QTL model selection: key players

Transcription

1 QTL Model Selection. Bayesian strategy. Markov chain sampling 3. sampling genetic architectures 4. criteria for model selection Model Selection Seattle SISG: Yandell 0 QTL model selection: key players observed measurements y phenotypic trait m markers & linkage map i individual index (,,n missing i data missing marker data QT genotypes alleles QQ, Q, or at locus unknown uantities λ QT locus (or loci μ phenotype model parameters γ QTL model/genetic architecture pr( m,λ,γ genotype model observed mx Yy missing grounded ddby lik linkage map, experimental tlcross recombination yields multinomial for given m pr(y,γ phenotype model distribution shape (assumed normal here unknown parameters μ (could be non-parametric Q unknown λλ θμ γ after Sen Churchill (00 Model Selection Seattle SISG: Yandell 0

2 QTL mapping (from ZB Zeng phenotype model pr(y,γ genotypes Q pr( m,λ,γ λ markers M Model Selection Seattle SISG: Yandell 0 3 classical likelihood approach genotype model pr( m,λ,γ missing genotypes depend on observed markers m across genome phenotype model pr(y,γ link phenotypes y to genotypes LOD( λ log 0{max pr( y m, μ, λ } + c ( 0 μ likelihood mixes over missing QTL genotypes: pr( y m, λ pr( y, μpr( m, λ Model Selection Seattle SISG: Yandell 0 4

3 EM approach Iterate E and M steps expectation (E: geno prob s pr( m,λ,γ maximization (M: pheno model parameters mean, effects, variance careful attention when many QTL present Multiple papers by Zhao-Bang Zeng and others Start with simple initial model Add QTL, epistatic effects seuentially Model Selection Seattle SISG: Yandell 0 5 classic model search initial model from single QTL analysis search for additional QTL search for epistasis between pairs of QTL Both in model? One in model? Neither? Refine model Update QTL positions Check if existing QTL can be dropped Analogous to stepwise regression Model Selection Seattle SISG: Yandell 0 6

4 comparing models (details later balance model fit against model complexity want to fit data well (maximum likelihood without getting too complicated a model smaller model bigger model fit model miss key features fits better estimate phenotype may be biased no bias predict new data may be biased no bias interpret model easier more complicated estimate effects low variance high variance SysGen: Overview Seattle SISG: Yandell 0 7. Bayesian strategy for QTL study augment data (y,m with missing genotypes study unknowns (λ,γ given augmented data (y,m, find better genetic architectures γ find most likely genomic regions QTL λ estimate phenotype parameters genotype means μ sample from posterior in some clever way multiple imputation (Sen Churchill 00 Markov chain Monte Carlo (MCMC (Satagopan et al. 996; Yi et al. 005, 007 likelihood* prior posterior constant phenotype likelihood*[prior for, λ, γ ] posterior for, λ, γ constant pr( y, γ *[pr( m, λ, γ pr( μ γ pr( λ m, γ pr( γ ] pr(, λ, γ y, m pr( y m Model Selection Seattle SISG: Yandell 0 8

5 Bayes posterior for normal data n lar rge n small prior r prior mean actual mean y phenotype values prior rge n small n lar prior mean actual me an y phenotype values small prior variance large prior variance Model Selection Seattle SISG: Yandell 0 9 Posterior on genotypic means? phenotype model pr(y,μ data means prior mean data mean n small prior posterior means n large Q QQ y phenotype values Model Selection Seattle SISG: Yandell 0 0

6 posterior centered on sample genotypic mean but shrunken slightly toward overall mean Bayes posterior QTL means phenotype mean: genotypic prior: posterior: / ( ( ( ( ( ( ( + n b y V y b y b y E V y E y V y E σ μ μ κσ μ μ σ μ QTL : Bayes Seattle SISG: Yandell 00 shrinkage: / sum } count{ } { + i i n n b n y y n i κ κ pr( m,λ recombination model pr( m,λ pr(geno map, locus pr(geno flanking markers, locus m m 3 m 4 m 5 m 6 m? markers Model Selection Seattle SISG: Yandell 0 λ distance along chromosome

7 Model Selection Seattle SISG: Yandell 0 3 what are likely QTL genotypes? how does phenotype y improve guess? D4Mit4 D4Mit4 bp what are probabilities for genotype between markers? recombinants AA:AB all : if ignore y and if we use y? AA AA AB AA AA AB AB AB Genotype Model Selection Seattle SISG: Yandell 0 4

8 posterior on QTL genotypes full conditional of given data, parameters proportional to prior pr( m, λ weight toward that agrees with flanking markers proportional to likelihood pr(y, μ weight toward with similar phenotype values posterior recombination model balances these two this is the E-step of EM computations pr( y, m, λ pr( y, μ * pr( m, λ pr( y m, λ Model Selection Seattle SISG: Yandell 0 5 Where are the loci λ on the genome? prior over genome for QTL positions flat prior no prior idea of loci or use prior studies to give more weight to some regions posterior depends on QTL genotypes pr(λ m, pr(λ pr( m,λ / constant constant determined by averaging over all possible genotypes over all possible loci λ on entire map no easy way to write down posterior Model Selection Seattle SISG: Yandell 0 6

9 what is the genetic architecture γ? which positions correspond to QTLs? priors on loci (previous slide which QTL have main effects? priors for presence/absence of main effects same prior for all QTL can put prior on each d.f. ( for BC, for F which pairs of QTL have epistatic interactions? prior for presence/absence of epistatic pairs depends on whether 0,, QTL have main effects epistatic effects less probable than main effects Model Selection Seattle SISG: Yandell 0 7 γ genetic architecture: loci: main QTL epistatic pairs effects: add, dom aa, ad, dd Model Selection Seattle SISG: Yandell 0 8

10 Bayesian priors & posteriors augmenting with missing genotypes prior is recombination model posterior is (formally E step of EM algorithm sampling phenotype model parameters μ prior is flat normal at grand mean (no information posterior shrinks genotypic means toward grand mean (details for unexplained variance omitted here sampling QTL loci λ prior is flat across genome (all loci eually likely sampling QTL genetic architecture model γ number of QTL prior is Poisson with mean from previous IM study genetic architecture of main effects and epistatic interactions priors on epistasis depend on presence/absence of main effects Model Selection Seattle SISG: Yandell 0 9. Markov chain sampling 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 sample QTL model components from full conditionals sample locus λ given,γ (using Metropolis-Hastings step sample genotypes given λ,y,γ (using Gibbs sampler sample effects μ given,y,γ (using Gibbs sampler sample QTL model γ given λ,y, (using Gibbs or M-H ( λ,, γ ~ pr( λ,, γ y, m ( λ,, γ ( λ,, γ L ( λ,, γ N Model Selection Seattle SISG: Yandell 0 0

11 MCMC sampling of unknowns (,µ,λ for given genetic architecture γ Gibbs sampler genotypes effects µ not loci λ ~ pr( y, m, λ i pr( y, μpr( μ μ ~ pr( y pr( m, λpr( λ m λ ~ pr( m i Metropolis-Hastings sampler extension of Gibbs sampler does not reuire normalization pr( m sum λ pr( m, λ pr(λ Model Selection Seattle SISG: Yandell 0 Gibbs sampler for two genotypic means want to study two correlated effects could sample directly from their bivariate distribution assume correlation ρ is known instead use Gibbs sampler: sample each effect from its full conditional given the other pick order of sampling at random repeat many times μ 0 ρ ~ N, μ 0 ρ μ ~ N μ ~ N ( ρ ρ ( ρμ, ρ Model Selection Seattle SISG: Yandell 0

12 Gibbs sampler samples: ρ 0.6 N 50 samples N 00 samples Gibbs: mean Markov chain index Gibbs: mean 3 n 3 n Gibbs: mean Gibbs: mean Markov chain index Gibbs: mean Gibbs: mea Gibbs: mea Gibbs: mean Gibbs: mean n n Gibbs: mean Markov chain index Gibbs: mean Markov chain index Gibbs: mean Model Selection Seattle SISG: Yandell 0 3 full conditional for locus cannot easily sample from locus full conditional pr(λ ymµ y,m,µ, pr(λ λ m m, pr( m, λ pr(λ / constant constant is very difficult to compute explicitly must average over all possible loci λ over genome must do this for every possible genotype Gibbs sampler will not work in general but can use method based on ratios of probabilities Metropolis-Hastings is extension of Gibbs sampler Model Selection Seattle SISG: Yandell 0 4

13 Metropolis-Hastings idea want to study distribution f(λ take Monte Carlo samples unless too complicated take samples using ratios of f Metropolis-Hastings samples: propose new value λ * near (? current value λ from some distribution g accept new value with prob a Gibbs sampler: a always f(λ g(λ λ λ * a * * f ( λ g( λ λ min, * f ( λ g( λ λ Model Selection Seattle SISG: Yandell mcmc seuence Metropolis-Hastings for locus λ 0.6 pr(θ Y θ θ λ λ added twist: occasionally propose from entire genome Model Selection Seattle SISG: Yandell 0 6

14 Metropolis-Hastings samples mcmc seuence N 00 samples N 000 samples narrow g wide g narrow g wide g mcmc seuence θ λ θλ θ λ θ λ. mcmc seuence mcmc seuence histogram pr(θ Y 0 4 histogram pr(θ Y histogram pr(θ Y histogram pr(θ Y θ θ θ θ Model Selection Seattle SISG: Yandell sampling genetic architectures search across genetic architectures γ of various sizes allow change in number of QTL allow change in types of epistatic interactions methods for search reversible jump MCMC Gibbs sampler with loci indicators complexity of epistasis Fisher-Cockerham effects model general multi-qtl interaction & limits of inference Model Selection Seattle SISG: Yandell 0 8

15 reversible jump MCMC consider known genotypes at known loci λ models with or QTL M-H step between -QTL and -QTL models model changes dimension (via careful bookkeeping consider mixture over QTL models H γ QTL : Y β 0 + β ( + e γ QTL : Y β 0 + β ( + β ( + e Model Selection Seattle SISG: Yandell 0 9 geometry of reversible jump Move Between Models Reversible Jump Seuence b m c 0.7 b m β b b β Model Selection Seattle SISG: Yandell 0 30

16 geometry allowing and λ to change a short seuence first 000 with m<3 b b b b Model Selection Seattle SISG: Yandell β β collinear QTL correlated effects 4-week 8-week 0.0 cor cor -0.7 additive additive additive additive effect effect linked QTL collinear genotypes correlated estimates of effects (negative if in coupling phase sum of linked effects usually fairly constant Model Selection Seattle SISG: Yandell 0 3

17 sampling across QTL models γ 0 λ λ m+ λ λ m L action steps: draw one of three choices update QTL model γ with probability -b(γ-d(γ update current model using full conditionals sample QTL loci, effects, and genotypes add a locus with probability b(γ 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(γ propose dropping one of existing loci decide whether to accept the death of locus Model Selection Seattle SISG: Yandell 0 33 Gibbs sampler with loci indicators consider only QTL at pseudomarkers every - cm modest approximation with little bias use loci indicators in each pseudomarker γ if QTL present γ 0 if no QTL present Gibbs sampler on loci indicators γ relatively easy to incorporate epistasis Yi, Yandell, Churchill, hill Allison, Eisen, Pomp (005 Genetics (see earlier work of Nengjun Yi and Ina Hoeschele μ μ + γ β ( + γ β (, γ k 0, Model Selection Seattle SISG: Yandell 0 34

18 Bayesian shrinkage estimation soft loci indicators strength of evidence for λ j depends on γ 0 γ (grey scale shrink most γs to zero Wang et al. (005 Genetics Shizhong Xu group at U CA Riverside μ β 0 + γ β ( + γ β (, 0 γ k Model Selection Seattle SISG: Yandell 0 35 other model selection approaches include all potential loci in model assume true model is sparse in some sense Sparse partial least suares Chun, Keles (009 Genetics; 00 JRSSB LASSO model selection Foster (006; Foster Verbyla Pitchford (007 JABES Xu (007 Biometrics; Yi Xu (007 Genetics Shi Wahba Wright Klein Klein (008 Stat & Infer Model Selection Seattle SISG: Yandell 0 36

19 4. criteria for model selection balance fit against complexity classical information criteria penalize likelihood L by model size γ IC log L(γ y + penalty(γ maximize over unknowns Bayes factors marginal posteriors pr(y γ average over unknowns Model Selection Seattle SISG: Yandell 0 37 classical information criteria start with likelihood L(γ y, m measures fit of architecture (γ to phenotype (y given marker data (m genetic architecture (γ depends on parameters have to estimate loci (µ and effects (λ complexity related to number of parameters γ size of genetic architecture BC: γ + n.tl + n.tl(n.tl F: γ + n.tl +4n.tl(n.tl Model Selection Seattle SISG: Yandell 0 38

20 classical information criteria construct information criteria balance fit to complexity Akaike AIC log(l + γ Bayes/Schwartz BIC log(l + γ log(n Broman BIC δ log(l + δ γ log(n general form: IC log(l + γ D(n compare models hypothesis testing: designed for one comparison log[lr(γ, γ ] L(y m, γ L(y m, γ model selection: penalize complexity IC(γ, γ log[lr(γ, γ ] + ( γ γ D(n Model Selection Seattle SISG: Yandell 0 39 information criteria vs. model size WinQTL.0 SCD data on F AAIC BIC( BIC( dbic(δ models,,3,4 QTL epistasis : AD information criteria model parameters p epistasis Model Selection Seattle SISG: Yandell 0 40 d d d d d A A A A A A A A d d d

21 Bayes factors ratio of model likelihoods ratio of posterior to prior odds for architectures averaged over unknowns pr( γ y, m / pr( γ y, m pr( y m, γ B pr( γ / pr( γ pr( y m, γ roughly euivalent to BIC BIC maximizes over unknowns BF averages over unknowns log( B log( LR ( γ γ log( n Model Selection Seattle SISG: Yandell 0 4 scan of marginal Bayes factor & effect Model Selection Seattle SISG: Yandell 0 4

22 issues in computing Bayes factors BF insensitive to shape of prior on γ geometric, Poisson, uniform precision improves when prior mimics posterior BF sensitivity to prior variance on effects θ prior variance should reflect data variability resolved by using hyper-priors automatic algorithm; no need for user tuning easy to compute Bayes factors from samples sample posterior using MCMC posterior pr(γ y, m is marginal histogram Model Selection Seattle SISG: Yandell 0 43 Bayes factors & genetic architecture γ γ number of QTL prior pr(γ chosen by user posterior pr(γ y,m sampled marginal histogram shape affected by prior pr(a pr( γ y, m / pr( γ BF γ, γ pr( γ y, m / pr( γ bability prior prob pattern of QTL across genome gene action and epistasis e e p p e p u exponential Poisson uniform u u p eu u pu u u e e p p e p e p e e p e u u pu e pu m number of QTL Model Selection Seattle SISG: Yandell 0 44

23 BF sensitivity to fixed prior for effects Bayes factors B45 B34 B3 B 4 3 ( σ / m, β ~ N 0, σ h σ j hyper-prior heritability h G G, h fixed Model Selection Seattle SISG: Yandell 0 45 total BF insensitivity to random effects prior hyper-prior density *Beta(a,b insensitivity to hyper-prior 3.0 density , ,9.5,9,0,3,.0 Bayes factors B34 B3 B hyper-parameter heritability h E(h ( σ / m, β ~ N 0, σ h σ j G G total, h ~ Beta( a, b Model Selection Seattle SISG: Yandell 0 46