Properties of transition matrices between IBD configurations are derived for four general classes of unilineal relative pairs obtained from the grand-parent/ grand-child, half-sib, avuncular, and cousin relationships. In this setting, IBD configurations are defined as orbits of groups acting on a set of inheritance vectors. Properties of the transition matrix between IBD configurations at two linked loci are derived by relating its infinitesimal generator to the adjacency matrix of a quotient graph. The second largest eigenvalue of the infinitesimal generator and its multiplicity are key in determining the form of the transition matrix and of likelihood-based linkage tests such as score tests.
Genetics | Statistical Methodology | Statistical Theory
Dudoit, Sandrine, "IBD Configuration Transition Matrices and Linkage Score Tests for Unilineal Relative Pairs" (February 2003). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 128.