## The Annals of Statistics

- Ann. Statist.
- Volume 27, Number 3 (1999), 943-986.

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

Sandrine Dudoit and Terence P. Speed

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 27, Number 3 (1999), 943-986.

**Dates**

First available in Project Euclid: 5 April 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1018031264

**Digital Object Identifier**

doi:10.1214/aos/1018031264

**Mathematical Reviews number (MathSciNet)**

MR1724037

**Zentralblatt MATH identifier**

0957.62101

**Subjects**

Primary: 62F03: Hypothesis testing

Secondary: 92D30: Epidemiology 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 15A18: Eigenvalues, singular values, and eigenvectors 05C20: Directed graphs (digraphs), tournaments 05C30: Enumeration in graph theory

**Keywords**

Score test Markov chain infinitesimal generator quotient graph adjacency matrix eigenvalues orbits Pólya's theory linkage analysis

#### Citation

Dudoit, Sandrine; Speed, Terence P. A score test for linkage using identity by descent data from sibships. Ann. Statist. 27 (1999), no. 3, 943--986. doi:10.1214/aos/1018031264. https://projecteuclid.org/euclid.aos/1018031264