QTL model selection: key players

Transcription

1 Bayesian Interval Mapping. Bayesian strategy -9. Markov chain sampling 0-7. sampling genetic architectures criteria for model selection 6-44 QTL : Bayes Seattle SISG: Yandell 008 QTL model selection: key players observed measurements y phenotypic trait m markers & linkage map i individual index,,n missing data missing marker data missing 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 grounded by linkage map, experimental cross recombination yields multinomial for given m pry,µ,γ phenotype model distribution shape assumed normal here unknown parameters µ could be non-parametric observed mx Yy Q unknown λ θµ γ after Sen Churchill 00 QTL : Bayes Seattle SISG: Yandell 008

2 . Bayesian strategy for QTL study augment data y,m with missing genotypes study unknowns µ,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 posterior likelihood* prior constant phenotype likelihood*[prior for, µ, λ, γ ] posterior for, µ, λ, γ constant pr y, µ, γ *[pr m, λ, γ pr µ γ pr λ m, γ pr γ ] pr, µ, λ, γ y, m pr y m QTL : Bayes Seattle SISG: Yandell 008 Bayes posterior for normal data n large n small prior prior mean actual mean y phenotype values n large n small prior prior mean actual mean y phenotype values small prior variance large prior variance QTL : Bayes Seattle SISG: Yandell 008 4

3 Bayes posterior for normal data model y i µ e i environment e ~ N 0,, known likelihood y~n µ, prior µ ~N µ 0, κ, κ known posterior: mean tends to sample mean single individual µ ~N µ 0 b y µ 0, b sample of n individuals µ ~ N with y bn y bn µ 0, bn / n sum y / n { i,..., n} i shrinkage factor shrinks to κn bn κn QTL : Bayes Seattle SISG: Yandell what values are the genotypic means? phenotype model pry,µ data means prior mean data mean n large n small prior posterior means Q QQ y phenotype values QTL : Bayes Seattle SISG: Yandell 008 6

4 QTL : Bayes Seattle SISG: Yandell posterior centered on sample genotypic mean but shrunken slightly toward overall mean phenotype mean: genotypic prior: posterior: shrinkage: Bayes posterior QTL means / sum } count{ / } { i i n n b n y y n n b y V y b y b y E V y E y V y E i κ κ µ µ κ µ µ µ QTL : Bayes Seattle SISG: Yandell 008 8, ; 0 Y E µ partition genotypic effects on phenotype phenotype depends on genotype genotypic value partitioned into main effects of single QTL epistasis interaction between pairs of QTL

5 QTL : Bayes Seattle SISG: Yandell consider same QTL epistasis centering variance genotypic variance heritability 0 0 h h h h V s V κ κ partitition genotypic variance QTL : Bayes Seattle SISG: Yandell / shrinkage sum ˆ variance sum ] ˆ sum [sum ˆ LS estimate, ˆ, ˆ ~ i i i i i ij j i C b C w V y w C N C b b N y κ κ posterior mean LS estimate

6 pr m,λ recombination model pr m,λ prgeno map, locus prgeno flanking markers, locus? m m m m m 4 5 m6 markers λ distance along chromosome QTL : Bayes Seattle SISG: Yandell 008 QTL : Bayes Seattle SISG: Yandell 008

7 what are likely QTL genotypes? how does phenotype y improve guess? D4Mit4 D4Mit4 bp what are probabilities for genotype between markers? recombinants AA:AB 90 all : if ignore y and if we use y? AA AA AB AA AA AB AB AB Genotype QTL : Bayes Seattle SISG: Yandell 008 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 pry, µ weight toward with similar phenotype values posterior recombination model balances these two this is the E-step of EM computations pr y, µ * pr m, λ pr y, m, µ, λ pr y m, µ, λ QTL : Bayes Seattle SISG: Yandell 008 4

8 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 prλ m, prλ 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 QTL : Bayes Seattle SISG: Yandell 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 QTL : Bayes Seattle SISG: Yandell 008 6

9 γ genetic architecture: loci: main QTL epistatic pairs effects: add, dom aa, ad, dd QTL : Bayes Seattle SISG: Yandell 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 QTL : Bayes Seattle SISG: Yandell 008 8

10 . 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,µ,y,γ using Gibbs sampler sample effects µ given,y,γ using Gibbs sampler sample QTL model γ given λ,µ,y, using Gibbs or M-H λ,, µ, γ ~ pr λ,, µ, γ y, m λ,, µ, γ λ,, µ, γ L λ,, µ, γ N QTL : Bayes Seattle SISG: Yandell MCMC sampling of unknowns,µ,λ for given genetic architecture γ Gibbs sampler genotypes effects µ not loci λ ~ pr y, m, µ, λ i pr y, µ pr 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 prλ QTL : Bayes Seattle SISG: Yandell 008 0

11 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 ~ N, µ 0 ρ µ ~ N µ ~ N ρµ, ρ ρµ, ρ QTL : Bayes Seattle SISG: Yandell 008 ρ Gibbs sampler samples: ρ 0.6 N 50 samples N 00 samples Gibbs: mean Markov chain index Gibbs: mean Gibbs: mean Gibbs: mean Gibbs: mean Gibbs: mean Markov chain index Gibbs: mean Gibbs: mean Gibbs: mean Gibbs: mean Markov chain index Gibbs: mean Markov chain index Gibbs: mean QTL : Bayes Seattle SISG: Yandell 008

12 full conditional for locus cannot easily sample from locus full conditional prλ y,m,µ, pr λ m, pr m, λ prλ / 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 QTL : Bayes Seattle SISG: Yandell 008 Metropolis-Hastings idea want to study distribution fλ 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 a * * f λ g λ λ min, * f λ g λ λ QTL : Bayes Seattle SISG: Yandell fλ gλ λ *

13 mcmc seuence Metropolis-Hastings for locus λ prθ Y θ θ λ λ added twist: occasionally propose from entire genome QTL : Bayes Seattle SISG: Yandell Metropolis-Hastings samples mcmc seuence N 00 samples N 000 samples narrow g wide g narrow g wide g mcmc seuence mcmc seuence mcmc seuence histogram prθ Y θ λ θλ θ λ θ λ histogram prθ Y histogram prθ Y histogram prθ Y θ θ θ θ QTL : Bayes Seattle SISG: Yandell 008 6

14 . sampling genetic architectures search across genetic architectures A 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 QTL : Bayes Seattle SISG: Yandell 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 0 e γ QTL : Y 0 e QTL : Bayes Seattle SISG: Yandell 008 8

15 geometry of reversible jump Move Between Models Reversible Jump Seuence b m m c 0.7 b b b QTL : Bayes Seattle SISG: Yandell geometry allowing and λ to change a short seuence first 000 with m< b b b b QTL : Bayes Seattle SISG: Yandell 008 0

16 collinear QTL correlated effects 4-week 8-week additive effect cor -0.8 additive effect cor additive effect additive effect linked QTL collinear genotypes correlated estimates of effects negative if in coupling phase sum of linked effects usually fairly constant QTL : Bayes Seattle SISG: Yandell 008 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 -bγ-dγ update current model using full conditionals sample QTL loci, effects, and genotypes add a locus with probability bγ 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 dγ propose dropping one of existing loci decide whether to accept the death of locus QTL : Bayes Seattle SISG: Yandell 008

17 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, Allison, Eisen, Pomp 005 Genetics see earlier work of Nengjun Yi and Ina Hoeschele µ µ γ γ, γ k 0, QTL : Bayes Seattle SISG: Yandell 008 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 0 γ γ, 0 γ k QTL : Bayes Seattle SISG: Yandell 008 4

18 4. criteria for model selection balance fit against complexity classical information criteria penalize likelihood L by model size γ IC log Lγ y penaltyγ maximize over unknowns Bayes factors marginal posteriors pry γ average over unknowns QTL : Bayes Seattle SISG: Yandell classical information criteria start with likelihood Lγ 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.tln.tl F: γ n.tl 4n.tln.tl QTL : Bayes Seattle SISG: Yandell 008 6

19 classical information criteria construct information criteria balance fit to complexity Akaike AIC logl γ Bayes/Schwartz BIC logl γ logn Broman BIC δ logl δ γ logn general form: IC logl γ Dn compare models hypothesis testing: designed for one comparison log[lrγ, γ ] Ly m, γ Ly m, γ model selection: penalize complexity ICγ, γ log[lrγ, γ ] γ γ Dn QTL : Bayes Seattle SISG: Yandell information criteria vs. model size WinQTL.0 SCD data on F AAIC BIC BIC dbicδ models,,,4 QTL 59 epistasis : AD information criteria d d d A A A A A A A A model parameters p epistasis QTL : Bayes Seattle SISG: Yandell d d d d d

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

21 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 prγ y, m is marginal histogram QTL : Bayes Seattle SISG: Yandell Bayes factors & genetic architecture γ γ number of QTL prior prγ chosen by user posterior prγ y,m sampled marginal histogram shape affected by prior pra pr γ y, m / pr γ BF γ, γ pr γ y, m / pr γ pattern of QTL across genome gene action and epistasis prior probability e e exponential p Poisson e u uniform p p u u p eu u pu u u e e p p e e p e p e p e u u pu p e u m number of QTL QTL : Bayes Seattle SISG: Yandell 008 4

22 BF sensitivity to fixed prior for effects Bayes factors B45 B4 B B 4 / m, ~ N 0, h j hyper-prior heritability h G G, h fixed QTL : Bayes Seattle SISG: Yandell total BF insensitivity to random effects prior hyper-prior density *Betaa,b insensitivity to hyper-prior density , ,9.5,9,0,, Bayes factors B4 B B hyper-parameter heritability h Eh / m, ~ N 0, h j G G total, h ~ Beta a, b QTL : Bayes Seattle SISG: Yandell