Statistical Papers

, Volume 60, Issue 1, pp 239–271 | Cite as

Chi-square processes for gene mapping in a population with family structure

  • Charles-Elie RabierEmail author
  • Jean-Marc Azaïs
  • Jean-Michel Elsen
  • Céline Delmas
Regular Article


Detecting a quantitative trait locus, so-called QTL (a gene influencing a quantitative trait which is able to be measured), on a given chromosome is a major problem in Genetics. We study a population structured in families and we assume that the QTL location is the same for all the families. We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL on the interval [0, T] representing a chromosome. We give the asymptotic distribution of the LRT process under the null hypothesis that there is no QTL in any families and under local alternative with a QTL at \(t^{\star }\in [0, T]\) in at least one family. We show that the LRT is asymptotically the supremum of the sum of the square of independent interpolated Gaussian processes. The number of processes corresponds to the number of families. We propose several new methods to compute critical values for QTL detection. Since all these methods rely on asymptotic results, the validity of the asymptotic assumption is checked using simulated data. Finally we show how to optimize the QTL detecting process.


Chi-square process Gaussian process Likelihood ratio test Mixture models QTL detection MCQMC 

Mathematics Subject Classification

62M86 65C05 62P10 



This work has been supported by the the National Center for Scientific Research (CNRS), the Animal Genetic Department of the French National Institute for Agricultural Research, and SABRE. We thank Simon de Givry for help with human data.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Charles-Elie Rabier
    • 1
    • 2
    Email author
  • Jean-Marc Azaïs
    • 2
  • Jean-Michel Elsen
    • 3
  • Céline Delmas
    • 1
  1. 1.MIAT, Université de Toulouse, INRACastanet-TolosanFrance
  2. 2.Institut de Mathématiques de Toulouse, CNRS UMR 5219Université Paul SabatierToulouseFrance
  3. 3.GenPhyse, Université de Toulouse, INRACastanet-TolosanFrance

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