Model Selection for Multiple QTL

Transcription

1 Model Selection for Multiple TL 1. reality of multiple TL 3-8. selecting a class of TL models comparing TL models 16-4 TL model selection criteria issues of detecting epistasis 4. simulations and data studies 5-40 simulation with 8 TL plant BC, animal F studies searching through TL models Model NCSU TL II: Yandell 004 1

2 what is the goal of TL study? uncover underlying biochemistry identify how networks function, break down find useful candidates for (medical) intervention epistasis may play key role statistical goal: maximize number of correctly identified TL basic science/evolution how is the genome organized? identify units of natural selection additive effects may be most important (Wright/Fisher debate) statistical goal: maximize number of correctly identified TL select elite individuals predict phenotype (breeding value) using suite of characteristics (phenotypes) translated into a few TL statistical goal: mimimize prediction error Model NCSU TL II: Yandell 004

3 1. reality of multiple TL evaluate some objective for model given data classical likelihood Bayesian posterior search over possible genetic architectures (models) number and positions of loci gene action: additive, dominance, epistasis estimate features of model means, variances & covariances, confidence regions marginal or conditional distributions art of model selection how select best or better model(s)? how to search over useful subset of possible models? Model NCSU TL II: Yandell 004 3

4 advantages of multiple TL approach improve statistical power, precision increase number of TL detected better estimates of loci: less bias, smaller intervals improve inference of complex genetic architecture patterns and individual elements of epistasis appropriate estimates of means, variances, covariances asymptotically unbiased, efficient assess relative contributions of different TL improve estimates of genotypic values less bias (more accurate) and smaller variance (more precise) mean squared error = MSE = (bias) + variance Model NCSU TL II: Yandell 004 4

5 major TL on linkage map 3 Pareto diagram of TL effects additive effect major TL (modifiers) minor TL polygenes rank order of TL Model NCSU TL II: Yandell 004 5

6 limits of multiple TL? limits of statistical inference power depends on sample size, heritability, environmental variation best model balances fit to data and complexity (model size) genetic linkage = correlated estimates of gene effects limits of biological utility sampling: only see some patterns with many TL marker assisted selection (Bernardo 001 Crop Sci) 10 TL ok, 50 TL are too many phenotype better predictor than genotype when too many TL increasing sample size may not give multiple TL any advantage hard to select many TL simultaneously 3 m possible genotypes to choose from Model NCSU TL II: Yandell 004 6

7 TL below detection level? problem of selection bias TL of modest effect only detected sometimes their effects are biased upwards when detected probability that TL detected avoids sharp in/out dichotomy avoid pitfalls of one best model examine better models with more probable TL build m = number of TL detected into TL model directly allow uncertainty in genetic architecture model selection over genetic architecture Model NCSU TL II: Yandell 004 7

8 . selecting a class of TL models phenotype distribution normal (usual), binomial, Poisson, exponential family, semi-parametric, nonparametric θ = gene action additive (A) or general (A+D) effects epistatic interactions (AA, AD,..., or other types?) λ = location of TL known locations? widely spaced (no in marker interval) or arbitrarily close? m = number of TL single TL? multiple TL: known or unknown number? Model NCSU TL II: Yandell 004 8

9 normal phenotype trait = mean + genetic + environment genetic effect uncorrelated with environment pr( trait Y genotype, effects θ ) Y = G var( G + e heritability G effectsθ = ( G ) = σ,var( e) = σ h, σ ) σ G = σ + σ G qq q Y G Model NCSU TL II: Yandell 004 9

10 Model NCSU TL II: Yandell ) var( ) ( ) ( ) ( ),var( σ σ σ σ β β β µ σ + + = = = = + = G G G e e G Y two TL with epistasis same phenotype model overview partition of genotypic value with epistasis partition of genetic variance

11 epistasis examples (Doebley Stec Gustus 1995; Zeng pers. comm.) genotypic value genotypic value effect +/- se genotypic value genotypic value aa Aa AA aa Aa AA BB Bb bb bb Bb BB bb Bb BB bb Bb BB AA effect +/- se genotypic value genotypic value effect +/- se Aa aa aa Aa AA a1 d1 a d iaa iad ida idd a1 d1 a d iaa iad ida idd aa Aa AA bb Bb BB bb Bb BB aa Aa AA traits 1,4,9 Fisher-Cockerham effects a1 d1 a d iaa iad ida idd Model NCSU TL II: Yandell

12 Model NCSU TL II: Yandell } { } { sum ) var( ) ( sum ),var( j M j G j M j G G e e G Y σ σ β µ σ = = + = = + = multiple TL with epistasis same overview model sum over multiple TL in model M = {1,,1, } partition genetic variance in same manner could restrict attention to -TL interactions

13 model selection with epistasis additive by additive -TL interaction adds only 1 model degree of freedom (df) per pair but could miss important kinds of interaction full epistasis adds many model df TL in BC: 1 df (one interaction) TL in F: 4 df (AA, AD, DA, DD) 3 TL in F: 0 df (3 4 d.f. -TL, 8 d.f. 3-TL) data-driven interactions (tree-structured) contrasts comparing subsets of genotypes double recessive or double dominant vs other genotypes discriminant analysis based contrasts (Gilbert and Le Roy 003, 004) some issues in model search epistasis between significant TL check all possible pairs when TL included? allow higher order epistasis? epistasis with non-significant TL whole genome paired with each significant TL? pairs of non-significant TL? Yi Xu (000) Genetics; Yi, Xu, Allison (003) Genetics; Yi (004) Model NCSU TL II: Yandell

14 3. comparing TL models balance model fit with model "complexity want maximum likelihood without too complicated a model information criteria quantifies the balance Bayes information criteria (BIC) for likelihood Bayes factors for Bayesian approach Model NCSU TL II: Yandell

15 TL likelihoods and parameters LOD or likelihood ratio compares model L(p) = log likelihood for a particular model with p parameters log(lr) = L(p ) L(p 1 ) LOD = log 10 (LR) = log(lr)/log(10) p = number of model degrees of freedom consider models with m TL and all -TL epistasis terms BC: p = 1 + m + m(m-1) F: p = 1 + m +4m(m-1) Bayesian information criterion balances complexity BIC(δ) = log[l(p)] + δ p log(n) n = number of individuals in study δ = Broman s BIC adjustment Model NCSU TL II: Yandell

16 information criteria: likelihoods L(p) = likelihood for model with p parameters common information criteria: Akaike AIC = log[l(p)] + p Bayes/Schwartz BIC = log[l(p)] + p log(n) BIC-delta BIC δ = log[l(p)] + δ p log(n) general form: IC = log[l(p)] + p D(n) comparison of models hypothesis testing: designed for one comparison log[lr(p 1,p )] = L(p ) L(p 1 ) model selection: penalize complexity IC(p 1,p ) = log[lr(p 1,p )] + (p p 1 ) D(n) Model NCSU TL II: Yandell

17 information criteria vs. model size WinTL.0 SCD data on F A=AIC 1=BIC(1) =BIC() d=bic(δ) models 1,,3,4 TL epistasis : AD information criteria d d 1 A model parameters p epistasis Model NCSU TL II: Yandell d d d A A A A A A A d d d

18 Bayes factors & BIC pr( model1 Y ) / pr( model Y ) B 1 = = pr( model ) / pr( model ) 1 pr( Y pr( Y model model what is a Bayes factor? ratio of posterior odds to prior odds ratio of model likelihoods BF is equivalent to LR statistic when comparing two nested models simple hypotheses (e.g. 1 vs TL) BF is equivalent to Bayes Information Criteria (BIC) for general comparison of any models want Bayes factor to be substantially larger than 1 (say 10 or more) 1 ) ) 1 log( B1) = log( LR) ( p p ) log( n) Model NCSU TL II: Yandell

19 m = number of TL TL Bayes factors prior pr(m) chosen by user posterior pr(m Y,X) sampled marginal histogram shape affected by prior pr(m) pattern of TL across genome more complicated prior posterior easily sampled BF m,m + 1 = prior probability pr( m Y, X )/ pr( m) pr( m + 1 Y, X )/ pr( m + 1) e e p p e p u exponential Poisson uniform u u eu u pu u u p e p e p e e p e p e p e u u pu ep u m = number of TL Model NCSU TL II: Yandell

20 issues in computing Bayes factors BF insensitive to shape of prior on m 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(m Y,X) is marginal histogram Model NCSU TL II: Yandell 004 0

21 multiple TL priors phenotype influenced by genotype & environment pr(y,θ) ~ N(G, σ ), or Y = G + environment partition genotype-specific mean into TL effects G = mean + main effects + epistatic interactions G = µ + β() = µ +sum j in M β j () priors on mean and effects µ ~ N(µ 0, κ 0 σ ) grand mean β() ~ N(0, κ 1 σ ) model-independent genotypic effect β j () ~ N(0, κ 1 σ / M ) effects down-weighted by size of M determine hyper-parameters via Empirical Bayes h 1 h G µ 0 Y and κ1 = Model NCSU TL II: Yandell σ σ

22 Model NCSU TL II: Yandell 004 phenotype influenced by genotype & environment pr(y,θ) ~ N(G, σ ), or Y = µ + G + environment relation of posterior mean to LS estimate 1 ) /( shrinkage sum ) ˆ ( variance sum )] ( ˆ sum [sum ˆ LS estimate ), ˆ ( ), ˆ ( ~, + = = = = = i i i i i j M j i C B C w G V Y w G C G N C B G B N m Y G κ κ σ σ β σ σ multiple TL posteriors

23 4. simulations and data studies simulated F intercross, 8 TL (Stephens, Fisch 1998) n=00, heritability = 50% detected 3 TL increase to detect all 8 n=500, heritability to 97% TL chr loci effect ch1 ch ch3 ch4 ch5 ch6 ch7 ch8 ch9 ch10 frequency in % Genetic map posterior number of TL Model NCSU TL II: Yandell 004 3

24 loci pattern across genome notice which chromosomes have persistent loci best pattern found 4% of the time Chromosome m Count of Model NCSU TL II: Yandell 004 4

25 B. napus 8-week vernalization whole genome study 108 plants from double haploid similar genetics to backcross: follow 1 gamete parents are Major (biennial) and Stellar (annual) 300 markers across genome 19 chromosomes average 6cM between markers median 3.8cM, max 34cM 83% markers genotyped phenotype is days to flowering after 8 weeks of vernalization (cooling) Stellar parent requires vernalization to flower Ferreira et al. (1994); Kole et al. (001); Schranz et al. (00) Model NCSU TL II: Yandell 004 5

26 Bayesian model assessment TL posterior Bayes factor ratios TL posterior posterior / prior row 1: # TL row : pattern col 1: posterior col : Bayes factor note error bars on bf evidence suggests 4-5 TL N(-3),N3,N number of TL pattern posterior model index strong moderate weak number of TL Bayes factor ratios model posterior * :,1 3:*,1 :,13 :,3 :,16 :,11 3:*,3 :,15 4:*,3,16 3:*,13 :,14 posterior / prior 5 e-01 1 e+01 5 e strong moderate weak model index Model NCSU TL II: Yandell 004 6

27 Bayesian estimates of loci & effects histogram of loci blue line is density red lines at estimates loci histogram napus8 summaries with pattern 1,1,,3 and m 4 N N3 N estimate additive effects (red circles) grey points sampled from posterior blue line is cubic spline dashed line for SD additive N N3 N Model NCSU TL II: Yandell 004 7

28 Bayesian model diagnostics pattern: N(),N3,N16 col 1: density col : boxplots by m environmental variance σ =.008, σ =.09 heritability h = 5% LOD = 16 (highly significant) but note change with m density density density marginal envvar, m marginal heritability, m 4 LOD heritability envvar envvar conditional on number of TL heritability conditional on number of T marginal LOD, m LOD conditional on number of TL Model NCSU TL II: Yandell 004 8

29 studying diabetes in an F segregating cross of inbred lines B6.ob x BTBR.ob F1 F selected mice with ob/ob alleles at leptin gene (chr 6) measured and mapped body weight, insulin, glucose at various ages (Stoehr et al. 000 Diabetes) sacrificed at 14 weeks, tissues preserved gene expression data Affymetrix microarrays on parental strains, F1 key tissues: adipose, liver, muscle, β-cells novel discoveries of differential expression (Nadler et al. 000 PNAS; Lan et al. 00 in review; Ntambi et al. 00 PNAS) RT-PCR on 108 F mice liver tissues 15 genes, selected as important in diabetes pathways SCD1, PEPCK, ACO, FAS, GPAT, PPARgamma, PPARalpha, G6Pase, PDI, Model NCSU TL II: Yandell 004 9

30 Multiple Interval Mapping SCD1: multiple TL plus epistasis! effect (add=blue, dom=red) LOD chr chr5 chr chr chr5 chr9 Model NCSU TL II: Yandell

31 Bayesian model assessment: number of TL for SCD1 TL posterior Bayes factor ratios TL posterior number of TL posterior / prior strong moderate weak number of TL Model NCSU TL II: Yandell

32 Bayesian LOD and h for SCD1 density marginal LOD, m LOD conditional on number of TL density heritability LOD marginal heritability, m heritability conditional on number of TL Model NCSU TL II: Yandell 004 3

33 Bayesian model assessment: chromosome TL pattern for SCD1 pattern posterior Bayes factor ratios model posterior :1,,*3 4:1,*,3 5:3*1,,3 5:*1,,*3 5:*1,*,3 6:3*1,,*3 6:3*1,*,3 5:1,*,*3 6:4*1,,3 6:*1,*,*3 :1,3 3:*1, :1, 4:*1,,3 posterior / prior :1,, moderate 6 weak model index model index Model NCSU TL II: Yandell

34 trans-acting TL for SCD1 (no epistasis yet: see Yi, Xu, Allison 003) dominance? Model NCSU TL II: Yandell

35 -D scan: assumes only TL! epistasis LOD peaks joint LOD peaks Model NCSU TL II: Yandell

36 sub-peaks can be easily overlooked! Model NCSU TL II: Yandell

37 1-D and -D marginals pr(tl at λ Y,X, m) unlinked loci linked loci Model NCSU TL II: Yandell

38 false detection rates and thresholds multiple comparisons: test TL across genome size = pr( LOD(λ) > threshold no TL at λ ) threshold guards against a single false detection very conservative on genome-wide basis difficult to extend to multiple TL positive false discovery rate (Storey 001) pfdr = pr( no TL at λ LOD(λ) > threshold ) Bayesian posterior HPD region based on threshold Λ ={λ LOD(λ) > threshold } {λ pr(λ Y,X,m ) large } extends naturally to multiple TL Model NCSU TL II: Yandell

39 pfdr and TL posterior positive false detection rate pfdr = pr( no TL at λ Y,X, λ in Λ ) pfdr = pr(h=0)*size pr(m=0)*size+pr(m>0)*power power = posterior = pr(tl in Λ Y,X, m>0 ) size = (length of Λ) / (length of genome) extends to other model comparisons m = 1 vs. m = or more TL pattern = ch1,ch,ch3 vs. pattern > *ch1,ch,ch3 Model NCSU TL II: Yandell

40 pfdr for SCD1 analysis Storey pfdr(-) relative size of HPD region pr( locus in HPD m>0 ) Model NCSU TL II: Yandell pr( H=0 p>size ) BH pfdr(-) and size(.) prior probability fraction of posterior found in tails