Model Selection for Multiple QTL
1. reality of multiple QTL
2. selecting a class of QTL models
3. comparing QTL models
•
•
QTL model selection criteria
issues of detecting epistasis
4. simulations and data studies
•
•
•
Model
3-8
9-15
16-24
25-40
simulation with 8 QTL
plant BC, animal F2 studies
searching through QTL models
NCSU QTL II: Yandell © 2004
1
what is the goal of QTL study?
• uncover underlying biochemistry
–
–
–
–
identify how networks function, break down
find useful candidates for (medical) intervention
epistasis may play key role
statistical goal: maximize number of correctly identified QTL
• basic science/evolution
–
–
–
–
how is the genome organized?
identify units of natural selection
additive effects may be most important (Wright/Fisher debate)
statistical goal: maximize number of correctly identified QTL
• select “elite” individuals
– predict phenotype (breeding value) using suite of characteristics
(phenotypes) translated into a few QTL
– statistical goal: mimimize prediction error
Model
NCSU QTL II: Yandell © 2004
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1. reality of multiple QTL
• evaluate some objective for model given data
– classical likelihood
– Bayesian posterior
• search over possible genetic architectures (models)
– number and positions of loci
– gene action: additive, dominance, epistasis
• estimate “features” of model
– means, variances & covariances, confidence regions
– marginal or conditional distributions
• art of model selection
– how select “best” or “better” model(s)?
– how to search over useful subset of possible models?
Model
NCSU QTL II: Yandell © 2004
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advantages of multiple QTL approach
• improve statistical power, precision
– increase number of QTL detected
– better estimates of loci: less bias, smaller intervals
• improve inference of complex genetic architecture
– patterns and individual elements of epistasis
– appropriate estimates of means, variances, covariances
• asymptotically unbiased, efficient
– assess relative contributions of different QTL
• improve estimates of genotypic values
– less bias (more accurate) and smaller variance (more precise)
– mean squared error = MSE = (bias)2 + variance
Model
NCSU QTL II: Yandell © 2004
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Pareto diagram of QTL effects
3
(modifiers)
minor
QTL
polygenes
1
2
major
QTL
0
3
additive effect
major QTL on
linkage map
2
1
Model
0
4
5
5
10
15
20
25
30
rank order of QTL
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limits of multiple QTL?
• limits of statistical inference
– power depends on sample size, heritability, environmental
variation
– “best” model balances fit to data and complexity (model size)
– genetic linkage = correlated estimates of gene effects
• limits of biological utility
– sampling: only see some patterns with many QTL
– marker assisted selection (Bernardo 2001 Crop Sci)
• 10 QTL ok, 50 QTL are too many
• phenotype better predictor than genotype when too many QTL
• increasing sample size may not give multiple QTL any advantage
– hard to select many QTL simultaneously
• 3m possible genotypes to choose from
Model
NCSU QTL II: Yandell © 2004
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QTL below detection level?
• problem of selection bias
– QTL of modest effect only detected sometimes
– their effects are biased upwards when detected
• probability that QTL detected
– avoids sharp in/out dichotomy
– avoid pitfalls of one “best” model
– examine “better” models with more probable QTL
• build m = number of QTL detected into QTL model
– directly allow uncertainty in genetic architecture
– model selection over genetic architecture
Model
NCSU QTL II: Yandell © 2004
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2. selecting a class of QTL models
• phenotype distribution
– normal (usual), binomial, Poisson, …
– exponential family, semi-parametric, nonparametric
•  = gene action
– additive (A) or general (A+D) effects
– epistatic interactions (AA, AD, ..., or other types?)
•  = location of QTL
– known locations?
– widely spaced (no 2 in marker interval) or arbitrarily close?
• m = number of QTL
– single QTL?
– multiple QTL: known or unknown number?
Model
NCSU QTL II: Yandell © 2004
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normal phenotype
• trait = mean + genetic + environment
• genetic effect uncorrelated with environment
• pr( trait Y | genotype Q, effects  )
Y  GQ  e
Y
var( GQ )   G2 , var( e)   2
Qq
effects   (GQ ,  2 )
qq

heritabili ty h  2
G  2
2
8
Model
QQ
2
G
9
GQQ
10
11
12
13
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15
16
17
18
19
20
9
two QTL with epistasis
• same phenotype model overview
Y  GQ  e, var( e)   2
• partition of genotypic value with epistasis
GQ    1 (Q)   2 (Q)  12 (Q)
• partition of genetic variance
2
2
2
2
var( GQ )   G   1   2   12
Model
NCSU QTL II: Yandell © 2004
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epistasis examples
(Doebley Stec Gustus 1995; Zeng pers. comm.)
3
1
effect +/- 2 se
0
4
bb
Bb
BB
a1
d1
a2
d2
iaa iad ida idd
aa
Aa
AA
Aa
AA
bb
Bb
BB
-2
0
AA
a1
d1
a2
d2
iaa iad ida idd
0.5
4.0
traits 1,4,9
Fisher-Cockerham effects
Aa
Model
AA
0.0
-1.0
-0.5
3.0
2.5
aa
1.5
2.0
aa
Aa
AA
1.0
2.0
1.5
1.0
bb
Bb
BB
effect +/- 2 se
3.5
genotypic value
genotypic
value 3.5
2.5
3.0
4.0
4.5
Aa
4.5
aa
aa
0
bb
BB
Bb
-40
20
20
-1
2
2
-20
genotypic value
2
6
6
20
0
60
aa
40
40
Bb
bb
genotypic4value
80
Aa
effect +/- 2 se
100
AA
genotypic value
genotypic
value
60
80
100
BB
bb
Bb
BB
a1
d1
a2
d2
iaa iad ida idd
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multiple QTL with epistasis
• same overview model
Y  GQ  e, var( e)   2
• sum over multiple QTL in model M = {1,2,12,…}
GQ    sum { jM }  j (Q )
• partition genetic variance in same manner
var( GQ )   G2  sum { jM } 2j
• could restrict attention to 2-QTL interactions
Model
NCSU QTL II: Yandell © 2004
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model selection with epistasis
•
additive by additive 2-QTL interaction
–
–
•
full epistasis adds many model df
–
–
–
•
2 QTL in BC: 1 df
2 QTL in F2: 4 df
3 QTL in F2: 20 df
(one interaction)
(AA, AD, DA, DD)
(34 d.f. 2-QTL, 8 d.f. 3-QTL)
data-driven interactions (tree-structured)
–
–
–
•
adds only 1 model degree of freedom (df) per pair
but could miss important kinds of interaction
contrasts comparing subsets of genotypes
double recessive or double dominant vs other genotypes
discriminant analysis based contrasts (Gilbert and Le Roy 2003, 2004)
some issues in model search
–
epistasis between significant QTL
•
•
–
epistasis with non-significant QTL
•
•
–
Model
check all possible pairs when QTL included?
allow higher order epistasis?
whole genome paired with each significant QTL?
pairs of non-significant QTL?
Yi Xu (2000) Genetics; Yi, Xu, Allison (2003) Genetics; Yi (2004)
NCSU QTL II: Yandell © 2004
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3. comparing QTL models
• balance model fit with model "complexity“
– want maximum likelihood
– without too complicated a model
• information criteria quantifies the balance
– Bayes information criteria (BIC) for likelihood
– Bayes factors for Bayesian approach
Model
NCSU QTL II: Yandell © 2004
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QTL likelihoods and parameters
• LOD or likelihood ratio compares model
– L(p) = log likelihood for a particular model with p parameters
– log(LR) = L(p2) – L(p1)
– LOD = log10 (LR) = log(LR)/log(10)
• p = number of model degrees of freedom
– consider models with m QTL and all 2-QTL epistasis terms
– BC:
p = 1 + m + m(m-1)
– F2:
p = 1 + 2m +4m(m-1)
• Bayesian information criterion balances complexity
– BIC() = –2 log[L(p)] +  p log(n)
– n = number of individuals in study
–  = Broman’s BIC adjustment
Model
NCSU QTL II: Yandell © 2004
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information criteria: likelihoods
• L(p) = likelihood for model with p parameters
• common information criteria:
– Akaike
AIC
– Bayes/Schwartz BIC
– BIC-delta BIC
– general form: IC
= –2 log[L(p)] + 2 p
= –2 log[L(p)] + p log(n)
= –2 log[L(p)] +  p log(n)
= –2 log[L(p)] + p D(n)
• comparison of models
– hypothesis testing: designed for one comparison
• 2 log[LR(p1,p2)] = L(p2) – L(p1)
– model selection: penalize complexity
• IC(p1,p2) = 2 log[LR(p1,p2)] + (p2 – p1) D(n)
Model
NCSU QTL II: Yandell © 2004
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information criteria vs. model size
WinQTL 2.0
SCD data on F2
A=AIC
1=BIC(1)
2=BIC(2)
d=BIC()
models
d
d
information criteria
300
320
340
•
•
•
•
•
•
•
360
d
d
d
1
A
1
1
3
1
1
1A
A
2
2
2
• 2+5+9+2
• 2:2 AD
2
2
d2
d
2
– 1,2,3,4 QTL
– epistasis
2
d
2
A
A
A
4
5
6
7
model parameters p
1
1
A
A
8
9
epistasis
Model
NCSU QTL II: Yandell © 2004
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Bayes factors & BIC
B12 
pr ( model 1 | Y ) / pr ( model 2 | Y ) pr (Y | model 1 )

pr ( model 1 ) / pr ( model 2 )
pr (Y | model 2 )
• what is a Bayes factor?
–
–
ratio of posterior odds to prior odds
ratio of model likelihoods
• BF is equivalent to LR statistic when
– comparing two nested models
– simple hypotheses (e.g. 1 vs 2 QTL)
• BF is equivalent to Bayes Information Criteria (BIC)
– for general comparison of any models
– want Bayes factor to be substantially larger than 1 (say 10 or more)
 2 log( B12 )  2 log( LR)  ( p2  p1 ) log( n)
Model
NCSU QTL II: Yandell © 2004
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QTL Bayes factors
• pattern of QTL across genome
– more complicated prior
– posterior easily sampled
prior probability
0.10
0.20
• sampled marginal histogram
• shape affected by prior pr(m)
e
e
e
p
u
exponential
Poisson
uniform
p p
u u e
u u p
u u u
p
e
p
e
e p
p
e
0.00
– prior pr(m) chosen by user
– posterior pr(m|Y,X)
0.30
• m = number of QTL
0
e
p e e
p
u p
u p
u e
u
2
4
6
8
m = number of QTL
pr (m|Y , X ) /pr (m)
BFm,m1 
pr (m  1|Y , X ) /pr (m  1)
Model
NCSU QTL II: Yandell © 2004
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10
issues in computing Bayes factors
• BF insensitive to shape of prior on m
– geometric, Poisson, uniform
– precision improves when prior mimics posterior
• BF sensitivity to prior variance on effects 
– prior variance should reflect data variability
– resolved by using hyper-priors
• automatic algorithm; no need for user tuning
• easy to compute Bayes factors from samples
– sample posterior using MCMC
– posterior pr(m|Y,X) is marginal histogram
Model
NCSU QTL II: Yandell © 2004
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multiple QTL priors
•
phenotype influenced by genotype & environment
pr(Y|Q,) ~ N(GQ , 2), or Y = GQ + environment
•
partition genotype-specific mean into QTL effects
GQ = mean + main effects + epistatic interactions
GQ =  + (Q) =  + sumj in M j(Q)
•
priors on mean and effects

(Q)
j(Q)
•
~ N(0, 02)
grand mean
~ N(0, 12)
model-independent genotypic effect
~ N(0, 12/|M|) effects down-weighted by size of M
determine hyper-parameters via Empirical Bayes
h2
 G2
0  Y and  1 
 2
2
1 h

Model
NCSU QTL II: Yandell © 2004
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multiple QTL posteriors
• phenotype influenced by genotype & environment
pr(Y|Q,) ~ N(GQ , 2), or Y =  + GQ + environment
• relation of posterior mean to LS estimate
GQ | Y , m ~ N ( BQ Gˆ Q , BQ CQ 2 )
 N (Gˆ Q , CQ 2 )
LS estimate Gˆ Q  sum i [sum jM ˆ j (Qi )]  sum i wiQY
variance
2
V (Gˆ Q )  sum i wiQ
 2  CQ 2
shrinkage
BQ   /(  CQ )  1
Model
NCSU QTL II: Yandell © 2004
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4. simulations and data studies
40
•simulated F2 intercross, 8 QTL
•increase to detect all 8
loci
effect
1
2
3
4
5
6
7
8
11
50
62
107
152
32
54
195
–3
–5
+2
–3
+3
–4
+1
+2
1
1
3
6
6
8
8
9
30
20
8
9
10
11
12
13
Genetic map
ch1
ch2
ch3
ch4
ch5
ch6
ch7
ch8
ch9
ch10
0
Model
7
number of QTL
Chromosome
QTL chr
0
– n=500, heritability to 97%
10
frequency in %
– (Stephens, Fisch 1998)
– n=200, heritability = 50%
– detected 3 QTL
posterior
50
100
NCSU QTL II: Yandell © 2004
150
200
23
loci pattern across genome
• notice which chromosomes have persistent loci
• best pattern found 42% of the time
m
8
9
7
9
9
9
Chromosome
1 2 3 4
2 0 1 0
3 0 1 0
2 0 1 0
2 0 1 0
2 0 1 0
2 0 1 0
Model
5
0
0
0
0
0
0
6
2
2
2
2
3
2
7
0
0
0
0
0
0
8
2
2
1
2
2
2
9
1
1
1
1
1
2
10
0
0
0
0
0
0
NCSU QTL II: Yandell © 2004
Count of 8000
3371
751
377
218
218
198
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B. napus 8-week vernalization
whole genome study
• 108 plants from double haploid
– similar genetics to backcross: follow 1 gamete
– parents are Major (biennial) and Stellar (annual)
• 300 markers across genome
– 19 chromosomes
– average 6cM between markers
• median 3.8cM, max 34cM
– 83% markers genotyped
• phenotype is days to flowering
– after 8 weeks of vernalization (cooling)
– Stellar parent requires vernalization to flower
• Ferreira et al. (1994); Kole et al. (2001); Schranz et al. (2002)
Model
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Bayesian model assessment
Bayes factor ratios
posterior / prior
0.3
0.2
50
QTL posterior
moderate
weak
1
0.0
3
number of QTL
Bayes factor ratios
1
Model
3
5
7
9
model index
11
13
NCSU QTL II: Yandell © 2004
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3
moderate
1
2
weak
2
2
strong
2
2
3
1 e+01
3
posterior / prior
5 e-01
0.0
0.1
2*2
2:2,12
3:2*2,12
2:2,13
2:2,3
2:2,16
2:2,11
3:2*2,3
2:2,15
4:2*2,3,16
3:2*2,13
2:2,14
0.2
model posterior
0.3
2
5 e+02
pattern posterior
evidence suggests
4-5 QTL
N2(2-3),N3,N16
11
9
7
5
3
1
11
9
7
5
number of QTL
2
1
2
2
col 1: posterior
col 2: Bayes factor
note error bars on bf
strong
5
0.1
row 1: # QTL
row 2: pattern
500
QTL posterior
1
3
5
7
9
model index
11
13
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Bayesian estimates of loci & effects
4
0.00
histogram of loci
blue line is density
red lines at estimates
loci histogram
0.02
0.04
0.06
napus8 summaries with pattern 1,1,2,3 and m
Model
0
50
N2
100
150
N3
200
250
100
150
N3
200
250
N16
additive
-0.02 0.02
50
-0.08
estimate additive effects
(red circles)
grey points sampled
from posterior
blue line is cubic spline
dashed line for 2 SD
0
N2
NCSU QTL II: Yandell © 2004
N16
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0.008
envvar
200
100
density
0.006
50
0
pattern: N2(2),N3,N16
col 1: density
col 2: boxplots by m
0.010
Bayesian model diagnostics
0.004
0.006
0.008
0.010
0.012
4
4
5
6
7
8
9
11
12
envvar conditional on number of QTL
0.5
0.6
0.7
4
4
5
6
7
8
9
11
12
LOD
14 16
18
20
0.12
heritability conditional on number of QTL
0.08
density
0.50
0.30
0.4
0.04
5
10
15
marginal LOD, m
Model
0.40
heritability
4
3
2
density
1
0
0.3
marginal heritability, m
10 12
but note change with m
0.2
0.00
environmental variance
2 = .008,  = .09
heritability
h2 = 52%
LOD = 16
(highly significant)
0.60
5
marginal envvar, m
20
25
4
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LOD conditional on number of QTL
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studying diabetes in an F2
• segregating cross of inbred lines
– B6.ob x BTBR.ob  F1  F2
– selected mice with ob/ob alleles at leptin gene (chr 6)
– measured and mapped body weight, insulin, glucose at various ages
(Stoehr et al. 2000 Diabetes)
– sacrificed at 14 weeks, tissues preserved
•
gene expression data
– Affymetrix microarrays on parental strains, F1
• key tissues: adipose, liver, muscle, -cells
• novel discoveries of differential expression (Nadler et al. 2000 PNAS; Lan et
al. 2002 in review; Ntambi et al. 2002 PNAS)
– RT-PCR on 108 F2 mice liver tissues
• 15 genes, selected as important in diabetes pathways
• SCD1, PEPCK, ACO, FAS, GPAT, PPARgamma, PPARalpha, G6Pase, PDI,…
Model
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effect (add=blue, dom=red)
-0.5 0.0 0.5 1.0
0
LOD
2
4
6
8
Multiple Interval Mapping
SCD1: multiple QTL plus epistasis!
0
0
Model
50
chr2
100
50
chr2
100
150
200
250
chr9
300
200
250
chr9
300
chr5
150
chr5
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Bayesian model assessment:
number of QTL for SCD1
Bayes factor ratios
0.05
posterior / prior
5 10
50
QTL posterior
0.10 0.15 0.20
500
0.25
QTL posterior
moderate
1
0.00
weak
1 2 3 4 5 6 7 8 9
11
number of QTL
Model
strong
13
1 2 3 4 5 6 7 8 9
number of QTL
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13
31
15
10
0.00
5
LOD
0.10
density
20
Bayesian LOD and h2 for SCD1
0
5
10
15
1
1
2
3
4
5
6
7
8
9 10
12
14
LOD conditional on number of QTL
0.5
0.1
0.3
heritability
3
2
1
0
density
4
marginal LOD, m
20
0.0
0.1
0.2
0.3
0.4
0.5
marginal heritability, m
Model
0.6
1
0.7
1
2
3
4
5
6
7
8
9 10
12
14
heritability conditional on number of QTL
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Model
1
3
5
7
9
11
model index
13
15
1
NCSU QTL II: Yandell © 2004
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5
7
9
model index
11
2
3
5
6
moderate
6
6
5
5
4
4
6
5
3
4
3:1,2,3
0.15
posterior / prior
0.2
0.4
0.6 0.8
4:2*1,2,3
4:1,2,2*3
4:1,2*2,3
5:3*1,2,3
5:2*1,2,2*3
5:2*1,2*2,3
6:3*1,2,2*3
6:3*1,2*2,3
5:1,2*2,2*3
6:4*1,2,3
6:2*1,2*2,2*3
2:1,3
3:2*1,2
2:1,2
model posterior
0.05
0.10
pattern posterior
2
0.00
Bayesian model assessment:
chromosome QTL pattern for SCD1
Bayes factor ratios
weak
13
15
33
trans-acting QTL for SCD1
(no epistasis yet: see Yi, Xu, Allison 2003)
dominance?
Model
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2-D scan: assumes only 2 QTL!
epistasis
LOD
peaks
Model
joint
LOD
peaks
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sub-peaks can be easily overlooked!
Model
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1-D and 2-D marginals
pr(QTL at  | Y,X, m)
unlinked loci
Model
linked loci
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false detection rates and thresholds
• multiple comparisons: test QTL across genome
– size = pr( LOD() > threshold | no QTL at  )
– threshold guards against a single false detection
• very conservative on genome-wide basis
– difficult to extend to multiple QTL
• positive false discovery rate (Storey 2001)
– pFDR = pr( no QTL at  | LOD() > threshold )
– Bayesian posterior HPD region based on threshold
•  ={ | LOD() > threshold }  { | pr( | Y,X,m ) large }
– extends naturally to multiple QTL
Model
NCSU QTL II: Yandell © 2004
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pFDR and QTL posterior
• positive false detection rate
– pFDR = pr( no QTL at  | Y,X,  in  )
– pFDR =
pr(H=0)*size
pr(m=0)*size+pr(m>0)*power
– power = posterior = pr(QTL in  | Y,X, m>0 )
– size = (length of ) / (length of genome)
• extends to other model comparisons
– m = 1 vs. m = 2 or more QTL
– pattern = ch1,ch2,ch3 vs. pattern > 2*ch1,ch2,ch3
Model
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0.3
1.0
0.0
Model
0.2
0.4
0.6
0.8
relative size of HPD region
1.0
0.2
Storey pFDR(-)
0.1
0
0.0
0.0
0.2
pr( H=0 | p>size )
0.4
0.6
0.8
prior probability
fraction of posterior
found in tails
BH pFDR(-) and size(.)
0.2
0.4
0.6
0.8
1.0
pFDR for SCD1 analysis
0.0
NCSU QTL II: Yandell © 2004
0.2
0.4
0.6
0.8
pr( locus in HPD | m>0 )
1.0
40