Abstract
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.
Highlights
Detecting a Quantitative Trait Locus, so-called QTL, on a given chromosome is a major problem in Genetics
In this paper, we address the problem of the asymptotic distribution of the likelihood ratio test (LRT) process when a few families are considered (I 1)
We propose several new methods, as a function of the map considered, to compute thresholds for the supremum of the LRT process under H0
Summary
The limiting process ∑Ii=1 Zi(.) 2 of Theorem 1 is a Chi-Square process with I degrees of freedom where the Zi(.) are independent and identically distributed (a particular case of formula 7). In the definition of α(t) and β (t), t1 becomes t and t2 becomes tr under the null hypothesis, the process Zi(.) considered at marker positions is the ”squeleton” of an Ornstein-Uhlenbeck process: the stationary Gaussian process with covariance ρ(tk,tk ) = exp(−2|tk − tk |) at the other positions, Zi(.) is obtained from Zi(t ) and Zi(tr) by interpolation. As a consequence, when the number of genetic markers is infinite, ∑Ii=1 Zi(.) 2 is an Ornstein-Uhlenbeck Chi-Square process with I degrees of freedom (OUCS(I)) since the processes Zi(.) are independent. We can notice that the path of the OUCS(3) is very jerky whereas the path of the process corresponding to the sparse map is smooth due to the interpolation between markers
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