On quantitative trait locus mapping with an interference phenomenon

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a gene with quantitative effect on a trait) on the interval [0, T] representing a chromosome. The observation is the trait and the composition of the genome at some locations called “markers”. We focus on the interference phenomenon, i.e. a recombination event inhibits the formation of another recombination event nearby. We give the asymptotic distribution of the LRT process under the null hypothesis that there is no QTL on [0, T] and under local alternatives with a QTL at <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$t^{\star}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mi>t</mi> <mo>⋆</mo> </msup> </math> </EquationSource> </InlineEquation> on [0, T]. We show that the LRT process is asymptotically the square of a “linear interpolated and normalized process”. We prove that under the null hypothesis, the distribution of the maximum of the LRT process is the same for a model with or without interference. However, the powers of detection are totally different between the two models. Copyright Sociedad de Estadística e Investigación Operativa 2014