Testing of Gender Differences on Sib-Sib Correlations for Binary Traits: Likelihood Based Inference with Application to Arterial Blood Pressures Data

Estimation of measures of familial aggregation is considered the first step in establishing whether a specified disease has a genetic component. Population based family study designs areusually used to estimate correlations among siblings. When the trait of interest is quantitative (e.g. blood pressure, body mass index, blood glucose level) testing the effect of gender differences on sib-sib correlations is achieved using the likelihood method of estimation under the assumption of multivariate normality. When the trait of interest is measured on the binary scale testing the equality of a brother-brother and sister-sister correlation is more complex. In this paper we develop likelihood-based inference procedures for this purpose which may beapplied to nuclear family data. Citation: Shoukri M, Collison K, Al-Mohanna F (2014) Testing of Gender Differences on Sib-Sib Correlations for Binary Traits: Likelihood Based Inference with Application to Arterial Blood Pressures Data. J Biomet Biostat 5: 186. doi:10.4172/2155-6180.1000186 J Biomet Biostat ISSN: 2155-6180 JBMBS, an open access journal Page 2 of 6 Volume 5 • Issue 1 • 1000186 sib correlations characterizing males and female have not yet been developed. This paper has a threefold objective. First, we develop a multivariate probability distribution for the vector of binary observations based on a random sample of independent sib-ships. The vector will be split into two sub-clusters, separating female responses from male responses. Second we construct the likelihood function of the sample as based on the joint distribution of the created sub-clusters. This allows us to develop score and Wald chi-square-tests of significance that compare the levels of similarity among males and females from the same family. Finally, we illustrate our procedures using published arterial blood pressures data. Models Suppose that we have a random sample of k sib-ships, where each sib-ship constitutes a cluster. Let yi=(yi1,yi2,..., i ib y ,xi1, xi2,... i is x )T denote the vector of observations from the ith cluster, where bi=number of brothers in the i th family, si=number of sisters in the i th family, ni=bi+si=sibship size of the i th family, 1 = =∑ k i b bi= number of brothers in the sample of k families, 1 = =∑ k i s si=number of sisters in the sample of k families, and N=b+s= number of siblings in the k families. It is clear then that each cluster (sib-ship) is naturally divided into two sub-clusters, one cluster represents brothers and the other sub-cluster represents sisters. Let yij=1(0) denote the presence (absence) of a trait observed on the jth brother from the ith family (j=1,2,...bi;i=1,2,...k). Similarly, let xij=1(0) denote the presence (absence) of this trait as observed on the jth sister in the ith family (j=1,2,...si;i=1,2,...k). Let ( 1| ) λ λ = = ib ij ib p y denote the probability that a randomly selected brother from the ith family is classified as having the trait of interest, and let 1 ( 0 | ) λ λ − = = ib ij ib p y . Moreover let ( 1| ) λ λ = = ij is is P x , and ( 0 | ) 1 λ λ = = − ij is is P x . We initially assume that the distribution of the brothers’ scores is conditionally independent of the distribution of the sisters’ scores. To introduce the correlation among brothers within the ith family we shall assume that λib is an element of a random sample obtained from a beta distribution with parameters (αb,βb) so that ( ) α μ λ α β = = + b b ib b b E , 2 ( ) (1 ) ( ) (1 ) α β λ ρ μ μ α β α β = = − + + + b b ib b b b b b b b Var , Where ρb=(1+αb+βb) -1. Similarly 2 ( ) ( ) (1 ) ( ) (1 ) α α μ λ λ ρ μ μ α β α β α β = = = = − + + + + s s s is is s s s s s s s s s E Var , where ρs=(1+αs+βs) -1. We can show that the population common intraclass correlation among brothers in the samesub-cluster is: ij ij ' Corr (y , y ') = ρb and the common intraclass correlation among sisters in the other subcluster is: ij ij Corr (x , x ') = ρs i≠j׳ =a,2,...bi, and m≠l=1,2,...si for all i=1,2,...k. We further define the interclass correlation among brothers and sisters as: Corr (yij, xil)=ρ12 i=1,2,...k, j=1,2,...bi and l = 1,2,...si. Note that, because of the exchangeability condition, the unconditional distribution of 1 = =∑ i b ib ij j y y is that of a betabinomial distribution with: E(yib)=biμb (1) 2 ( ) (1 )[1 ( 1) ] σ μ μ ρ = = − + − i b ib i b b i b Var y b b (2) Similarly, the unconditional distribution of 1 = =∑ i s is ij j x x will be that of a beta-binomial distribution with E(xis)=siμs (3) 2 ( ) (1 )[1 ( 1) ] σ μ μ ρ = = − + − si is i s s i s Var x s s (4) Details may be found in references [9-11]. The beta-binomial probability distributions of yib and xis are given respectively as: