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June 1999 NCSU QTL Workshop © Brian S. Yandell
1
Bayesian Inference for QTLs in Inbred Lines II
Brian S YandellUniversity of Wisconsin-Madisonwww.stat.wisc.edu/~yandell
NCSU Statistical GeneticsJune 1999
June 1999 NCSU QTL Workshop © Brian S. Yandell
2
Many Thanks
Michael NewtonDaniel SorensenDaniel GianolaJaya SatagopanPatrick GaffneyFei Zou
Tom OsbornDavid ButruilleMarcio FerreraJosh UdahlPablo Quijada
USDA Hatch Grants
June 1999 NCSU QTL Workshop © Brian S. Yandell
3
Overview II
• quick review of trait model– single & multiple QTL– details of Gibbs sampler full conditionals– vector notation
• reversible jump MCMC– multiple regression– number of QTLs
• deconstructing Bayesian LODs
June 1999 NCSU QTL Workshop © Brian S. Yandell
4
Quick Review of trait Model
• single QTL details of Gibbs sampler– normal priors & likelihoods
• mean, additive effects
– inverse gamma prior for variance• or inverse chi-square
– vague priors lead to usual estimates as posterior means
• multiple QTL trait model– model with vector notation
June 1999 NCSU QTL Workshop © Brian S. Yandell
5
Single QTL trait Model• trait = mean + additive + error• trait = effect_of_geno + error• prob( trait | geno, effects )
**
2**
**
),,;|(
jj
jj
jjj
xby
bxy
exby
10 12 14
x=-1x=0
x=1
June 1999 NCSU QTL Workshop © Brian S. Yandell
6
Gibbs Sampler updates
variancemean
traits
additive
genos
June 1999 NCSU QTL Workshop © Brian S. Yandell
7
Full Conditional for mean
nnbV
n
xby
n
xby
bE
xbyb
n
jjj
n
jjj
n
j
jj
222**
1
**
1
**
2**
1
**2**
2
2
2
2
2
2
),;,|(
)()(
),;,|(
),;,|(
xy
xy
xy• normal prior
with large variance
• leads to normal posterior
• posterior mean
• posterior variance
2
June 1999 NCSU QTL Workshop © Brian S. Yandell
8
Full Conditional for additive Effect
n
jj
n
jj
n
jj
n
jjj
n
jj
n
jjj
n
j
jj
xxbV
x
yx
x
yx
bE
xbybb
1
2*
2
1
2*
22**
1
2*
1
*
1
2*
1
*
2**
1
***2**
)()(),;,|(
)(
)(
)(
)(
),;,|(
),;,|(
2
2
2
2
xy
xy
xy• normal prior with large variance
• leads to normal posterior
• posterior mean
• posterior variance
2
June 1999 NCSU QTL Workshop © Brian S. Yandell
9
Full Conditional for variance
n
xby
a
vbE
vaInvb
evaInv
n
jjj
n
n
nn
va
1
2**
2
2
22**2
222
**2
/)1(22
)(
ˆ1
ˆ),;,|(
ˆ,~,;,|
)(),(~2
xy
xy
• inverse gamma prior with large v/a
• posterior distribution
• posterior mean
June 1999 NCSU QTL Workshop © Brian S. Yandell
10
MCMC run for variance
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0 200 400 600 800 1000
1.0
1.2
1.4
1.6
1.8
2.0
MCMC run
1.0
1.2
1.4
1.6
1.8
2.0
0 10 20 30 40
varia
nce
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
11
Alternative for Variance:use Inverse Chi-square
nd
xbyvd
ndInvb
vdvdvdInv
n
jjj
dd
1
2**
2**2
222
22
)(
,~,;,|
~or ,),(~
xy
• inverse chi-square prior with large d,v
• posterior distribution
June 1999 NCSU QTL Workshop © Brian S. Yandell
12
Multiple QTL model
• trait = mean + add1 + add2 + error• trait = effect_of_genos + error• prob( trait | genos, effects )
j
m
rjrrj
jjjj
exby
exbxby
1
**
*2
*2
*1
*1
June 1999 NCSU QTL Workshop © Brian S. Yandell
13
Vector Notation for QTLs• inner product for sum• condense notation
**1
*
*
*1
*
*
*1
*
**
1
**
,,,,
,
n
jm
j
j
m
j
m
rjrr
x
x
b
b
xb
xxXxb
xb
June 1999 NCSU QTL Workshop © Brian S. Yandell
14
Multiple loci• vector of loci across linkage map• careful bookkeeping during update
– identifiability & bump hunting– possibility of two loci in one marker interval
• ordered loci are sufficient
n
jrjrrr
m
m
rr
x1
**
11
**
)|()()|(
),,(,)|()|(
X
XX
June 1999 NCSU QTL Workshop © Brian S. Yandell
15
Posterior: Multiple QTLs• posterior = likelihood * prior / constant• posterior( paramaters | data )
prob( genos, effects, loci | traits, map )
is proportional to
)|;,,;( 2** yX b
m
r
n
jrjrrr
n
jjj
xb
y
1 1
**2
1
2**
)|()()()()(
),,;|(
bx
June 1999 NCSU QTL Workshop © Brian S. Yandell
16
MCMC for Multiple QTLs• construct Markov chain around posterior• update one (or several) components at a time
– update effects given genotypes & traits– update loci given genotypes & traits– update genotypes give loci & effects
• update all terms for each locus at one time?– open questions of efficient mixing
N
21
2**2** )|;,,;(~);,,;( ybXbX
June 1999 NCSU QTL Workshop © Brian S. Yandell
17
MCMC Updates
effectsloci
traits
genos
June 1999 NCSU QTL Workshop © Brian S. Yandell
18
MCMC Conditions• construct Markov chain with stable distribution• ergodic Markov chain
– reversibile (detailed balance)– irreducible (can get from any value to any other)– aperiodic (no fixed pattern)– positive recurrent (chance to visit all possible values)
z
N
zzzz
zzzzzz
zzzz
all22
212121
10
)|()()(0
1)|()()|()(0
)(~,,,
June 1999 NCSU QTL Workshop © Brian S. Yandell
19
Reversible Jump MCMC
• basic idea of Green(1995)• model selection in regression• how many QTLs?
– number of QTL is random– estimate the number m
• RJ-MCMC vs. Bayes factors• other similar ideas
June 1999 NCSU QTL Workshop © Brian S. Yandell
20
Jumping the Number of QTL • model changes with number of QTL
– almost analogous to stepwise regression– use reversible jump MCMC to change number
• book keeping helps in comparing models• change of variables between models
• prior on number of QTL– uniform over some range– Poisson with prior mean
!)|(
m
em
m
June 1999 NCSU QTL Workshop © Brian S. Yandell
21
Posterior: Number of QTL• posterior = likelihood * prior / constant• posterior( paramaters | data )
prob( genos, effects, loci, m | traits, map )
is proportional to
)|,;,,;( 2** yX mb
m
r
n
jrjrrr
n
jjj
xbm
my
1 1
**2
1
2**
)|()()()()()(
);,,;|(
bx
June 1999 NCSU QTL Workshop © Brian S. Yandell
22
Reversible Jump Choicesaction step: draw one of three choices• update step with probability 1-b(m+1)-d(m)
– update current model– loci, effects, genotypes as before
• add a locus with probability b(m+1)– propose a new locus– innovate effect and genotypes at new locus– decide whether to accept the “birth” of new locus
• drop a locus with probability d(m)– pick one of existing loci to drop– decide whether to accept the “death” of locus
June 1999 NCSU QTL Workshop © Brian S. Yandell
23
Markov chain for number m
• add a new locus• drop a locus• update current model
0 1 mm-1 m+1...
m
June 1999 NCSU QTL Workshop © Brian S. Yandell
24
Jumping QTL number & loci
111111111111111111111111
1
11111
111111
11111
1111
111
1111
11111
11111
11111111
111111111
11
111111111
1111111
111
22222222222222222222222
2 22222222222 22
2222 22222
22222222
222222222 222222222
22 22
23333
333333333333 333
3
0 20 40 60 80 100
20
40
60
80
dist
ance
(cM
)
MCMC run
0 20 40 60 80 1000
1
2
3
num
ber
of Q
TL
June 1999 NCSU QTL Workshop © Brian S. Yandell
25
RJ-MCMC Updates
effectsloci
traits
genos
addlocus
drop locus
b(m+1)
d(m)
1-b(m+1)-d(m)
June 1999 NCSU QTL Workshop © Brian S. Yandell
26
Propose to Add a locus• propose a new locus
– similar proposal to ordinary update• uniform chance over genome• easier to avoid interval with another QTL
– need genotypes at locus & model effect
• innovate effect & genotypes at new locus– draw genotypes based on recombination (prior)
• no dependence on trait model yet
– draw effect as in Green’s reversible jump• adjust for collinearity• modify other parameters accordingly
• check acceptance ...
Dqb /1)(
June 1999 NCSU QTL Workshop © Brian S. Yandell
27
Propose to Drop a locus
• choose an existing locus– equal weight for all loci ?– more weight to loci with small effects?
• “drop” effect & genotypes at old locus– adjust effects at other loci for collinearity– this is reverse jump of Green (1995)
• check acceptance …– do not drop locus, effects & genotypes– until move is accepted
mmrqd /1);(
June 1999 NCSU QTL Workshop © Brian S. Yandell
28
Acceptance of Reversible Jump
• accept birth of new locus with probabilitymin(1,A)
• accept death of old locus with probabilitymin(1,1/A)
m
Jmrq
q
mb
md
m
mA
m
d
mb
m
m
,;,,;
1
)1;(
)(
)(
)1(
)|,(
)|1,(
2**
11
bX
yy
June 1999 NCSU QTL Workshop © Brian S. Yandell
29
Acceptance of Reversible Jump
• move probabilities
• birth & death proposals
• Jacobian between models–fudge factor–see stepwise regression example
nsJ
mrq
q
mb
md
othersr
d
mb
|
1
)1;(
)(
)(
)1(
m m+1
June 1999 NCSU QTL Workshop © Brian S. Yandell
30
RJ-MCMC: Number of QTL
MCMC run0 200 400 600 800
01
23
45
6
MCMC run/10
num
ber
of Q
TL
0 200 400 600 800
01
23
45
6
MCMC run/1000 200 400 600 800
01
23
45
6
MCMC run/1000
num
ber
of Q
TL
0 200 400 600 800
01
23
45
6
June 1999 NCSU QTL Workshop © Brian S. Yandell
31
Posterior # QTL for 8-week Data
0 1 2 3 4 5 6
0.0
0.2
0.4
98% credible region for m: (1,3)based on 1 million steps
with prior mean of 3
June 1999 NCSU QTL Workshop © Brian S. Yandell
32
How Good is RJ-MCMC?
• simulations with 0, 1 or 2 QTL– strong effects (additive = 2, variance = 1)– linked loci 36cM apart
• differences with number of QTL– clear differences by actual number– works well with 100,000, better with 1M
• effect of Poisson prior mean– larger prior mean shifts posterior up– but prior does not take over
June 1999 NCSU QTL Workshop © Brian S. Yandell
33
Posterior for Simulated Data
• 0,1 or 2 large QTL• prior Poisson mean of 2• 100,000 RJ-MCMC runs
0 1 2 3 4 5
0.0
0.4
0.8
no QTL present0 1 2 3 4 5
0.0
0.4
1 QTL present0 1 2 3 4 5
0.0
0.4
0.8
2 QTL present
June 1999 NCSU QTL Workshop © Brian S. Yandell
34
Effect of Prior Mean
0 1 2 3 4 5
0.0
0.4
0.8
prio
r m
ean
= 2
priorpost.
0 QTL present0 1 2 3 4 5
0.0
0.4
1 QTL present0 1 2 3 4 5
0.0
0.4
0.8
2 QTL present
0 1 2 3 4 5
0.0
0.2
0.4
prio
r m
ean
= 4
0 QTL present0 1 2 3 4 5
0.0
0.2
0.4
0.6
1 QTL present0 1 2 3 4 5
0.0
0.2
0.4
2 QTL present
June 1999 NCSU QTL Workshop © Brian S. Yandell
35
# QTL in Brassica Data
• 4-week & 8-week vernalization– log( days to flower)– 105 lines, 10 markers– modest effects– evidence of 1 or 2 QTL using Bayes factors
• histograms of posterior number of QTL– depends somewhat on prior– mode is 1 or 2 QTL
• 90% credible sets– all include 2 QTL– include 1 QTL if prior not huge
June 1999 NCSU QTL Workshop © Brian S. Yandell
36
#QTL for Brassica 8-week
0 1 2 3 4 5 6 7 8
0.0
0.4
0.8
priorpost.
prior mean = 10 1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
prior mean = 20 1 2 3 4 5 6 7 8
0.0
0.2
0.4
prior mean = 3
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
prior mean = 40 1 2 3 4 5 6 7 8
0.0
0.2
prior mean = 60 1 2 3 4 5 6 7 8
0.0
0.15
0.30
prior mean = 10
June 1999 NCSU QTL Workshop © Brian S. Yandell
37
Brassica #QTL 90% Credible Sets
prior lo hi level lo hi level
1 1 2 0.98 1 2 0.99
2 1 2 0.95 1 2 0.94
3 1 3 0.98 1 3 0.98
4 1 3 0.95 1 3 0.93
6 1 4 0.96 1 4 0.94
10 2 5 0.90 2 6 0.97
8-week 4-week
June 1999 NCSU QTL Workshop © Brian S. Yandell
38
Brassica #QTL Comparison
0 1 2 3 4 5 6
0.0
0.4
0.8
4-w
eek
data
prior mean = 10 1 2 3 4 5 6
0.0
0.2
0.4
prior mean = 20 1 2 3 4 5 6
0.0
0.2
0.4
prior mean = 3
0 1 2 3 4 5 6
0.0
0.4
0.8
8-w
eek
data
prior mean = 10 1 2 3 4 5 6
0.0
0.2
0.4
0.6
prior mean = 20 1 2 3 4 5 6
0.0
0.2
0.4
prior mean = 3
June 1999 NCSU QTL Workshop © Brian S. Yandell
39
Reversible Jump II
• reversible jump MCMC details– can update model with m QTL– have basic idea of jumping models– now: careful bookkeeping between models
• RJ-MCMC & Bayes factors– Bayes factors from RJ-MCMC chain– components of Bayes factors
June 1999 NCSU QTL Workshop © Brian S. Yandell
40
RJ-MCMC Updates
effectsloci
traits
genos
addlocus
drop locus
b(m+1)
d(m)
1-b(m+1)-d(m)
June 1999 NCSU QTL Workshop © Brian S. Yandell
41
Reversible Jump Idea• expand idea of MCMC to compare models• adjust for parameters in different models
– augment smaller model with innovations– constraints on larger model
• calculus “change of variables” is key– add or drop parameter(s)– carefully compute the Jacobian
• consider stepwise regression– Mallick (1995) & Green (1995)– efficient calculation with Hausholder decomposition
June 1999 NCSU QTL Workshop © Brian S. Yandell
42
Model Selection in Regression
• known regressors (e.g. markers)– models with 1 or 2 regressors
• jump between models– centering regressors simplifies calculations
jjjj
jjj
exxbxxbym
exxbym
)()(:2
)(:1
222111
11
June 1999 NCSU QTL Workshop © Brian S. Yandell
43
Slope Estimate for 1 Regressor
recall least squares estimate of slopenote relation of slope to correlation
n
jjy
n
jj
y
n
jjj
yyy
nyysnxxs
ss
nyyxx
rs
srb
1
22
1
211
21
1
111
11
1
/)(,/)(
/))((
,ˆ
June 1999 NCSU QTL Workshop © Brian S. Yandell
44
2 Correlated Regressors
slopes adjusted for other regressors
2
11222
1
2122121
2
2
1
21211
)(ˆ
,ˆ)(ˆ
s
srrrb
s
srcc
s
srb
s
srrrb
yyy
yyyyy
June 1999 NCSU QTL Workshop © Brian S. Yandell
45
Gibbs Sampler for Model 1
• mean
• slope
• variance
n
jjj
n
n
jjj
xxbyvaInv
nsns
yxx
b
nn
yn
1
2112
12
2
21
2
21
111
2
)(,~
,
))((
~
,~
2
2
2
2
2
2
2
2
2
2
June 1999 NCSU QTL Workshop © Brian S. Yandell
46
Gibbs Sampler for Model 2
• mean
• slopes
• variance
n
j kkjkkj
n
n
jjj
n
jjjj
xxbyvaInv
nxxcxxs
nsns
xxbyxx
b
nn
yn
1
22
121
22
1
2112122
21|2
21|2
2
21|2
111122
2
2
)(,~
/))((
,
))()((
~
,~
2
2
2
2
2
2
2
2
2
2
June 1999 NCSU QTL Workshop © Brian S. Yandell
47
Updates from 2->1
• drop 2nd regressor• adjust other regressor
02
2121
b
cbbb
June 1999 NCSU QTL Workshop © Brian S. Yandell
48
Updates from 1->2• add 2nd slope, adjusting for collinearity• adjust other slope & variance
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ˆ
)(ˆˆ)(ˆ,ˆ
),1,0(~
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n
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June 1999 NCSU QTL Workshop © Brian S. Yandell
49
Model Selection in Regression
• known regressors (e.g. markers)– models with 1 or 2 regressors
• jump between models– augment with new innovation z
0),,,(12
ˆ,1)0(~);,,(21
2121221
2121
222
z
cbbbbb
cbbb
Jzbbzzb
tionstransformasinnovationparametersm
June 1999 NCSU QTL Workshop © Brian S. Yandell
50
Change of Variables• change variables from model 1 to model 2• calculus issues for integration
– need to formally account for change of variables– infinitessimal steps in integration (db)– involves partial derivatives (next page)
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zbg
b
z
b
J
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),,|;(),,|,(
),,|;(ˆ10
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2
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xxy
June 1999 NCSU QTL Workshop © Brian S. Yandell
51
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Jacobian & the Calculus
• Jacobian sorts out change of variables– careful: easy to mess up here!
June 1999 NCSU QTL Workshop © Brian S. Yandell
52
Geometry of Reversible Jump
0.0 0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
b1
b2
c21 = 0.7
Move Between Models
m=1
m=2
0.0 0.2 0.4 0.6 0.8
0.0
0.2
0.4
0.6
0.8
b1
b2
Reversible Jump Sequence
June 1999 NCSU QTL Workshop © Brian S. Yandell
53
QT additive Reversible Jump
0.05 0.10 0.15
0.0
0.05
0.10
0.15
b1
b2
a short sequence
-0.3 -0.1 0.1
0.0
0.1
0.2
0.3
0.4
first 1000 with m<3
b1
b2
-0.2 0.0 0.2
June 1999 NCSU QTL Workshop © Brian S. Yandell
54
Credible Set for additive
-0.1 0.0 0.1 0.2
-0.1
0.0
0.1
0.2
0.3
b1
b2
90% & 95% setsbased on normal
regression linecorresponds toslope of updates
June 1999 NCSU QTL Workshop © Brian S. Yandell
55
Efficient Updating of additive
• more computations when m > 2• want to avoid matrix inverses
– decompose matrix instead– solve linear system of equations
• use linear algebra– Hausholder (QR) decomposition– LAPACK User’s Guide (1995, 2nd ed)
Anderson et al., SIAM.
June 1999 NCSU QTL Workshop © Brian S. Yandell
56
Hausholder (QR) Decomposition
• decomposition– G is upper triangular– F is orthogonal
• orthogonality
• design matrix 0XF
0FF0,FF
IFF,IFF
IFFFF
FFFFFF
GF0
GFFFGX
T
TT
mnT
mT
nTT
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2
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June 1999 NCSU QTL Workshop © Brian S. Yandell
57
QR & Regression
• model
• error piece
• model piece
• estimators 1
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)(ˆr
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eFeFXbFyF
eXby
June 1999 NCSU QTL Workshop © Brian S. Yandell
58
Absorbing Old Model
• old model– m regressors– QR decomposition
• new model– m+1 regressor– use QR to absorb
old model
eFxFyF
exXby
GFFGX
eXby
Tmm
TT
mmold
old
b
b
21122
11
11
June 1999 NCSU QTL Workshop © Brian S. Yandell
59
Adjusted Slope Estimators• old slopes
– note m=1 case
• added slope– note sum of squares
• variance– note Jacobian
• new slopes 1111
1
1111
1
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21|21221
2
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b
b
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sr
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yFFx
yFGb
June 1999 NCSU QTL Workshop © Brian S. Yandell
60
How To Infer loci?
• if m is known, use fixed MCMC– histogram of loci– issue of bump hunting
• combining loci estimates in RJ-MCMC– some steps are from wrong model
• too few loci (bias)• too many loci (variance/identifiability)
– condition on number of loci• subsets of Markov chain
June 1999 NCSU QTL Workshop © Brian S. Yandell
61
Brassica 8-week Data locus MCMC with m=2
11
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0 200 400 600 800 1000
20
40
60
80
MCMC run
20
40
60
80
0 50 100 150 200 250 300
dist
ance
(cM
)
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
62
Jumping QTL number & loci
111111111111111111111111
1
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0 20 40 60 80 100
20
40
60
80
dist
ance
(cM
)
MCMC run
0 20 40 60 80 1000
1
2
3
num
ber
of Q
TL
June 1999 NCSU QTL Workshop © Brian S. Yandell
63
RJ-MCMC loci chain
111111111111111111111111
1
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11
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0
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22
22
2
222
2
2
2
2
2
2
2
2
2
22
2222
22
22
2
2
222
2
222222
2
22
2
222
2
22
2
2
222
2
2
2
22
22
2
2
2
2
2222
222
22
22
222222
22
2
2
2
22
2
2222
22
2
22
22
222222
2
2
22
2
22
2
2222222
2222
22
222
2
22
2
2
2
222
2
2
2
22
2
2
22
222
2
222
2
2222
22
2
2
22
2
2
2
2
22222
2222
2
2
2
222
2
2
22
22
2
22
2
2
2
2
2
2
2
22
2
22
2
22
22
2
2222
2
2
2
2
2
2
22
22
2222222222
2
22
22
222
22
2
2
22
2
2222
22
2
22222
2
222
222
2
2
22
222
2
2
2
222
2
222
2
2
2
222
222222
222
2
222
22
22
222
2
2222
2
222
2
2
2
2
2
222
2
2
22222
222
2
22
22
22
2
2
2
222
22
222
2
22
22
2
2
2
22
2
2
2
22
2
22222
2
22
2
222
222
2
22
22
22
2
2
2
2
2
222
2
2
222222
2
2
2222
2
2
2
22
2
2
2
22
2222
2222
2222
2
2
22
22
22
222
22
2
222
2
22
22
2
222
2
2
2
2
2
2
2
2
22
2222
222222
22
2222
2
2222222
22
2
2
22222
22
2
2222222
2
333333
33
3
3
3
33333
3
333
3
333
33
3
3
33
333
3
3
33
3
3
333
3
3
333
33333
3
3
33333333333
3
3
3
33333
3
3
3
3
333
33
33
33
33333
3
33
33
3
33
333
33333
33
3
333
333
3
33
33
3
3
3
33333
3
3
3
3
33
33
33
33
3
3
3
3
3
33
3
3
3
33
333
3
3333
3333
3
3
3
333333333
3
3
3
3
3
3
3
3
3
3
3
3333
3
3
3
3
3
33333333
3
33
3
33
3
3
333
3
33
3
3
3
3333
3
3
3333
333
33
333
3333
33
3
3
33
333333
4
4
44
4
444
4
4
4 444 4
4
4 4 44
4
44
4
4
44
4
4
44
4444
4
444
444
444
4
4444
5 555 5 5
55 55 55
5
5
5
MCMC run/10
num
ber
of Q
TL
0 400 8000
20
40
60
80
11
1
11
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
111
1
11
1
1
11
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
111
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
11
1
1
11
1
1
1
111
1
1
1
1
11
1
1
1
11
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
1
1
1
1
1
1
1
11
1
11
1
1
1
1
11
1
11
1
1
11
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
11
111
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
11
11
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11
1
1
11
1
1
11
1
1
1
11
11
1
1
11
1
1
11
1
1
1
1
1
1
1
11
1
1
1
111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
111
11
1
11
1
1
1
1
1
1
1
1
1
1
11
11
1
1
1
1
1
11
1
1
1
1
1
1
111
1
1
1
1
111
1
111111
1
1
1
11
1
11
1
1
11
1
1
11
11
11
1
1
1
11
1
1
1
1
1
1
11
1
1
1
1
1
1
11
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11
1
1
1
1
1
1
1
1
1
11
1
1
11
11
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
111
1
1
1
111
1
1
11
1
1
11
1
1
11
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
111
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
1
1
1
1
111
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
11
1
1
1
1
1
1
1
1
1
1
1
111
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
111
1
1
1
11
1
1
1
1
1
1
11
11
1
1
1
1
1
11
1
1
1
1
1
1
11
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11
1
1
1
1
1
1
1
1
1111
1
1
1
1
1
1
111
11
1
1
11
1
11
1
1
11
1
1111
1
1
1
11
1
1
1
1
1
1
1
1
11
1
1
1
11
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
11
111
1
1
11
11
1
11
1
1
1
111
11
11
11
1
1
11
1
1
11
11
1
11
1
1111
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
11
11
1
11
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11
1
1
1
1
1
1
11
11
1
1
11
1
11
1
11
1
1
1
1
1
1
11
11
1
1
1
111
1
11
2
22
2
222
2
2
2
2
2
2
22
2
2
22
2
2
2
2
2
2
2
2
2
2
2
222
2
2
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2222
2
2
2
2
2222
2
2
2
22
222
22
2
2
2
2
2
2
2
2
2
2
22
2
2
2
2
22
2
2
2
22
22
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2222
2
2
2
22
2
2
2
2
2
2
2
222
2
222
2
2
2
2
2
2
2
22
2
22
22
2
2
222
2
2
2
2
2
22
22
2
222
2
22
2
2
2
2
2
2
2
22
2
22
22
2
22
22
2
2
2
22
2
2
2
2
22
2
22
2
2
2
2
2
2
2
2222
2
22
22
22
2
22
2
22
22
2
2
2
2
2
2
2
22
2
22
22
2
22
2222
2
22
2
2
2
22
2
2
2
2
2
2
2
2
22
22
2
2
2
2
2
2
2
2
22
2
2
2
2
22
2
2
2
2
2
22
2
22
2
2
2
2
2
2
22
22
2
2
2
2
22
2
2
22
22
2
2
2
2
2
22
2
22
2
222
2
2
222
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
222
2
2
222
2
2
2
2
2
2
2
2
2
2
22
22
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
222
22
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
2
22
2
2
2
2
22
2
22
2
22
22
22
2
2
22
2
22
2
22
2
2
2
2
2
2
2
2
2
22
2
2
22
2
2
2
22
2
22222
2
2
2
2
2
22
22
2
22
2
2
22222
2
2
2
2
2
2
22
22
2
2
2
22
22
2
2
2
2
22
2
2
2
2
22
2
2
2
2
222
2
2
222
2
2
2
2
2
2
2
2
2
2
2
222
22
2
2
2
2
2
22
2
22
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
22
222
2
222
2
22
2
2
2
22
22
22
2
2
2
2
2
2
22
2
2
2
22
2
2
2
22
2
2
2
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
22
222
2
2
2
2
2
222
2
22
222
2
222
2
2
22
2
2
2
2
2
22
2
22
22
2
2
3
3
3
3
33
3
333
33
33
333
3
3
3
3
3
3
33
3
33333
33
33
3
33
3
3
3333
3
3
33
3
3
333
33
3
3
3
3
33
3
3
3
33333
3
333
3
3
3
3
3
3
3
3
33
333
3
3
3
3
3
3
33
3
3
3
3
3
3
3
3
3333
3
33
33
3
333
3333
33
3
3
3
3
3
3
33
33
3
333
3
33
3
3
33
3
3
3
333333
3
3
33
3
3
3
3
3
33
3
33
3
33333
33
333
3
3
3
333
3
33
3
3
3
3
33
3
33
3
3
3
33
33
3
33333
3
33
333
3
3
3
33
3
33
333
3
33
3
3
3
333333
33
33
3
3
3
33
3
33
3
3
3
3
3
3
3
3
3
33
44
444444
4
4
4
4
444
4
4
4
4444
444
4
44
4
4
4
44
4
4 44444
4
4
4
444
4444
4
4
4
444
5 5
5
55
55
6
MCMC run/1000 400 800
0
20
40
60
80
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11
1
11
1
1
111
1
1
11
1
1
11
1
11
1
1
1
1
11
1
1
11
1
1
1
1
1
1
1
1
11111
1
11
1
1
1
1
1
1
11
1
11
1
11
1
1
1
1
1
1
11
11
1
11
1
1
1
1
1
11
1
1
111
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
11
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
11
1
111
1
1
11
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
111
1
1
1
1
1
1
1
11
1
1
11
1
1
1
11
1
1
11
1
1
11
1
1
1
1
1
1
1
1
1
11
1
111
1
1
11
11
1
11
11
1
1
1
11
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
11
11
1
1
1
1
1
1
111
1
1
1
1
1
1
1
11
1
1
1
1
1
1
11
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
1
1
1
1
11
1
1
1
1
1
1
111
1
11
1
1
11
11
1
1
1
1
1
1
1
11
1
1111
1
1
1
1
11
1
1
111
11
1
1
1
11
11
1
1
1
1
1
1
1
1
1
1
1
11111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11111
1
1
1
11
1
1
111
11
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
11
1
1
1
1
1
1
11
11
1
1
1
111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
111
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
11
1
1
1
1
1
11
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
1
11
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
11
111
1
1
11
1
111
1
1
1
1
11
1
11
1
1
1
1
11111
1
111
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
111
1
1
1
1
11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
1
1
1
1
1
1
11
1
1
1
1
1
11
1
1
1
11111
1
11
1
1
1
11
1
11111
1
1
1
1
1
1
1
1
1
11
11
1
11
1
1
1
1
1
1
1
1
1
1
1
11
1
1111
1
1
11
1
11
1111
11
1
1
1111
1
1
1
1
11
11
11
1
11
1
1
1
1
11
1
1
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2
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334
4
44
4
44 444 4 44
4 44 4
4
4
MCMC run/1000
num
ber
of Q
TL
0 400 8000
20
40
60
80
June 1999 NCSU QTL Workshop © Brian S. Yandell
64
Raw Histogram of loci
0 20 40 60 80
0.0
0.01
0.02
0.03
0.04
distance (cM)
June 1999 NCSU QTL Workshop © Brian S. Yandell
65
Conditional Histograms
20 40 60 80
0.0
0.02
0.05
distance (cM)
m =
1
0 20 40 60 80
0.0
0.02
0.04
distance (cM)
m =
2
0 20 40 60 80
0.0
0.02
0.04
distance (cM)
m =
3
0 20 40 60 80
0.0
0.02
0.04
distance (cM)
m >
3
June 1999 NCSU QTL Workshop © Brian S. Yandell
66
Bayes Factors•ratio of posterior odds to prior odds
– RJ-MCMC gives posterior on number of QTL– prior is Poisson
BF(1:2) 2log(BF) evidence for 1st
< 1 < 0 negative
1 to 3 0 to 2 negligible
3 to 12 2 to 5 positive
12 to 150 5 to 10 strong
> 150 > 10 very strong
)|(
)|(
)(/)(
)|(/)|(
2
1
21
2112 model
model model model model model
yyyy
B
June 1999 NCSU QTL Workshop © Brian S. Yandell
67
#QTL for Brassica 8-week
0 1 2 3 4 5 6 7 8
0.0
0.4
0.8
priorpost.
prior mean = 10 1 2 3 4 5 6 7 8
0.0
0.2
0.4
0.6
prior mean = 20 1 2 3 4 5 6 7 8
0.0
0.2
0.4
prior mean = 3
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
prior mean = 40 1 2 3 4 5 6 7 8
0.0
0.2
prior mean = 60 1 2 3 4 5 6 7 8
0.0
0.15
0.30
prior mean = 10
June 1999 NCSU QTL Workshop © Brian S. Yandell
68
RJ-Bayes Factors(8-week Brassica data)
prior mean
ratio
1 2 3 4 6 10
1:2 2.87 1.91 1.51 1.45 1.12 0.85
1:3 27.62 9.10 5.06 4.22 2.28 1.28
1:4 1743.29 81.30 28.85 18.51 7.17 2.51
2:3 9.63 4.76 3.35 2.91 2.04 1.5
2:4 608.00 42.51 19.09 12.75 6.41 2.95
3:4 63.13 8.93 5.70 4.39 3.15 1.96
June 1999 NCSU QTL Workshop © Brian S. Yandell
69
Simulation Study of Prior Effect
• how dramatic is the effect of prior?• simulations of 0, 1 or 2 QTL
– QTL have large effect • additive = 2, variance = 1
– 2 QTL spaced 36cM apart– sample sized of 105
• RJ-MCMC runs of 100,000
June 1999 NCSU QTL Workshop © Brian S. Yandell
70
Effect of Prior Mean
0 1 2 3 4 5
0.0
0.4
0.8
prio
r m
ean
= 2
priorpost.
0 QTL present0 1 2 3 4 5
0.0
0.4
1 QTL present0 1 2 3 4 5
0.0
0.4
0.8
2 QTL present
0 1 2 3 4 5
0.0
0.2
0.4
prio
r m
ean
= 4
0 QTL present0 1 2 3 4 5
0.0
0.2
0.4
0.6
1 QTL present0 1 2 3 4 5
0.0
0.2
0.4
2 QTL present
June 1999 NCSU QTL Workshop © Brian S. Yandell
71
Bayes Factor
m
ratio
0 1 2
0:1 3.85 0 0
0:2 50.93 0 0
0:3 569.11 0.03 0
1:2 13.22 1.87 0
1:3 147.75 30.09 0
2:3 11.17 16.05 2.38
m
ratio
0 1 2
0:1 0.97 0 0
0:2 3.02 0 0
0:3 15.07 0 0
1:2 3.12 1.32 0
1:3 15.54 3.04 0
2:3 4.99 2.58 0.75
prior of 2 prior of 4
June 1999 NCSU QTL Workshop © Brian S. Yandell
72
Computing Bayes Factors• arithmetic mean
– using samples from prior– mean across Monte Carlo or MCMC runs– can be inefficient if prior differs from posterior
• harmonic mean– using samples from posterior– more efficient but less stable– careful choice of weight h() close to posterior
1
1 )()(
)()|(ˆ
G
g kkkk
kk
hG
model |model ;|model
yy
kkkkkk d )()|()|( model |model ;model yy
June 1999 NCSU QTL Workshop © Brian S. Yandell
73
Stable Bayes Factors• Satagopan, Raftery & Newton (1999)
– weighted harmonic mean
– absorb variance (normal to t dist) replace
);,,|( *2* Xby by
);,|( ** Xby
June 1999 NCSU QTL Workshop © Brian S. Yandell
74
Bayes Factors & LODs
• others have tried arithmetic & harmonic mean• why not geometric mean?• terms that are averaged are log likelihoods...
runsMCMCGg
G
G
gkk
k
,,1
)(log
exp)|(ˆ 1
model ;|
model
y
y
June 1999 NCSU QTL Workshop © Brian S. Yandell
75
Bayesian LOD• Bayesian “LOD” computed at each step
– based on LR given sampled genotypes and effects– can be larger or smaller than profile LOD– informal diagnostic of fit– combine to for geometric estimates of Bayes factors
n
j j
jj
n
j j
xj
by
bxyeBLOD
by
xbxy
eLOD
12*
2**
10
12*
1,0,1
2*
10
),0,|(
),,;|(ln)(log
)ˆ̂,0,ˆ|(
)|()ˆ,ˆ,ˆ;|(
ln)(log)(
June 1999 NCSU QTL Workshop © Brian S. Yandell
76
Compare LODs• scatter plot of loci and Bayesian LODs
– same BLOD for all loci of step– overlay LOD from interval mapping (red)– overlay LOD from CIM (green)– vertical lines at true or inferred loci (blue)
• steps with higher BLOD– may have more likely genotypes– basis for MCMC step
• always up to higher likelihood• sometimes down to lower likelihood
June 1999 NCSU QTL Workshop © Brian S. Yandell
77
BLODs with no QTL• LOD profile roughly zero• most BLOD values should be negative
– no pattern across linkage group
• distribution similar to rescaled chi-square with 1 df– -2log(LR) approximately chi-square– assignment of genotypes “irrelevant”
• RJ-MCMC skewed with inferred QTL– numbers indicate #QTL
June 1999 NCSU QTL Workshop © Brian S. Yandell
78
Simulated Data with No QTL
78
91
011
12
78
91
011
12
0 2 4 6
June 1999 NCSU QTL Workshop © Brian S. Yandell
79
BLOD: no QTL-3
-2-1
0
distance (cM)0 10 20 30 40 50 60 70 80
-3-2
-10
0.0 0.2 0.4 0.6 0.8 1.0 1.2
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
80
RJ-MCMC BLOD: no QTL-3
-2-1
01
2
distance (cM)0 10 20 30 40 50 60 70 80
1
11
1
2
1
21
1
3
1
2
1
1
1
2
11
1
11
1
2
1
1 1
1
1
1
1
1
1
2
1
1
11
2
1
1
12
1
1
2
1
1
1
1
2
11
2
2
11
1 1
1
1
1
3
2
11
2
1
1
11
12
1
21
1
11
1
1
1
2
1
1
1
1
1
2
2
1
1
11
1
1
11
2
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1
1
1
1
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1
1
1
1
1
1
1
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1
1
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12
11
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1
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1
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1
3
1
11
1
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1
1
1
1
1
1
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1
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11
2
11
2
2
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1
1
1
1
1
2
1
1
1
1
2
1
3
4
2
2 1
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1
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12
1
1
2
2
1
1
1
1
1
2
1
1
2
2
1 2
1
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2
12
1
1
2
1
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1
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1
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1
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1
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1
2
2
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1
2
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3
12
1
1
2
1
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2
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1
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1 1
2
2
1
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1
3
1
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11
2
2 1
1
1
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1
1
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2
21
1
1
1
2
1
1
1
12
1
1
11
11
1
11
1
1
1
2
11
1
1
1
1
2
1
1 1
1
2
1
1
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1
2
12
111
12
1
1
2
11
2
1
1
3
1
1
1
2
11
2
2
1
1
1
3
1
1
12
1
1
1
1
1
1
1
1
1
1
1
1
1
12
1
1
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1 1
2
1
1
1
1
1
1
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1 1
2
1
1
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1
12
12 11 1
11
1
1
1
1
1
1 1
1
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1
1
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1
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1
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2
11
1
11
1
1
11
1
2
1
1
2
2
1
12
2
1
11
11
1
1
1 1
2
11
11
2
1
11
11
1 1
1
1
1 1
11
1
1
1
1
11
1
111
1
211
11
11
2
11
1
1
1
111
1
1
1
1
1
1
1
2
11
1
1 2
2
32
2
2
2
2
2
22
2
2
3
2
222
2
2
2
2
2
22
22
3
3
2
2
2
2
2
3
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
22
2
22
2
2
32
2
2
2
2 22
2
2
2
2
2
2
22
3
2
2
23 2
2
2
222
2
2
2
2
22
2
2
2
2
2
2
3
3
3
3
3
4
3
3
3
3
4
-3-2
-10
12
0.0 0.2 0.4 0.6 0.8 1.0 1.2
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
81
BLODs with 1 QTL• LOD peaks at correct locus• most BLOD values near locus
– some considerably larger than LOD– inferred genotype vs EM average
• non-central distribution of BLOD– rescaled non-central chi-square?
• RJ-MCMC dispersed from peak– locus proposal has local dispersion
June 1999 NCSU QTL Workshop © Brian S. Yandell
82
LOD for 1 QTL 5
10
15
20
25
30
35
distance (cM)0 10 20 30 40 50 60 70 80
51
01
52
02
53
03
5
0.0 0.1 0.2 0.3 0.4 0.5 0.6
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
83
RJ-MCMC BLOD: 1 QTL5
10
15
20
25
30
35
distance (cM)0 10 20 30 40 50 60 70 80
1
2
1
1
32
2
11
11
1
2
2 1
1
1
1
2
2
2
1
11
2
1
2
2
2
3
1
11
1
2
2
1
2
11
2
1
111
1
11
211
11
3
1
11
21
2
3
1
12
2
3
2
2
1
3 11
1
3
1
1
2
1
3
111
2
1112
2
2
3
11
1
2
2
1
1
11
4
1
11
1
21
2
1
1
1
3
12 1
31
12
11
1
12
2
1
111
1221
2 111
1
11
1
1
2
11
1
1 1
12
1
13
2
11
1
21
1
1
1
2
221
2
21
2 1
1
1
3
1
121
11
2
1
22
1
1
1
1
1
2
2
2
1
1
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2
1
1
1
3
21
11
1
1
2
12
1
1
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222
2
2
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1
1
1
1
2
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1
1
3
2
1
1
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1
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1
1
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2
2
2
2
3
1
1
12
2
1
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1
1 1
1
1
1
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1
1
1
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2
112
111
1
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11
2
1
1
2
2
2
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1
2
2
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1
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1
2
2
1
1
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1
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22
2
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1
2
2
2
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1
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1
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1
1
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1
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1
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1
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1
1
1
2
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1
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3
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111
22
1
1
1
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1
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2
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1
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1
2
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1
4
2
1
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2
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1
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1
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1
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1
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2
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3
1
1
1
2
1
2 2
1
1
12
2
1
1
1
2
1
2
1
1
2 12
122
32
2
2
22
2
2
22
2
2
3
2
2
22
2
3
2
2
3
2
2
3
2
23
3
2
3
2
2
2
2
3
2
2
4
22
3
2
3
2
2
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2
23
2
2
2
22
2
22
3
2
2
22
2
2
2
2
23
2
2
22
22
2
2
2
2
2 2
3
2
2
2
32
2
2
2
3
2
2
2
2
2
22
2
2
2
2
322
2
2
2
2
2
3
3
222
22
2
22
2
2
2
2
22
2
2
2
2
2
3
2222
2
2
2
2
2
3
2
23
2
2
2
3
4
2
2
2 2
2
2
22
2
2
2
2
2
22
2
2 2
2
2
2
2 2 2
2
2
2
2
2
2
2
2
2
2
2
2
4
2
2
2
3
2
2
2
2
2
3
24
2
2
2
2
22
3
2
2
3
2
2
2
3
2
2
2
2
22
3
2
2
2 22 2
2
2
2
2
2
23
2
4
2
4
2 232
3
2
2
2
2
3
22 2
2
3
23
2
2
2
2
32
2
3
2
22
2
32
2 2
2
2
2
2
2
2
22
3
3
2
22
32 23
2
2
2
2
2
2
32
2
2
22
3
22
2
2
3
2
2
2 22
2
232
2
3
2
3
3
2
22
2
2
2
2
4
2
2
2
3
2
2
2
2
3
2
2
2
3
2
2 2
2
2
2
2
222
3
3
3
3
3
3
3
33
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
3 3
4
33
3
3
3
4
4
3
3
3
3
3
3
33
3
3
33
3
3
3
3
3
3
34 3
33
4
4
4
4
4
4
4
51
01
52
02
53
03
5
0.0 0.05 0.10 0.15 0.20 0.25
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
84
BLODs with 2 QTL• incorrect fit of 1 QTL model
– LOD peaks at ghost locus– BLOD values near LOD peaks
• correct 2-QTL model– IM misses, CIM gets loci– BLOD at loci– BLOD approx. 2x LOD
• simultaneous fit of both loci
• RJ-MCMC dispersed from peak– need to look conditional on m=2
June 1999 NCSU QTL Workshop © Brian S. Yandell
85
LOD for 2 QTL with 1 Fit5
10
15
20
25
30
35
distance (cM)0 10 20 30 40 50 60 70 80
51
01
52
02
53
03
5
0.0 0.1 0.2 0.3 0.4
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
86
LOD for 2 QTL with 2 Fit1
02
03
04
05
06
0
distance (cM)0 10 20 30 40 50 60 70 80
10
20
30
40
50
60
0.0 0.1 0.2 0.3
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
87
RJ-MCMC BLOD: 2 QTL0
10
20
30
40
50
60
distance (cM)0 10 20 30 40 50 60 70 80
22
333
2
232
3
23
42 3
22
2
423
4
2
2
3
2223
2
2
2
22
22
3
222
23 2
23
2
3
22
333
4
22
322
2
2
3
2
2
2
3
222
22332
232 2
2
3 2
2
42 2
222
33
333223 3
2
2
3 3354
2
3 2
3
4
32
2
3
3
32
2
3
2
2
3
3
2
3
233
33
2
3
3222
24
5
3
2 2323
2
3
32
2
4
3
2
3
522
2
22
2
2
2
22
22
43 43
2
3
2
2
42
2
32
22
2
222
3 4 2
3
3 3
35
2
3 2333
2
2
2
3
3
2
2
2
3
22
2
3
4
3
2322
2
3
2
3
32
22
2
2
2
2 22
33
32
2
2
3
333
2
2
2
23
2 2
22 22
4
23
4
333
3
2
3
2
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3
2
33
2
4
43
3 2
2
23222
33
3
2
2
2
22 322
2
2222
2
232
2
2
243
2 33
3 2
2
22
332 232
2
33
3
322
3
23
3323
22
3
3
22
3 24 2
2
3
3
2
2
22
3
2
32
2
2
233
3
3
3
2
2
3244 22
22
3
2
3
2
4
2
4
42
2
2334
33
2
2
2
232233 22
3
5
2
3
23
3
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2
4
22
3
2
2
22
2
2
32
23
2
2
3
22
2
23 22
3
2
3
3233
3
2222
32
22
2
2
3
3
23
223
2
3
2323
3
2
3
2
3
4 3
3
23
422
2323
24
222
3
33
223
2
3
4
4
222
3
3
2
3 2
232
3
322
2
3 233
33 2
33
22323
243
23
3
3
3
3
2
2
4
2
22
22
3
22
2
2
2
32
3
2
3
2
2324
3
3
3
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322
24
2
3
22
325
23
3
3
2
222
433
2
2
2
2
33
2
22
3
3
2
4
3
222
3
33
32
3
33
2
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3
223
23
3
33
222
332332
323
3
2
33
2
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22
2
2
3
4
3
33
33
2223
223 2
3 3
3
3
2
3
2
2
22
3
32
3
32
2
23
3
3
3
3
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3
23
3
33
22222222
222
43
3
2
25332
3 222
3
23
2
222
2
2
2
2
322
2
3
22
3
3
2
3
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33
3
3
3
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232
3
2
2
2
3
333
33
33
22
2
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2
32
22
3
22
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32
4
3
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2
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332 2
2
332
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2
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3
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3 33
2
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3 32
2
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3
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4
2
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222
3
3
3
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2
2
2
2
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3
3
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3
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22
32
2
3
3
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3
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2
4 2
2
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2
2
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332
232
3
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3
3
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23
2
3
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3
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3
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2
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222 2 2
333
2
232
3
23
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22
2
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2
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23 2
23
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3
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4
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2
3
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32
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2
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22 2
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223 3
2
2
3 3 354
2
3 2
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3
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5
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2
3
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2
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2
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2
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2
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34 2
3
3 3
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2
3 2333
2
2
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3
3
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3
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2
3
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3
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2
3
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3
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22
2
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33
3 2
2
2
3
333
2
2
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22
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4
23
4
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3
2
3
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3
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2
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22
33
3
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2
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2
22
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233
3 2
2
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33 2 23 22
33
3
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3
23
33
23
22
3
3
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2
3
3
2
2
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3
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322
2
233
3
3
3
2
2
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3
2
3
2
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2
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2
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33
2
2
2
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3
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2
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2
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2
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3
3
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23 23
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2
3
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2
3
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2
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3
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33
223 23
243
23
3
3
3
3
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2
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2
22
22
3
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2
2
2
32
3
2
3
2
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24
3
3
3
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322
24
2
3
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3 25
23
3
3
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222
43
3
2
2
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33
2
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3
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32
3
33
2
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3
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3 3 233
2
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3
2
33
2
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2
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33
22
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33
3
3
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3
3
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33
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2 22222
2 243
3
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3
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23 2
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33
33
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3
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3 322
2
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4
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3
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2
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3
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3
3
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3
34
3
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3
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3 333
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4
33
33
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43
3
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33
3
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5
3
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3
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54
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3
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3
33
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3
33
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3
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3
33
3
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33
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3
3
3
33
3
3
3
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3
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3
3
3
344
3
3
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4
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333 3
3
3
5
3
3
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33
33
43
33
33
33
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3
33
3
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3
33
3
3
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3
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3
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33
33
4
4
3
333
3
33
33
33
33
33
43
3
3
3
3
3
43
3
3
3
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3
3
3
5
34
3
35 3
3
343
3
33
3
3
4
3
3
33
3
3
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33
3
33
33 33
33
33
33
4
3
33
33
3
3
33
3
3
3
3
3
3
3
3
3
3
3
3
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334 3
3
53
33
3
3
33
3
3
3
333
3
3
3
34
3
3
3
333
33
33
4 3
3
3
4
3
3
3
33
333
3
4
4
3
3
343
3
3
3
4
33
334
33 33
3
3
34
3
34
3
4
33
333
3
3
33
4
33
3
3
3
3
3
4
3
33
33
3
3
3
33
4
33333
33
3
3
3
33
3
3
3
33
3
34
4
4
4454
4
4
5
4
54
44
4
5
44
44
44444
4
4
445
4
44
4
4
4
44
454
5 4
4
44 45
4
4
44
4
4
4
4
4
44
4
4455
555
555
01
02
03
04
05
06
0
0.0 0.05 0.10 0.15 0.20
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
88
Brassica BLODs• 4-week clearer than 8-week
– ghost QTL or smear for 8-week?
• MCMC with m=2 fairly clear• RJ-MCMC dispersed
– conditioning on m=2 similar to MCMC• not shown
– mixing models together– local proposal moves hamper mixing
June 1999 NCSU QTL Workshop © Brian S. Yandell
89
Brassica 4-week BLOD Map0
51
01
52
02
5
distance (cM)0 10 20 30 40 50 60 70 80
05
10
15
20
25
0.0 0.05 0.10 0.15 0.20 0.25 0.30
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
90
Brassica 8-week BLOD Map0
24
68
10
distance (cM)0 10 20 30 40 50 60 70 80
02
46
81
0
0.0 0.05 0.10 0.15 0.20 0.25 0.30
LOD
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
91
4-week RJ-MCMC BLOD0
51
01
52
02
5
distance (cM)0 10 20 30 40 50 60 70 80
2 2
23
1
3
11
2
12
1
11
212
1
2
2 2
1
11
3
2
2
2
2
2
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2
1
1
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111
1
1
22
1
1
112
2
11
1
1 1
3
1
2
222
1
11
11
1
1
2
1
2
2
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2
1
2
1
2
122
2
1
1
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222
2
2
2
2
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1
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12
31
1
1
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22
21
21
3 1
2
21
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1
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2
3
21
2
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11
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1
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1
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1
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1
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3
1
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1
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2
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1
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3
2
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1 2
2
241
1
1
1
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11
1
1
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1
2
2
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2
1
2
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1
12
11
2
1
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2
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221
2
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1
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1
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11
2
1
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1
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1
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2
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13 12111
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1
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2
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2
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2
1
11
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121
2
2
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1
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2
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1
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2 1
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June 1999 NCSU QTL Workshop © Brian S. Yandell
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8-week RJ-MCMC BLOD0
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June 1999 NCSU QTL Workshop © Brian S. Yandell
93
The Art of MCMC• convergence issues
– burn-in period & when to stop
• proper mixing of the chain– smart proposals & smart updates
• frequentist approach– simulated annealing: reaching the peak– simulated tempering: heating & cooling the chain
• Bayesian approach– influence of priors on posterior– Rao-Blackwell smoothing
• bump-hunting for mixtures (e.g. QTL)
June 1999 NCSU QTL Workshop © Brian S. Yandell
94
RJ-MCMC Software• General MCMC software
– U Bristol links
• http://www.stats.bris.ac.uk/MCMC/pages/links.html– BUGS (Bayesian inference Using Gibbs Sampling)
• http://www.mrc-bsu.cam.ac.uk/bugs/
• Our MCMC software for QTLs– C code using LAPACK
• ftp://ftp.stat.wisc.edu/pub/yandell/revjump.tar.gz– coming soon:
• perl preprocessing (to/from QtlCart format)• Splus post processing• Bayes factor computation
June 1999 NCSU QTL Workshop © Brian S. Yandell
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RJ-MCMC Software Details
• input files– marker.dist
– marker.mark
– trait.y
• output file– result.write– result.error
• distances between loci (cM)
• marker genotypes (-1,1)– one line per marker
• trait phenotypes
• results• errors if any
June 1999 NCSU QTL Workshop © Brian S. Yandell
96
trait.y file
missing value = -99.000000 8.097789 7.928476 12.538893 8.370929 8.347937 11.252023 10.687911 6.961958 12.318708 11.985510 11.841586 8.339165 9.347868 11.933833 7.593694 12.248663 13.183897 9.928043 8.326296 ...
June 1999 NCSU QTL Workshop © Brian S. Yandell
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marker.dist file
8.8
11.8
6.8
6.8
8.7
10.7
10.5
5.1
14.7
June 1999 NCSU QTL Workshop © Brian S. Yandell
98
marker.mark file
one row per marker, one column per line-1 -1 -1 1 -1 1 1 ...
-1 -1 -1 1 -1 1 1 ...
-1 -1 1 1 -1 1 1 ...
-1 -1 1 1 -1 1 1 ...
-1 -1 1 -1 -1 1 1 ...
-1 -1 1 -1 -1 1 1 ...
-1 -1 1 -1 -1 1 1 ...
-1 -1 1 -1 -1 1 1 ...
-1 -1 1 -1 -1 1 -1 ...
-1 1 1 1 -1 1 -1 ...
June 1999 NCSU QTL Workshop © Brian S. Yandell
99
nval.dat file
1 # 1=revjump,0=no
105 10 # n=individuals markers
10000 10 # N=MCMC_runs skips
2 # m
0 0.5 # mu sigmasq
0 0 # initial b’s
3.5 7.5 # initial loci
0 10 1 1 # prior(mu)~N(0,10) prior(sigmasq)~IG(1,1)
0 10 # prior(b)~N(0,1)
4 # prior(m)~Poisson(4)
June 1999 NCSU QTL Workshop © Brian S. Yandell
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result.write filem mu b(1)..b(m) sigmasq lambda(1)..lambda(m) move LOD2 3.0702 -0.0820 0.0975 0.2085 10.5301 83.7381 1 -0.39433 3.0512 0.0343 -0.1166 -0.0600 0.0853 18.4030 51.2329 76...2 3.0400 -0.0415 -0.0946 0.0619 28.6062 74.0904 3 6.1284...1 3.1045 -0.1412 0.0851 76.9420 3 6.13102 3.0669 -0.0434 -0.1576 0.1091 17.4566 80.7973 3 4.3184
propose acceptbirth 384474 100625death 227999 100625locus 387527 259705
m=0 m=1 m=2 m=3 m=4 m=5 ...1936 428306 425105 126871 16703 1047 32 0 0 0
June 1999 NCSU QTL Workshop © Brian S. Yandell
101
Bayes Factor References• MA Newton & AE Raftery (1994) “Approximate Bayesian
inference with the weighted likelihood bootstrap”, J Royal Statist Soc B 56: 3-48.
• RE Kass & AE Raftery (1995) “Bayes factors”, J Amer Statist Assoc 90: 773-795.
• JM Satagopan, MA Newton & AE Rafter (1999) ms in prep, mailto:[email protected].
June 1999 NCSU QTL Workshop © Brian S. Yandell
102
Reversible JumpMCMC References
• PJ Green (1995) “Reversible jump Markov chain Monte Carlo computation and Bayesian model determination”, Biometrika 82: 711-732.
• S Richardson & PJ Green (1997) “On Bayesian analysis of mixture with an unknown of components”, J Royal Statist Soc B 59: 731-792.
• BK Mallick (1995) “Bayesian curve estimation by polynomials of random order”, TR 95-19, Math Dept, Imperial College London.
• L Kuo & B Mallick (1996) “Bayesian variable selection for regression models”, ASA Proc Section on Bayesian Statistical Science, 170-175.
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Reversible JumpMCMC: Inbred Lines
• JM Satagopan & BS Yandell (1996) “Estimating the number of quantitative trait loci via Bayesian model determination”, Proc JSM Biometrics Section.
• DA Stephens & RD Fisch (1998) “Bayesian analysis of quantitative trait locus data using reversible jump Markov chain Monte Carlo”, Biometrics 54: 1334-1347.
• MJ Sillanpaa & E Arjas (1998) “Bayesian mapping of multiple quantitative trait loci from incomplete inbred line cross data”, Genetics 148: 1373-1388.
• R Waagepetersen & D Sorensen (1999) “Understanding reversible jump MCMC”, mailto:[email protected].
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Reversible JumpMCMC: Pedigrees
• S Heath (1997) “Markov chain Monte Carlo segregation and linkage analysis for oligenic models”, Am J Hum Genet 61: 748-760.
• I Hoeschele, P Uimari , FE Grignola, Q Zhang & KM Gage (1997) “Advances in statistical methods to map quantitative trait loci in outbred populations”, Genetics 147:1445-1457.
• P Uimari and I Hoeschele (1997) “Mapping linked quantitative trait loci using Bayesian analysis and Markov chain Monte Carlo algorithms”, Genetics 146: 735-743.
• MJ Sillanpaa & E Arjas (1999) “Bayesian mapping of multiple quantitative trait loci from incomplete outbred offspring data”, Genetics 151, 1605-1619.
June 1999 NCSU QTL Workshop © Brian S. Yandell
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June 1999 NCSU QTL Workshop © Brian S. Yandell
106
MCMC run of loci for 2 QTL
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Simulated Data effect Correlation
1.8 2.0 2.2 2.4 2.6
1.8
2.0
2.2
2.4
effect 1
effect
2
cor = -0.58
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June 1999 NCSU QTL Workshop © Brian S. Yandell
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Samples from MCMC• ergodic Markov chain
– sample averages agree tend to mean of stable dist– but samples are dependent
• ordinary sample summaries – means, quantiles, histograms– high probability density (HPD) regions
• sample chain for long run (100,000-1,000,000)• use time series methods to study chain
– standard error of estimates– Monte Carlo error
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Bayesian Posterior• posterior = likelihood * prior / constant• posterior distribution is proportional to
– likelihood of traits & genos given effects & locus – prior distribution of effects & locus
• posterior distribution of genotypes, effects & locus given traits and linkage map
n
jjjj
n
jjj
bxyxb
bxybb
1
2***2*
1
2**2*2**
),,;|()|()()()()(
);,,|,();,,()|;,,;(
yx
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Brassica napus Experimental Results
• two QTLs inferred on LG9• corroborated by Butruille (1998)• now exploiting synteny with
Arabidopsis thaliana
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Brassica napus Experimental Design
• 3 vernalization treatments– none– 4 week– 8 week
• only consider 8 week Vernalization here• non-flowering lines for none, 4-week• skew suggests log transformation• data average over 3 blocks (EU = line)
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Scatter Plot of 2 loci Subset
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June 1999 NCSU QTL Workshop © Brian S. Yandell
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RJ-MCMC Changing loci
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June 1999 NCSU QTL Workshop © Brian S. Yandell
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Bayesian Inference for QTLs in Inbred Lines I
Brian S YandellUniversity of Wisconsin-Madisonwww.stat.wisc.edu/~yandell
NCSU Statistical GeneticsJune 1999
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Overview• Part I: Single QTL
– Modeling a trait with a QTL– Likelihoods, Frequentists & Bayesians– Profile LODs & Bayesian Posteriors – Graphical Summaries– Testing & Estimation
• Part II: Multiple QTLs – Sampling from a Markov Chain – Issues for Multiple QTLs– Bayes Factors & Model Selection– Brassica data analysis
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Part I: Single QTL• Modelling a trait with a QTL
– linear model for trait given genotype– recombination near loci for genotype
• Bayesian Posterior• Likelihoods
– EM & MCMC– Frequentists & Bayesians
• Review Interval Maps & Profile LODs• Case Study: Simulated Single QTL
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Components• observed data on individual
– trait: field or lab measurement• log( days to flowering ) , yield, …
– markers: from wet lab (RFLPs, etc.)• linkage map of markers assumed known
• unobserved data on individual– geno: genotype (QQ=1/Qq=0/qq=-1)
• unknown model parameters– effects: mean, difference, variance– locus: quantitative trait locus (QTL)
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Single QTL trait Model• trait = mean + additive + error• trait = effect_of_geno + error• prob( trait | geno, effects )
**
2**
**
),,;|(
jj
jj
jjj
xby
bxy
exby
10 12 14
x=-1x=0
x=1
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Simulated Data with 1 QTL6
810
1214
x=-1 x=1
68
1012
14
0 2 4 6 8frequency
trai
t
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Recombination and Distance• no interference--easy approximation
– Haldane map function– no interference with recombination
• all computations consistent in approximation– rely on given map
• marker loci assumed known
– 1-to-1 relation of distance to recombination– all map functions are approximate
• assume marker positions along map are known
221 1 er
June 1999 NCSU QTL Workshop © Brian S. Yandell
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*x
2r
1M 2M 3M 4M 5M 6M
5r4r3r1r
markers, QTL & recombination rates
distance along chromosome
?
?
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Interval Mapping of QT genotype
• can express probabilities in terms of distance– locus is distance along linkage map– flanking markers sufficient if no missing data– could consider more complicated relationship
prob( geno | locus, map )= prob( geno | locus, flanking markers )
),,|()|( 1,,**
kjkjjj MMxx
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Basic Idea of Likelihood Use
• build likelihood in steps– build from trait & genotypes at locus– likelihood for individual i– log likelihood over individuals
• maximum likelihood (interval mapping)– EM method (Lander & Botstein 1989)– MCMC method (Guo & Thompson 1994)
• posterior distribution (Bayesian)– analytical methods (e.g. Carlin & Louis 1998)– MCMC method (Satagopan et al 1996)
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Bayes Theorem• posteriors and priors
– prior: prob( parameters )– posterior: prob( parameters | data )
• posterior = likelihood * prior / constant• posterior distribution is proportional to
– likelihood of parameters given data– prior distribution of parameters
)()|()|(
)(
)()|(
)(
),()|(
paramparamdatadataparam
data
paramparamdata
data
dataparamdataparam
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Bayesian Prior• “prior” belief used to infer “posterior” estimates
– higher weight for more probable parameter values• based on prior knowledge
– use previous study to inform current study• weather prediction: tomorrow is much like today• previous QTL studies on related organisms
– historical criticism: can get “religious” about priors
• often want negligible effect of prior on posterior– pick non-informative priors
• all parameter values equally likely• large variance on priors
– always check sensitivity to prior
June 1999 NCSU QTL Workshop © Brian S. Yandell
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How to Study Posterior?
• exact methods– exact if possible– can be difficult or
impossible to analyze
• approximate methods– importance sampling– numerical integration– Monte Carlo & other
• Monte Carlo methods– easy to implement– independent samples
• MCMC methods– handle hard problems– art to efficient use– correlated samples
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Bayesian Inference
• study posterior distribution of locus & effects– sample joint distribution
• locus, effects & genotypes
– study marginal distribution of• locus• effects
– overall mean, genotype difference, variance
• locus & effects together
• estimates & confidence regions– histograms, boxplots & scatter plots– HPD regions
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Posterior for locus & effect
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2.0
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distance (cM)
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0.0
0.1
0.2
0.3
distance (cM)
1.8
2.0
2.2
QTL 1
addi
tive
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Marginal Posterior• posterior = likelihood * prior / constant
• posterior distribution is proportional to– likelihood of traits & genos given effects & locus – prior distribution of effects & locus
is proportional to
n
jjj bxyb
1
2*** ),,;|()(
)|( * yb
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Marginal Posterior Summary
• marginal posterior for locus & effects• highest probability density (HPD) region
– smallest region with highest probability– credible region for locus & effects
• HPD with 50,80,90,95%– range of credible levels can be useful– marginal bars and bounding boxes– joint regions (harder to draw)
June 1999 NCSU QTL Workshop © Brian S. Yandell
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HPD Region for locus & effect
36 38 40 42
1.8
2.0
2.2
distance (cM)
addi
tive
June 1999 NCSU QTL Workshop © Brian S. Yandell
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QTL Bayesian Posterior• posterior = likelihood * prior / constant• posterior( paramaters | data )
prob( genos, effects, loci | trait, map )
is proportional to
)|;,,;( 2** yx b
n
jj
n
jjj
xb
bxy
1
*2*
1
2**
)|()()()()(
),,;|(
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Frequentist or Bayesian?
• Frequentist approach– fixed parameters
• range of values
– maximize likelihood• ML estimates• find the peak
– confidence regions• random region• invert a test
– hypothesis testing• 2 nested models
• Bayesian approach– random parameters
• distribution
– posterior distribution• posterior mean• sample from dist
– credible sets• fixed region given data• HPD regions
– model selection/critique• Bayes factors
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Frequentist or Bayesian?
• Frequentist approach– maximize over mixture
of QT genotypes– locus profile likelihood
• max over effects
– HPD region for locus• natural for locus
– 1-2 LOD drop
• work to get effects– approximate shape
of likelihood peak
• Bayesian approach– joint distribution over
QT genotypes– sample distribution
• joint effects & loci
– HPD regions for• joint locus & effects• use density estimator
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Interval Mapping forQuantitative Trait Loci
• profile likelihood (LOD) across QTL– scan whole genome locus by locus
• use flanking markers for interval mapping
– maximize likelihood ratio (LOD) at locus• best estimates of effects for each locus• EM method (Lander & Botstein 1989)
n
j j
xj
by
xbxy
eLOD1
2*
1,0,1
2*
10)ˆ̂,0,ˆ|(
)|()ˆ,ˆ,ˆ;|(
ln)(log)(
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Profile LOD for 1 QTL0
51
01
52
02
5
distance (cM)
LOD
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QTLIM
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Building trait Likelihood
• likelihood is mixture across possible genotypes• sum over all possible genotypes at locus
like( effects, locus | trait )
= sum of prob( trait, genos | effects, locus )
1,0,1
2*
2*2*
)|(),,;|(
);,,|()|,,,(
xj
jj
xbxy
byybL
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Likelihood over Individuals
• product of trait probabilities across individuals– product of sum across possible genotypes
like( effects, locus | traits, map )
= product of prob( trait | effects, locus, map )
n
j xj
n
jj
xbxy
bybL
1 1,0,1
2*
1
2*2*
)|(),,;|(
);,,|()|;,,(
y
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Interval Mapping Tests• profile LOD across possible loci in genome
– maximum likelihood estimates of effects at locus– LOD is rescaling of L(effects, locus|y)
• test for evidence of QTL at each locus– LOD score (LR test)– adjust (?) for multiple comparisons
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Interval Mapping Estimates
• confidence region for locus– based on inverting test of no QTL– 2 LODs down gives approximate CI for locus– based on chi-square approximation to LR
• confidence region for effects– approximate CI for effect based on normal– point estimate from profile LOD
)ˆ(96.1ˆ
2)()ˆ(|** bsebCIeffect
LODLODCIlocus
June 1999 NCSU QTL Workshop © Brian S. Yandell
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IM Confidence Region
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1.8
2.0
2.2
2.4
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addi
tive
June 1999 NCSU QTL Workshop © Brian S. Yandell
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How to Proceed?
• figure out how to construct posterior– Markov chain Monte Carlo
• run Markov chain• summarize results• diagnostics
June 1999 NCSU QTL Workshop © Brian S. Yandell
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MCMC Run for 1 locus Data
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June 1999 NCSU QTL Workshop © Brian S. Yandell
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Part II: Multiple QTL• Sampling from the Posterior
– Markov chain Monte Carlo• Gibbs sampler for effects• Metropolis-Hastings for loci
• Issues for 2 QTL• Bayes factors & Model Selection• Simulated data for 0,1,2 QTL• Brassica data on days to flowering
June 1999 NCSU QTL Workshop © Brian S. Yandell
148
Posterior for effects
•start with joint distribution of effects– does not depend on locus given genotypes– standard linear model problem
n
jjj bxyb
1
2**2* ),,;|()()()(
),|,,( *2* xy b
is proportional to
June 1999 NCSU QTL Workshop © Brian S. Yandell
149
How to Study Posterior?
• exact methods– exact if possible– can be difficult or
impossible to analyze
• approximate methods– importance sampling– numerical integration– Monte Carlo & other
• Monte Carlo methods– easy to implement– independent samples
• MCMC methods– handle hard problems– art to efficient use– correlated samples
June 1999 NCSU QTL Workshop © Brian S. Yandell
150
Ordinary Monte Carlo• independent samples of joint distribution• chaining (or peeling) of effects• requires numerical integration
– possible analytically here– very messy in general
),|,,(),|(
),|();,|(),;,|(
),|,,(
*2*
),(
*
*****2
*2*
2* xyxy
xyxyxy
xy
bE
bb
b
b
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Gibbs Sampler for effects• set up Markov chain around posterior for effects• sample from posterior by sampling from full conditionals
– conditional posterior of each parameter given the other– update parameter by sampling full conditional
),,;|()(),;,|(
),,;|()(),;,|(
),,;|()(),;,|(
2**2**2
2***2**
2**2**
bb
bbb
bb
xyxy
xyxy
xyxy
update mean
update additive
update variance
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Markov chain updates
variancemean
traits
additive
genos
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Markov chain idea• future events given the present state are
independent of past events• update chain based on current value• can make chain arbitrarily complicated• chain converges to stable pattern
0 1 1-q
q
p
1-p
)/()1( qpp
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Markov chain Monte Carlo• can study arbitrarily complex models
– need only specify how parameters affect each other– can reduce to specifying full conditionals
• construct Markov chain with “right” model– update some parameters given data and others– can fudge on “right” (importance sampling)– next step depends only on current estimates
• nice Markov chains have nice properties– sample summaries make sense– consider almost as random sample from distribution
June 1999 NCSU QTL Workshop © Brian S. Yandell
155
MCMC run of mean & effect
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addi
tive
frequency
June 1999 NCSU QTL Workshop © Brian S. Yandell
156
Care & Use of MCMC• sample chain for long run (100,000-1,000,000)
– longer for more complicated likelihoods– use diagnostic plots to assess “mixing”
• standard error of estimates– use histogram of posterior– compute variance of posterior--just another
summary
• studying the Markov chain– Monte Carlo error of series (Geyer 1992)
• time series estimate based on lagged auto-covariances
– convergence diagnostics for “proper mixing”
June 1999 NCSU QTL Workshop © Brian S. Yandell
157
MCMC Idea for QTLs• construct Markov chain around posterior
– want posterior as stable distribution of Markov chain– in practice, the chain tends toward stable distribution
• initial values may have low posterior probability• burn-in period to get chain mixing well
• update one (or several) components at a time– update effects given genotypes & traits– update locus given genotypes & traits– update genotypes give locus & effects
N
bb
21
2**2** )|;,,;(~);,,;( yxx
June 1999 NCSU QTL Workshop © Brian S. Yandell
158
Markov chain updates
effectslocus
traits
genos
June 1999 NCSU QTL Workshop © Brian S. Yandell
159
Full Conditional for genos• full conditional for genotype depends on
– effects via trait model– locus via recombination model
• can explicitly decompose by individual j– binomial (or trinomial) probability
1,0,1
2*
*2**2**
*
)|(),,;|(
)|(),,;|();,,;|(
1,0,1
xj
jjjjj
j
xbxy
xbxybyx
orx
June 1999 NCSU QTL Workshop © Brian S. Yandell
160
Prior for locus• no prior information on
locus– uniform prior over genome– use framework map
•choose interval proportional to length
•then pick uniform position within interval
0 20 40 60 80
0.0
0.2
distance (cM)
• prior information from other studies
•concentrate on credible regions•use posterior of previous study as new prior
June 1999 NCSU QTL Workshop © Brian S. Yandell
161
Full Conditional for locus• cannot easily sample from locus full conditional• cannot explicitly determine full conditional
– difficult to normalize– need to consider all possible genotypes over entire map
• Gibbs sampler will not work – but can get something proportional ...
n
jjx
b
1
*
*2**
)|()(
)|(),,;,|(
xxy
June 1999 NCSU QTL Workshop © Brian S. Yandell
162
Metropolis-Hastings Step• pick new locus based upon current locus
– propose new locus from distribution q( )• pick value near current one?• pick uniformly across genome?
– accept new locus with probability a()
• Gibbs sampler is special case of M-H– always accept new proposal
• acceptance insures right stable distribution
),()|(
),()|(,1min),(
*
*
newoldold
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qa
xx
June 1999 NCSU QTL Workshop © Brian S. Yandell
163
MCMC run: 2 loci assuming only 1
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40
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MCMC run/1000
40
50
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dist
ance
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)
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June 1999 NCSU QTL Workshop © Brian S. Yandell
164
Multiple QTL model
• trait = mean + add1 + add2 + error• trait = effect_of_genos + error• prob( trait | genos, effects )
j
m
rjrrj
jjjj
exby
exbxby
1
**
*2
*2
*1
*1
June 1999 NCSU QTL Workshop © Brian S. Yandell
165
Simulated Data with 2 QTL4
68
1012
1416
-1,-1 -1,1 1,-1 1,1recombinants
46
810
1214
16
0 1 2 3 4 5 6frequency
trai
t
June 1999 NCSU QTL Workshop © Brian S. Yandell
166
Issues for Multiple QTL
• how many QTL influence a trait?– 1, several (oligogenic) or many (polygenic)?– how many are supported by the data?
• searching for 2 or more QTL– conditional search (IM, CIM)– simultaneous search (MIM)
• epistasis (inter-loci interaction)– many more parameters to estimate– effects of ignored QTL
June 1999 NCSU QTL Workshop © Brian S. Yandell
167
LOD for 2 QTL0
51
01
52
02
53
0
distance (cM)
LOD
0 10 20 30 40 50 60 70 80 90
QTLIMCIM
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Interval Mapping Approach
• interval mapping (IM)– scan genome for 1 QTL
• composite interval mapping (CIM)– scan for 1 QTL while adjusting for others– use markers as surrogates for other QTL
• multiple interval mapping (MIM)– search for multiple QTL
June 1999 NCSU QTL Workshop © Brian S. Yandell
169
MCMC run with 2 loci
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0 200 400 600 800 1000
40
50
60
70
80
MCMC run/100
40
50
60
70
80
0 50 100 150 200 250 300
dist
ance
(cM
)
frequency
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Bayesian Approach
• simultaneous search for multiple QTL• use Bayesian paradigm
– easy to consider joint distributions– easy to modify later for other types of data
• counts, proportions, etc.
– employ MCMC to estimate posterior dist
• study estimates of loci & effects• use Bayes factors for model selection
– number of QTL
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Effects for 2 Simulated QTL
distance (cM)
effe
ct
40 50 60 70 80
1.8
2.0
2.2
2.4
2.6
40 50 60 70 80
0.0
0.0
50
.10
0.1
5
distance (cM)
1.8
2.0
2.2
2.4
2.6
effect 1 effect 2
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Brassica napus Data• 4-week & 8-week vernalization effect
– log(days to flower)
• genetic cross of– Stellar (annual canola)– Major (biennial rapeseed)
• 105 F1-derived double haploid (DH) lines– homozygous at every locus (QQ or qq)
• 10 molecular markers (RFLPs) on LG9– two QTLs inferred on LG9 (now chromosome N2)– corroborated by Butruille (1998)– exploiting synteny with Arabidopsis thaliana
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Brassica 4- & 8-week Data
2.5 3.0 3.5 4.0
2.5
3.0
3.5
4-week
8-w
eek
2.5
3.5
0 2 4 6 8 108-week vernalization
2.5 3.0 3.5 4.0
02
46
8
4-week vernalization
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Brassica Data LOD Maps0
24
68
distance (cM)
LOD
0 10 20 30 40 50 60 70 80 90
QTLIMCIM
8-w
eek
05
10
15
distance (cM)
LOD
0 10 20 30 40 50 60 70 80 90
QTLIMCIM
4-w
eek
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4-week vs 8-week vernalization
4-week vernalization• longer time to flower• larger LOD at 40cM• modest LOD at 80cM• loci well determined
cM add40 .3080 .16
8-week vernalization• shorter time to flower• larger LOD at 80cM• modest LOD at 40cM• loci poorly determined
cM add40 .0680 .13
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Brassica Credible Regions
20 40 60 80
-0.6
-0.4
-0.2
0.0
0.2
distance (cM)
addi
tive
4-week
20 40 60 80
-0.3
-0.2
-0.1
0.0
0.1
0.2
distance (cM)
addi
tive
8-week
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Collinearity of QTLs• multiple QT genotypes are correlated
– QTL linked on same chromosome– difficult to distinguish if close
• estimates of QT effects are correlated– poor identifiability of effects parameters– correlations give clue of how much to trust
• which QTL to go after in breeding?– largest effect?– may be biased by nearby QTL
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Brassica effect Correlations
-0.6 -0.4 -0.2 0.0 0.2
-0.6
-0.4
-0.2
0.0
additive 1
addi
tive
2
cor = -0.81
4-week
-0.2 -0.1 0.0 0.1 0.2
-0.3
-0.2
-0.1
0.0
additive 1
addi
tive
2
cor = -0.7
8-week
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Bayes FactorsWhich model (1 or 2 or 3 QTLs?) has higher probability of supporting the data?
– ratio of posterior odds to prior odds– ratio of model likelihoods– related to likelihood ratio test of simple hypotheses
)|(
)|(
)(/)(
)|(/)|(
2
112
21
2112
model model
model model model model
yy
yy
B
B
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Bayes Factors & LR
• equivalent to LR statistic when– comparing two nested models– simple hypotheses (e.g. 1 vs 2 QTL)
• Bayes Information Criteria (BIC) in general– Schwartz introduced for model selection– penalty for different number of parameters p
)log()()log(2)log(2 1212 nppLRB
June 1999 NCSU QTL Workshop © Brian S. Yandell
181
Model Determinationusing Bayes Factors
• pick most plausible model– histogram for range of models– posterior distribution of models– use Bayes theorem– often assume flat prior across models
• posterior distribution of number of QTLs
June 1999 NCSU QTL Workshop © Brian S. Yandell
182
Brassica Bayes Factors
i vs. j Bayes Factor -log(LR)
2 vs. 1 2.49 7.82
3 vs. 1 .005 7.41
3 vs. 2 .002 4.17
•compare models for 1, 2, 3 QTL•Bayes factor and -2log(LR)•large value favors first model•8-week vernalization only here
June 1999 NCSU QTL Workshop © Brian S. Yandell
183
Computing Bayes Factors• using samples from prior
– mean across Monte Carlo or MCMC runs– can be inefficient if prior differs from posterior
• using samples from posterior– weighted harmonic mean more efficient but unstable– care in choosing weight h() close to posterior– increase stability: absorb variance (normal to t dist)
1
1 )()(
)()|(ˆ
G
g kkkk
kk
hG
model |model ;|model
yy
kkkkkk d )()|()|( model |model ;model yy
June 1999 NCSU QTL Workshop © Brian S. Yandell
184
MCMC Software• General MCMC software
– U Bristol links
• http://www.stats.bris.ac.uk/MCMC/pages/links.html– BUGS (Bayesian inference Using Gibbs Sampling)
• http://www.mrc-bsu.cam.ac.uk/bugs/
• Our MCMC software for QTLs– C code using LAPACK
• ftp://ftp.stat.wisc.edu/pub/yandell/revjump.tar.gz– coming soon:
• perl preprocessing (to/from QtlCart format)• Splus post processing• Bayes factor computation
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Bayes & MCMC References• CJ Geyer (1992) “Practical Markov chain Monte Carlo”,
Statistical Science 7: 473-511• L Tierney (1994) “Markov Chains for exploring posterior
distributions”, The Annals of Statistics 22: 1701-1728 (with disc:1728-1762).
• A Gelman, J Carlin, H Stern & D Rubin (1995) Bayesian Data Analysis, CRC/Chapman & Hall.
• BP Carlin & TA Louis (1996) Bayes and Empirical Bayes Methods for Data Analysis, CRC/Chapman & Hall.
• WR Gilks, S Richardson, & DJ Spiegelhalter (Ed 1996) Markov Chain Monte Carlo in Practice, CRC/Chapman & Hall.
June 1999 NCSU QTL Workshop © Brian S. Yandell
186
QTL References• D Thomas & V Cortessis (1992) “A Gibbs sampling
approach to linkage analysis”, Hum. Hered. 42: 63-76.• I Hoeschele & P vanRanden (1993) “Bayesian analysis of
linkage between genetic markers and quantitative trait loci. I. Prior knowledge”, Theor. Appl. Genet. 85:953-960.
• I Hoeschele & P vanRanden (1993) “Bayesian analysis of linkage between genetic markers and quantitative trait loci. II. Combining prior knowledge with experimental evidence”, Theor. Appl. Genet. 85:946-952.
• SW Guo & EA Thompson (1994) “Monte Carlo estimation of mixed models for large complex pedigrees”, Biometrics 50: 417-432.
• JM Satagopan, BS Yandell, MA Newton & TC Osborn (1996) “A Bayesian approach to detect quantitative trait loci using Markov chain Monte Carlo”, Genetics 144: 805-816.
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Brassica Data: 1 QTL
distance (cM)
effe
ct
20 30 40 50 60 70 80
-0.2
5-0
.15
-0.0
5
20 30 40 50 60 70 80
0.0
0.0
20
.04
0.0
6
distance (cM)
-0.2
5-0
.15
-0.0
5
effect 1
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Brassica 8-week Data: 2 QTL
20 40 60 80
-0.3
-0.1
0.1
0.2
distance (cM)
20 40 60 80
0.0
0.0
20
.04
0.0
60
.08
distance (cM)
-0.3
-0.1
0.1
0.2
QTL 1 QTL 2
addi
tive
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Brassica 8-week Data: 2 QTL
20 40 60 80
-0.6
-0.2
0.0
0.2
distance (cM)
20 40 60 80
0.0
0.0
20
.04
0.0
60
.08
distance (cM)
-0.6
-0.2
0.0
0.2
QTL 1 QTL 2
addi
tive
June 1999 NCSU QTL Workshop © Brian S. Yandell
190
Geometry of Reversible Jump
2/)(
12:
21:
21
2
1
bbb
m
zbb
zbb
m
0 b b+z
0b
b-z
b1
b2 b
1
2
June 1999 NCSU QTL Workshop © Brian S. Yandell
191
Simulation of Jumps
b1
b2 b
June 1999 NCSU QTL Workshop © Brian S. Yandell
192
Mean & Variance Estimates
n
jjjj
n
jjj
n
jj
nxxbxxbym
nxxbym
nyym
1
2
2221112
1
2
112
1
/)(ˆ)(ˆˆˆ̂:2
/)(ˆˆˆ:1
/ˆ:2,1
June 1999 NCSU QTL Workshop © Brian S. Yandell
193
Another Way to Jump
• many ways to parameterize models• can augment both models
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zzzzbbvzzzzb
tionstransformasinnovationparametersm
rr
rr
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,)(0,~,),;,,(21
212
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2121212
June 1999 NCSU QTL Workshop © Brian S. Yandell
194
Simulated Data
• two loci, estimates from data• Bayes factors for model selection
2.78ˆ,26.ˆ
2.42ˆ,10.ˆ
08.0ˆ,78.2ˆ
2*2
1*1
2
b
b
June 1999 NCSU QTL Workshop © Brian S. Yandell
195
Posterior Number of loci (simulated after 8-week)
m 3 4 5 60 .05 .02 .00 .001 .44 .31 .26 .182 .38 .41 .44 .403 .12 .20 .23 .304 .02 .05 .06 .105 .00 .01 .01 .02
distribution across m by prior mean
June 1999 NCSU QTL Workshop © Brian S. Yandell
196
Bayes Factors(simulated after 8-week)
prior mean
models
3 4 5 6
1:2 1.74 1.51 1.48 1.35
1:3 5.50 4.13 4.71 3.60
1:4 24.75 16.53 22.57 16.20
2:3 3.17 2.73 3.19 2.67
2:4 14.25 10.93 15.28 12.00
3:4 4.50 4.00 4.79 4.50
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Sensitivity of Number to Prior
0 1 2 3 4 5 6 7 8
010
2030
4050
234610
4-week vernalization
0 1 2 3 4 5 6 7 8
010
2030
4050
60
234610
8-week vernalization
June 1999 NCSU QTL Workshop © Brian S. Yandell
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Number of lociBrassica 8-week
m 3 4 5 61 .43 .28 .21 .122 .43 .44 .41 .343 .12 .22 .28 .344 .02 .05 .09 .165 .00 .00 .01 .046 .00 .00 .00 .005
distribution across m by prior mean
June 1999 NCSU QTL Workshop © Brian S. Yandell
199
Sensitivity of Prior: Simulations
0 1 2 3 4 5
020
4060
80012
Prior of 2
0 1 2 3 4 5 6
010
2030
4050
60
012
Prior of 4
June 1999 NCSU QTL Workshop © Brian S. Yandell
200
Profile LOD & LR• profile LOD related to LR test at a locus
– maximum likelihood estimates of effects– weighted average over all possible genotypes
n
j j
xj
by
xbxy
eLOD
bL
bLLRLReLOD
12*
1,0,1
2*
10
2*
2*
1021
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)|()ˆ,ˆ,ˆ;|(
ln)(log)(
)|;ˆ,ˆ,ˆ(
)|ˆ̂,0,ˆ(ln2)(),()(log)(
y
y
June 1999 NCSU QTL Workshop © Brian S. Yandell
201
Bayes FactorsWhich model (1 or 2 or 3 QTLs?) has higher probability of supporting the data?
– ratio of posterior odds to prior odds– ratio of model likelihoods– related to likelihood ratio test of simple hypotheses
)|(
)|(
)(/)(
)|(/)|(
2
112
21
2112
model model
model model model model
yy
yy
B
B
June 1999 NCSU QTL Workshop © Brian S. Yandell
202
Bayes Factors & LR
• equivalent to LR statistic when– comparing two nested models– simple hypotheses (e.g. 1 vs 2 QTL)
• Bayes Information Criteria (BIC) in general– Schwartz introduced for model selection– penalty for different number of parameters p
)log()()log(2)log(2 1212 nppLRB
June 1999 NCSU QTL Workshop © Brian S. Yandell
203
Model Determinationusing Bayes Factors
• pick most plausible model– histogram for range of models– posterior distribution of models– use Bayes theorem– often assume flat prior across models
• posterior distribution of number of QTLs
June 1999 NCSU QTL Workshop © Brian S. Yandell
204
Computing Bayes Factors• samples from prior distribution
– mean across Monte Carlo or MCMC runs– can be inefficient if prior differs from posterior
• samples from posterior distribution– weighted harmonic mean more efficient but unstable– care in choosing weight h() close to posterior– increase stability absorbing variance (normal to t dist)
1
1 )()(
)()|(ˆ
)()|()|(
G
g kkkk
kk
kkkkkk
hG
d
model |model ;|model
model |model ;model
yy
yy
June 1999 NCSU QTL Workshop © Brian S. Yandell
205
Brassica Bayes Factors
)17.4(
002.0
)41.7(
005.0)82.7(
49.2
QTLthree
QTLtwo
QTLtwoQTLone
• compare models for 1, 2, 3 QTL• Bayes factor (and -2log(LR)) • arrows point to favored model
June 1999 NCSU QTL Workshop © Brian S. Yandell
206
QTLs and Polygenesphenotype = design + QTLs + polygenes + error
QTLs: quantitative trait locipolygenes: many genes of small effect
spread throughout genome
distinction is arbitrary, depending on• sample size• magnitude of effects• design/cross/marker polymorphism
analogy to multiple regression
June 1999 NCSU QTL Workshop © Brian S. Yandell
207
Polygenes and Inbred Lines
• same (raw) genetic correlation across cohort:– 1/2 for DH– 2/3 for F2– 4/7 for F3– 1/2 for RI
• modified by specific information:– major & minor QTLs– marker surrogates for polygenes (CIM)
June 1999 NCSU QTL Workshop © Brian S. Yandell
208
Composite QTL model
• trait = mean + add + dom + other + error• trait = effect_of_geno + other + error
• prob( trait | genos, effects, other )
• other ( ): other linked and unlinked QTLs, and polygenes
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j
m
rjrrj
jjjj
exby
exbxby
1
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*
2
*
2
*
1
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