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

Abstract

We consider score tests of the null hypothesis $H_0: \theta = 1/2$ against the alternative hypothesis $H_1: 0 \leq \theta < 1/2$, based upon counts multinomially distributed with parameters $n$ and $\rho(\theta,\pi)_{1 \times m} = \pi_{1\times m}T(\theta)_{m \times m}$, where $T(\theta)$ is a transition matrix with $T(0)=I$ , the identity matrix, and $T(1/2)=(1,\dots,1)^T (\alpha_1,\dots,\alpha_m)$. 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 $\pi$ 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.