A Bayesian Method For Estimating The Degree Of The Skewness Of X Chromosome Inactivation Based On The General Pedigrees And Unrelated Females

Background:Skewed X chromosome inactivation(XCI-S)is an important epigenetic phenomenon in organisms,which means that in female somatic cells,more than 75%of the cells keep the alleles on the same X chromosome inactive.It has been reported that the XCI-S is associated with many genetic diseases,and the degree of the XCI-S can affect the severity of genetic diseases in heterozygous females.Estimating the degree of the XCI-S can not only improve the test power of association analysis,but also be indicative of the disease affection status in heterozygous females.Several methods have been proposed to estimate the degree of the XCI-S.These methods encode the three female genotypes(dd,Dd and DD,where D is the deleterious allele)as 0,γ and 2,respectively,where γ E[0,2]is an unknown parameter and can be used to measure the degree of the XCI-S,and then use the frequentist or Bayesian methods to conduct the point estimation and the interval estimation of γ.The Bayesian methods usually show better performance than the frequentist methods because they can incorporate the prior information of γ ∈[0,2].However,the existing methods are only based on family trios or unrelated females,and there has been no method available to estimate γ based on general pedigrees or the mixture of general pedigrees and unrelated females(called mixed data for brevity)currently.Objective:To propose Bayesian methods for estimating the degree of the XCI-S,based on the mixed data or only general pedigrees,where the proposed methods are suitable for both quantitative and qualitative traits.Methods:For the mixed data,firstly,we use the double kinship matrix to represent the genetic correlation among females within general pedigrees,and use the identity matrix to represent the genetic correlation among unrelated females.Then,the form of a block matrix is used to combine the above two correlation matrices.Further,we obtain the likelihood function based on the generalized linear mixed model,set the prior of the target parameter γ as the truncated normal distribution and the uniform distribution respectively,and choose the samples from the posterior distribution of γ by the Hamiltonian Monte Carlo algorithm.Note that the Bayesian methods are timeconsuming,so we use the eigenvalue decomposition and Cholesky decomposition to speed up the posterior sampling process for quantitative and qualitative traits,respectively.Finally,the point estimate of γ is calculated by the mode of the samples,and the credible interval of γ is calculated by the highest posterior probability density interval of the samples.The methods can also be applied to only general pedigrees when the block matrix degenerates to the double kinship matrix.For the mixed data,we denote the method based on the truncated normal prior as BNM and the method based on the uniform prior as BUM.Correspondingly,for only general pedigrees,the methods based on the two priors are denoted as BUP and BNP,respectively.Moreover,we conduct extensive simulations to compare the performances of these four proposed methods with the existing BN and BU methods which are based on only unrelated females.Moreover,we apply our proposed methods to the Minnesota Center for Twin and Family Research(MCTFR)data for their practical use.Results:Simulation results show that the mean squared errors of the point estimates of γobtained by the BNM and BUM methods can be 0.0147 to 0.1853 lower than those yielded by the BN and BU methods.The mean squared errors of the point estimates ofγ generated by the BNP and BUP methods are 0.0003 to 0.1449 higher than those obtained by the BN and BU methods.For the interval estimation,all the six methods can control the coverage probability of the credible intervals around 95%.The median,the mean,the interquartile range and the standard deviation of the widths of the credible intervals obtained by the BNM and BUM methods are smaller under most of the simulated scenarios.When the total number of the females in all the pedigrees and that of the unrelated females are the same,the BNP and BUP methods perform slightly worse than the BN and BU methods in the interval estimation(e.g.,the medians of the widths of the credible intervals obtained by these two kinds of methods have the difference of about 0.0002 to 0.0708).For the two priors of γ,the methods based on the truncated normal prior(BNM,BNP and BN)slightly outperform over the methods based on the uniform prior(BUM,BUP and BU)in general.In the application of the MCTFR data,we found three single nucleotide polymorphisms(SNPs)rs10522027,rs12860832 and rs12849233 which are statistically significantly associated with the alcohol dependence composite score.The point estimates and the credible intervals ofγ at these three SNPs yielded by the BNM method are 0.6922,0.8371 and 0.7633,and(0.2451,1.3518),(0.3266,1.4935),and(0.2236,1.2934),respectively,meanwhile the credible intervals obtained by other methods all contain 1,indicating that the three SNPs undergo random X chromosome inactivation(XCI)or escape from the XCI.Conclusion:Our four proposed methods are suitable for the estimation of the degree of the XCI-S based on the mixed data or only general pedigrees.Among these methods,the BNM and BUM methods based on the mixed data have better performance in both the point estimation and the interval estimation,which means that it is more efficient to estimate the degree of the XCI-S by simultaneously using general pedigrees and additional unrelated females.