A score test for linkage using identity by descent data from sibships

Abstract

We consider score tests of the null hypothesis H0: θ = 1/2 against the alternative hypothesis H1: 0 ≤ θ ≤ 1/2, based upon counts multinomially distributed with parameters n and ρ(θ, π)1×m = π1×mT(θ)m×m, where T(θ) is a transition matrix with T(0) = I, the identity matrix, and T(1/2) = (1,..., 1)T(α1,..., αm). This type of testing problem arises in human genetics when testing the null hypothesis of no linkage between a marker and a disease susceptibility gene, using identity by descent data from families with affected members. In important cases in this genetic context, the score test is independent of the nuisance parameter tt and based on a widely used test statistic in linkage analysis. The proof of this result involves embedding the states of the multinomial distribution into a continuous-time Markov chain with infinitesimal generator Q. The second largest eigenvalue of Q and its multiplicity are key in determining the form of the score statistic. We relate Q to the adjacency matrix of a quotient graph in order to derive its eigenvalues and eigenvectors.