Page 1

Copyright © 2005 by the Genetics Society of America

DOI: 10.1534/genetics.104.040386

Bayesian Model Selection for Genome-Wide Epistatic Quantitative

Trait Loci Analysis

Nengjun Yi,*,†,1Brian S. Yandell,‡Gary A. Churchill,§David B. Allison,*,†

Eugene J. Eisen** and Daniel Pomp††

*Department of Biostatistics, Section on Statistical Genetics,†Clinical Nutrition Research Center, University of Alabama, Birmingham,

Alabama 35294,‡Departments of Statistics and Horticulture, University of Wisconsin, Madison, Wisconsin 53706,

§The Jackson Laboratory, Bar Harbor, Maine 04609, **Department of Animal Science, North Carolina

State University, Raleigh, North Carolina 27695 and††Department of Animal Science,

University of Nebraska, Lincoln, Nebraska 68583

Manuscript received December 29, 2004

Accepted for publication April 4, 2005

ABSTRACT

The problem of identifying complex epistatic quantitative trait loci (QTL) across the entire genome

continues to be a formidable challenge for geneticists. The complexity of genome-wide epistatic analysis

results mainly from the number of QTL being unknown and the number of possible epistatic effects being

huge. In this article, we use a composite model space approach to develop a Bayesian model selection

framework for identifying epistatic QTL for complex traits in experimental crosses from two inbred lines.

By placing a liberal constraint on the upper bound of the number of detectable QTL we restrict attention

to models of fixed dimension, greatly simplifying calculations. Indicators specify which main and epistatic

effects of putative QTL are included. We detail how to use prior knowledge to bound the number of

detectable QTL and to specify prior distributions for indicators of genetic effects. We develop a computa-

tionally efficient Markov chain Monte Carlo (MCMC) algorithm using the Gibbs sampler and Metropolis-

Hastings algorithm to explore the posterior distribution. We illustrate the proposed method by detecting

new epistatic QTL for obesity in a backcross of CAST/Ei mice onto M16i.

M

mined by multiple genetic and environmental influ-

ences (Lynch and Walsh 1998). Mounting evidence

suggests that interactions among genes (epistasis) play

an important role in the genetic control and evolu-

tion of complex traits (Cheverud 2000; Carlborg and

Haley 2004). Mapping quantitative trait loci (QTL) is

a process of inferring the number of QTL, their geno-

micpositions,andgeneticeffectsgivenobservedpheno-

type and marker genotype data. From a statistical per-

spective, two key problems in QTL mapping are model

search and selection (e.g., Broman and Speed 2002;

Sillanpa ¨a ¨ and Corander 2002; Yi 2004). Traditional

QTL mapping methods utilize a statistical model, which

estimates the effects of only one QTL whose putative

positionisscannedacrossthegenome(e.g.,Landerand

Botstein 1989; Jansen and Stam 1994; Zeng 1994).

Extensions of this approach can allow for main and epi-

static effects at two or perhaps a few QTL at a time and

employ a multidimensional scan to detect QTL. How-

ever, such an approach neglects potential confound-

ing effects from additional QTL and requires prohibi-

ANY complex human diseases and traits of bio-

logical and/or economic importance are deter-

tive corrections for multiple testing. Non-Bayesian model

selection methods combine simultaneous search with a

sequential procedure such as forward or stepwise selec-

tion and apply criteria such as P-values or modified Baye-

sian information criterion (BIC) to identify well-fitting

multiple-QTL models (Kao et al. 1999; Carlborg et al.

2000;Reifsnyderetal.2000;Bogdanetal.2004).These

methods, although appealing in their simplicity and pop-

ularity,haveseveraldrawbacks,including:(1)theuncer-

tainty about the model itself is ignored in the final in-

ference, (2) they involve a complex sequential testing

strategy that includes a dynamically changing null hy-

pothesis, and (3) the selection procedure is heavily in-

fluenced by the quantity of data (Raftery et al. 1997;

George 2000; Gelman et al. 2004; Kadane and Lazar

2004).

Bayesian model selection methods provide a power-

fulandconceptuallysimpleapproachtomappingmulti-

ple QTL (Satagopan et al. 1996; Hoeschele 2001; Sen

and Churchill 2001). The Bayesian approach pro-

ceeds by setting up a likelihood function for the pheno-

type and assigning prior distributions to all unknowns

in the problem. These induce a posterior distribution

on the unknown quantities that contains all of the avail-

able information for inference of the genetic architec-

ture of the trait. Bayesian mapping methods can treat

the unknown number of QTL as a random variable,

1Corresponding author: Department of Biostatistics, University of Ala-

bama, Ryals Public Health Bldg., 1665 University Blvd., Birmingham,

AL 35294-0022.E-mail: nyi@ms.soph.uab.edu

Genetics 170: 1333–1344 (July 2005)

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1334N. Yi et al.

A BAYESIAN MODEL SELECTION FRAMEWORK

FOR QTL MAPPING

whichhasseveraladvantagesbutresultsinthecomplica-

tion of varying the dimension of the model space. The

reversible jump Markov chain Monte Carlo (MCMC)

algorithm, introduced by Green (1995), offers a power-

ful and general approach to exploring posterior distri-

butions in this setting. However, the ability to “move”

between models of different dimension requires a care-

ful construction of proposal distributions. Despite the

challenges of implementation of reversible jump algo-

rithms, effective approaches for mapping multiple non-

interacting QTL have been developed (Satagopan and

Yandell1996;Heath1997;Thomasetal.1997;Uimari

and Hoeschele 1997; Sillanpa ¨a ¨ and Arjas 1998; Ste-

phens and Fisch 1998; Yi and Xu 2000; Gaffney 2001).

Bayesian model selection methods employing the re-

versible jump MCMC algorithm have been proposed to

mapepistaticQTLininbredlinecrossesandoutbredpop-

ulations (Yi and Xu 2002; Yi et al. 2003, 2004a,b; Narita

and Sasaki 2004). However, the complexity of the reversi-

ble jump steps increases computational demand and may

prohibit improvements of the algorithms.

Recently, Yi (2004) proposed a unified Bayesian

model selection framework to identify multiple nonepi-

static QTL for complex traits in experimental designs,

based upon a composite space representation of the

problem. The composite space approach, which is a

modification of the product space concept developed

by Carlin and Chib (1995), provides an interesting

viewpoint on a wide variety of model selection prob-

lems (Godsill 2001). The key feature of the composite

model space is that the dimension remains fixed,

allowing for MCMC simulation to be performed on a

space of fixed dimension, thus avoiding the complexi-

ties of reversible jump. In Yi (2004), the varying dimen-

sional space is augmented to a fixed dimensional space

(the composite model space) by placing an upper bound

on the number of detectable QTL. In the composite

model space, latent binary variables indicate whether

each putative QTL has a nonzero effect. The result-

ing hierarchical model can vastly simplify the MCMC

search strategy.

In this work we extend the composite model space

approach to include epistatic effects. We develop a frame-

work of Bayesian model selection for mapping epistatic

QTL in experimental crosses from two inbred lines. We

show how to incorporate prior knowledge to select an

upper bound on the number of detectable QTL and

prior distributions for indicator variables of genetic ef-

fects and other parameters. A computationally efficient

MCMC algorithm using a Gibbs sampler or Metropolis-

Hastings (M-H) algorithm is developed to explore the

posterior distribution on the parameters. The proposed

algorithm is easy to implement and allows more com-

plete and rapid exploration of the model space. We first

describe the implementation of this algorithm and then

illustrate the method by analyzing a mouse backcross

population.

We consider experimental crosses derived from two

inbred lines. In QTL studies, the observed data consist

of phenotypic trait values, y, and marker genotypes, m,

forindividualsinamappingpopulation.Weassumethat

markers are organized into a linkage map and restrict

attention to models with, at most, pairwise interactions.

We partition the entire genome into H loci, ? ? {?1,

. . . , ?H}, and assume that the possible QTL occur at

these fixed positions. This introduces only a minor bias

inestimatingthepositionofQTLwhenHislarge.When

the markers are densely and regularly spaced, we set ?

to the marker positions; otherwise, ? includes not only

the marker positions but also points between markers.

In general, the genotypes, g, at loci ? are unobservable

except at completely informative markers, but their

probabilitydistribution,p(g|?,m),canbeinferredfrom

the observed marker data using the multipoint method

(Jiang and Zeng 1997). This probability distribution is

used as the prior distribution of QTL genotypes in our

Bayesian framework.

The problem of inferring the number and locations

of multiple QTL is equivalent to the problem of select-

ing a subset of ? that fully explains the phenotypic varia-

tion. Although a complex trait may be influenced by

multitudes of loci, our emphasis is on a set of at most

L QTL with detectable effects. Typically L will be much

smaller than H. Let ? ? {?1, . . . , ?L} (?{?1, . . . , ?H})

be the current positions of L putative QTL. Each locus

may affect the trait through its marginal (main) effects

and/or interactions with other loci (epistasis). The phe-

notype distribution is assumed to follow a linear model,

y ? ? ? X? ? e, (1)

where ? is the overall mean, ? denotes the vector of

all possible main effects and pairwise interactions of L

potential QTL, X is the design matrix, and e is the vec-

tor of independent normal errors with mean zero and

variance ?2. The number of genetic effects depends on

the experimental design, and the design matrix X is

determined from those genotypes g at the current loci

? by using a particular genetic model (see appendix a

for details of the Cockerham genetic model used here).

There is prior uncertainty about which genetic effects

should be included in the model. As in Bayesian vari-

able selection for linear regression (e.g., George and

McCulloch 1997; Kuo and Mallick 1998; Chipman

et al. 2001), we introduce a binary variable ? for each

effect, indicating that the corresponding effect is in-

cluded (? ? 1) or excluded (? ? 0) from a model.

Letting ? ? diag(?), the model becomes

y ? ? ? X?? ? e. (2)

This linear model defines the likelihood, p(y|?, X, ?),

with ? ? (?, ?, ?2), and the full posterior can be writ-

ten as

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1335 Bayesian Analysis of Genome-Wide Epistasis

p(?, ?, g, ?|y, m) ? p(y|?, X, ?) p(?, ?, g, ?|m).wmand we, it may be better to first determine the prior

expected numbers of main-effect QTL, lm, and all QTL,

l0? lm(i.e., main-effect and epistatic QTL), and then

solve for wmand wefrom the expressions of the prior ex-

pected numbers. It is reasonable to require that wm? we,

which requires some adjustment below when lm? 0.

As shown in appendix b, the prior expected number

of main-effect QTL can be expressed as

(3)

Specifications of priors p(?, ?, g, ?|m) and posterior

calculation are given in subsequent sections.

The vector ? determines the number of QTL (see

appendix b). Hereafter, we denote the included po-

sitions of QTL by ??. The vector (?, ??) comprises a

model index that identifies the genetic architecture of

the trait. A natural model selection strategy is to choose

the most probable model (?, ??) on the basis of its

marginal posterior, p(?, ??|y, m) (George and Foster

2000). For genome-wide epistatic analysis, however, no

single model may stand out, and thus we average over

possible models when assessing characteristics of ge-

netic architecture, with the various models weighted by

their posterior probability (Raftery et al. 1997; Ball

2001; Sillanpa ¨a ¨ and Corander 2002).

lm? L[1 ? (1 ? wm)K],(5)

and the prior expected number of all QTL as

l0? L[1 ? (1 ? wm)K(1 ? we)K2(L?1)],(6)

where K is the number of possible main effects for each

QTL and K2is the number of possible epistatic effects

for any two QTL.

The prior expected number of main-effect QTL, lm,

could be set to the number of QTL detected by tra-

ditional nonepistatic mapping methods, e.g., interval

mapping or composite interval mapping (Lander and

Botstein 1989; Zeng 1994). The prior expected num-

ber of all QTL, l0, should be chosen to be at least lm.

The number of QTL detected by traditional epistatic

mapping methods, e.g., two-dimensional genome scan,

could provide a rough guide for choosing l0. From

Equations 5 and 6, we obtain

PRIOR DISTRIBUTIONS

The above Bayesian model selection framework pro-

videsaconceptuallysimpleandgeneralmethodtoiden-

tify complex epistatic QTL across the entire genome.

However, its practical implementation entails two chal-

lenges: prior specification and posterior calculation. In

this section, we first propose a method to choose an

upper bound for the number of QTL and then describe

the prior specifications for the model index and other

unknowns.

Choice of the upper bound L: We suggest first speci-

fying the prior expected number of QTL, l0, on the

basis of initial investigations with traditional methods,

and then determining a reasonably large upper bound,

L. We assign the prior probability distribution for the

number of QTL, l, to be a Poisson distribution with

mean l0. The value of L can be selected to be large

enough that the probability Pr(l ? L) is very small. On

the basis of a normal approximation to the Poisson

distribution, we could take L as l0? 3√l0.

Prior on ?: For the indicator vector ?, we use an

independence prior of the form

p(?) ??w?j

wm? 1 ??1 ?lm

L?

1/K

(7)

and

we? 1 ??

1 ? (l0/L)

(1 ? wm)K?

1/K2(L?1)

.(8)

We note above that if no main-effect QTL is detected

by traditional nonepistatic mapping methods and lm?

0, then wm ? 0. In this case, we suggest making all

weights equal, wm? we ?

?w, and using (6) to obtain

w ? 1 ??1 ?l0

L?

1/(K?K2(L?1))

. (9)

Prior on ?: When there is no prior information con-

cerning QTL locations, these could be assumed to be

independent and uniformly distributed over the H pos-

sible loci. Thus, given l0the prior probability that any

locus is included becomes l0/H. In practice, it may be

reasonable to assume that any intervals of a given length

(e.g., 10cM) contain atmost one QTL. Althoughthis as-

sumptionisnotnecessary,itcansubstantiallyreducethe

model space and thus accelerate the search procedure.

Prior on ?: We propose the following hierarchical

mixture prior for each genetic effect,

j(1 ? wj)1??j,(4)

where wj? p(?j? 1) is the prior inclusion probability

for the jth effect. We assume that wjequals the predeter-

mined hyperparameter wmor we, depending on the jth

effect being main effect or epistatic effect, respectively.

Under this prior, the importance of any effect is inde-

pendent of the importance of any other effect and the

prior inclusion probability of main effect is different

from that of epistatic effect.

Thehyperparameterswmand wecontroltheexpected

numbers of main and epistatic effects included in the

model, respectively; small wmand wewould concentrate

the priors on parsimonious models with few main ef-

fects and epistatic effects. Instead of directly specifying

?j|(?j, ?2, x•j) ? N(0, ?jc?2(xT

•jx•j)?1),(10)

wherex•j?(x1j,. . . ,xnj)Tisthevectorofthecoefficients

of ?j, and c is a positive scale factor. Many suggestions

have been proposed for choice of c for variable selec-

Page 4

1336N. Yi et al.

tion problems of linear regression (e.g., Chipman et al.

2001; Fernandez et al. 2001). In this study, we take c ?

n, which is a popular choice and yields the BIC if the

prior inclusion probability for each effect equals 0.5

(e.g., George and Foster 2000; Chipman et al. 2001).

In this prior setup, a point mass prior at 0 is used for

the genetic effect ?jwhen ?j? 0, effectively removing

?jfrom the model. If ?j? 1, the prior variances reflect

the precision of each ?j and are invariant to scales

changes in the phenotype and the coefficients. The

value (xT

fects. For a large backcross population with no segrega-

tion distortion, for example, (xT

ginal effects and [1 ? (1 ? 2r)2]/16 for epistatic effects,

with r the recombination fraction between two QTL,

under Cockerham’s model (Zeng et al. 2000).

Priors on ? and ?2: The prior for the overall mean

? is N(?0, ?2

p(??, g?, ??|?, y) ? p(y|?, X?, ??)p(??, g?, ??|?),

(14)

p(???, g??, ???|?, y) ? p(???, g??, ???|?), (15)

and

p(?|?, g, ?, y) ? p(y|?, X?, ??)p(?)p(??, g?, ??|?)

? p(???, g??, ???|?).(16)

It can be seen that the unused parameters do not affect

the conditional posterior of (??, g?, ??) and thus do

not need to be updated conditional on ?. Since the

unused parameters do not contribute to the likelihood,

the posterior of (???, g??, ???) is identical to its prior.

From (16), the conditional posterior of ? depends on

(???,g??,???)andthustheupdateof?requiresgenera-

tion of the corresponding unused parameters in the

current model. These properties lead us to develop

MCMC algorithms as described below. We first briefly

describe the algorithms for updating ??, g?, and ??and

then develop a novel Gibbs sampler and Metropolis-

Hastings algorithm to update the indicator variables for

main and epistatic effects, respectively.

Conditional on ?, X?, and ??, the parameters ?, ?2,

and ?? can be sampled directly from their posterior

distributions, which have standard form (Gelman et al.

2004). Conditional on ?, ??, and ??, the posterior distri-

bution of each element of g?is multinomial and thus

can be sampled directly as well (Yi and Xu 2002). We

adapt the algorithm of Yi et al. (2003) to our model to

update locations ??: (1) ? is restricted to the discrete

space ? ? {?1, . . . , ?H}, and (2) any intervals of some

length ? include at most one QTL. To update ?q, there-

fore, we propose a new location ?*

uniformly from 2d most flanking loci of ?q, where d is

apredeterminedinteger(e.g.,d?2),andthengenerate

genotypes at the new location for all individuals. The

proposals for the new location and the genotypes are

then jointly accepted or rejected using the Metropolis-

Hastings algorithm.

At each iteration of the MCMC simulation, we update

all elements of ? in some fixed or random order. For

the indicator variable of a main effect, we need to con-

sider two different cases: a QTL is currently (1) in or

(2) out of the model. For (1), the QTL position and

genotypes were generated at the preceding iteration.

For (2), we sample a new QTL position from its prior

distribution and generate its genotypes for all individu-

als. An epistatic effect involves two QTL, hence three

different cases: (1) both QTL are in, (2) only one QTL

is in, and (3) both QTL are out of the model. Again,

the new QTL position(s) and genotypes are sampled as

needed.

We update ?j, the indicator variable for an effect,

using its conditional posterior distribution of ?j, which

is Bernoulli,

•jx•j)?1varies for different types of genetic ef-

•jx•j)?1/n ?1⁄4for mar-

0). We could empirically set

?0? y ?1

n?

i?1

n

yi

and

?2

0? s2

y?

1

n ? 1?

n

i?1

(yi? y)2.

We take the noninformative prior for the residual vari-

ance, p(?2) ? 1/?2(Gelman et al. 2004). Although this

prior is improper, it yields a proper posterior distri-

bution for the unknowns and so can be used formally

(Chipman et al. 2001).

MARKOV CHAIN MONTE CARLO ALGORITHM

To develop our MCMC algorithm, we first partition

the vector of unknowns (?, g, ?) into (??, g?, ??) and

(???, g??, ???), representing the unknowns included

or excluded from the model, respectively, where ??and

g?(???and g??) are the positions and the genotypes

of QTL included (excluded), respectively, ??(???) rep-

resent the genetic effects included (excluded), ? ? (?,

?, ?2), ??? (??, ?, ?2), and ???? ???. Similarly, X?

(X??) represent the model coefficients included (ex-

cluded), which are determined by g and ?.

We suppressthe dependenceon theobserved marker

data below. For a particular ? the likelihood function

depends onlyupon theparameters (X?,??) usedby that

model, i.e.,

q for the qth QTL

p(y|?, X, ?) ? p(y|?, X?, ??).(11)

The prior distribution of (?, ?, g, ?) can be partitionedas

p(?, ?, g, ?) ? p(?)p(??, g?, ??|?)p(???, g??, ???|?).

(12)

The full posterior distribution for (?, ?, g, ?) can now

be expressed as

p(?, ?, g, ?|y) ? p(y|?, X?, ??)p(?)p(??, g?, ??|?)

? p(???, g??, ???|?).(13)

From (13), we can derive the conditional posterior dis-

tributions

Page 5

1337 Bayesian Analysis of Genome-Wide Epistasis

p(?j? 1|???j, X, ???j, y) ? 1 ? p(?j? 0|???j, X, ???j, y)

p(?h|y) ?1

N?

N

t?1?

L

q?1

1(?(t)

q ? ?h, ?(t)

q ? 1),h ? 0, 1, . . . , H,

(19)

?

wR

(1 ? w) ? wR

,

(17)

where ?qis the binary indicator that QTL q is included

or excluded from the model. Thus, we can obtain the

cumulative distribution function per chromosome, de-

finedasFc(x|y) ? ?

mosome c. It is worth noting that the cumulative distribu-

tionfunctiondefinedherecanbe?1ifthecorresponding

chromosome contains more than one QTL. Both p(?h|y)

and Fc(x|y) can be graphically displayed and show evi-

dence of QTL activity across the whole genome. Com-

monly used summaries include the posterior probabil-

ity that a chromosomal region contains QTL, the most

likely position of QTL (the mode of QTL positions),

and the region of highest posterior density (HPD) (e.g.,

Gelman et al. 2004). To take the prior specifications,

p(?h), into consideration, we can use the Bayes factor

to show evidence for inclusion of ?hagainst exclusion

of ?h(Kass and Raftery 1995),

where

R ?

p(y|?j? 1, ???j, X, ???j)

p(y|?j? 0, ???j, X, ???j)??

(?

??2

?j? ??2?

??2

n

i?1x2

ij

?j

?

?0.5

x

?h?0p(?h|y)foranypositionxonchro-

? exp?

1

2

n

i?1xij(yi? ? ? xi·? ? xij?j)??2)2

??2

?j? ??2?

n

i?1x2

ij

?,

xi•isthevectorofthecoefficientsof?fortheithindivid-

ual,w?pr(?j?1)isthepriorprobabilitythat?jappears

inthemodel,?2

10), ???jmeans all the elements of ? except for ?j, and

???jrepresents all the elements of ? except for ?j. We

cansample ?jdirectly from(17)or update?jwithproba-

bility min(1, r), where r ? ((w/1 ? w)R)1?2?j.

Theeffect?jwasintegratedfrom(17).Wecangenerate

?jas follows. If ?jis sampled to be zero, ?j? 0. Otherwise,

?jis generated from its conditional posterior

?jisthepriorvarianceof?j(seeEquation

BF(?h) ?

p(?h|y)

1 ? p(?h|y)·1 ? p(?h)

p(?h)

.

(20)

p(?j|?j? 1, ???j, X, ???j, y) ? N(? ˜j, ? ˜2

j),(18)

where

In a similar fashion, we can compute the Bayes factor

comparing a chromosomal region containing QTL to

that excluding QTL.

We can estimate the main effects at any locus or chro-

mosomal intervals ?,

? ˜j? (?2??2

?j??

n

i?1

x2

ij)?1?

n

i?1

xij(yi? ? ? xi•? ? xij?j)

and

? ˜?2

j

? ??2

?j? ??2?

n

i?1

x2

ij.

?k(?) ?1

N?

N

t?1?

L

q?1

1(?(t)

q ? ?, ?(t)

q ? 1)?(t)

qk,k ? 1, 2, . . . , K.

(21)

The heritabilities explained by the main effects can also

be estimated. In epistatic analysis, we need to estimate

two types of additional parameters, the posterior inclu-

sion probability and the size of epistatic effects, both

involving pairs of loci. These two types of unknowns can

be estimated with natural extensions of (19) and (21),

respectively.

POSTERIOR ANALYSIS

TheMCMCalgorithmdescribedabovestartsfromini-

tial values and updates each group of unknowns in turn.

Initial iterations are discarded as “burn-in.” To reduce

serial correlation, we thin the subsequent samples by

keeping every kth simulation draw and discarding the

rest, where k is an integer. The MCMC sampler se-

quence {(?(t), ?(t)

draw from the joint posterior distribution p(?, ??, g?,

??|y), and thus the embedded subsequence {(?(t), ?(t)

t ? 1, . . . , N} is a random sample from its marginal

posterior distribution p(?, ??|y), which is used to infer

the genetic architecture of the complex trait. For ge-

nome-wide epistatic analysis, no single model may stand

out, and we may average over all possible models to as-

sessgeneticarchitecture.Bayesianmodelaveragingpro-

videsmorerobustinferencesaboutquantitiesofinterest

than any single model since it incorporates model un-

certainty (Raftery et al. 1997; Ball 2001; Sillanpa ¨a ¨

and Corander 2002).

The most important characteristic may be the poste-

rior inclusion probability of each possible locus ?h, esti-

mated as

?, g(t)

?, ?(t)

?); t ? 1, . . . , N} is a random

EXAMPLE

?);We illustrate the application of our Bayesian model

selection approach by an analysis of a mouse cross pro-

duced from two highly divergent strains: M16i, consist-

ing of large and moderately obese mice, and CAST/Ei,

a wild strain of small mice with lean bodies (Leamy et al.

2002). CAST/Ei maleswere mated to M16ifemales, and

F1males were backcrossed to M16i females, resulting in

54 litters and 421 mice (213 males, 208 females) reach-

ing 12 weeks of age. All mice were genotyped for 92

microsatellitemarkerslocatedon19autosomalchromo-

somes. The marker linkage map covered 1214 cM with

average spacing of 13 cM. In this study, we analyze FAT,

the sum of right gonadal and hindlimb subcutaneous

fatpads.PriortoQTLanalysis,thephenotypicdatawere