Spectral decomposition of score functions in linkage analysis

Abstract

We consider stochastic processes occurring in nonparametric linkage analysis for mapping disease susceptibility genes in the human genome. Under the null hypothesis that no disease gene is located in the chromosomal region of interest, we prove that the linkage process Z converges weakly to a mixture of Ornstein-Uhlenbeck processes as the number of families N tends to infinity. Under a sequence of contiguous alternatives, we prove weak convergence towards the same Gaussian process with a deterministic non-zero mean function added to it. The results are applied to power calculations for chromosome- and genome-wide scans, and are valid for arbitrary family structures. Our main tool is the inheritance vector process v, which is a stationary and continuous-time Markov process with state space the set of binary vectors w of given length. Certain score functions are expanded as a linear combination of an orthonormal system of basis functions which are eigenvectors of the intensity matrix of v.