On stochastic processes for quantitative trait locus mapping under selective genotyping

Abstract

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval [0, T] representing a chromosome. The originality of this study is that we are under selective genotyping: only the individuals with extreme phenotypes are genotyped. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on [0, T] and under local alternatives with a QTL at t^☆ on [0, T]. We show that the LRT process is asymptotically the square of a ‘non-linear interpolated and normalized Gaussian process’. We have an easy formula in order to compute the supremum of the square of this interpolated process. We prove that we have to genotype symmetrically and that the threshold is exactly the same as in the situation where all the individuals are genotyped.

Keywords:

• Gaussian process
• likelihood ratio test
• mixture models
• nuisance parameters present only under the alternative
• QTL detection
• selective genotyping

AMS Subject Classification:

• 62M86
• 65C05
• 62P10

Acknowledgements

I thank Professor Jean-Marc Azaïs from university Paul-Sabatier Toulouse (FR) and Céline Delmas, Researcher at ‘Institut National de la Recherche Agronomique’ Toulouse (FR) for fruitful discussions. I thank Jean-Michel Elsen for having proposed this subject of research.