Bayesian Inference for QTLs
in Inbred Lines
Brian S. Yandell
University of Wisconsin-Madison
www.stat.wisc.edu/~yandell
with Jaya M. Satagopan, Sloan-Kettering
and Patrick J. Gaffney, Lubrizol
NCSU Statistical Genetics
June 2001
June 2001
NCSU QTL II Workshop © Brian S.
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Many Thanks
Michael Newton
Daniel Sorensen
Daniel Gianola
Jaya Satagopan
Patrick Gaffney
Fei Zou
Liang Li
Tom Osborn
David Butruille
Marcio Ferrera
Josh Udahl
Pablo Quijada
Alan Attie
Jonathan Stoehr
USDA Hatch Grants
June 2001
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What is the Goal Today?
• resampling from data
– permutation tests
– bootstrap, jackknife
– MCMC
• special Markov chain
• Monte Carlo sampling
• show MCMC ideas
– Gibbs sampler
– Metropolis-Hastings
– reversible jump MCMC
June 2001
• Bayesian perspective
– common in animal model
– use “prior” information
• previous experiments
• related genomes
• inbred lines “easy”
– can check against *IM
– ready extension
•
•
•
•
NCSU QTL II Workshop © Brian S.
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multiple experiments
pedigrees
non-normal data
epistasis
3
Note on Outbred Studies
• Interval Mapping for Outbred Populations
–
–
–
–
Haley, Knott & Elsen (1994) Genetics
Thomas & Cortessis (1992) Hum. Hered.
Hoeschele & vanRanden (1993ab) Theor. Appl. Genet. (etc.)
Guo & Thompson (1994) Biometrics
• Experimental Outbred Crosses (BC, F2, RI)
– collapse markers from 4 to 2 alleles
• Multiple Cross Pedigrees
– polygenic effects not modeled here
– related individuals are correlated (via coancestry)
– Liu & Zeng (2000) Genetics
– Zou, Yandell & Fine (2001) Genetics
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Overview
• I: Single QTL
• II: Bayesian Idea
• V: How many QTL?
– Reversible Jump
– analog to regression
– Bayes rule
– posterior & likelihood • VI: RJ-MCMC Details
• III: MCMC Samples
– Monte Carlo idea
– study posterior
• IV: Multiple QTL
June 2001
• VII: Bayes Factors
• VIII: References
– Software
– Articles
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Part I: Interval Mapping
• Modelling a trait with a QTL
– linear model for trait given genotype
– recombination near loci for genotype
• Likelihoods
• Review Interval Maps & Profile LODs
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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)
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Single QTL trait Model
• trait = mean + additive + error
• trait = effect_of_geno + error
• prob( trait | geno, effects )
y j    b* x*j  e j
x=1
x=-1
x=0
 ( y j | x*j ;  , b* ,  2 )
 y j    b* x*j 

 





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trait
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Simulated Data with 1 QTL
x=-1
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frequency
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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
r
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1
2
1  e 
2 
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markers, QTL & recombination rates
r3
r2
r1
r5
r4
*
x
?
M1 M 2
M3
M4
M5
M6
?

June 2001
distance along chromosome
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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 )
 ( x*j |  )   ( x*j | , M j ,k , M j ,k 1 )
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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 )
L (  , b* ,  2 ,  | y j )   ( y j |  , b* ,  2 ;  )

*
2

(
y
|
x
;

,
b
,

) ( x |  )
 j
x  1, 0 ,1
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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
L(  , b* ,  2 ;  | y )    ( y j |  , b* ,  2 ;  )
j 1
n

*
2

(
y
|
x
;

,
b
,

) ( x |  )
 j
j 1 x  1, 0 ,1
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Profile LOD for 1 QTL
0
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LOD
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QTL
IM
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Interval Mapping for
Quantitative 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

LOD ( )  (log 10 e) ln 
j 1


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  (y
| x; ˆ , bˆ* , ˆ 2 ) ( x |  ) 
x  1, 0 ,1

*
2
ˆ
 ( y j | ˆ , b  0, ˆ )


j
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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
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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


locus CI   | LOD (ˆ )  LOD ( )  2
effect CI  bˆ*  1.96se(bˆ* )
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Part II: Bayesian Idea
• joint distribution of known & unknown
– known: trait, markers, linkage map
– unknown: locus, genotype, effect, variance
• Use Same Likelihood Components
– trait given genotype
• follows linear model
• depends on size of effect, variance
– genotype given locus, markers & map
• depends on recombination near locus
• Inference about unknowns
– Bayes theorem
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What is Probability?
Frequentist Analysis
repeat experiment
– many times
– hypothetical
long term frequency
– Type I error rate
– reject null when true
June 2001
Bayesian Analysis
uncertainty about true value
prior
– uncertainty before
examining data
– incorporate prior
knowledge/experience
posterior
– uncertainty after
analyzing current data
– balance prior & current
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Bayes Theorem
• posteriors and priors
– prior:
– posterior:
prob( parameters )
prob( parameters | data )
• posterior = likelihood * prior / constant
• posterior distribution is proportional to
– likelihood of parameters given data
– prior distribution of parameters
*
*
*

(
b
and
y
)

(
y
|
b
)

(
b
)
*
 (b | y ) 

 (y )
 (y )
 (b* | y )   (y | b* ) (b* ) / constant
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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
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parameter
June 2001
0.2
parameter = 6
012345678910
Poisson data : y  1,3,8
posterior
0.3
0
0.0
0.1
0.0002
0.2
likelihood
0 2 4 6 8
parameter = 4
012345678910
0.0
0.2
0.0
parameter = 2
012345678910
0.0004
0.0
0.2
Likelihood & Posterior Example
parameter : t  ?
t k e t
prob{ y  k | t} 
k!
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Bayesian Idea for QTLs
• Model a trait with a QTL
– linear model for trait given genotype
– recombination near loci for genotype
• Find Bayesian Posterior
– compute likelihood using MCMC
• Case Studies
– simulated single QTL
– simulated two QTL
– flowering time data for Brassica napus
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Posterior for QTL Effect
• posterior = likelihood * prior / constant
• posterior distribution is proportional to
– prior distribution of effect
– likelihood of traits given effect & genos
 (b* | y)
is proportional to
n
 (b )   ( y j | x*j ;  , b* ,  2 )
*
j 1
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Full Posterior for QTL
• posterior = likelihood * prior / constant
• posterior( paramaters | data )
prob( genos, effects, loci | trait, map )
 ( x * ;  , b* ,  2 ;  | y )
is proportional to
n
 (  ) (b* ) ( 2 ) ( )   ( x*j |  )
j 1
n
   ( y j | x*j ;  , b* ,  2 )
j 1
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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
June 2001
• Monte Carlo methods
– easy to implement
– independent samples
• MCMC methods
– handle hard problems
– art to efficient use
– correlated samples
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2.0
2.2
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1.8
1.8
2.0
additive
2.2
Posterior for locus & effect
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QTL 1
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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)
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1.8
2.0
additive
2.2
HPD Region for locus & effect
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40
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distance (cM)
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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
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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
June 2001
• 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
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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
• Bayesian approach
– joint distribution over
QT genotypes
– sample distribution
• joint effects & loci
– HPD regions for
• joint locus & effects
• use density estimator
– approximate shape of
likelihood peak
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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
• maximize likelihood (interval mapping)
– EM method (Lander & Botstein 1989)
– MCMC method (Guo & Thompson 1994)
• study whole likelihood as posterior (Bayesian)
– analytical methods (e.g. Carlin & Louis 1998)
– MCMC method (Satagopan et al 1996)
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Studying the Likelihood
• maximize (*IM)
– find the peak
– avoid local maxima
– profile LOD
• across locus
• max for effects
• sample (Bayes)
– get whole curve
– summarize later
– posterior
• EM method
– always go up
– steepest ascent
• MCMC method
–
–
–
–
jump around
go up if you can
sometimes go down
cool down to find peak
• simulated annealing
• simulated tempering
• locus & effects together
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EM-MCMC duality
• EM approach can be redone with MCMC
– EM estimates & maximizes
– MCMC draws random samples
– both can address same problem
• sometimes EM is hard (impossible) to use
• MCMC is tool of “last resort”
–
–
–
–
June 2001
use exact methods if you can
try other approximate methods
be clever!
very handy for hard problems in genetics
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Simulation Study
• 200 simulation runs
• n = 100, 200; h^2 = 10, 20%
• 1 QTL at 15cM
• markers at 0, 10, 20, 40, 60, 80
• effect = 1
• variance depends on h^2
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200 Simulations: Effect
1.5
0.2 0.6 1.0 1.4
2.0
0.6
1.0
1.4
n = 200 h^2 = 10
n = 200 h^2 = 20
1.2
0.8
0.6
1.0
1.4
QTL Cartographer
0.6 0.8 1.0 1.2
QTL Cartographer
Bayesian MCMC
QTL Cartographer
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1.0
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Bayesian MCMC
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Bayesian MCMC
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QTL Cartographer
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200 Simulations: Locus
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10 20 30 40
0
10 20 30 40
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n = 200 h^2 = 20
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Bayesian MCMC
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QTL Cartographer
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Part III: MCMC Sampling
• Study the Bayesian Posterior
– use Markov chain to sample
• Markov chain Monte Carlo
• Gibbs sampler for effects
• Metropolis-Hastings for loci
• Brassica data on days to flowering
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How to Proceed?
• want to study π(parameters|data)
• run Markov chain with stable pattern π()
• study properties of Markov chain to learn
about posterior π(parameters|data)
– Markov chain Monte Carlo
• summarize results in graphical form
• diagnostics
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Markov chain idea
• future given present is independent of past
• update chain based on current value
– can make chain arbitrarily complicated
– chain converges to stable pattern π() we wish to study
 (1)  p /( p  q)
p
1-p
0
1
1-q
q
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Markov chain idea
p
1-p
Mitch’s
0
 (1)  p /( p  q)
q
1
1-q
1
Series1
Series2
Other
0
1
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5
7
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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
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MCMC Run for 1 locus Data
distance (cM)
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1 1
1 11 1 1 1 111 1 11
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1 1 1 1 11
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1 1 11 1
1
1111 1 11 111 1
111
11 1
11 1
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1 11
11 1 111 11 1 1 1 11111 1 1 1111 1
111 1 1
1
1
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1
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1 1 1 11 11 1 1 11 1 1 11
1
1
11 1 11111111 1 1111 1
11
1 111
1 11 111 11 11
1 11111111
1
1
1
1
11 1 1 11
111 11 1
1 11 1
11 1
1
1 11 111 111 11111
11
1
1 1 1 1111 1 11
11111 111 1111 1 1
1
1
1
11111 1 1
1
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1 11 11111111
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1 1 1 11 1 1111 1 1 11 1 11 111 1 111 111 1
1 111 11111
11111
1 1 1 1 11 11111 111 111111 11 1
11111 111 1 1
1
1
1
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1
1
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111111 11
1 1 1 1 11
11 1 111
1 1 1 1 1 111111 1 1 1 1111111 11 1 11 11 11 1 11
1
111111111 111111111111 1 1 11 1111 1 11111 111 11 111 11 1 1
1 111
1
1
11 111111 1111 11 1 11111 1 1
1 11 1
1
1
1
1
1
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1
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1 1111
1 1111 11
1
1 1 1 1 11
1 111
111 11 1 11 111 11111 1 111 11 1111 11 11 11 1111111
1
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111 11
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1 1 1 11 1111 11 11 1 1 1 1 1 11 1 1111 1 111111 11
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1 1
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1 111 111
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1 1 1 1 111 111 11 11 11 111 1 1 1 11111111111 11 111
1
1 11 11 1
11 1
11 11 1 1 1
111 1 1111
111
1
1 1 11
111 1 11 1
1
11 11 1
1
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11 1 11 1 11 1 1111
1
1 1 1 1 1 11 111
1
1
1
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11 11 11 1 1
11 1
1 1 1 11 11 11 1 1 11
1
1 1 11 11 111 1 1 11 1 1 1
1
1
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1
1 1 1 11
11 1 111 1 1 111
1
1
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1 1 1 11
1 11 1
1
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1 1
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1 11
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1 1 11 1
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111
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1 1
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1000
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MCMC run/100
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Why not
Ordinary Monte Carlo?
• independent samples of joint distribution
• chaining (or peeling) of effects
• requires numerical integration
– possible analytically here
– very messy in general
 (  , b* ,  2 | y, x * ) 
 ( 2 | y, x * ;  , b* )   (b* | y, x * ;  )   (  | y, x * )
 (  | y, x * )  E(b , )  (  , b* ,  2 | y, x * ) 
*
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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 one (or several) components at a time
– update effects given genotypes & traits
– update locus given genotypes & traits
– update genotypes give locus & effects
  ( x * ;  , b* ,  2 ;  ) ~  ( x * ;  , b* ,  2 ;  | y )
1   2     N
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MCMC Effect Updates
genos
mean
additive
variance
traits
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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
update mean
 (  | y, x * ; b* ,  2 )   (  ) (y | x * ;  , b* ,  2 ) / c
update additive
 (b* | y, x * ;  ,  2 )   (b* ) (y | x * ;  , b* ,  2 ) / c
update variance
 ( 2 | y, x * ;  , b* )   ( 2 ) (y | x * ;  , b* ,  2 ) / c
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
50
0
200
400
600
800
10.4
mean
10.0 10.2
1
1
1
1
1
1 11
11
1
1
1
11 11
1
1
1
111
11 11
1111 1111 1 1111 11 1111
1
1
1
1
1
1
1
1
1
1
1
1 111 111 11
1
11 1 1 1 111 1 111 1 1 1 11
1
11
1 1 1 1111111111 1
111
111
11111
11111111111111111111
1111 11 1
1111
11111111
11111111 111
11 1111
11
111111
111 11 1 11
111111
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
111
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
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 1111111
1 1111111 111111 1111111
1
1
1
1
1
1
1
1111 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 11111111111111111111
11111111 1111111
111
11111
1111111 11
1111111 111
11
111
1111111 111
1
1111
11 111111111
11
111
11111
11
1111
1111
11 1111111111
11 111111
11111111
111111111
1111
111111
11
111 11
11
1111111111
1111
11111
111111
1
1
1
1
1 111111
1
1
11
1
1
1
111111 1 11111 1111111
1
1
1
1
1
1
1
1
1
111111111
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
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11 1111 1111 111111
111111 1 1111111 1111 11 1111111 1111111
1
11 1111111 1 111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1 11
1 1 1 11 1 1 1111111 111 111
11
11 111
11 1
1
1
11 1 11 11 1 11
1 11 11 11 1 11 111
1
1
1
1 111 1 1 1
1 1
1
1
11
1
1
1
1
9.8
1
9.6
9.6
9.8
10.0
10.2
10.4
MCMC run of mean & additive
1000
0
20
MCMC run/100
40
60
frequency
11
1 1
1
1
1
1
11
111 1
1
1
11
1
1 1 11
11 1 1 1111 1 11 111 1 11 1
1
1
1
1
1
1
1
1
1 11 1111
11111 1 1
1 11 1 1 1 111 11 1 11 11 111 11111111
111 1 1111 1 11 11
111111111111111
1
11 111 111111111 1
1
1
1
1
1
1
1
1
11
11111 1 1 1111111111
11 11111111 1111 111 11111
1111111 11111111111
1 1111111
11
1111 11 111111111
1111 1111111 11111111111111111
11
111 1111111
1 11 111
1111111
111 11111 1111 111111
1111
1111
1
111111
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
1 111
1111
1 111
11111
1111 111111111 11 1 1111
11111111111111111111
1 1 1111111
111111111 111
11111
11111
1111
11111
11
11
1 111111
111
11111 111 111111
1
1
11
111 1111111 1 111
11111
1
11
1
1
1
1
1
1
1
1
1
1
11111 1111
11 111111 11111
11
1 111111
1111111
11111
111 11
11111
1111111 11 11111
1111
1111111
11111111
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
11 1 11 1
1111111111 111
1111 1 1 1 1111
11 11 1 1 1
111
1 111
1 1111111111
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
111 11 1
1 1111111111 11111
1 11 11 1 1 11 1 11 11
11 111111 11 11
11
11
1 11
1 11111111 111 11 1
11 1 1111111111 11
1
1
1 11
11111111 111 1 1 111 111
11111
1 1 1 11 1 11 1
1 111
1 1 11
1
1
1
1
1 1
1
0
200
400
600
800
1000
additive
2.0
2.2
1.8
1.8
2.0
2.2
1
0
MCMC run/100
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
20
40
60
frequency
51
Full Conditional for mean
• normal prior
with large
variance  2
• leads to normal
posterior
• posterior mean
• posterior
variance
June 2001
* *
n

y



b
xj 





j
* *
2

 (  | y, x ; b ,  )   
  


   j 1 

n
E (  | y, x * ; b* ,  2 ) 
 ( y j  b x )    2
* *
j
j 1
n   2
2
n
2

* *
(
y

b
 j xj)
j 1
n
2
2


V (  | y, x * ; b* ,  2 ) 


n
n 
2
2
NCSU QTL II Workshop © Brian S.
Yandell
52
Full Conditional for additive Effect
• normal prior with
large variance  2
• leads to normal
posterior
• posterior mean
* *
*
n

y



b
xj 


b
j
*
*
2


 (b | y, x ;  ,  )      


   j 1 

n
E (b* | y, x * ;  ,  2 ) 
 x (y
j 1
n
 (x )
j 1
• posterior variance
V (b | y, x ;  ,  ) 
*
*
j
* 2
j
n
 (x )
NCSU QTL II Workshop © Brian S.
Yandell
* 2
j
n
 )


 x (y
j 1
  2
j
 )
 (x )
j 1

2
*
j
n
2
2
2
2
j 1
June 2001
*
j
* 2
j
2
n
 (x )
j 1
* 2
j
53
Full Conditional for variance
• inverse gamma
 2 ~ Inv (a, v)  ( 2 )  ( a 1) e  v / 
prior with large v/a
• posterior
distribution
2
 2 | y, x * ;  , b* ~ Inv a  n2 , v  n2 ˆ 2 
n
• posterior mean
June 2001
v  n2 ˆ 2
2
ˆ
E ( | y, x ;  , b ) 



n
a  2 1
2
*
*
NCSU QTL II Workshop © Brian S.
Yandell
(y
j 1
* * 2



b
xj)
j
n
54
MCMC run for variance
1
1
1
1
1
1 1
11 1 11
1
1 1
1
1
1 1
1
1
1 1
11 1 11
1
1 1
1
1
1
1
1 111 1
1 11 1 11 1
11
11 1 1 1
1
1
1
1
1 1 11
1
1 1 1
1 11
11
11 1 1 1
1 111
11111 1111 11
1 1
1
1 1
1
1
1
1
1
1
1 1 1 11 11 11
1 11 1 11 11
1 1111 1 11 111 1 11 1 111111 1 1 1 1 1 1 1111 1 111 1
1 1 1 11 1
11 1 1
11 11
1 1 1 1 11
1 1 11 11111 1
1 11 1 11 111 1
11
1 1
1
1
1
1
1
1
1
1
11111111111 1 1
111 1 1 1 1
1 1 111111 11 1 11 1111 111
111111 11 11
1 11
1 1
1
1 11 1 111 11 11
1 1111 11 1 1 11
1 1 1 1 1111 11
1
1 111 11
1
1
1
1
1
1
1
1
1
1
1 11 1 111 1
1
1
11111 1
1
1
11 1 111 111 111 11 1 111 11111 111
1 11 1 11 11 1111
11 1 1111
1 11
1 1
111111111111111111 111 11 1 1 1111111
1111111111111 1 11 1 1111111 111 1
1 111 111
1
1
1
1
1
1
1
1
1
1
1
1 1 1111111 1 11 1111 1 111 111 1 11111 11 111 11111
1111 11 1 11 11
1
1111 1111 1 1
11111 1 1 1
111 1 1 1 11111 1 11
1
111 11111111111
11 11 111
111
1 1111111111111 11111 1
111 1 1
11111
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 1 111 1
1
1 11
1 11
1
1 11
11 1 11 11 11 1 1 1 1
11
1 1111 1 11 1111111 1 1 111111 11111111 1111111 11 1111 1
11111
1
1
1
1
1
1
1
1 11 1 11 1 1 1 1
1 1
1
1
1
1 1 11111 1 11
1
11 1 1111 1 1 11 1 1 1111 111111 11111 11111111 1 111
1
1 11 11 1 111 111 1
11 1
11
1 1111 11
1111 1 111 1111 1 1 11 111 1 11
1 1 111 1 1
1 1 1
1 1
11 1
1 1 1 11 1111 111 111 1 1 11 1
1
1
1
1
1
11
1 1 1 11 1
1
11
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 11
0
1
200
400
600
800
1000
variance
1.4
1.6
1
1 1
1.2
1.6
1.4
1
1
1
1.2
1
1
1.0
1.8
1
1.0
2.0
1
1.8
2.0
1
0
10
MCMC run
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
20
30
40
frequency
55
Alternative for Variance:
use Inverse Chi-square
vd
vd
• inverse chi-square
 2 ~ Inv  2 (d , v)  2 , or 2 ~  d2
prior with large d,v
d

• posterior
distribution
June 2001
n


vd   ( y j    b* x*j ) 2 

j 1

 2 | y, x * ;  , b* ~ Inv  2  d  n,

d n




NCSU QTL II Workshop © Brian S.
Yandell
56
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
57
MCMC locus & geno updates
genos
locus
effects
traits
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
58
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
x*j  1, 0, or 1
*
*
2
*

(
y
|
x
;

,
b
,

)

(
x
j
j
j | )
*
*
2
Pj   ( x j | y j ;  , b ,  ;  ) 
*
2

(
y
|
x
;

,
b
,

) ( x |  )
 j
x  1, 0 ,1
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
59
Missing marker Data
• sample missing marker data a la QT genotypes
• full conditional for missing markers depends on
– flanking markers
– possible flanking QTL
• can explicitly decompose by individual j
– binomial (or trinomial) probability
M kj  1, 0, or 1
 ( M kj | x*j , y j ;  , b* ,  2 ;  ; M j )   ( M kj | x*j ; M j )
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
60
Prior for locus
• prior information from
other studies
– uniform prior over genome
– use framework map
•choose interval proportional
to length
•then pick uniform position
within interval
0.0
0.2
•concentrate on credible
regions
•use posterior of previous
study as new prior
• no prior information on
locus
0
June 2001
20
40
60
distance (cM)
NCSU QTL II Workshop © Brian S.
Yandell
80
61
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 ...
 ( | y, x * ;  , b * ,  2 )   ( | x * )
n
  ( )   ( x |  ) / c
*
j
j 1
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
62
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
  (new | x * )q(new , old ) 

a(old , new )  min 1,
*
  (old | x )q(old , new ) 
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
63
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
64
Part IV: Multiple QTL
• Multiple QTL Model
• Sampling from the Posterior
• Issues for 2 QTL
• Bayes factors & Model Selection
• Simulated data for 0,1,2 QTL
• Brassica data on days to flowering
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
65
MCMC run: 2 loci assuming only 1
1
1
1
1 1 11
1 11 1 1
1
1
11
1 11 1 1 1 1 1 1 1 1
1
1
1
1
1 1
11
1 11 1 1
1 11 1 1 1
1
1 111 1 1 1
1
1 1
1
1
1 111 11
1
11
1
11
1111 11 1 11 11 1 1 1
1
1
1
11 1 11111 1 11 1
1
1
1 11 111 111 11 1 111 1111111 111 11111111
1
111 111
111 111
11
111111
11
111111
11111111111111111111
11111 111111111111111111
1111
1
1
11
111111111
1
1
111
1
11
1
1
1111111
11111
1
1 11
1
111111111111 11
1111111
1
1
1
1
111
1111
111111111
1
1111111
1111111
11
1111
11
1111111
111
11
111
11111111111111111111
11 11111111
1111
111 111111
1111
1
1
1
1
1
111
1
1
1
1
1
1
1
111
1
1
1
1
1
11
111111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 11111111 11111
11111111
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
11111111111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11
11 11111
1111 1 111 11111111111 11111 11111 111111111 1111111111111
1
1
1111111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1111 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 11
11 1 1 111 11
1 1 1
1
1
1111
1
1 11111 1 1 11 11111 111 1 1 1 111 1 1 111111111 1
1
1
1
1 11 111 11 1 111
1
1 1 11 11 1
1 111 1 1 1
1 11
1 1 1 11 11
1111111 1 11 11111
1
1 1
1
1
1
1 1 11 1
1
1
11
1
1 1 11 1
1 1 1
1 1 1 1 111
1
1
1 1
1
1
1
70
1
50
distance60(cM)
1
50
60
70
1
1
1
1
11
1
40
40
1
1
11
11
1
0
200
400
600
800
1000
0
50
MCMC run/1000
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
100
150
200
250
frequency
66
Multiple QTL model
• trait = mean + add1 + add2 + error
• trait = effect_of_genos + error
• prob( trait | genos, effects )
y j    b x  b x  ej
* *
1 j1
* *
2 j2
m
y j    b x  ej
r 1
June 2001
* *
r jr
NCSU QTL II Workshop © Brian S.
Yandell
67
4
4
6
6
8
8
10
trait
10
12
12
14
14
16
16
Simulated Data with 2 QTL
-1,-1 -1,1 1,-1 1,1
recombinants
June 2001
0
1
NCSU QTL II Workshop © Brian S.
Yandell
2 3 4
frequency
5
6
68
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
69
30
LOD for 2 QTL
0
5
10
15
LOD
20
25
QTL
IM
CIM
0
10
20
30
40
50
60
70
80
90
distance (cM)
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
70
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
71
Multiple QTL model
• trait = mean + add1 + add2 + error
• trait = effect_of_genos + error
• prob( trait | genos, effects )
y j    b x  b x  ej
* *
1 j1
* *
2 j2
m
y j    b x  ej
r 1
June 2001
* *
r jr
NCSU QTL II Workshop © Brian S.
Yandell
72
Vector Notation for Trait Model
• inner product for sum
• condense notation
m
y  X *b *  e or y j   br* x*jr  e j
r 1
y *  ( y j  yn* ) T , e *  (e j  en* ) T ,
 
X  x
*
June 2001
* n,m
jr j,r 1
and b *  (b1* bm* ) T
NCSU QTL II Workshop © Brian S.
Yandell
73
Vector Notation for 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
m
 ( | X * )    (r | X * )
r 1
n
 (r | X )   (r )  ( x*jr | r )
*
j 1
QT loci   (1  m ) T
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
74
Posterior: Multiple QTLs
• posterior = likelihood * prior / constant
• posterior( paramaters | data )
prob( genos, effects, loci | traits, map )
 (X * ;  , b * ,  2 ;  | y)
is proportional to
m 
n

2
*
*
 (  ) ( )    (br ) (r )  ( x jr | r ) 
r 1 
j 1

n
   ( y j | x *j ;  , b * ,  2 )
j 1
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
75
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
  (X * ;  , b * ,  2 ; ) ~  (X * ;  , b * ,  2 ;  | y)
1   2     N
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
76
70
distance
(cM)
60
60
1
1
111111 11 11111111 1 1111111 1 11 1 1111 11 1111111111 111 11 1 1111
1 111
11
11
1
1111
1111111111111
11111
1111
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1 11111111
11
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111
1111
11
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111111111
1111
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11
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1
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11
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1
11
1
111
11
11
11
11
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111
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11
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1
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11
111
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111
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11
11
11
11
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1
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111
11
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11
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1
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11
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1
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11
11
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1
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11
1
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1
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1
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11
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1
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1
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1
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11
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1
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1
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1
1
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1
1
1
1
1
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1
1
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1
1
1
1
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1
1
1
1
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1
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1 1
0
200
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40
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80
2
22 22 2 2 2
2
22
2
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2 22 22
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22
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2
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2
2
2
2 22
2 22 2
22 22 22
2 2
2
2 2
70
80
MCMC run with 2 loci
0
50
MCMC run/100
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
100
150
200
250
300
frequency
77
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
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
78
2.6
2.4
2.2
1.8
2.0
2.2
2.0
1.8
effect
2.4
2.6
Effects for 2 Simulated QTL
40
50
60
70
80
effect 1
effect 2
0.0
0.05
0.10
0.15
distance (cM)
40
50
60
70
80
distance (cM)
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
79
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
80
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
0
2
4
6
8 10
8-week vernalization
0
2
4
6
8
4-week
2.5 3.0 3.5 4.0
4-week vernalization
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
81
8
Brassica Data LOD Maps
8-week
0
2
LOD
4
6
QTL
IM
CIM
0
10
20
30
40
50
60
70
80
90
60
70
80
90
15
distance (cM)
4-week
0
5
LOD
10
QTL
IM
CIM
0
10
20
30
40
50
distance (cM)
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
82
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
40
80
June 2001
add
.30
.16
•
•
•
•
8-week vernalization
shorter time to flower
larger LOD at 80cM
modest LOD at 40cM
loci poorly determined
cM
40
80
NCSU QTL II Workshop © Brian S.
Yandell
add
.06
.13
83
Brassica Credible Regions
8-week
-0.3
-0.6
-0.2
-0.4
-0.2
additive
additive
-0.1
0.0
0.0
0.1
0.2
0.2
4-week
20
40
60
80
20
distance (cM)
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
40
60
80
distance (cM)
84
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
85
Brassica effect Correlations
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
additive 1
June 2001
-0.1
0.0
0.1
0.2
additive 1
NCSU QTL II Workshop © Brian S.
Yandell
86
Simulation Study
• 2 linked QTL
• QTL Cart vs. Bayesian QTL estimates
– locus: 15, 65cM
– effect: 1, 1
• n = 100, h^2 = 30
• also considered
– n = 200, h^2 = 25, 30, 40
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
87
2 QTL: Loci Estimates
locus 2: n = 100, h^2 = 30
10 20 30 40 50 60
70
80
90
100
0
60
Bayesian QTL Locus
50
40
30
20
10
Bayesian QTL Locus
locus 1: n = 100, h^2 = 30
20
QTL Cart Locus
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
40
60
80
100 120
QTL Cart Locus
88
2 QTL: Effect Estimates
locus 2: n = 100, h^2 = 30
1.0
0.0
0.5
Bayesian QTL Effect
1.4
1.0
0.6
Bayesian QTL Effect
1.8
1.5
locus 1: n = 100, h^2 = 30
0.8 1.0 1.2 1.4 1.6 1.8
QTL Cart Effect
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
-0.5
0.5 1.0 1.5 2.0
QTL Cart Effect
89
2 QTL: Loci & Effects
30
85
75
65
55
50
50
60
70
80
locus 1: n = 200, h^2 = 40
locus 2: n = 200, h^2 = 40
0.8
1.2
QTL Cart Effect
1.6
0.6
1.0
1.4
QTL Cart Locus
Bayesian QTL Effect
QTL Cart Locus
0.8 1.0 1.2 1.4
Bayesian QTL Effect
10
June 2001
locus 2: n = 200, h^2 = 40
Bayesian QTL Locus
40
30
20
10
Bayesian QTL Locus
locus 1: n = 200, h^2 = 40
0.8
1.2
1.6
QTL Cart Effect
NCSU QTL II Workshop © Brian S.
Yandell
90
Bayes Factors
Which 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
 ( model1 | y) /  ( model2 | y)  (y | model1 )
B12 

 ( model1 ) /  ( model2 )
 (y | model2 )
June 2001
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
NCSU QTL II Workshop © Brian S.
Yandell
91
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
 2 log( B12 )  2 log( LR)  ( p2  p1 ) log( n)
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
92
Model Determination
using 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 2001
NCSU QTL II Workshop © Brian S.
Yandell
93
Brassica Bayes Factors
•compare models for 1, 2, 3 QTL
•Bayes factor and -2log(LR)
•large value favors first model
•8-week vernalization only here
June 2001
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
NCSU QTL II Workshop © Brian S.
Yandell
94
Computing Bayes Factors
• arithmetic mean
– using samples from prior
– mean across Monte Carlo or MCMC runs
– can be inefficient if prior differs from posterior
 (y | modelk )    (y |  k ; modelk ) ( k | modelk )d k
• harmonic mean
– using samples from posterior
– more efficient but less stable
– careful choice of weight h() close to posterior


h( k )
ˆ
 (y | modelk )  G 


(
y
|

;
model
)

(

|
model
)
k
k
k
k 
 g 1
G
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
1
95
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
96
Part V: How many QTLs?
• 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 2001
NCSU QTL II Workshop © Brian S.
Yandell
97
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
 me
 ( m | ) 
m!
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
98
Posterior: Number of QTL
• posterior = likelihood * prior / constant
• posterior( paramaters | data )
prob( genos, effects, loci, m | traits, map )
 (X* ;  , b* ,  2 ; , m | y)
is proportional to
n
*
*
2

(
y
|
x
;

,
b
,

; m) 
 j j
j 1
n


2
*
*
 (m) (  ) ( )    (br ) (r )  ( x jr | r ) 
r 1 
j 1

m
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
99
Reversible Jump Choices
action 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 2001
NCSU QTL II Workshop © Brian S.
Yandell
100
Markov chain for number m
• add a new locus
• drop a locus
• update current model
0
June 2001
1
...
m-1
m
m
NCSU QTL II Workshop © Brian S.
Yandell
m+1
101
number of QTL
distance (cM)
Jumping QTL number & loci
112222111112222233333333222222222112222223
222211221
211111223
3
3
2
33332 1
22211
1
1
3
2
1
1
111
333 11122222211111
60
2
1111111
1
1
1
2
1
2
1
1
2222221
40 222222222222222222222221
111111111
111111
20 111111111111111111111111
11
111
11111 1
80
0
20
40
60
MCMC run
80
100
0
20
40
80
100
3
2
1
0
June 2001
60
NCSU QTL II Workshop © Brian S.
Yandell
102
RJ-MCMC Updates
1-b(m+1)-d(m)
add
locus
b(m+1)
genos
loci
d(m)
effects
traits
drop locus
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
103
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
qb ( )  1 / D
– 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 ...
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
104
Propose to Drop a locus
• choose an existing locus
qd (r; m)  1 / m
– 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
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
105
Acceptance of Reversible Jump
• accept birth of new locus with probability
min(1,A)
• accept death of old locus with probability
min(1,1/A)
 ( m1 , m  1 | y) d (m  1) qb (m 1 ) 1
A

 ( m , m | y)
b(m) qd (r ; m  1) J
 m  X * ;  , b * ,  2 ; , m 
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
106
Acceptance of Reversible Jump
• move probabilities
m
m+1
• birth & death proposals
• Jacobian between models
–fudge factor
–see stepwise regression example
June 2001
d (m  1)
b( m)
NCSU QTL II Workshop © Brian S.
Yandell
qb (m 1 )
qd (r ; m  1)
J

sr |others n
107
number of QTL
0 1 2 3 4 5 6
0 1 2 3 4 5 6
RJ-MCMC: Number of QTL
0
200 400 600 800
MCMC run/100
June 2001
0
200 400 600 800
MCMC run/10
0
200 400 600 800
MCMC run/1000
number of QTL
0 1 2 3 4 5 6
200 400 600 800
MCMC run
0 1 2 3 4 5 6
0
NCSU QTL II Workshop © Brian S.
Yandell
108
Posterior # QTL for 8-week Data
0.0
0.2
0.4
98% credible region for m: (1,3)
based on 1 million steps
with prior mean of 3
0
June 2001
1
2
3
4
NCSU QTL II Workshop © Brian S.
Yandell
5
6
109
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
110
VI: Reversible Jump Details
• 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 2001
NCSU QTL II Workshop © Brian S.
Yandell
111
RJ-MCMC Updates
1-b(m+1)-d(m)
add
locus
b(m+1)
genos
loci
d(m)
effects
traits
drop locus
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
112
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
113
Model Selection in Regression
• known regressors (e.g. markers)
– models with 1 or 2 regressors
• jump between models
– centering regressors simplifies calculations
m  1 : y j    b ( x j 1  x1 )  e j
m  2 : y j    b1 ( x j 1  x1 )  b2 ( x j 2  x 2 )  e j
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
114
Slope Estimate for 1 Regressor
recall least squares estimate of slope
note relation of slope to correlation
n
bˆ 
r1 y s y
s1
, r1 y 
 (x
j 1
j1
 x1 )( y j  y ) / n
s1 s y
n
n
j 1
j 1
s12   ( x j1  x1 ) 2 / n, s y2   ( y j  y ) 2 / n
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
115
2 Correlated Regressors
slopes adjusted for other regressors
(r1 y  r12r2 y ) s y ˆ r2 y s y
r12 s2
ˆ
b1 
b
c21, c21 
s1
s2
s1
(r2 y  r12r1 y ) s y
ˆ
b2 
s2
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
116
Gibbs Sampler for Model 1
• mean
 ny    2  2
 ~ 
,
 n   22 n   22



• slope
 n
  ( x j1  x1 )( y j   )
2
j 1

b ~ 
, 2 2
2
2
ns1   2
ns1   2


• variance
n

2
n
1
 ~ Inv a  2 , v  2   y j    b( x j1  x1 )  
j 1


2
June 2001










2
NCSU QTL II Workshop © Brian S.
Yandell
117
Gibbs Sampler for Model 2
 ny    2  2
 ~ 
,
 n   22 n   22



2
• mean
• slopes




 n
  ( x j 2  x2 )( y j    b1 ( x j1  x1 ))
2
j 1

b2 ~  
, 2 2
2
2
ns2|1   2
ns2|1   2








n
s   ( x j 2  x2  c21 ( x j1  x1 )) 2 / n
2
2|1
• variance
June 2001
j 1
2
n
2




2
n
1
 ~ Inv  a  2 , v  2   y j     bk ( x jk  xk )  


j 1 
k 1



NCSU QTL II Workshop © Brian S.
Yandell
118
Updates from 2->1
• drop 2nd regressor
• adjust other regressor
b  b1  b2c21
b2  0
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
119
Updates from 1->2
• add 2nd slope, adjusting for collinearity
• adjust other slope & variance
z ~  (0,1),
J

s21 n
n
b2  bˆ2  z  J , bˆ2 
 (x
j 1
j2

 x2 ) y j  ˆ  bˆ1 ( x j1  x1 )

ns221
b1  b  b2 c21  b  z  c21J  bˆ2 c21
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
120
Model Selection in Regression
• known regressors (e.g. markers)
– models with 1 or 2 regressors
• jump between models
– augment with new innovation z
m
parameters
innovations
1 2
(  , b,  2 ; z )
z ~  (0,1)
2  1 (  , b1 , b2 ,  )
2
June 2001
transformations
b2  bˆ2  z  J 


 b1  b  b2 c21 
b  b1  b2 c21 


z0


NCSU QTL II Workshop © Brian S.
Yandell
121
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)
 b1  1  c21 J
   
J
 b2  0
  (b , b
1
June 2001
2
b
 c21   
  z   g (b; z | y, x 1 , x 2 )

1  ˆ 
 b2 
| y, x 1 , x 2 )db1db2    (b; z | y, x 1 , x 2 ) Jdbdz
NCSU QTL II Workshop © Brian S.
Yandell
122
Jacobian & the Calculus
• Jacobian sorts out change of variables
– careful: easy to mess up here!
g (b; z ) 1  c21 J 
g (b; z )  (b1 , b2 ),


0
J
bz


 1  c21 J  
  1 J  0  (c21 J )  J
det  


0
J




 g (  , b,  2 ; z ) 
 db1db2  Jdbdz
db1db2  det 
bz


June 2001
NCSU QTL II Workshop © Brian S.
Yandell
123
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
June 2001
0.2
0.4
b1
0.6
0.8
0.0
NCSU QTL II Workshop © Brian S.
Yandell
0.2
0.4
b1
0.6
0.8
124
QT additive Reversible Jump
first 1000 with m<3
0.0
0.0
0.05
b2
0.1 0.2
b2
0.10
0.3
0.15
0.4
a short sequence
0.05
June 2001
0.10
b1
0.15
-0.3 -0.2 -0.1 0.0 0.1 0.2
b1
NCSU QTL II Workshop © Brian S.
Yandell
125
0.0
-0.1
regression line
corresponds to
slope of updates
b2
0.1
90% & 95% sets
based on normal
0.2
0.3
Credible Set for additive
-0.1
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
0.0
b1
0.1
0.2
126
Multivariate Updating of effects
• more computations when m > 2
• avoid matrix inverse
– Cholesky decomposition of matrix
• simultaneous updates
– effects at all loci
• accept new locus based on
– sampled new genos at locus
– sampled new effects at all loci
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
127
Multivariate Birth Acceptance
•sample effects around least squares estimator
•acceptance of locus involves effects update
•can combine with long distance locus updates
 (y | X ;  m 1 )  ( m 1 ) / q ( m 1 )
A
 ( y | X ;  m )  ( m ) / q ( m )
*
m 1
*
m
  (  , b ) ~ Normal ( , D)
ˆ , X T X   2 D-1 )
 | X, y,  2 ~ Normal (
*
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
*
128
VII: RJMCMC & Bayes Factors
• RJ-MCMC & Bayes factors
– Bayes factors from RJ-MCMC chain
– components of Bayes factors
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
129
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 2001
NCSU QTL II Workshop © Brian S.
Yandell
130
0
200
400
600
800
1000
80
distance
(cM)
40
60
222222222222
2
22222222222222222222222
2222
22222222
22222222222222222222
22
2222222
22222
2 2222
2222
22
222222
2
2222222
22222
222222
222222
22
2222222
2
2222222
2222
2222
22222222222
22
22
2
2
2
22
2222
22
2
22
2222
222222
222222
222222
22
2222
22
22
222
22
22
22
22
222
22222222222222222
21
22222
2222122
2222
2
2222
222222
2
222222
222222
2
2
2
2
2
2
2
2
2
222
2
2
2
2
2
2
2
2
2
222
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
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
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
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
2
2
2
2
2
2
2
2
2
2
1
2
2
1
1
1
2
2
2
2
2
2
2
2
2
2222
2
222 22222
2
22
1
2
2222
22
1222222222222 2222
222
1222221
2222212222222
22222212
2222
2
22212
222
222222
1
2222
222
1222222222
1
222222222222
1
2
222
1 2222
2
222221
22122
22222221
11
1
22212
212
1211222221222222 2221222222
2211 2
1
1
2
2
1
1
2
2
2
1
2
2
2
2
2
1
2
2
2
1
2
2
1
2
2
2
2
1
2
2 2 12 1 122 212 122
112
21 221 2 2 1 21
1 1211
21 221 2
1 2 2211 21 122121
12 1 2 2 11 1
1 22 2 2 1 1 1
1
1
1 1
21 1 1
1 11
12
11 111
112111
1
11111 12
1
1111
1
11
1 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
111111 211
11
11 1 11
1 1 11 111
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
111 1 1111 1111 11 11 1 1
11 111
11
1111 1 1 1 11 1111111
1 1 11 11 11 1 11 1 1 1
11 1 1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
11111 111 111 1
111111 1 1 11 111111 111 1 1111111 111
111 1 1111111111
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1 11 11
1 1111 1
111 1
1 11
1
1
1
1
11 11111
1
11 1 11 11111 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
1 111 111111 11 1111111111111 11 1 11 111111
111 11 1 111
1111111
1111111
11111 1111 1
111
1
11
1
111 1 111111
1111111111
1
1
1
1
11 11 1
11111 1
1
11111111
111 111111 1
11111 1111
1
1
1
1
1
1
1
1
1
1
1
1
1
11
1
1
11 1111
1
1
1
1
1
1
1
111 111 1 1 1111 1 1 1 11 1 1 1111 11
1
1
1
1
11 11 1111111
1
11
1111
1 111
111
111111
11111111
1
111 1111
111111
11
1111111
11 1
11111111111 11111111111
1
11
111
1
1
1
1
1
11
1
1
1111
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
1111
1111 111 1
11
1 1 11 1111 1
111111111 1111111111 111
1 111111 11111 111
111
11 1 11
1 1111111
1
1
1
1
1 11 1
1 11
1
1 1 1 1 11 1
1 1 1 1
11
1
1
1
1
1 11
1
1
1
1
1
1
1
1 1
1
1
11
1
1
1
11
1
1
1
20
20
40
60
80
Brassica 8-week Data locus
MCMC with m=2
0
50
MCMC run
June 2001
100
150
200
250
300
frequency
NCSU QTL II Workshop © Brian S.
Yandell
131
number of QTL
distance (cM)
Jumping QTL number & loci
112222111112222233333333222222222112222223
222211221
211111223
3
3
2
33332 1
22211
1
1
3
2
1
1
111
333 11122222211111
60
2
1111111
1
1
1
2
1
2
1
1
2222221
40 222222222222222222222221
111111111
111111
20 111111111111111111111111
11
111
11111 1
80
0
20
40
60
MCMC run
80
100
0
20
40
80
100
3
2
1
0
June 2001
60
NCSU QTL II Workshop © Brian S.
Yandell
132
RJ-MCMC loci chain
60
40
20
0
4
3
2
3
242
2
3
32
1
21322
3
1122
6
544
3
2
22
3
4
21
2114
321
5 222
3
3
123
33
22 5
323
2
1
4
54
3
4
5
32
32
13
2
2
3
2211
2
5
4
3
21
1
2
234
322
1
211
12
3
2 3
34
56
12
3
6
5
3
4
233
5
2
3321
23
1
21
23
31
43
4 22
22
2 2
22
2
5
3
2
5
1
2321
44
23
12
2
11
33
23
3224
32
22
225
2
3
3
2
4
232
21212 3
2
1
3
2
111
4
5
3
1
1
2
2
2
1
4
2
33
4
24
23
2 1 21
33
2
4
4
2
23
4
42
3
2
12
32 2
22
2
3
31
2
3
2
223
2
1
3 43 2
1
1 12
2
1
2
2
3
1
2
2
3
2
1
2
3
3
2
3
2
3
2
2
3
2
2
3
2
2
2
3
3
2
1
3
4
2
1
1
122 211
22
1
3
22 1 4 2
22 23
1211
1
21
2 3
1
1
112
2
3
2
111
23
433
2 2 2222 3
1
31
1
2
1
2
1221
1 2 2
1
22
2
12
11
11 233
2 1
2
2
1
112
53
12 4
1
2121
1
31
12
2
1
1
2
1
1
1
1
2
1
1
2
1
3
22 4
1
21
1 11 1
122
11212
2
21
223
2
1
1
3
1
2
1
1
1
1
1
2
2
1
1 4 1 222321
1
212
3
1 1112
222
121
3
2 2 111
1
3
3
2 21 2 34
1
2
12222
1111 111
4
3
1
12
31
221 2
2
4
2
1
4
23
1 221331 11
33
2
11
3
11
22
1 11
2
3
2
12
11 211211
3
22 111
1
1
3
2
2
1
1
2
1
2
2
1
1
1
3
2
1 11
2
1
2 1 12
1
2
2
3
1
1
1
1
2
3
1
2
1
3
2
3
1
2
2 1111
11
12
1
11
2
1 1111 2
11
1 1
11 1
2
33
1 2
11 1
1
2
2 1
11
1
1 1
1
2
2
1
2
1
111
1111
1 1
22
111 2 11
1
1
22
1
1 11121 1
11 1
122
11 1
11
1
1 1
1
111
1
1
1
1
1
1
2
1
11
2221 1
1 11
1
1 1
11
1 1
1
1
11121 11 11
11
1 11122 1 1
1
1
1111111
1
1
1
2
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
2
1
1
11
1 1
2211
11
1
11
1
1111
0
3323
2
3
22
13
3 22
32
4
122323
2
23
3
3
2
3
332
2
1
1
3
21
2
2
3
214
4
12
43
233
51
1
4
2
3323
22112
23
2123
3211
22131423
4
12
15
22
1
4
4
423
2
25
3
24
4
31
4
1
21
2215
2223
223
5
2
22
23
22
3
2323
3
32223
22
5
3
2335
322
3
222
22
1
2
number of QTL
80
0
2 323
24 23323
1 122311 24211432143 232
111213211 3213253
422112131
21
331
3222
23
1211222
14
1
222 11113122
12411
1323
2213232
222
3
321222
12
321
32
3
11111
22115
14321212
3
2
4
1
2
2
2
3
1
2
2
1
512
32
2
2
125
3
2
1
4
1
3
2
3
3
3
3
3
2
2
3
2
2
2
1
3
2
2
2
1
3
2
2
1
2
2
2
3
2
1
4
2
1
3
2
1
2
1
4
3
1
2
1
1
1
3
2
2
223
3
1
2
3
2
1
2
1
2
1
1
2
1
2
3
1
2
2
2
3
1
1
3
2
2
3
3
1
2
3232
2 11 212113232 22111
223 231211 21 3 22 1 211
1
11
1 2222 2 222 2
1
3
1
221222113
2
3
1
4
1
2
2
2
1
3
22
2 2114
121 31 222
21
3 2112211111
2114 2121 2221
2
2
12 2
2111 221213
22121 2
1
122 1211121 1132111222112213 32
11221 2
2
1
1
2
13
3
2
3
1
2
12
22421132
221
2
1
1
2
2
2
1
2
2
1
1
3
2
1
2
1
1
2
2
1
2
3
1
1
1
1
2
1
1
2
2
1
1
2
1
2
3
2
3
1
1
1
2
3
3
2
3
2
2
2
1
2
1
1
1
2
1
2
1111111111 131211
12112
2131
11
1 3
331
2221132
22
1
12
32
2
1
22
3
22
1
2
22
111
1111211
311121
112
2
3
1
2
11122
1
11
3
222
1
1
21
1211
1112221
22
224
11122
22
12
122
211113
12
321212
12
22
21
11132
322
1
11
2121
2
3
22111321
2111
22
111111
133
2
1211122
12
1
11
1242
11112
1
1
111
1
3
1
1
1
1
2
2
1
2
2
2
1
2
2
2
1
2
2
1
2
2
1
1
1
2
1
1
2
1
2
2
2
1
1
3
1
1
2
1
1
1
2
1
1
1
1
2
1
1
1
2
1
2
1
1
2
2
2
2
1
1
1
1
11112 1112
21 121322212
122 1
111 211122111
2
22211121
111
121 1111112
31 2
11
11231 1
212121
212212111222222
2 1 1111112112
111121212121 11
2
1
2
11121221 2
1
1
2
1
1
1
2
2
2
1
1
1
1
1
1
1
1
1 1111 12 111 13112
12
11 11111131
13
11211121
1121111211
1
1 11 12
11 11 1111
1
221
11 1111111
11
11113111 111121
11
111 11121111111112
11
1111 1 11
1
1
2
1
2
1
1
1
1
1
1
2
1
1
1
1
2
1
1
1
1
1
1 1 1111111 1 1113111 11211 1 1111 1 11 121 211
112 121 21
221 11
1211111 11 121
1
1 1111111 11
2
1
21112 1 1 1 1111
11 11 111 1122 11111112 11 111 1
1
1
1
1
1
1
1
1
1
1
1
2
11111 1111 111 1 1 11 1 1 1 111111 11 1 11111
1
1 11111111 1 111 11 1
1111111 13
11 1111
2
1111 1 12
11
121 111
11 1 1 1
1 111
1
111 1 1 1
1 111 12
11111 1 1 11 1 11111
0
June 2001
20
400
800
MCMC run/100
23
12112 2
33 22332 1
222 232 3
1
3 2
32
3 122
1 11
2321
1
2
411
11222
21
2
32
2
23 2
1
232
33
222
2
1
12342
2321
4
232
3
12
1
1
121
13
34
2
1
1
3
22
121
3241
22
3 32121
2
2321
1
3
1
1
1
1
2
2
1
11
1
1
1
3
2
1
2
3
12
1
2
322
3
221
1
1
1
3
1
13222
2
12
2
1
2
1
1
1
2
1
2
2
1
1
1
3
1
1
3
3
2
1
2
3
1
1
4
2
2
3
2
1
4
2
2
2
2
2
3
1
1
1
1
2
2
2
2
1322 22 2 21
11 113 22 2
1
2
2
1
2
3
2
1
1
2
3
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3
2
2
2
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21
3
31 22 141
2 1 22 31 143
11331
14 31 222
3 11
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2
22
21 2
22
1
1
1
1
4
3
1
2
2
1
4
1
1
2
1
2
3
2
1
1
1
1
3
2
1
4
1 133 1112
1 2 12 11
2 12 14
41
1
3
2 12
2
311
2222123121
2221 112
32
212
12
111 2
1
2
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112
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11 1
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3
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1
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1
111311121112211
3
12
31
2
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11
11
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3
211
21
1
3
221
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11
1
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2
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1
1
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1
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2
1
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1
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2
1
1
1
1
2
2
1
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3
3
1
2
11
2
1
1
1
2
2
2
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1
1
1
1
1
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2
2
2
2
2
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1
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2
1
112
1113
22
2
11
2
211
1
2
2
1
3
1
1
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1
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1
2
1
1
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1
1
2
3
1
2
2
2
2
1
2
1
2
1
1
1
2
2
2
1
2
2
2
1
1
2
2
1
2
1
3
1
2
2
1
2
2
1
1
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2
2
3
1
2
1
1
1
1
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1
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2
3
1
1
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1
3
2
1
2
1
2
3
2
2
1
2
1
1
1
1
1
1
2
1
2
1
2
1
1
3
2
2
1
1
1
1
1
2
1
3
2
1
4
2
1
1
2
1
1111
11111 111
2
23
3
2 12
21
312 11
111
2111122211
11
2
11112211
111
2
1 11
22
22111
1
2
2
1
2
112
3
11
2
1112
2
31111113
3
211112
2112
1211 11
11
11111
111
2
2
212
1
11 1111111 111
1
321111112
11
11
11
3
1
1
2
21
1
1
1
2
1
1
3
1
1
1
2
1
1
2
1
1
2
1
1
2 11
11 12
1
1
1
2
1
1
1
1
2
1
1
1
2
1
1
2
1
1
2
1
1
1
2
1
2
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1 2 11121
1111 1111111111
11 112
1111121 1 1 1
12
2 1 111
21 1111
1
1111112111111
1 211
11
1 11
1 11111 1
21
1 11
12111 112 1
2
1 1 111 111 1 21
1
1
1
2 1 1 1 111 1 1 11
1 11 1 1
21
11 11
22 11111 11
1111111 2
111
1111
11 11 1
111111 11
11
1 1112
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
1
111111
1
1
1
1
1
1
11 11
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1111 1 11 11 1 1 11 1
11
0
400
800
MCMC run/10
22221221321232
2
1241
313
22
23
1
22
22
1
12
3
1232223
44
3
21
21
3
22323
222
332
12
1
312323
2
22
22
3
1
21
221
112
24
23
1
2
12
3
131
31
21
22
33
1
11
11
3
22
2
32113
32
2
112123
2
33
32
321
2121
32
323
3
2222
2
231231
211133
332
1
231
323 24
3
number of QTL
20
40
0
241213222
3 1222333
21
1534123
232
2
3231
63
323
21
232423
33
2
223
232
13
233
2
322
2
43
3224
32
32
2142
2
133
42243
1223423
32212
1
332
21
2
22
322
31
1
21
3
3
1134321422112
24232122
1432
22
21
40
60
400
800
MCMC run
80 322423143223123231232231212132123321233223123422121142221323312311211132232123143323331221334232122312421112223221423422331223322142233314312323222123121521214231323243232132241232233221123242332412311233422241221222312322321321232131322112111233434122112232332221211424213522221231211122321341221211214223432
60
80 232322132122122232124224322113322231221233221212212222121113223324142312223521332123221132321331222233221213223211212313423322121222433122332211232334231245353122124231222232323123133222532112233212124225531212131322323322232112222224313212333411212212143112432131323232223222313321221331322122312334212353232112232231212221
80 321312121331121214211312123223121222123212231232222231133141231213141211211212424122211231222132323223213123112112141122212111223211112322332221131222122141122222121222222232111122121124123221134221212122221322211213122231123122231121112122123322111132121211211212211334311112222122211123222124122111113122111113111223231112
60
40
20
0
1 231 1 2 242 2
1 322221 212233221 221 332
1222122
11
1233
1
12 32
1 321311112133
2122
312312 12
211112
12 2 12
22
2
112211
1
2112
12131
11
2
111
11
122112
212
1
21
2
2
2
3
1
1
3122
31
131311
2
3
2
2
1
1
1
1
1
21
2
2
13
221 1 122122 2 222
3
1
1
2
1
1
1
1
2
1
1
3
2
1
2
1
1
2
2
2
2
3
2
1
1
2
3
1
122 33
1 21
1
1
1
2
2
1
2
1
1
2
3
1
2
1
2
1
2
1
2
2
1
3
21
1
2 1
1 12
2 11 32
1
2 11 2
1 1 2 13 131211 1 2
2 32213 2 2 2
2112
22
1 13
21 3 2 11121
211 24112 1111
2
1
2
2
11 1 2 21
2211 11 111
1111211 123
211 1 121 2211131111 11
2 211
11
3422
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111
22121
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221111
113 13
112
1
21 1
2
2221
11
1
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1
21321
12
213
211
112112
1
1
3
312
1121
111
1211
2
1
2
211
211
13
1
2
1 311
2
1
113121
1
1
1
1
1
1
212121121
1
3
1
2221
1
1
1
2
1
1
1
2
1
3
1
1
1
2
2
1
1
2
1
1
1
2
2
1
2
11
1
3
1
2
2
2
2
1
1
2
2
2
1
1
1
2
1
1
1
1
1
2
1
3
2
1
1
2
1
2
1
2
1
1
1
1
2
1
2
1
1
1
3
1
2
221
1
2
3
1
1
1
1
1
122
1
2
2
3
1
1
1
1
1
1
2
2
2
1
1
1
1
2
1
1
2
2
1
2
1
1
2
2
2
1
1
1
2
1
1
2
2222211 22
2
1
1
2
2
1
2
1
2
2
2
1
1
1
2
1
1
2
1
1
1
1
1
1
1
2
2
3
1
1
1
1
1
1
1221131 1 2 1111 11 111211111211 21 1 2
111212121
21 1 221 111 2 2
11
211111 11 11111111
1111
1111 11121121112
122111212
12 111 1111111111121111111211211
12111111
112 11 1 1 1 111
111 11 21111211
1112111 1
1
111 112111211
2111 11111111
11
111111112 1 1
111 111 11111 111
1
1
1
2
1
1
1
1 1 1 1 1 1 11 2112 1
1
1 11 11 1
1111
1211 1 11111 1 1 2 1 1
11 111 11 11 11 1111 21111 11111 1111 11 1 11 1
11
121111 1 1 1 1 11 111 1 11
1 1111111 1 11
1 1111
1 1 1 1 111
1
1 1111 11 1 1111111
111 11 1 1 1 112
11 11 1 1 1
11
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0.0
0.01
0.02
0.03
0.04
Raw Histogram of loci
0
June 2001
20
40
distance (cM)
60
NCSU QTL II Workshop © Brian S.
Yandell
80
134
0.0
0.0
m=2
0.02
0.04
m=1
0.02
0.05
Conditional Histograms
40
60
distance (cM)
80
20
40
60
distance (cM)
80
20
40
60
distance (cM)
80
0
20
40
60
distance (cM)
80
0.0
0.0
m=3
0.02
0
June 2001
0
m>3
0.02
0.04
0.04
20
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Yandell
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June 2001
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Yandell
136
Bayes Factors
•ratio of posterior odds to prior odds
– RJ-MCMC gives posterior on number of QTL
– prior is Poisson
B12 
June 2001
 ( model1 | y) /  ( model2 | y)  (y | model1 )

 ( model1 ) /  ( model2 )
 (y | model2 )
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
NCSU QTL II Workshop © Brian S.
Yandell
137
0.4
0.2
prior mean = 6
012345678
NCSU QTL II Workshop © Brian S.
Yandell
0.0
0.15
0.2
prior mean = 4
012345678
0.0
0.2
0.0
June 2001
prior mean = 3
012345678
0.30
prior mean = 2
012345678
0.0
0.2
0.0
prior mean = 1
012345678
0.4
0.0
0.6
0.4
prior
post.
0.4
0.8
#QTL for Brassica 8-week
prior mean = 10
012345678
138
Bayes Factors & LODs
• others have tried arithmetic & harmonic mean
• why not geometric mean?
• terms that are averaged are log likelihoods...
 G

  log  (y | k ; modelk ) 

ˆ (y | modelk )  exp  g 1

G




g  1, , G MCMC runs
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
139
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

LOD ( )  (log 10 e) ln 
j 1


ˆ* , ˆ 2 ) ( x |  ) 
ˆ

(
y
|
x
;

,
b
 j

x  1, 0 ,1
 ( y j | ˆ , b  0, ˆˆ 2 )
*



  ( y j | x*j ;  , b* ,  2 ) 

BLOD  (log 10 e) ln 
*
2
  ( y |  , b  0,  ) 
j 1
j


n
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
140
Compare LODs
•
•
•
•
200 simulations (only 100 for some)
n = 100, 200; h^2 = 25, 30, 40%
2 QTL at 15, 65cM
Bayesian vs. CIM-based LODs
– Bayesian for simultaneous fit
– classical for sum of CIM LODs at peaks
• plot symbol is number of CIM peaks
June 2001
NCSU QTL II Workshop © Brian S.
Yandell
141
Comparing LODs
6
15
10
5
RJ-MCMC mean LOD
10 15
5
0
2
2
32234 2333 3
4
3
212
2
3333243333
23
323232332
2
1
2
3
4
2
2
2
2
3
2
222
332333 4
2223
333
3
2
2
3
3
3
3
2
2
1
1
22
2
3
2
2
3
2
3 32
23
2
2
212
3232
22
22
3
2
2
12
11122
23 3
2
2223 3
5
10
10
20
n = 200, h^2 = 40
4
8
12
QT L Cart sum of LODs
June 2001
2
2 2
22222222 3
2
2
2
22222222
2 3
2
2
2
2
2
2
2
32
3
222
2
22222
2
2
2
1222222
2
2
2
5
10
20
QT L Cart sum of LODs
NCSU QTL II Workshop © Brian S.
Yandell
15
15
5
2
2
2
2
2
222
1 222
2
2
2222222
2
1222
2
2
2
1
2
2
2
1
2
2222 3
1 121
22222
1
2
2
2122222222 3 2 2
122
122
1 121121
2
1
2
222322222323
2
1
1
1
1
2222222
1
1 2
25
n = 200, h^2 = 30
RJ-MCMC mean LOD
n = 200, h^2 = 25
25
QT L Cart sum of LODs
RJ-MCMC mean LOD
QT L Cart sum of LODs
20
QT L Cart sum of LODs
5 10
RJ-MCMC mean LOD
2
22
2 3
1 2222
2222 3333
222
2
2
2
22
112121
22
2221
22232 3
23
22
2
2
1
1
2
2
2
1
2
2
2
1
222
111121
222 22
1
11111222
2
1
1 22
1
2
8 12
4
0
3
RJ-MCMC mean LOD
12
8
4
0
RJ-MCMC mean LOD
1
11 22 2222
1222 2
1
212
11
2
1
211222222222
11111112
1
2
2
2
2222232223
11212
1111
2222 3
2
1
1
22122
1111
1
222222232
1
1
2
1
1
0
0 11 222
0
n = 100, h^2 = 40
n = 100, h^2 = 30
n = 100, h^2 = 25
2
3
2 2222
2
2
22
22222222 2 3
3
2223223 33
2
2
2
2222
2323
22
23
22
222
22223 3
22
2
2
22
2222 3 2
2 2
4
2
10 20 30 40
QT L Cart sum of LODs
142
VIII: 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
• http://www.stat.wisc.edu/~yandell/statgen
– coming soon:
• perl preprocessing (to/from QtlCart format)
• R post processing
June 2001
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Yandell
143
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 2001
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Yandell
144
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 (2000) “On the
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Reversible Jump
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QTL Reversible Jump
MCMC: Inbred Lines
• JM Satagopan & BS Yandell (1996) “Estimating the number
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QTL Reversible Jump
MCMC: Pedigrees
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Bayes & MCMC References
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QTL References
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