiCaahabeaaaSqabaGccqGG7aWocqGF5bqEdaWgaaWcbaGaeeOrayeabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGgbGraeqaaOGaeiilaWIae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcMcaPaaaaaa@99F0@

= log ⁡ L ( Θ M p | S M ) + log ⁡ L ( Θ F p | S F ) + log ⁡ L ( Θ M q | y M , S M , Θ M p ) + log ⁡ L ( Θ F q | y F , S F , Θ F p ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabg2da9aqaaiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOGGabiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemiCaahabeaaaSqabaGccqGG8baFieqacqGFtbWudaWgaaWcbaGaeeyta0eabeaakiabcMcaPiabgUcaRiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOiab=H5arnaaBaaaleaacqqGgbGrdaWgaaadbaGaemiCaahabeaaaSqabaGccqGG8baFcqGFtbWudaWgaaWcbaGaeeOrayeabeaakiabcMcaPiabgUcaRiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemyCaehabeaaaSqabaGccqGG8baFcqGF5bqEdaWgaaWcbaGaeeyta0eabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGnbqtaeqaaOGaeiilaWIae8hMde1aaSbaaSqaaiabb2eannaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcMcaPiabgUcaRiGbcYgaSjabc+gaVjabcEgaNjabdYeamjabcIcaOiab=H5arnaaBaaaleaacqqGgbGrdaWgaaadbaGaemyCaehabeaaaSqabaGccqGG8baFcqGF5bqEdaWgaaWcbaGaeeOrayeabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGgbGraeqaaOGaeiilaWIae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcMcaPaaaaaa@7EF9@

where equation (4) is derived from equation (3) because the males and females are assumed to be independent, and

log ⁡ L ( Θ k p | S k ) = constant log ⁡ L ( Θ k q | y k , S k , Θ k p ) = + 2 n 11 / 11 k log ⁡ p k 11 ∑ i = 1 n 11 / 11 k log ⁡ f k 2 ( y k i ) + n 11 / 10 k log ⁡ ( 2 p k 11 p k 10 ) + ∑ i = 1 n 11 / 10 k log ⁡ f k 1 ( y k i ) + 2 n 11 / 00 k log ⁡ p k 10 + ∑ i = 1 n 11 / 00 k log ⁡ f k 0 ( y k i ) + n 10 / 11 k log ⁡ ( 2 p k 11 p k 01 ) + ∑ i = 1 n 10 / 11 k log ⁡ f k 1 ( y k i ) + n 10 / 10 k log ⁡ ( 2 p k 11 p k 00 + 2 p k 10 p k 01 ) + ∑ i = 1 n 10 / 10 k log ⁡ [ φ k f k 1 ( y k i ) + ( 1 − φ k ) f k 0 ( y k i ) ] + n 10 / 00 k log ⁡ ( 2 p k 10 p k 00 ) + ∑ i = 1 n 10 / 00 k log ⁡ f k 0 ( y k i ) + 2 n 00 / 11 k log ⁡ p k 01 + ∑ i = 1 n 00 / 11 k log ⁡ f k 0 ( y k i ) + n 00 / 10 k log ⁡ ( 2 p k 01 p k 00 ) + ∑ i = 1 n 00 / 10 k log ⁡ f k 0 ( y k i ) + 2 n 00 / 00 k log ⁡ p k 00 + ∑ i = 1 n 00 / 00 k log ⁡ f k 0 ( y k i ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabOGaaaaaaeaacyGGSbaBcqGGVbWBcqGGNbWzcqWGmbatcqGGOaakiiqacqWFyoqudaWgaaWcbaGaem4AaS2aaSbaaWqaaiabdchaWbqabaaaleqaaOGaeiiFaWhcbeGae43uam1aaSbaaSqaaiabdUgaRbqabaGccqGGPaqkcqGH9aqpcqqGJbWycqqGVbWBcqqGUbGBcqqGZbWCcqqG0baDcqqGHbqycqqGUbGBcqqG0baDaeaacyGGSbaBcqGGVbWBcqGGNbWzcqWGmbatcqGGOaakcqWFyoqudaWgaaWcbaGaem4AaS2aaSbaaWqaaiabdghaXbqabaaaleqaaOGaeiiFaWNae4xEaK3aaSbaaSqaaiabdUgaRbqabaGccqGGSaalcqGFtbWudaWgaaWcbaGaem4AaSgabeaakiabcYcaSiab=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@523F@

where fkj(yki) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aaSbaaSqaaiabdUgaRnaaBaaameaacqWGQbGAaeqaaaWcbeaakiabcIcaOiabdMha5naaBaaaleaacqWGRbWAdaWgaaadbaGaemyAaKgabeaaaSqabaGccqGGPaqkaaa@36A9@ is a normal distribution density function of composite diplotype j (j = 2, 1, 0) for sex k, and

φ k = p k 11 p k 00 p k 00 p k 00 + p k 10 p k 01 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqOXdy2aaSbaaSqaaiabdUgaRbqabaGccqGH9aqpjuaGdaWcaaqaaiabdchaWnaaBaaabaGaem4AaS2aaSbaaeaacqaIXaqmcqaIXaqmaeqaaaqabaGaemiCaa3aaSbaaeaacqWGRbWAdaWgaaqaaiabicdaWiabicdaWaqabaaabeaaaeaacqWGWbaCdaWgaaqaaiabdUgaRnaaBaaabaGaeGimaaJaeGimaadabeaaaeqaaiabdchaWnaaBaaabaGaem4AaS2aaSbaaeaacqaIWaamcqaIWaamaeqaaaqabaGaey4kaSIaemiCaa3aaSbaaeaacqWGRbWAdaWgaaqaaiabigdaXiabicdaWaqabaaabeaacqWGWbaCdaWgaaqaaiabdUgaRnaaBaaabaGaeGimaaJaeGymaedabeaaaeqaaaaaaaa@4F63@

is the relative proportion of diplotype [11] [00] within the double heterozygote for sex k.

It can be seen from equation (3) or (4) that maximizing L(ΘMp,ΘMq;ΘFp,ΘFq|yM,SM,yF,SF) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGccceGae8hMde1aaSbaaSqaaiabb2eannaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYcaSiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemyCaehabeaaaSqabaGccqGG7aWocqWFyoqudaWgaaWcbaGaeeOray0aaSbaaWqaaiabdchaWbqabaaaleqaaOGaeiilaWIae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGXbqCaeqaaaWcbeaakiabcYha8Hqabiab+Lha5naaBaaaleaacqqGnbqtaeqaaOGaeiilaWIae43uam1aaSbaaSqaaiabb2eanbqabaGccqGGSaalcqGF5bqEdaWgaaWcbaGaeeOrayeabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGgbGraeqaaOGaeiykaKcaaa@51A7@ is equivalent to maximizing log (Θkp|Sk) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGccceGae8hMde1aaSbaaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYha8Hqabiab+nfatnaaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaaa@3682@ and log L(Θkp|yk,Sk,Θkp) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGccceGae8hMde1aaSbaaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYha8Hqabiab+Lha5naaBaaaleaacqWGRbWAaeqaaOGaeiilaWIae43uam1aaSbaaSqaaiabdUgaRbqabaGccqGGSaalcqWFyoqudaWgaaWcbaGaem4AaS2aaSbaaWqaaiabdchaWbqabaaaleqaaOGaeiykaKcaaa@4115@ individually in equation (5).

A closed-form solution for the EM algorithm 14 has been derived to estimate the unknown parameters that maximize the two sex-specific likelihoods of (5). The estimates of sex-specific haplotype frequencies are based on the log-likelihood function (Θkp|Sk) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGccceGae8hMde1aaSbaaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYha8Hqabiab+nfatnaaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaaa@3682@, whereas the estimates of sex-specific genotypic values of composite diplotypes and the residual variance are based on the log-likelihood function L(Θkp|yk,Sk,Θ^kp) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGccceGae8hMde1aaSbaaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYha8Hqabiab+Lha5naaBaaaleaacqWGRbWAaeqaaOGaeiilaWIae43uam1aaSbaaSqaaiabdUgaRbqabaGccqGGSaalcuWFyoqugaqcamaaBaaaleaacqWGRbWAdaWgaaadbaGaemiCaahabeaaaSqabaGccqGGPaqkaaa@4125@. These two different types of parameters can be estimated using a two-stage hierarchical EM algorithm (see ref. 14 for a detailed implementation).

According to equation (5), the summed likelihood across the sexes, L(ΘMp,ΘMq|yM,SM)+L(ΘFp,ΘFq|yF,SF) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGccceGae8hMde1aaSbaaSqaaiabb2eannaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYcaSiab=H5arnaaBaaaleaacqqGnbqtdaWgaaadbaGaemyCaehabeaaaSqabaGccqGG8baFieqacqGF5bqEdaWgaaWcbaGaeeyta0eabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGnbqtaeqaaOGaeiykaKIae83kaSIaemitaWKaeiikaGIae8hMde1aaSbaaSqaaiabbAeagnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYcaSiab=H5arnaaBaaaleaacqqGgbGrdaWgaaadbaGaemyCaehabeaaaSqabaGccqGG8baFcqGF5bqEdaWgaaWcbaGaeeOrayeabeaakiabcYcaSiab+nfatnaaBaaaleaacqqGgbGraeqaaOGaeiykaKcaaa@54F7@, is formulated by assuming that haplotype 11 is a risk haplotype. However, a real risk haplotype is unknown from raw data (y_k, S_k). An additional step for choosing the most likely risk haplotype should be implemented. The simplest way to do so is to calculate the likelihood values by assuming that any one of the four haplotypes can be a risk haplotype (Table 1). Thus, we obtain four possible likelihood values as follows:

Risk No . Haplotype Likelihood 1 [ 11 ] L 1 ( Θ ^ M p , Θ ^ 1M q | y M , S M ) + L 1 ( Θ ^ F p , Θ ^ 1 F q | y F , S F ) 2 [ 10 ] L 2 ( Θ ^ M p , Θ ^ 2M q | y M , S M ) + L 2 ( Θ ^ F p , Θ ^ 2 F q | y F , S F ) 3 [ 01 ] L 3 ( Θ ^ M p , Θ ^ 3M q | y M , S M ) + L 3 ( Θ ^ F p , Θ ^ 3 F q | y F , S F ) 4 [ 00 ] L 4 ( Θ ^ M p , Θ ^ 4M q | y M , S M ) + L 4 ( Θ ^ F p , Θ ^ 4 F q | y F , S F ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@19F5@

The largest likelihood value calculated is thought to correspond to the most likely risk haplotype. Under an optimal risk haplotype, we estimate sex-specific quantitative genetic parameters Θ^Mq MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGaf8hMdeLbaKaadaWgaaWcbaGaeeyta00aaSbaaWqaaiabdghaXbqabaaaleqaaaaa@3051@ and Θ^Fq MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGaf8hMdeLbaKaadaWgaaWcbaGaeeOray0aaSbaaWqaaiabdghaXbqabaaaleqaaaaa@3043@.

Haplotype effects on the trait, i.e., the existence of functional haplotypes, in both male and female populations can be tested using the following hypotheses expressed as

H 0 : μ M j ≡ μ M  and  μ F j ≡ for  j = 2 , 1 , 0 H 1 : At least one equality in  H 0  does not hold MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabiGaaaqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdabaGaeqiVd02aaSbaaSqaaiabd2eannaaBaaameaacqWGQbGAaeqaaaWcbeaakiabggMi6kabeY7aTnaaBaaaleaacqWGnbqtaeqaaOGaeeiiaaIaeeyyaeMaeeOBa4MaeeizaqMaeeiiaaIaeqiVd02aaSbaaSqaaiabdAeagnaaBaaameaacqWGQbGAaeqaaaWcbeaakiabggMi6kabbAgaMjabb+gaVjabbkhaYjabbccaGiabdQgaQjabg2da9Gqabiab=jdaYiabcYcaSiab=fdaXiabcYcaSiab=bdaWaqaaiabdIeainaaBaaaleaacqaIXaqmaeqaaOGaeiOoaOdabaGaeeyqaeKaeeiDaqNaeeiiaaIaeeiBaWMaeeyzauMaeeyyaeMaee4CamNaeeiDaqNaeeiiaaIaee4Ba8MaeeOBa4MaeeyzauMaeeiiaaIaeeyzauMaeeyCaeNaeeyDauNaeeyyaeMaeeiBaWMaeeyAaKMaeeiDaqNaeeyEaKNaeeiiaaIaeeyAaKMaeeOBa4MaeeiiaaccbiGae4hsaG0aaSbaaSqaaiabbcdaWaqabaGccqqGGaaicqqGKbazcqqGVbWBcqqGLbqzcqqGZbWCcqqGGaaicqqGUbGBcqqGVbWBcqqG0baDcqqGGaaicqqGObaAcqqGVbWBcqqGSbaBcqqGKbazaaaaaa@8693@

The log-likelihood ratio test statistic (LR) under these two hypotheses can be similarly calculated,

LR = − 2 [ log ⁡ L 0 ( μ ˜ M ; μ ˜ F | y M , ; y F ) − log ⁡ L 1 ( Θ ^ M q ; Θ ^ F q | y M , S M , Θ ^ M p ; y F , S F , Θ ^ F p ) ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeeitaWKaeeOuaiLaeyypa0JaeyOeI0IaeGOmaiJaei4waSLagiiBaWMaei4Ba8Maei4zaCMaemitaW0aaSbaaSqaaiabicdaWaqabaGccqGGOaakcuaH8oqBgaacamaaBaaaleaacqqGnbqtaeqaaOGaei4oaSJafqiVd0MbaGaadaWgaaWcbaGaeeOrayeabeaakiabcYha8Hqabiab=Lha5naaBaaaleaacqqGnbqtaeqaaOGaeiilaWIaei4oaSJae8xEaK3aaSbaaSqaaiabbAeagbqabaGccqGGPaqkcqGHsislcyGGSbaBcqGGVbWBcqGGNbWzcqWGmbatdaWgaaWcbaGaeGymaedabeaakiabcIcaOGGabiqb+H5arzaajaWaaSbaaSqaaiabb2eannaaBaaameaacqWGXbqCaeqaaaWcbeaakiabcUda7iqb+H5arzaajaWaaSbaaSqaaiabbAeagnaaBaaameaacqWGXbqCaeqaaaWcbeaakiabcYha8jab=Lha5naaBaaaleaacqqGnbqtaeqaaOGaeiilaWIae83uam1aaSbaaSqaaiabb2eanbqabaGccqGGSaalcuGFyoqugaqcamaaBaaaleaacqqGnbqtdaWgaaadbaGaemiCaahabeaaaSqabaGccqGG7aWocqWF5bqEdaWgaaWcbaGaeeOrayeabeaakiabcYcaSiab=nfatnaaBaaaleaacqqGgbGraeqaaOGaeiilaWIaf4hMdeLbaKaadaWgaaWcbaGaeeOray0aaSbaaWqaaiabdchaWbqabaaaleqaaOGaeiykaKIaeiyxa0LaeiilaWcaaa@793C@

where the L₀ and L₁ are the plug-in likelihood values under the null and alternative hypotheses of (8), respectively. Although the critical threshold for determining the existence of a functional haplotype can be based on empirical permutation tests, the LR may asymptotically follow a χ² distribution with four degrees of freedom, so that the threshold can be obtained from the χ²distribution table.

The male and female populations may be different in terms of population genetic parameters. Such sex-specific differences can be tested by formulating the following hypotheses

H 0 : p M = p F H 1 : p M ≠ p F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabiGaaaqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdabaGaemiCaa3aaSbaaSqaaiabb2eanbqabaGccqGH9aqpcqWGWbaCdaWgaaWcbaGaeeOrayeabeaaaOqaaiabdIeainaaBaaaleaacqaIXaqmaeqaaOGaeiOoaOdabaGaemiCaa3aaSbaaSqaaiabb2eanbqabaGccqGHGjsUcqWGWbaCdaWgaaWcbaGaeeOrayeabeaaaaaaaa@404D@

for allele frequency at SNP 1,

H 0 : q M = q F H 1 : q M ≠ q F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabiGaaaqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdabaGaemyCae3aaSbaaSqaaiabb2eanbqabaGccqGH9aqpcqWGXbqCdaWgaaWcbaGaeeOrayeabeaaaOqaaiabdIeainaaBaaaleaacqaIXaqmaeqaaOGaeiOoaOdabaGaemyCae3aaSbaaSqaaiabb2eanbqabaGccqGHGjsUcqWGXbqCdaWgaaWcbaGaeeOrayeabeaaaaaaaa@4055@

for allele frequency at SNP 2, and

H 0 : D M = D F H 1 : D M ≠ D F MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabiGaaaqaaiabdIeainaaBaaaleaacqaIWaamaeqaaOGaeiOoaOdabaGaemiraq0aaSbaaSqaaiabb2eanbqabaGccqGH9aqpcqWGebardaWgaaWcbaGaeeOrayeabeaaaOqaaiabdIeainaaBaaaleaacqaIXaqmaeqaaOGaeiOoaOdabaGaemiraq0aaSbaaSqaaiabb2eanbqabaGccqGHGjsUcqWGebardaWgaaWcbaGaeeOrayeabeaaaaaaaa@3EED@

for the linkage disequilibrium between the two SNPs.

For each of the hypotheses (10)–(12), the LR values are calculated, which are each thought to asymptotically follow a χ²-distribution with one degree of freedom. Sex-specific differences in overall population genetic architecture can be tested with the null hypothesis H₀: p_M = p_F, q_M = q_F, and D_M = D_F, with the corresponding LR value to be χ ²-distributed with three degrees of freedom.

Sex-specific differences in overall quantitative genetic architecture can be tested by formulating the hypotheses

H 0 : a M = a F  and  d M = d F H 1 : At least one equality in  H 0  does not hold MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7771@

The LR value calculated under the null and alternative hypotheses is suggested to follow a χ²-distribution with two degrees of freedom. The rejection of the null hypothesis implies that the effects of the same haplotype are different between the two sexes. If there exists a sex-specific difference, the next step is to test whether this difference is due to the additive or dominant genetic effects, or both.

In the preceding sections, the same risk haplotype was assumed between the male and female populations. It is possible that the two sexes have different risk haplotypes. Let μMjm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiVd02aaSbaaSqaaiabb2eannaaBaaameaacqWGQbGAdaWgaaqaaiabd2gaTbqabaaabeaaaSqabaaaaa@31F1@ (j_m = 2, 1, 0) and μFjf MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiVd02aaSbaaSqaaiabbAeagnaaBaaameaacqWGQbGAdaWgaaqaaiabdAgaMbqabaaabeaaaSqabaaaaa@31D5@ (j_f = 2, 1, 0) be the genotypic values of composite diplotypes for the males and females constructed by a sex-specific rick haplotype. By reformulating the likelihood log L(Θkp|yk,Sk,Θkp) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGccceGae8hMde1aaSbaaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabcYha8Hqabiab+Lha5naaBaaaleaacqWGRbWAaeqaaOGaeiilaWIae43uam1aaSbaaSqaaiabdUgaRbqabaGccqGGSaalcqWFyoqudaWgaaWcbaGaem4AaS2aaSbaaWqaaiabdchaWbqabaaaleqaaOGaeiykaKcaaa@4115@ of equation (5) based on sex-specific composite diplotypes, these genotypic values can be estimated with the EM algorithm. A best combination of risk haplotypes between the two sexes can be determined from the AIC values.

Consider three associated SNPs, S₁, S₂, and S₃, each with two alleles denoted by 1 and 0. Let p, q and r, and D₁₂, D₁₃, D₂₃ and D₁₂₃ be the 1-allele frequencies for the three SNPs, and the linkage disequilibria between SNPs 1 and 2, SNPs 1 and 3, SNPs 2 and 3 and among the three SNPs, respectively. Eight haplotypes, [111], [110], [101], [100], [011], [010], [001] and [000], formed by these three SNPs, have sex-specific frequencies arrayed in Θkp=(pk111,pk110,pk101,pk100,pk011,pk010,pk001,pk000) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGae8hMde1aaSbaaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaaWcbeaakiabg2da9iabcIcaOiabdchaWnaaBaaaleaacqWGRbWAdaWgaaadbaGaeGymaeJaeGymaeJaeGymaedabeaaaSqabaGccqGGSaalcqWGWbaCdaWgaaWcbaGaem4AaS2aaSbaaWqaaiabigdaXiabigdaXiabicdaWaqabaaaleqaaOGaeiilaWIaemiCaa3aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqaIWaamcqaIXaqmaeqaaaWcbeaakiabcYcaSiabdchaWnaaBaaaleaacqWGRbWAdaWgaaadbaGaeGymaeJaeGimaaJaeGimaadabeaaaSqabaGccqGGSaalcqWGWbaCdaWgaaWcbaGaem4AaS2aaSbaaWqaaiabicdaWiabigdaXiabigdaXaqabaaaleqaaOGaeiilaWIaemiCaa3aaSbaaSqaaiabdUgaRnaaBaaameaacqaIWaamcqaIXaqmcqaIWaamaeqaaaWcbeaakiabcYcaSiabdchaWnaaBaaaleaacqWGRbWAdaWgaaadbaGaeGimaaJaeGimaaJaeGymaedabeaaaSqabaGccqGGSaalcqWGWbaCdaWgaaWcbaGaem4AaS2aaSbaaWqaaiabicdaWiabicdaWiabicdaWaqabaaaleqaaOGaeiykaKcaaa@6977@ for sex k. Each of these haplotype frequencies is constructed by allele frequencies at different SNPs and their linkage disequilibria of different orders, expressed as

p k 111 = p k q k r k + p k D k 23 + q k D k 13 + r k D k 12 + D k 123 p k 110 = p k q k ( 1 − r k ) − p k D k 23 − q k D k 13 + ( 1 − r k ) D k 12 − D k 123 p k 101 = p k ( 1 − q k ) r k − p k D k 23 + ( 1 − q k ) D k 13 − r k D k 12 − D k 123 p k 100 = p 1 ( 1 − q k ) ( 1 − r k ) + p k D k 23 − ( 1 − q k ) D k 13 − ( 1 − r k ) D k 12 + D k 123 p k 011 = ( 1 − p k ) q k r k + ( 1 − p k ) D k 23 − q k D k 13 − r k D k 12 − D k 123 p k 010 = ( 1 − p k ) q k ( 1 − r k ) − ( 1 − p k ) D k 23 + q k D k 13 − ( 1 − r k ) D k 12 + D k 123 p k 001 = ( 1 − p k ) ( 1 − q k ) r k − ( 1 − p k ) D k 23 − ( 1 − q k ) D k 13 − r k D k 12 + D k 123 p k 000 = ( 1 − p k ) ( 1 − q k ) ( 1 − r k ) + ( 1 − p k ) D k 23 + ( 1 − q k ) D k 13 + ( 1 − r k ) D k 12 + D k 123 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@071C@

for sex k.

Sex-specific population genetic architecture can be tested by comparing the differences in allele frequencies (p_k, q_k, r_k) and linkage disequilibria of different orders (Dk12,Dk13,Dk23,Dk123) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaemiraq0aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqaIYaGmaeqaaaWcbeaakiabcYcaSiabdseaenaaBaaaleaacqWGRbWAdaWgaaadbaGaeGymaeJaeG4mamdabeaaaSqabaGccqGGSaalcqWGebardaWgaaWcbaGaem4AaS2aaSbaaWqaaiabikdaYiabiodaZaqabaaaleqaaOGaeiilaWIaemiraq0aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqaIYaGmcqaIZaWmaeqaaaWcbeaakiabcMcaPaaa@4421@ between the males and females.

In a natural population, there are 27 genotypes for the three SNPs. The frequency of each genotype is expressed in terms of haplotype frequencies. Some genotypes are consistent with diplotypes, whereas the others that are heterozygous at two or more SNPs are not. Each double heterozygote contains two different diplotypes. One triple heterozygote, i.e., 10/10/10, contains four different diplotypes, [111] [000] (in a probability of 2pk111pk000 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeGOmaiJaemiCaa3aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqaIXaqmcqaIXaqmaeqaaaWcbeaakiabdchaWnaaBaaaleaacqWGRbWAdaWgaaadbaGaeGimaaJaeGimaaJaeGimaadabeaaaSqabaaaaa@38C3@), [110] [001] (in a probability of 2pk110pk001 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeGOmaiJaemiCaa3aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqaIXaqmcqaIWaamaeqaaaWcbeaakiabdchaWnaaBaaaleaacqWGRbWAdaWgaaadbaGaeGimaaJaeGimaaJaeGymaedabeaaaSqabaaaaa@38C3@), [101] [010] (in a probability of 2pk101pk010 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeGOmaiJaemiCaa3aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqaIWaamcqaIXaqmaeqaaaWcbeaakiabdchaWnaaBaaaleaacqWGRbWAdaWgaaadbaGaeGimaaJaeGymaeJaeGimaadabeaaaSqabaaaaa@38C3@) and [100] [011] (in a probability of 2p₁₀₀p₀₁₁). The relative frequencies of different diplotypes for this double or triple heterozygote are a function of haplotype frequencies (Supporting Information Table 1). The integrative EM algorithm can be employed to estimate the MLEs of haplotype frequencies. A general formula for estimating haplotype frequencies can be derived.

By assuming [111] as a risk haplotype (labeled by A) and all the others as non-risk haplotypes (labelled by A¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebaaaa@2CF8@), the formulation of genotypic values for three composite diplotypes, μ₂ for AA, μ₁ for AA¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqaeKafmyqaeKbaebaaaa@2E03@ and μ₀ for A¯A¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyqaeKbaebacuWGbbqqgaqeaaaa@2E1B@ can be derived. Similar procedures described for the two-SNP model can be obtained to estimate and test sex-specific additive and dominance genetic effects when a haplotype contains three SNPs.

It is possible that the two- and three-SNP models are too simple to characterize genetic variants for quantitative variation. We can develop a model that includes an arbitrary number of SNPs whose sequences are associated with the phenotypic variation. A key issue for the multi-SNP sequencing model is how to distinguish among 2^ℓ-1 different diplotypes for the same genotype heterozygous at ℓ loci. The relative frequencies of these diplotypes can be expressed in terms of haplotype frequencies.

Consider a a functional haplotype that contains L SNPs among which there exist linkage disequilibria of different orders. The two alleles, 1 and 0, at each of these SNPs are symbolized by r₁, ..., r_L, respectively. Let pr1k,...,prLk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiCaa3aa0baaSqaaiabdkhaYnaaBaaameaacqaIXaqmaeqaaaWcbaGaem4AaSgaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemiCaa3aa0baaSqaaiabdkhaYnaaBaaameaacqWGmbataeqaaaWcbaGaem4AaSgaaaaa@3B90@ be the allele frequencies for these different SNPs within sex k. A haplotype frequency, denoted as pr1r2⋯rL MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiCaa3aaSbaaSqaaiabdkhaYnaaBaaameaacqaIXaqmaeqaaSGaemOCai3aaSbaaWqaaiabikdaYaqabaWccqWIVlctcqWGYbGCdaWgaaadbaGaemitaWeabeaaaSqabaaaaa@374A@, is decomposed into the following components:

p r 1 r 2 … r L k = p r 1 k p r 2 k … p r L k No LD + ( − 1 ) r L − 1 + r L p r 1 k … p r L − 2 k D k ( L − 1 ) L + … + ( − 1 ) r 1 + r 2 p r 3 k ⋯ p r L k D k 12 Digenic LD + ( − 1 ) r L − 2 + r L − 1 + r L p r 1 k … p r L − 3 k D k ( L − 2 ) ( L − 1 ) L + … + ( − 1 ) r 1 + r 2 + r 3 p r 4 k … p r L k D k 123 Trigenic LD + … + ( − 1 ) L ( − 1 ) r 1 + … + r L D k 1 … L L -genic LD MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@2345@

where D_k's are the linkage disequilibria of different orders among particular SNPs for sex k.

Sex-specific difference in terms of allele frequencies and linkage disequilibria between different SNPs as well as haplotype additive and dominance effects can be tested by formulating the corresponding hypotheses.

The model proposed was used to detect differences in the genetic architecture of pain sensitivity between men and women. Genetic and phenotypic data were from a pain genetics project in which 237 subjects (including 143 men and 94 women) from five different races were sampled for six SNPs at three candidate genes. As a demonstration of the utilization of the model, we will focus on two SNPs, OPRDT80G (with two alleles T and G) and OPRDT921C (with two alleles T and C), at the delta opioid receptor. Pain testing procedures followed Fillingim et al. 16. The phenotypic values of traits were subtracted by the means for each race to remove the effect due to races.

These two SNPs construct four haplotypes, [TC], [TT], [GC], and [GT], which yield 10 diplotypes, [TC] [TC], [TC] [TT], [TT] [TT], [TC] [GC], [TC] [GT], [TT] [GC], [TT] [GT], [GC] [GC], [GC] [GT], and [GT] [GT] and nine genotypes, TT/CC, TT/CT, TT/TT, TG/CC, TG/CT, TG/TT, GG/CC, GG/CT and GG/TT. Based on the observed numbers of each genotype in the male and female populations, we estimated sex-specific haplotype frequencies (Table 1). The pattern of haplotype distribution is consistent between the two sexes, with haplotypes [TC] and [TT] jointly occupying a majority proportion in the populations. Haplotype [GT] is very rare, with the frequency close to zero. SNP OPRDT80G has a low heterozygosity because the frequency of its commoner allele (T) is closer to 0.90, whereas there is a high heterozygosity for SNP OPRDT921C in terms of its averaged allele frequencies. The two SNPs are highly significantly associated at p = 3.41 × 10^-5 for males and p = 9.63 × 10^-5 for females, with a normalized linkage disequilibrium of D' = 1.00, because alleles T from OPRDT80G and T from OPRDT921C as well alleles G from OPRDT80G and C from OPRDT921C tend to form the same haplotypes more frequently than at random. There is no sex-specific difference in allele frequencies at the two SNPs and their linkage disequilibrium.

By assuming that one of the haplotypes is a risk haplotype, we estimated the effects of each haplotype on a pain sensitivity trait, assessed with a baseline pressure pain threshold measured at the ulna, in the pooled male and female population. The likelihoods of haplotype [TC], [TT] and [GC] as a risk haplotype are -594.8, -594.5, and -596.1, and thus the most likely risk haplotype is [TT]. The genotypic values of composite diplotypes constructed by this risk haplotype and its non-risk haplotype counterpart were estimated and compared between different sexes. In both males and female, the three composite diplotypes do not display significant genetic differences in the pain trait studied, but the directions of the additive and dominance effects are different between the two sexes (Table 2). In males, the non-risk haplotype tends to increase pressure pain thresholds, and it is overdominant to the risk haplotype, leading to increased pressure pain thresholds at a marginal significance level (p = 0.058) (Fig. 1). By contrast, in females, the non-risk haplotype tends to reduce pressure pain thresholds, and it also tends to be overdominant to the risk haplotype by reducing pressure pain thresholds. These discrepancies in both effect size and direction (Fig. 2) make the overall quantitative genetic architecture of the pain sensitivity trait significantly different between the two sexes (p = 1.49 × 10^-7) (Table 2). Although the additive genetic effect displays a gene by sex interaction at the p = 0.03 significance level, a gene by sex interaction for the dominance effect is highly significant at p = 6.16 × 10^-7. No significant difference was observed in inheritance mode between males and females.

Simulation studies were performed to test the statistical properties of the model proposed. Given a certain sample size (n), we simulated two SNPs by assuming different allele frequencies and linkage disequilibria between two sexes. The hypothesized allele frequencies at the two SNPs are p_M = 0.5, q_M = 0.6 and D_M = 0.1 for males and p_F = 0.8, q_F = 0.9 and D_F = 0.06 for females. By postulating one of the four haplotypes constructed by the two SNPs as a risk haplotype, we calculated the genetic variance among three composite diplotypes using the additive effect a = 0.6 and dominance effect d = 0.8, from which the residual variance was calculated when the heritability (H²) of a trait is given. The phenotypic values of the trait were simulated by assuming that they follow a normal distribution under four different simulation designs, (1) n = 100 and H² = 0.1, (2) n = 400 and H² = 0.1, (3) n = 100 and H² = 0.4, and (4) n = 400 and H² = 0.4.

The new model was used to analyze the simulated SNP and phenotypic data with the results tabulated in Table 3. Population genetic parameters including allele frequencies (p and q) and linkage disequilibrium (D) can well be estimated, with increasing precision when the sample size increases from 100 to 400. The power to detect the given sex-specific differences in allele frequencies and linkage disequilibrium is as high as 0.95 even with a modest sample size (100). Although the traditional model that does not implement sex-specific differences can provide precise estimates of these parameters, the estimates are generally between the true values of males and females.