Detecting variance-controlling genes

Transkript

1 Detecting variance-controlling genes Lars Rönnegård Högskolan Dalarna Swedish University of Agricultural Sciences NordStat 2012

2 Outline Are there genes controlling phenotypic robustness? Methods to nd genes for phenotypic robustness The non-parametric Levene's test A full parametric method using Double Generalized Linear Models hglm: A package for tting hierarchical generalized linear models (HGLM) Double hierarchical generalized linear models (DHGLM) Modelling phenotypic robustness in animal models

3 Mean-controlling genes

4 Variance-controlling genes

5 Figure 1: Substitution eect of the MOT1 allele. (From Shen et al. 2012)

6 Non-parametric test Parametric test Levene's test for inequality of variance We have observations Y with sample size N divided into k subgroups, then Levene's test statistic is W = (N k) k i=1 N i( Zi. Z.. ) 2 (k 1) k i=1 N i j=1 ( Zij Zi.) 2 where Z ij = Y ij Ȳ i. with Y ij the value of Y for the j:th observation of the i:th subgroup, and with bars indicating subgroup means. The Levene test rejects the hypothesis that the variances are equal if: W > F (α,k 1,N k) Pros: Not sensitive to deviations from normality Cons: Not possible to include covariates and random eects (to adjust for family structure).

7 Non-parametric test Parametric test Levene's test for inequality of variance We have observations Y with sample size N divided into k subgroups, then Levene's test statistic is W = (N k) k i=1 N i( Zi. Z.. ) 2 (k 1) k i=1 N i j=1 ( Zij Zi.) 2 where Z ij = Y ij Ȳ i. with Y ij the value of Y for the j:th observation of the i:th subgroup, and with bars indicating subgroup means. The Levene test rejects the hypothesis that the variances are equal if: W > F (α,k 1,N k) Pros: Not sensitive to deviations from normality Cons: Not possible to include covariates and random eects (to adjust for family structure).

8 Results from Arabidopsis Non-parametric test Parametric test Shen, Pettersson, Rönnegård and Carlborg (2012). Inheritance beyond plain heritability: variance controlling genes in Arabidopsis thaliana. PLoS Genetics (accepted) Arabidopsis thaliana study including 199 individuals Trait: molybdenum content 216,130 SNP Most signicant SNP located within the gene: ion transporter gene MOT1

9 Figure 2: A gene controlling robustness of molybdenum contents in Arabidopsis. Top gure (a) shows logp values for mean-controlling SNP (yellow) and variance-controlling SNP (dierent colours for dierent chromosomes). Bottom gure (b) shows substitution eect of the MOT1 allele. (Shen et al PLoS Genetics)

10 Results from human studies Using Levene's test to nd interactions Non-parametric test Parametric test Paré et al (2010) On the Use of Variance per Genotype as a Tool to Identify Quantitative Trait Interaction Eects: A Report from the Women's Genome Health Study. PLoS Genetics 6(6): e Women's Genome Health Study including 21,799 American women Trait: Inammatory marker, C-reactive protein (CRP) 339,596 SNP The SNP rs had P-value Interaction eects tested using a linear model SNP x BMI (with P-value 10 9 ) Their conclusion: detection of variance heterogeneity is a way to prioritize search for interaction eects

11 Results from human studies Using Levene's test to nd interactions Non-parametric test Parametric test Paré et al (2010) On the Use of Variance per Genotype as a Tool to Identify Quantitative Trait Interaction Eects: A Report from the Women's Genome Health Study. PLoS Genetics 6(6): e Women's Genome Health Study including 21,799 American women Trait: Inammatory marker, C-reactive protein (CRP) 339,596 SNP The SNP rs had P-value Interaction eects tested using a linear model SNP x BMI (with P-value 10 9 ) Their conclusion: detection of variance heterogeneity is a way to prioritize search for interaction eects

12 Results from human studies Using Levene's test to nd interactions Non-parametric test Parametric test Paré et al (2010) On the Use of Variance per Genotype as a Tool to Identify Quantitative Trait Interaction Eects: A Report from the Women's Genome Health Study. PLoS Genetics 6(6): e Women's Genome Health Study including 21,799 American women Trait: Inammatory marker, C-reactive protein (CRP) 339,596 SNP The SNP rs had P-value Interaction eects tested using a linear model SNP x BMI (with P-value 10 9 ) Their conclusion: detection of variance heterogeneity is a way to prioritize search for interaction eects

13 Non-parametric test Parametric test Using Double GLM to t a parametric model

14 Non-parametric test Parametric test Introduction to genome-wide association studies

15 GWAS Non-parametric test Parametric test Example: 3 individuals and 5 SNPs Linear model y = µ + x j b + e with y = and x1 = and ˆb = 6.5 (P = 0.56) Calculate P-value for each b at dierent positions and plot log 10 P

16 GWAS Non-parametric test Parametric test Example: 3 individuals and 5 SNPs Linear model y = µ + x j b + e with y = and x1 = and ˆb = 6.5 (P = 0.56) Calculate P-value for each b at dierent positions and plot log 10 P

17 GWAS Non-parametric test Parametric test Example: 3 individuals and 5 SNPs Linear model y = µ + x j b + e with y = and x1 = and ˆb = 6.5 (P = 0.56) Calculate P-value for each b at dierent positions and plot log 10 P

18 A GWAS plot Non-parametric test Parametric test Example from: Weedon et al. 2008, Nature Genetics 40,

19 Model Non-parametric test Parametric test Traditional model for single SNP regression y = µ + x j b + e e N(0,σ 2 e ) Model to detect variance-controlling genes y = µ + x j b + e e i N(0,σ 2 e,i) ; log(σ 2 e ) = c + x j v

20 Model Non-parametric test Parametric test Traditional model for single SNP regression y = µ + x j b + e e N(0,σ 2 e ) Model to detect variance-controlling genes y = µ + x j b + e e i N(0,σ 2 e,i) ; log(σ 2 e ) = c + x j v

21 Using GLM Non-parametric test Parametric test Traditional model for single SNP regression y = µ + x j b + e e N(0,σ 2 e ) model1 <- glm(y ~ SNP, family = gaussian(link = identity)) Model to detect variance-controlling genes y = µ + x j b + e e i N(0,σe,i) 2 ; log(σ 2 e ) = c + x j v Model easy to t using Gordon K. Smyth's dglm package in R library(dglm) model2 <- dglm(y ~ SNP, ~ SNP, family = gaussian(link = identity))

22 Using GLM Non-parametric test Parametric test Traditional model for single SNP regression y = µ + x j b + e e N(0,σ 2 e ) model1 <- glm(y ~ SNP, family = gaussian(link = identity)) Model to detect variance-controlling genes y = µ + x j b + e e i N(0,σe,i) 2 ; log(σ 2 e ) = c + x j v Model easy to t using Gordon K. Smyth's dglm package in R library(dglm) model2 <- dglm(y ~ SNP, ~ SNP, family = gaussian(link = identity))

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24 The hglm package Example vqtl: library(hglm) model2 <- hglm(fixed = y ~ SNP, disp = ~ SNP, random = ~ 1 ID, family = gaussian(link = identity) ) Design matrix notation: model2 <- hglm(x, y, Z, X.disp=X, family = gaussian(link = identity) )

25 The hglm package Example vqtl: library(hglm) model2 <- hglm(fixed = y ~ SNP, disp = ~ SNP, random = ~ 1 ID, family = gaussian(link = identity) ) Design matrix notation: model2 <- hglm(x, y, Z, X.disp=X, family = gaussian(link = identity) )

26 Books on HGLM and the extended likelihood principle

27 Rönnegård, Felleki, Fikse, Mulder and Strandberg (2010) Genetic heterogeneity of residual variance - estimation of variance components using double hierarchical generalized linear models. Genetics Selection Evolution 42:8.

28 Milk yield (liter/day) Daughter average Bull 1 Bull 2

29 Milk yield (liter/day) x x 30 x x x Daughter average Bull 1 Bull 2

30 Linear mixed model using pedigree information Animal model y = Xβ + Za + e a N(0, Aσ 2 a ) e N(0,σ 2 e ) a i = additive genetic eect for individual i A= relationship matrix (calculated from pedigree information) Estimated Breeding Values = Best Linear Unbiased Predictor (BLUP) of a i

31 Extending the animal model y = Xβ + Za + e a N(0, Aσ 2 a ) e i N(0,σ 2 e,i), log(σ 2 e ) = X d β d + Za d ad N(0, Aσ 2 a d ) In Rönnegård et al. (2010 GSE 42:8) we t a DHGLM for pig litter size data (previously studied in Sorensen and Waagepetersen (2003) using MCMC) DHGLM possible to t using existing variance-component estimation software (eg ASReml).

32 Extending the animal model y = Xβ + Za + e a N(0, Aσ 2 a ) e i N(0,σ 2 e,i), log(σ 2 e ) = X d β d + Za d ad N(0, Aσ 2 a d ) In Rönnegård et al. (2010 GSE 42:8) we t a DHGLM for pig litter size data (previously studied in Sorensen and Waagepetersen (2003) using MCMC) DHGLM possible to t using existing variance-component estimation software (eg ASReml).

33 Data Description Data from Danish Pig Production. Pig litter size from 4,149 sows mean litter size 10.3 The data includes 10,060 records from these 4,149 sows in 82 farms. Fixed eects: farm, season, type of insemination, parity

34 Results on pig litter size Table 1 - DHGLM and MCMC estimates σ 2 a σ 2 a d ρ DHGLM (-0.52) MCMC Felleki, Lee, Lee, Gilmour and Rönnegård (2012) Estimation of breeding values for mean and dispersion, their variance and correlation using double hierarchical generalized linear models (manuscript)

35 Results on pig litter size Table 1 - DHGLM and MCMC estimates σ 2 a σ 2 a d ρ DHGLM (-0.52) MCMC Felleki, Lee, Lee, Gilmour and Rönnegård (2012) Estimation of breeding values for mean and dispersion, their variance and correlation using double hierarchical generalized linear models (manuscript)

36 Method recently applied on a large dairy cow data set Swedish Holstein Cows ( ) including 1,693,154 observations from 177,411 animals Trait: milk yield (liter/day) rst lactation only where each cow had on average 9.5 records Rönnegård, Felleki, Fikse, Mulder and Strandberg (2012) Variance component and breeding value estimation for environmental sensitivity in Swedish Holstein dairy cattle (submitted manuscript)

37 Method recently applied on a large dairy cow data set Swedish Holstein Cows ( ) including 1,693,154 observations from 177,411 animals Trait: milk yield (liter/day) rst lactation only where each cow had on average 9.5 records Rönnegård, Felleki, Fikse, Mulder and Strandberg (2012) Variance component and breeding value estimation for environmental sensitivity in Swedish Holstein dairy cattle (submitted manuscript)

38 Thank you! Lars Rönnegård Special thanks to my students Majbritt Felleki and Xia Shen. Financial support from FORMAS and the EU-nanced RobustMilk project.