An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 755-778.

Nous considérons le processus de test de rapport de vraisemblance (LRT) relatif au test d’absence de QTL (un QTL désigne un gène à effet quantitatif sur un trait) sur un intervalle [0,T] représentant un chromosome. L’originalité de cette étude vient du fait que certains génotypes s’avèrent manquants. Nous donnons la distribution asymptotique du processus de LRT, sous l’hypothèse nulle d’absence de QTL sur [0,T], et sous des alternatives locales où le QTL se situe en t sur [0,T]. Nous montrons que le processus de LRT est asymptotiquement le carré d’un “processus Gaussien d’interpolation non linéaire et renormalisé”. Nous présentons une formule simple permettant le calcul du maximum du carré du processus interpolé. Pour finir, nous prouvons que la valeur critique est exactement la même que dans la configuration classique sans génotypes manquants.

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 is in the fact that some genotypes are missing. 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 the threshold is exactly the same as in the classical situation without missing genotypes.

@article{AFST_2014_6_23_4_755_0,
     author = {Rabier, Charles-Elie},
     title = {An asymptotic test for {Quantitative} {Trait} {Locus} detection in presence of missing genotypes},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {755--778},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 23},
     number = {4},
     year = {2014},
     doi = {10.5802/afst.1423},
     mrnumber = {3270422},
     zbl = {06374887},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1423/}
}
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Rabier, Charles-Elie. An asymptotic test for Quantitative Trait Locus detection in presence of missing genotypes. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 755-778. doi : 10.5802/afst.1423. http://archive.numdam.org/articles/10.5802/afst.1423/

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