Bayes factors for detection of Quantitative Trait Loci

Varona Luis, García-Cortés Luis, Pérez-Enciso, Pérez-Enciso Miguel

Description

A fundamental issue in quantitative trait locus (QTL) mapping is to determine the plausibility of the presence of a QTL at a given genome location. Bayesian analysis offers an attractive way of testing alternative models (here, QTL vs. no-QTL) via the Bayes factor. There have been several numerical approaches to computing the Bayes factor, mostly based on Markov Chain Monte Carlo (MCMC), but these strategies are subject to numerical or stability problems. We propose a simple and stable approach to calculating the Bayes factor between nested models. The procedure is based on a reparameterization of a variance component model in terms of intra-class correlation. The Bayes factor can then be easily calculated from the output of a MCMC scheme by averaging conditional densities at the null intra-class correlation. We studied the performance of the method using simulation. We applied this approach to QTL analysis in an outbred population. We also compared it with the Likelihood Ratio Test and we analyzed its stability. Simulation results were very similar to the simulated parameters. The posterior probability of the QTL model increases as the QTL effect does. The location of the QTL was also correctly obtained. The use of meta-analysis is suggested from the properties of the Bayes factor.

Sujets

• Bayes factor
• Quantitative trait locus
• Statistical hypothesis testing
• Markov chain Monte Carlo

Informations

• Publié par: biomed
• Publié le: 01 janvier 2001
• Nombre de lectures: 12
• Langue: English

Extrait

Genet. Sel. Evol. 33 (2001) 133 152 133
? INRA, EDP Sciences, 2001
Original article
Bayes factors for detection
of Quantitative Trait Loci
a; bLuis VARONA , Luis Alberto GARC˝A CORT? S ,
aMiguel P? REZ ENCISO
a Area de Producci Animal, Centre UdL-IRTA, c/ Rovira Roure 177,
25198 Lleida, Spain
b Unidad de GenØtica Cuantitativa y Mejora Animal,
Universidad de Zaragoza, 50013 Zaragoza, Spain
(Received 8 November 1999; accepted 24 October 2000)
Abstract A fundamental issue in quantitative trait locus (QTL) mapping is to determine the
plausibility of the presence of a QTL at a given genome location. Bayesian analysis offers
an attractive way of testing alternative models (here, QTL vs. no-QTL) via the Bayes factor.
There have been several numerical approaches to computing the Bayes factor, mostly based on
Markov Chain Monte Carlo (MCMC), but these strategies are subject to numerical or stability
problems. We propose a simple and stable approach to calculating the Bayes factor between
nested models. The procedure is based on a reparameterization of a variance component model
in terms of intra-class correlation. The Bayes factor can then be easily calculated from the
output of a MCMC scheme by averaging conditional densities at the null intra-class correlation.
We studied the performance of the method using simulation. We applied this approach to QTL
analysis in an outbred population. We also compared it with the Likelihood Ratio Test and we
analyzed its stability. Simulation results were very similar to the simulated parameters. The
posterior probability of the QTL model increases as the QTL effect does. The location of the
QTL was also correctly obtained. The use of meta-analysis is suggested from the properties of
the Bayes factor.
Bayes factor / Quantitative Trait Loci / hypothesis testing / Markov Chain Monte Carlo
1. INTRODUCTION
Mapping of quantitative trait loci (QTLs) is a rapidly evolving topic in
Statistical Genomics. Several procedures have been described for mapping
QTLs in experimental crosses [10,20,21] and in outbred populations [1,14,
33]. In all these settings, hypothesis testing is one of the most delicate and
controversial issues.
Correspondence and reprints
E-mail: [e-mail]134 L. Varona et al.
From a Bayesian perspective, a procedure was described by Hoeschele
and van Raden [16,17]. It allows the estimation of QTL effects, and it
has been implemented using Monte Carlo methods in crosses [27,29] and
in outbred populations [18,28]. In a Bayesian setting, QTL detection involves
the calculation of the Bayes factor (BF) or the posterior probability of the
models [19,22]. The factor provides a rigorous framework for model
testing in terms of probability, and it does not require assuming any asymptotic
property as it does for the Likelihood Ratio Test (LRT). Unfortunately, the exact
calculation of general BF is not feasible for relatively complex models [19]. For
this reason, Monte Carlo methods, such as the Harmonic Mean Estimation [24]
or the Monte Carlo marginal likelihood [3], have been developed, as reviewed
by Gelman and Meng [7] and Han and Carlin [11]. Moreover, some other
alternatives for providing posterior probabilities have been suggested [4,8].
Among these methods, the Reversible Jump Markov Chain Monte Carlo [8]
has been used in the scope of QTL detection [13,18,28,30,32]. This method
provides a useful tool for calculating the posterior probability of each model,
although it becomes more dif cult as the complexity of the models increases
(multiple markers or multiple alleles at the QTL).
Following the point null Bayes factor approach [2], Garc a-CortØset al. [6]
described a procedure to compare nested variance component models from the
perspective of a Dirac Delta approach. The objective of the present paper is
to describe a point null approach to calculate the Bayes factor using a Markov
Chain Monte Carlo method. The method was compared with LRT and its
performance and stability in QTL mapping.
2. MATERIAL AND METHODS
2.1. Theory
We compare models that only differ by the presence of a QTL. These are
considered as nested models because the parameters of the simple model (o)
are a subset of the parameters of the complex model (h;o). Following the
procedure described in the Appendix, if we compare two nested models, one
complete (A), and one reduced (B), BF can be calculated from the following
simple expression:
p .hD 0/A
BFD (1)
p .hD 0jy/A
where p .hD 0/ and p .hD 0jy/ are the prior and posterior densities of h.A A
First, we will apply this procedure to a simple QTL model, and, later on, we
will analyze a mixed QTL model which also includes polygenic effects.Bayes factors for QTL detection 135
2.1.1. Simple QTL model
Calculation of Bayes factor
Now, we present the Bayes factor for a model containing a QTL effect over
a no-QTL model. Consider the following (model 1):
yD mC ZqC e
where y contains the phenotypic records, m is the overall mean, Z is the
incidence matrix relating observations to QTL effects (q) and e is the vector of
residuals, q and e are assumed to be normally distributed:
2q N.0; Qs /q
2e N.0; Is /e
2 2with s being the variance explained by the QTL, s , the residual variance, andq e
Q, the relationship matrix between QTL effects. Model 1 can be reparameter-
ized as:
yD mC e
where:
e D ZqC e.
Consequently,
e N.0; V/

0 2 2 2 0 2 2VD ZQZ s C Is D s ZQZ h C I.1 h /q e p q q
2 2 2where h D s =s is the proportion of phenotypic variation explained by theq q p
2 2 2QTL, and s D s C s is the phenotypic variance.p q e
The joint distribution of all variables in model 1 is:
2 2 2 2 2 2p .y;m;s ; h /D p .yj m;s ; h /p .m/p .s /p .h /1 1 1 1 1p q p q p q
where: 2 2p .yj m;s ; h / N.m; V/1 p q

1 1
p .m/D k if m2 ; and 0 otherwise, (2)1 1
2k 2k1 1
2 2p .h /D 1 if h 2 [0; 1] and 0 otherwise,1 q q

12 2p .s /D k if s 2 0; and otherwise, (3)1 2p p k2136 L. Varona et al.
where k and k are two small enough values to ensure a at distribution over1 2
the parametric space.
The null hypothesis model is the no-QTL model (model 2):
yD mC e
where:
2e N.0; Is /.p
Then, the joint distribution of records and parameters is:
2 2 2p .y;m;s /D p .yjm;s /p .m/p .s /2 2 2 2p p p
2where we can assume that prior distributions p .m/ and p .s / are identical to2 2 p
equations (2) and (3), respectively, and
2 2p .yjm;s / N.m; Is /.2 p p
From equation (1):
2p .h D 0/1 1q BF D D (4)12 2 2 p .h D 0 y/ p .h D 0 y/1 1q q
2because p .h D 0/D 1.1 q
2.1.2. Mixed QTL model
Let us now consider a mixed inheritance model (model 3) that includes
polygenic effects (u):
yD mC Z uC Z qC e1 2
2 2where u N.0; As /, A being the polygenic relationship matrix and s theu u
polygenic genetic variance, Z and Z are incidence matrices. Notation and1 2
distribution of random QTL effects (q) and residuals (e) are assumed to be the
same as in model 1.
This model can again be reparameterized as:
yD mC e
where:
e D Z uC Z qC e;1 2
consequently,
e N.0; V/
0 2 0 2 2VD Z QZ s C Z AZ s C Is1 21 q 2 u e

2 0 2 0 2 2 2D s Z QZ h C Z AZ h C I.1 h h /1 2p 1 q 2 u q uBayes factors for QTL detection 137
2 2 2where h D s =s is the proportion of phenotypic variation explained byu u p
2 2 2 2polygenes and s is the phenotypic variance s C s C s .p u q e
Records and parameters are jointly distributed as:
2 2 2 2 2 2 2 2 2p .y;m;s ; h ; h // p .yjm;s ; h ; h /p .m/p .s /p .h ; h /3 3 3 3 3p q u p q u p q u
where:

1 1
; and 0 otherwise, (5)p .m/D k if m23 1
2k 2k1 1
2 2 2 2p .h ; h /D 2 if h C h 2 [0; 1] and 0 otherwise,3 q u q u

12 2p .s /D k if s 2 0; and otherwise. (6)3 2p p k2
2 2Note that, assuming prior independence, marginal priors of h and h are:q u
2 2p .h /D 2 2h D Beta.1; 2/3 q q
2 2p .h /D 2 2h D Beta.1; 2/.3 u u
Model 3 will be compared to the following null hypothesis model (model 4):
yD mC Z uC e1
which reduces to:
yD mC e
where:
e D Z uC e;1
consequently
e N.0; V/

0 2 2 2 0 2 2VD Z AZ s C Is D s Z AZ h C I.1 h /1 11 u e p 1 u u
2 2 2 2 2 2p .y;m;s ; h // p .yjm;s ; h /p .m/p .s /p .h /4 4 4 4 4p u p u p u
2where priors for m and s are the same as in model 3, equations (5) and (6),p
2respectively. Prior distribution for h isu

2 2 2p h D U.0; 1/D p hjh D 0 .4 3u u q
U denotes a uniform distribution. As before, model 4 is a particular case of
2model 3 when h D 0.q
The BF of model 3 versus model 4:
2p .h D 0/ 23 q