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June 1999 A score test for linkage using identity by descent data from sibships
Sandrine Dudoit, Terence P. Speed
Ann. Statist. 27(3): 943-986 (June 1999). DOI: 10.1214/aos/1018031264

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.

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Sandrine Dudoit. Terence P. Speed. "A score test for linkage using identity by descent data from sibships." Ann. Statist. 27 (3) 943 - 986, June 1999. https://doi.org/10.1214/aos/1018031264

Information

Published: June 1999
First available in Project Euclid: 5 April 2002

zbMATH: 0957.62101
MathSciNet: MR1724037
Digital Object Identifier: 10.1214/aos/1018031264

Subjects:
Primary: 62F03
Secondary: 05C20‎ , 05C30 , 15A18‎ , 60J20 , 92D30

Keywords: ‎adjacency matrix , Eigenvalues , infinitesimal generator , linkage analysis , Markov chain , orbits , Pólya's theory , quotient graph , score test

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 3 • June 1999
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