Bayesian Model Selection
for Quantitative Trait Loci
using Markov chain Monte Carlo
in Experimental Crosses
Brian S. Yandell
University of Wisconsin-Madison
www.stat.wisc.edu/~yandell/statgen
with Chunfang “Amy” Jin, UW-Madison,
Patrick J. Gaffney, Lubrizol,
and Jaya M. Satagopan, Sloan-Kettering
Jackson Laboratory, September 2002
September 2002
Jax Workshop © Brian S. Yandell
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minor
QTL
polygenes
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2
major
QTL
0
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additive effect
major QTL on
linkage map
3
Pareto diagram of QTL effects
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1
September 2002
0
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5
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rank order of QTL
Jax Workshop © Brian S. Yandell
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how many (detectable) QTL?
• build m = number of QTL detected into model
– directly allow uncertainty in genetic architecture
– model selection over number of QTL, architecture
– use Bayes factors and model averaging
• to identify “better” models
• many, many QTL may affect most any trait
– how many QTL are detectable with these data?
• limits to useful detection (Bernardo 2000)
• depends on sample size, heritability, environmental variation
– consider probability that a QTL is in the model
• avoid sharp in/out dichotomy
• major QTL usually selected, minor QTL sampled infrequently
September 2002
Jax Workshop © Brian S. Yandell
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interval mapping basics
•
observed measurements
– Y = phenotypic trait
– X = markers & linkage map
observed
• i = individual index 1,…,n
•
missing
• alleles QQ, Qq, or qq at locus
•
Q
unknown quantities
–  = QT locus (or loci)
–  = phenotype model parameters
– m = number of QTL
unknown
pr(Q|X,,m) recombination model
– grounded by linkage map, experimental cross
– recombination yields multinomial for Q given X
•
Y
missing data
– missing marker data
– Q = QT genotypes
•
X
after
Sen Churchill (2001)
pr(Y|Q,,m) phenotype model
– distribution shape (assumed normal here)
– unknown parameters  (could be non-parametric)
September 2002
Jax Workshop © Brian S. Yandell
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recombination model pr(Q|X,)
• locus  is distance along linkage map
– identifies flanking marker region
• flanking markers provide good approximation
– map assumed known from earlier study
– inaccuracy slight using only flanking markers
• extend to next flanking markers if missing data
– could consider more complicated relationship
• but little change in results
pr(Q|X,) = pr(geno | map, locus) 

pr(geno | flanking markers, locus)
Xk
September 2002
Q?
X k 1
Jax Workshop © Brian S. Yandell
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idealized phenotype model
• trait = mean + additive + error
• trait = effect_of_geno + error
• pr( trait | geno, effects )
Y  GQ  E
Qq
pr (Y | Q, ) 
qq
QQ
normal( GQ ,  )
2
8
September 2002
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who was Bayes?
• Reverend Thomas Bayes (1702-1761)
–
–
–
–
part-time mathematician
buried in Bunhill Cemetary, Moongate, London
famous paper in 1763 Phil Trans Roy Soc London
was Bayes the first with this idea? (Laplace)
• billiard balls on rectangular table
– two balls tossed at random (uniform) on table
– where is first ball if the second is to its left (right)?

first
second
Y=1
September 2002
Y=0
prior
pr() = 1
likelihood pr(Y | ) = Y(1- )1-Y
posterior pr( |Y) = ?
Jax Workshop © Brian S. Yandell
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what is Bayes theorem?
• before and after observing data
– prior:
– posterior:
pr() = pr(parameters)
pr(|Y) = pr(parameters|data)
• posterior = likelihood * prior / constant
– usual likelihood of parameters given data
– normalizing constant pr(Y) depends only on data
• constant often drops out of calculation
pr ( , Y ) pr (Y |  )  pr ( )
pr ( | Y ) 

pr (Y )
pr (Y )
September 2002
Jax Workshop © Brian S. Yandell
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Bayesian interval mapping
• likelihood is mixture over genotypes Q
L(,|Y) = pr(Y|X,,) = producti [sumQ pr(Q|Xi,) pr(Yi|Q,)]
• Bayesian posterior includes Q as missing data
– sample unknown data instead of averaging
• sample unknown genotypes Q
• prior on unknown loci  and effects  of interest
pr(,Q,|Y,X) = [producti pr(Qi|Xi,) pr(Yi|Qi,)] pr(,|X)
– marginal summaries provide key information
• loci:
• effects:
September 2002
pr(|Y,X) = sumQ, pr(,Q,|Y,X)
pr(|Y,X) = sumQ, pr(,Q,|Y,X)
Jax Workshop © Brian S. Yandell
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2.5
3.0
2.5
8-week
3.5
3.5
Brassica 4- & 8-week Data
2.5
3.0
3.5
4.0
4-week
0
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8 10
8-week vernalization
•4-week & 8-week vernalization
8
–log(days to flower)
6
•genetic cross of
2
4
–Stellar (annual canola)
–Major (biennial rapeseed)
0
•105 double haploid (DH) lines
2.5 3.0 3.5 4.0
4-week vernalization
September 2002
–homozygous at every locus
•10 markers on chromosome N2
Jax Workshop © Brian S. Yandell
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Brassica Data LOD Maps
8-week
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LOD
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QTL
IM
CIM
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distance (cM)
4-week
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LOD
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QTL
IM
CIM
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distance (cM)
September 2002
Jax Workshop © Brian S. Yandell
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Bayesian
Brassica
Brassica samples
Crediblefor
Regions
8-week
-0.3
-0.6
-0.2
-0.4
-0.2
additive
additive
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0.0
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4-week
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distance (cM)
loci (cM)
September 2002
Jax Workshop © Brian S. Yandell
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60
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distance (cM)
loci (cM)
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multiple QTL phenotype model
•
phenotype influenced by genotype & environment
pr(Y|Q,) ~ N(GQ , 2), or Y = GQ + environment
•
partition mean into separate QTL effects
GQ = mean + main effects
GQ = 
+ 1Q + . . . + mQ
•
priors on mean and effects
GQ ~
 ~
jQ ~
j2Q ~
•
+ epistatic interactions
+ 12Q + . . .
N(0 , 2)
N(0, 02)
N(0, 12/m)
N(0, 22/m2)
model independent genotypic value
grand mean
effects down-weighted by m
interactions down-weighted by m2
determine hyper-parameters
via Empirical Bayes
2
2
h
G
0  Y ,    0 
 2 ,    0  1   2
2
1 h

September 2002
Jax Workshop © Brian S. Yandell
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phenotype posterior mean
• 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 ( 0  BQ (Gˆ Q  0 ), BQCQ 2 )
 N (Gˆ Q , CQ 2 )
LS estimate Gˆ Q  ˆ  ˆijQ   wiQYi
i
variance
j
i
2
V (Gˆ Q )   wiQ
 2  CQ 2
i
shrinkage
September 2002
BQ   /(  CQ )  1
Jax Workshop © Brian S. Yandell
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effect of prior variance on posterior
0.5
2.0
2 3 4 5 6 7 8 9
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=0.52
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2 3 4 5 6 7 8 9
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normal prior, posterior for n = 1, posterior for n = 5 , true mean
(solid black) (dotted blue)
(dashed red)
(green arrow)
September 2002
Jax Workshop © Brian S. Yandell
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prior & posterior for genotypes Q
• prior is recombination model
pr(Q|Xi ,)
• can explicitly decompose by individual i
– binomial (or trinomial) probability
• posterior for genotype depends on
– effects via trait model
– locus via recombination model
• posterior agrees exactly with interval mapping
– used in EM: estimation step
– but need to know locus  and effects 
pr (Yi | Q, )pr (Q | X i ,  )
PQi  pr (Q | Yi , X i ,  , ) 
sum Q pr (Yi | Q, )pr (Q | X i ,  )
September 2002
Jax Workshop © Brian S. Yandell
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prior & posterior for QT locus
• prior information from other
studies
0.2
•concentrate on credible regions
•use posterior of previous study
as new prior
• no prior information on locus
– uniform prior over genome
– use framework map
• choose interval proportional to
length
• then pick uniform position within
interval
posterior
0.0
prior
0
September 2002
20
40
60
distance (cM)
Jax Workshop © Brian S. Yandell
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prior & posterior on number of QTL
• what prior on number of QTL?
• push data in discovery process
– bad: skeptic revolts!
• “answer” depends on “guess”
e
e
0
September 2002
Jax Workshop © Brian S. Yandell
e
p
u
exponential
Poisson
uniform
p p
e u u
p u u
u u u
p
e
p
e
e p
p
e
0.00
– good: reflects prior belief
prior probability
0.10
0.20
• prior influences posterior
0.30
– uniform over some range
– Poisson with prior mean
– geometric with prior mean
e
p e e
p
u p
u p
u e
u
2
4
6
8
m = number of QTL
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10
0.1
propose new  “nearby”
accept if more probable
toss coin if less probable
based on relative heights
(Metropolis-Hastings)
0.0
propose  ~ pr( |Y)
(ideal: Gibbs sample)
pr( |Y)
0.2
0.3
have posterior pr( |Y)
want to draw samples
0.4
Markov chain Monte Carlo idea
0
September 2002
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pr( |Y)
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mcmc sequence
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MCMC realization
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added twist: occasionally propose from whole domain
September 2002
Jax Workshop © Brian S. Yandell
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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 m-QTL model components from full conditionals
– update effects  given genotypes & traits
– update locus  given genotypes & marker map
– update genotypes Q given traits, marker map, locus & effects
( , Q, , m) ~ pr ( , Q, , m | Y , X )
( , Q, , m)1  ( , Q, , m)2    ( , Q, , m) N
September 2002
Jax Workshop © Brian S. Yandell
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sample from full conditionals
for model with m QTL
• hard to sample from joint posterior
observed
X
missing
Y
Q
unknown
– pr(,Q, |Y,X) = pr()pr()pr(Q|X,)pr(Y|Q,) /constant
• easy to sample parameters from full conditionals
– full conditional for genetic effects
• pr( |Y,X,,Q) = pr( |Y,Q) = pr() pr(Y|Q,) /constant
– full conditional for QTL locus
• pr( |Y,X,,Q) = pr( |X,Q) = pr() pr(Q|X,) /constant
– full conditional for QTL genotypes
• pr(Q|Y,X,, ) = pr(Q|X,) pr(Y|Q,) /constant
September 2002
Jax Workshop © Brian S. Yandell
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reversible jump MCMC
0
1
m+1 2 … m
L
action steps: draw one of three choices
• update m-QTL model with probability 1-b(m+1)-d(m)
– update current model using full conditionals
– sample m QTL loci, effects, and genotypes
• add a locus with probability b(m+1)
– propose a new locus along genome
– innovate new genotypes at locus and phenotype effect
– decide whether to accept the “birth” of new locus
• drop a locus with probability d(m)
– propose dropping one of existing loci
– decide whether to accept the “death” of locus
September 2002
Jax Workshop © Brian S. Yandell
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sampling the number of QTL
• use reversible jump MCMC to change m
–
–
–
–
bookkeeping helps in comparing models
adjust to change of variables between models
Green (1995); Richardson Green (1997)
other approaches out there these days…
• think model selection in multiple regression
– but regressors (QT genotypes) are unknown
– linked loci = collinear regressors = correlated effects
– consider additive effects with coding Qij= -1,0,1
 ijQ  j ( Qij  Q j )
September 2002
Jax Workshop © Brian S. Yandell
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Model Selection in Regression
• consider known genotypes (Q)
– models with 1 or 2 QTL at known loci
• jump between 1-QTL and 2-QTL models
– adjust posteriors when model changes
– due to collinearity of QTL genotypes
m  1 : Yi    ( Qi 1  Q1 )  ei
m  2 : Yi   1 ( Qi 1  Q1 )  2 ( Qi 1  Q1 )  ei
September 2002
Jax Workshop © Brian S. Yandell
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collinear QTL = correlated effects
8-week
additive
2
-0.2
-0.1
cor = -0.7
-0.6
-0.3
-0.4
-0.2
additive 2
cor = -0.81
0.0
0.0
4-week
-0.6
-0.4
-0.2
0.0
0.2
-0.2
-0.1
additive 1
0.0
0.1
0.2
additive 1
effect 1
effect 1
•linked QTL: collinear genotypes & correlated effect estimates
–sum of linked effects usually well determined
•which QTL to go after in breeding, genome walking?
September 2002
Jax Workshop © Brian S. Yandell
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Geometry of Reversible Jump
0.6
0.6
0.8
Reversible Jump Sequence
0.8
Move Between Models
b2
0.2 0.4
b2
0.2 0.4
c21 = 0.7
0.0
0.0
m=2
m=1
0.0
0.2
September 2002
0.4
1b1
0.6
0.8
0.0
0.2
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b1
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1
Jax Workshop © Brian S. Yandell
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QT additive Reversible Jump
first 1000 with m<3
0.0
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0.05
b2
0.1 0.2
b2
0.10
0.3
0.15
0.4
a short sequence
0.05
September 2002
0.10
 b1
1
0.15
-0.3 -0.2 -0.1 0.0 0.1 0.2
b1

Jax Workshop © Brian S. Yandell
1
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a complicated simulation
•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
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8
8
9
30
20
8
9
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11
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Genetic map
ch1
ch2
ch3
ch4
ch5
ch6
ch7
ch8
ch9
ch10
0
September 2002
7
number of QTL
Chromosome
QTL chr
0
– n=500, heritability to 97%
posterior
10
frequency in %
– (Stephens, Fisch 1998)
– n=200, heritability = 50%
– detected 3 QTL
40
•simulated F2 intercross, 8 QTL
50
100
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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
September 2002
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Jax Workshop © Brian S. Yandell
Count of 8000
3371
751
377
218
218
198
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Bayes factors to assess models
•Bayes factor: which model best supports the data?
– ratio of posterior odds to prior odds
– ratio of model likelihoods
•equivalent to LR statistic when
– comparing two nested models
– simple hypotheses (e.g. 1 vs 2 QTL)
•Bayes Information Criteria (BIC)
– Schwartz introduced for model selection in general settings
– penalty to balance model size (p = number of parameters)
pr ( model 1 | Y ) / pr ( model 2 | Y ) pr (Y | model 1 )
B12 

pr ( model 1 ) / pr ( model 2 )
pr (Y | model 2 )
 2 log( B12 )  2 log( LR)  ( p2  p1 ) log( n )
September 2002
Jax Workshop © Brian S. Yandell
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QTL Bayes factors & RJ-MCMC
• easy to compute Bayes factors from samples
– posterior pr(m|Y,X) is marginal histogram
– posterior affected by prior pr(m)
pr (m|Y , X ) /pr (m)
BFm,m1 
pr (m  1|Y , X ) /pr (m  1)
• BF insensitive to shape of prior
– 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 tuning by user
September 2002
Jax Workshop © Brian S. Yandell
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3 4
BF sensitivity to fixed prior for effects
Bayes factors
0.5
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2.00 5.00 2
hyper-prior heritability h
4
3
2
1
B45
B34
B23
B12
20.00 50.00
2
 jQ ~ N0,1 2 / m,1 2  h2 total
, h2 fixed
September 2002
Jax Workshop © Brian S. Yandell
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BF insensitivity to random effects prior
insensitivity to hyper-prior
1.0
3.0
hyper-prior density 2*Beta(a,b)
0.0
 jQ
0.5
1.0
1.5
2
hyper-parameter heritability h
3
2
3
2
3
2
3
2
Bayes factors
0.2
0.4 0.6
0.0
density
1.0
2.0
0.25,9.75
0.5,9.5
1,9
2,10
1,3
1,1
2.0
3
2
1
1
1
1
0.05
0.10
0.20
2
Eh
1
3
3
2
2
B34
B23
B12
1
1
0.50
1.00
2
h
2
~ N 0, 1 2 / m , 1 2  h 2 total
,
~ Beta (a, b)
2
September 2002
Jax Workshop © Brian S. Yandell
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RJ-MCMC software
• General MCMC software
– U Bristol links
• www.stats.bris.ac.uk/MCMC/pages/links.html
– BUGS (Bayesian inference Using Gibbs Sampling)
• www.mrc-bsu.cam.ac.uk/bugs/
• MCMC software for QTLs
– Bmapqtl (Satagopan Yandell 1996; Gaffney 2001)
• www.stat.wisc.edu/~yandell/qtl/software/Bmapqtl
– Bayesian QTL / Multimapper (Sillanpää Arjas 1998)
• www.rni.helsinki.fi/~mjs
– Yi, Xu (shxu@citrus.ucr.edu)
– Stephens & Fisch (email)
June 2002
NCSU QTL II © Brian S. Yandell
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Bmapqtl: our RJ-MCMC software
• www.stat.wisc.edu/~yandell/qtl/software/Bmapqtl
–
–
–
–
module using QtlCart format
compiled in C for Windows/NT
extensions in progress
R post-processing graphics
• library(bim) is cross-compatible with library(qtl)
• Bayes factor and reversible jump MCMC computation
• enhances MCMCQTL and revjump software
– initially designed by JM Satagopan (1996)
– major revision and extension by PJ Gaffney (2001)
•
•
•
•
June 2002
whole genome
multivariate update of effects; long range position updates
substantial improvements in speed, efficiency
pre-burnin: initial prior number of QTL very large
NCSU QTL II © Brian S. Yandell
36
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
September 2002
Jax Workshop © Brian S. Yandell
37
-50000
-40000
-30000
-20000
-10000
0
-50000
-40000
-30000
-20000
-10000
0
-50000
-40000
-30000
-20000
-10000
0
-50000
-40000
-30000
-20000
-10000
0
8 10
6
0 e+00 2 e+05 4 e+05 6 e+05 8 e+05 1 e+06
mcmc sequence
0.5
herit
mcmc sequence
0.1
0 e+00 2 e+05 4 e+05 6 e+05 8 e+05 1 e+06
mcmc sequence
15
0
5
10
10
LOD
15
20
20
burnin sequence
5
LOD
0.010
0 e+00 2 e+05 4 e+05 6 e+05 8 e+05 1 e+06
0.3
0.4
burnin sequence
0.2
herit
0.6
0.004
0.006
0.010
envvar
0.014
0.016
burnin sequence
0.0
number of QTL
environmental variance
h2 = heritability
(genetic/total variance)
LOD = likelihood
envvar
0
2
5
4
nqtl
nqtl
burnin (sets up chain)
mcmc sequence
10 15 20 25
Markov chain Monte Carlo sequence
September 2002
burnin sequence
Jax Workshop © Brian S. Yandell
0 e+00 2 e+05 4 e+05 6 e+05 8 e+05 1 e+06
mcmc sequence
38
MCMC sampled loci
120
tg2f12
100
wg6b2
E35M59.117
wg9c7
E38M50.133
Aca1
E33M62.140
E33M50.183
wg4a4b
tg6a12
wg5a5
wg8g1b
E32M61.218
ec4e7a
wg5b1a
wg6b10
E35M62.117
E33M59.59
wg7f3a
ec3a8
wg7a11
ec4g7b
wg6c6
E35M48.148
wg4f4c
E33M49.165
wg3g11
E32M49.73
wg1g8a
20
note concentration
on chromosome N2
40
points jittered for view
blue lines at markers
isoIdh
wg2e12b
E33M49.211
ec2d1a
MCMC sampled loci
60
80
subset of chromosomes
N2, N3, N16
wg3f7c
tg2h10b
wg1a10
ec2d8a
E33M47.338
wg2d11b
ec3b12
E38M50.159
E33M49.117
E32M50.409
E35M59.85
slg6
tg5d9b
wg4d10
wg7f5b
E32M47.159
ec5e12a
0
ec4h7
wg1g10b
E38M62.229
E33M49.491
ec2e5a
N2
September 2002
Jax Workshop © Brian S. Yandell
N3
chromosome
wg2a3c
wg5a1b
ec3e12b
N16
39
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
September 2002
3
5
7
9
model index
11
13
Jax Workshop © Brian S. Yandell
4
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
40
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
September 2002
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
Jax Workshop © Brian S. Yandell
N16
41
0.008
100
0
0.006
50
density
pattern: N2(2),N3,N16
col 1: density
col 2: boxplots by m
envvar
200
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
September 2002
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
Jax Workshop © Brian S. Yandell
4
5
6
7
8
9
11
12
LOD conditional on number of QTL
42
some QTL references
• Bernardo R (2000) What if we knew all the genes for a quantitative
trait in hybrid crops? Crop Sci. (submitted).
• Gaffney PJ (2001) An efficient reversible jump Markov chain Monte
Carlo approach to detect multiple loci and their effects in inbred
crosses. PhD thesis, UW-Madison Statistics.
• Heath S (1997) Markov chain Monte Carlo segregation and linkage
analysis for oligenic models, Am J Hum Genet 61: 748-760.
• Satagopan JM, Yandell BS, Newton MA, Osborn TC (1996) A
Bayesian approach to detect quantitative trait loci using Markov chain
Monte Carlo. Genetics 144: 805-816.
• Satagopan JM, Yandell BS (1996) Estimating the number of
quantitative trait loci via Bayesian model determination. Proc JSM
Biometrics Section.
September 2002
Jax Workshop © Brian S. Yandell
43
more QTL references
• Sillanpaa MJ, Arjas E (1998) Bayesian mapping of multiple
quantitative trait loci from incomplete inbred line cross data., Genetics
148: 1373-1388.
• Stephens DA, Fisch RD (1998) Bayesian analysis of quantitative trait
locus data using reversible jump Markov chain Monte Carlo.
Biometrics 54: 1334-1347.
• Uimari P and Hoeschele I (1997) Mapping linked quantitative trait loci
using Bayesian analysis and Markov chain Monte Carlo algorithms,
Genetics 146: 735-743.
• Zou F, Fine JP, Yandell BS (2001) On empirical likelihood for a
semiparametric mixture model. Biometrika 00: 000-000.
• Zou F, Yandell BS, Fine JP (2001) Threshold and power calculations
for QTL analysis of combined lines. Genetics 00: 000-000.
September 2002
Jax Workshop © Brian S. Yandell
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reversible jump MCMC references
• Green PJ (1995) Reversible jump Markov chain Monte Carlo
computation and Bayesian model determination. Biometrika 82: 711732.
• Kuo L, Mallick B (1998) Variable selection for regression models.
Sankhya, Series B, Indian J Statistics 60: 65-81
• Mallick BK (1998) Bayesian curve estimation by polynomial of
random order. J Statistical Planning and Inference 70: 91-109
• Richardson S, Green PJ (1997) On Bayesian analysis of mixture with
an unknown of components. J Royal Statist Soc B 59: 731-792.
September 2002
Jax Workshop © Brian S. Yandell
45
many thanks
Michael Newton
Daniel Sorensen
Daniel Gianola
Yang Song
Fei Zou
Liang Li
Hong Lan
Tom Osborn
David Butruille
Marcio Ferrera
Josh Udahl
Pablo Quijada
Alan Attie
Jonathan Stoehr
USDA Hatch Grants
September 2002
Jax Workshop © Brian S. Yandell
46