Seminario - Testing gene-environment interaction in generalized … · Breslow, N. and Clayton, D, 1993. Approximate inference in generalizedlinearmixedmodels.J. Am. Stat. Assoc.,88,9-25

Seminario Escuela de Estadística UNALMED

SeminarioTesting gene-environment interaction in generalized linear mixed

models with family data

20 de noviembre de 2017

Seminario Introduction

Sección 1

Introduction


Gene, Environment and Disease


Gene-environment interaction


Family data and kinship matrix


Notation

Consider a sample of N independent families.ni : number of members in the ith family (i = 1, . . . ,N).Yij : reponse variable for the phenotype (discrete orcontinuous).Xij = (X 1

ij , . . . ,Xpij )T : p non-environmental covariates.

Gij = (G1ij , . . . ,G

qij )T : q observed genotypes at certain

targeted genetic marker loci.Eij : some environmental exposure factor of interest.Sij = (EijG1

ij , . . . ,EijGqij )T : GE interaction.


Observed genotypes - Gij -

Each column in Gij is a Single Nucleotide Polymorphism (SNP).The genetic information is represented according to the followingcodification:

Gkij =

2, subject j at ith family is homozygous BB1, subject j at ith family is heterozygous Bb or bB0, subject j at ith family is homozygous bb,

with k = 1, . . . , q. B and b represent the dominant and recessivealleles, respectively. In addition, B is the allele that occurs at minorfrequency.


Observed genotypes - Gij -

Seminario Model

Sección 2

Model

Seminario Model

Generalized linear mixed model (GLMM) for Subject j at ith family

g [E (Yij |αij)] = XTij β1 + Eijβ2 + GT

ij θ + STij γ + αij , (1)

Var(Yij |αij) = φω−1ij ν [E (Yij |αij)] ,

αi = (αi1, . . . , αini )T ∼ N(0, 2σ2Φi),where,

g(.): monotone known function.Yij |αij follows a distribution in the exponential family.ν(.): known function.φ: a scale parameter that may be known or may need to beestimated.ωij : known weights (commonly equal to 1).Φi : the kinship matrix and σ2 is a parameter to be estimated.

Seminario Model

Examples of link functions

Family Link g(·) Trait ν(·)Binomial Logit ln

(µ

1−µ

)Binary µ(1 − µ)

Gaussian Identity µ Continuous φGamma Inverse 1/µ Continuous φµ2

Inverse.gaussian Inverse squared 1/µ2 Continuous φµ3

Poisson Log ln(µ) Count µQuasi Identity µ Continuous φ

Quasibinomial Logit ln(

µ1−µ

)Binary φµ(1 − µ)

Quasipoisson Log ln(µ) Count φµ

Seminario Model

GLMM for ith Family

g(µb

i

)= Xiβ1 + Eiβ2 + Giθ + Siγ + Kibi , (2)

where,2Φi = KiKT

i (Cholesky decomposition).αi = Kibi , with bi ∼ N(0, σ2Ini ) and Ini : identity matrix.

µbi = E (Yi |bi).

g(µbi ) = (g(µb

i1), . . . , g(µbini ))

T .Xi = [Xi1 . . .Xini ]T .

Gi = [Gi1 . . .Gini ]T .Ei = (Ei1, . . . ,Eini )T .Si = [Si1 . . .Sini ]T .

Seminario Model

General GLMM

g(µb)

= Xβ + Gθ + Sγ + Kb, (3)

whereµb = E (Y |b).X = [X1 . . .XN ]T .G = [G1 . . .GN ]T .E = (E1, . . . ,EN)T

S = [S1 . . .SN ]T .

K = diag{K1 . . .KN}.b = [b1 . . .bN ]T .X = [X E ]T .β = (βT

1 , β2)T .

We are interested in testing the hypthotesis H0 : γ = 0.

Seminario Model

Important facts

If γ is treated as a fixed vector and the null hyphotesis is testedwith a q degrees of freedom score test, it can result in loss ofpower (Lin et. al., 2013).

Another common strategy is to use a single SNP at time to testGE interaction.Assume γ as a random vector following a multivariate normaldistribution N(0, τ Iq) and to test the equivalent null hypothesisH0 : τ = 0.

= Xβ + Gθ

Seminario Model

Important facts

If γ is treated as a fixed vector and the null hyphotesis is testedwith a q degrees of freedom score test, it can result in loss ofpower (Lin et. al., 2013).Another common strategy is to use a single SNP at time to testGE interaction.

Assume γ as a random vector following a multivariate normaldistribution N(0, τ Iq) and to test the equivalent null hypothesisH0 : τ = 0.

= Xβ + Gθ

Seminario Model

Important facts

If γ is treated as a fixed vector and the null hyphotesis is testedwith a q degrees of freedom score test, it can result in loss ofpower (Lin et. al., 2013).Another common strategy is to use a single SNP at time to testGE interaction.Assume γ as a random vector following a multivariate normaldistribution N(0, τ Iq) and to test the equivalent null hypothesisH0 : τ = 0.

= Xβ + Gθ

Seminario Model

Important facts


g(µb,γ

)= Xβ + Gθ + Sγ + Kb︸︷︷︸

random

Seminario Model

Important facts


g(µb)

= Xβ + Gθ+Sγ + Kb︸︷︷︸random

Seminario Model

Linkage disequilibrium (LD) -PPARG-

Seminario Model

Linkage disequilibrium (LD) -CDKAL1-

Seminario Model


Seminario Model


Seminario Model


Seminario Model

Remedial considerations to face LD

Given the high correlation among the SNPs in G, Lin et. al.(2013) proposed (for independet subjects) to penalize the esti-mation of parameter θ by using ridge regression and introducinga penalization term λ in the quasi-likelihood function. The tun-ning parameter λ is selected using generalized cross validation(Fu, 2005).

Shen et. al. (2013) proposed for generalized linear models (GLM)that ridge regression is equivalent to assume the penalized pa-rameters as independent random variables.

Seminario Model


Given the high correlation among the SNPs in G, Lin et. al.(2013) proposed (for independet subjects) to penalize the esti-mation of parameter θ by using ridge regression and introducinga penalization term λ in the quasi-likelihood function. The tun-ning parameter λ is selected using generalized cross validation(Fu, 2005).Shen et. al. (2013) proposed for generalized linear models (GLM)that ridge regression is equivalent to assume the penalized pa-rameters as independent random variables.

Seminario Model


It is also equivalent for GLMM and assuming θ ∼ N(0, σ2θ Iq), it is

possible to show that λ = φ/σ2θ .

= Xβ +

Seminario Model




g(µb,γ,θ

)= Xβ + Gθ + Sγ + Kb︸︷︷︸

random

Seminario Model




g(µb,θ

)= Xβ + Gθ︸︷︷︸

random

+Sγ + Kb︸︷︷︸random

Seminario Null model estimation

Sección 3

Null model estimation



g(µd1,d2

)= Xβ + d1 + d2

with d1 = Gθ, d2 = Kb.

Breslow and Clayton (1993) proposed aFisher scoring solution that may be expressed as the iterative solutionto the system XT W X XT W XT W

W X φ

σ2θ

(GGT )−1 + W WW X W φ

σ2 (KKT )−1 + W

( βd1d2

)=

XT W YW YW Y

Y = Xβ + d1 + d2 + ε: working vector; ε ∼ N(0, φW−1);

W = diag{ωij/

[ν(µd

ij )g ′(µdij )2]}

.Ridge



g(µd1,d2

)= Xβ + d1 + d2

with d1 = Gθ, d2 = Kb. Breslow and Clayton (1993) proposed aFisher scoring solution that may be expressed as the iterative solutionto the system XT W X XT W XT W

W X φ

σ2θ

(GGT )−1 + W WW X W φ

σ2 (KKT )−1 + W

( βd1d2

)=

XT W YW YW Y

Y = Xβ + d1 + d2 + ε: working vector; ε ∼ N(0, φW−1);

W = diag{ωij/

[ν(µd

ij )g ′(µdij )2]}

.Ridge



β = (XT Σ−1X)−1XT Σ−1Y ,

(d1d2

)=

σ2θ

(GGT

)Σ−1(Y − Xβ)

σ2(KKT

)Σ−1(Y − Xβ)

,with Σ = σ2

θGGT + σ2KKT + φW−1.

Seminario GE interaction test

Sección 4

GE interaction test


Testing GE interaction

Following the approach developed by Zhang and Lin (2003), wepropose as score statitistic the quadratic form

Uτ = Uτ(β, π

)= 1

2

{(Y − Xβ

)TΣ−1SST Σ−1

(Y − Xβ

)} ∣∣∣∣β.π

.

with π is the estimator of π = (σ2θ , σ

2, φ)T .

To correct for bias, weuse the restricted maximum likelihood (REML) estimators (Breslowand Clayton, 1993) in the GLMM framework to obtain β and πunder the null hypothesis.



Following the approach developed by Zhang and Lin (2003), wepropose as score statitistic the quadratic form

Uτ = Uτ(β, π

)= 1

2

{(Y − Xβ

)TΣ−1SST Σ−1

(Y − Xβ

)} ∣∣∣∣β.π

.

with π is the estimator of π = (σ2θ , σ

2, φ)T .To correct for bias, weuse the restricted maximum likelihood (REML) estimators (Breslowand Clayton, 1993) in the GLMM framework to obtain β and πunder the null hypothesis.



Zhang and Lin (2003) showed that under H0 : τ = 0, Uτ followsapproximately a mixture of one degree of freedom independent chi-square distributions. However, for computational ease, we use theSatterthwaite method (Satterthwaite, 1941) to approximate the dis-tribution of Uτ by a scaled chi-square distribution κχ2

ξ .

When REML estimates are used to calculate Uτ showed that themean and variance of Uτ can be approximated, respectively, by

tr(PSST

) ∣∣∣∣π

and Iτ = 12{tr(PSST PSST )− JT M−1J

} ∣∣∣∣β,π



Zhang and Lin (2003) showed that under H0 : τ = 0, Uτ followsapproximately a mixture of one degree of freedom independent chi-square distributions. However, for computational ease, we use theSatterthwaite method (Satterthwaite, 1941) to approximate the dis-tribution of Uτ by a scaled chi-square distribution κχ2

ξ .When REML estimates are used to calculate Uτ showed that themean and variance of Uτ can be approximated, respectively, by

tr(PSST

) ∣∣∣∣π

and Iτ = 12{tr(PSST PSST )− JT M−1J

} ∣∣∣∣β,π



J =(

tr[PSST PGGT ] tr[PSST PKKT ] tr[PSST PW−1])T

M =

tr[PGGT PGGT ] tr[PGGT PKKT ] tr[PGGT PW−1]

tr[PKKT PGT G] tr[PKKT PKKT ] tr[PKKT PW−1]

tr[PW−1PGT G] tr[PW−1PKKT ] tr[PW−1PW−1]

,

and P = Σ−1 −Σ−1X(XT Σ−1X

)−1XT Σ−1.



Since the mean and variance of κχ2ξ are given by κξ and 2κ2ξ,

respectively, we obtain the equations tr(PSST ) = κξ and Iτ =2κ2ξ, where P denotes the matrix P evaluated in π. By solvingthese equations, we demonstrate that

κ = Iτ/[2 tr(PSST )]

andξ = 2

[tr(PSST )

]2/Iτ .

Uτκ ∼ χ

2ξ



Since the mean and variance of κχ2ξ are given by κξ and 2κ2ξ,

respectively, we obtain the equations tr(PSST ) = κξ and Iτ =2κ2ξ, where P denotes the matrix P evaluated in π. By solvingthese equations, we demonstrate that

κ = Iτ/[2 tr(PSST )]

andξ = 2

[tr(PSST )

]2/Iτ .

Uτκ ∼ χ

2ξ

Seminario Simulations

Sección 5

Simulations


Simulated model

Eij = 2 + 0,01Ageij + 0,1I(Femaleij) + γi + εij ;

logit [P (Yij = 1|Ageij ,Femaleij ,Eij ,G1ij ,G2ij)]= 0,1 + 0,01Ageij + 0,1I(Femaleij) + 0,1Eij + 0,3G1ij

+0,3G2ij + γ1(G1ij × Eij) + γ2(G2ij × Eij) + αij

withI(·) is theindicatorfunction;

εi = (εi1, . . . , εi10)T ∼ N(0, 4I10);αi = (αi1, . . . , αi10)T ∼ N(0, 2Φi);γi ∼ N(0, 4)

Φi is the kinship matrix corresponding to the aforementionedfamily pedigree .


Correlation of the 50 simulated SNPs in LD

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

123456789

1011121314151617181920212223242526272829303132333435363738394041424344454647484950


Type I error

We first compared the empirical type I error of the different methodsat 0.05 α-level. To evaluate type I error, we set γ1 = γ2 = 0 andvaried the number of SNPs q in the gene. The SNPs were eitherindependent or in LD.

SNPs category q σ2 σ2θ λ = 1/σ2

θ Score MinP VCT

Independent5 1.247 0.034 29.4 0.020 0.031 0.034

10 1.240 0.017 58.8 0.023 0.026 0.02450 1.222 0.003 333 0.004 0.022 0.014

LD5 1.243 0.021 47.6 0.025 0.026 0.031

10 1.239 0.009 111 0.004 0.034 0.03050 1.222 0.002 500 0.000 0.028 0.022


Type I error

We first compared the empirical type I error of the different methodsat 0.05 α-level. To evaluate type I error, we set γ1 = γ2 = 0 andvaried the number of SNPs q in the gene. The SNPs were eitherindependent or in LD.

SNPs category q σ2 σ2θ λ = 1/σ2

θ Score MinP VCT

Independent5 1.247 0.034 29.4 0.020 0.031 0.034

10 1.240 0.017 58.8 0.023 0.026 0.02450 1.222 0.003 333 0.004 0.022 0.014

LD5 1.243 0.021 47.6 0.025 0.026 0.031

10 1.239 0.009 111 0.004 0.034 0.03050 1.222 0.002 500 0.000 0.028 0.022


Empirical power for independent SNPs

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

q = 5

γ1 = γ2

Em

piric

al P

ower

ScoreMinPVCT

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

q = 10

γ1 = γ2

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

q = 50

γ1 = γ2


Empirical power for dependent SNPs

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

q = 5

γ1 = γ2

Em

piric

al P

ower

ScoreMinPVCT

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

q = 10

γ1 = γ2

0.00 0.02 0.04 0.06 0.08 0.10

0.0

0.2

0.4

0.6

0.8

1.0

q = 50

γ1 = γ2

Seminario Application

Sección 6

Application


Baependi Heart Study


Variables

Phenotype: Type II diabetesEnvironmental variable: Body Mass Index (BMI)Genotype: Were considered the following three genetic regions:

Peroxisome-Proliferator-Activated Receptors gamma (PPARG)with 16 variants genotyped;Fat Mass and Obesity associated protein (FTO) with 149 va-riants genotyped;Cyclin-dependent kinase 5 regulatory subunit associated protein1-like 1 (CDKAL1) with 186 variants genotyped.

Covariates: Age, sex, and the first two principal components ofthe entire genotype data of Baependi.


Summary of cases per subjects and families

Gene Subjects FamiliesControl Cases Total Control Cases Total

PPARG 845 83 928 43 42 85FTO 712 71 783 47 38 85

CDKAL1 661 69 730 47 38 85

bits.3.20GHz with a RAM memory 8.00 GB and operating system 64-R version 3.3.1 and a processor Intel(R) Core(TM) i5-6500 CPU @


Summary of cases per subjects and families

Gene Subjects FamiliesControl Cases Total Control Cases Total

PPARG 845 83 928 43 42 85FTO 712 71 783 47 38 85

CDKAL1 661 69 730 47 38 85

bits.3.20GHz with a RAM memory 8.00 GB and operating system 64-R version 3.3.1 and a processor Intel(R) Core(TM) i5-6500 CPU @


Sample size, GLMM parameters, p-values and execution times

Seminario References

Sección 7

References


References

Breslow, N. and Clayton, D, 1993. Approximate inference ingeneralized linear mixed models. J. Am. Stat. Assoc., 88, 9-25.Coombes, B., Basu, S. and Mcgue, M. , 2017. A combinationtest for detection of gene-environment interaction in cohort stu-dies. Genet. Epidemiol., 41, 396-412.Chen, H. and Conomos, M., 2016. GMMAT: Generalized Li-near Mixed Model Association Tests; R Package Version 0.7.Available online:https://content.sph.harvard.edu/xlin/dat/GMMAT_user_manual_v0.7.pdf(accessed on 16 January 2017).Lin, X., Lee, S., Chistiani, D. and Lin, X., 2013. Test for in-teractions between a genetic marker set and environment ingeneralized linear models. Biostatistics, 14, 667-681.


References

Lin, X., Lee, S., Wu, M., Wang, C., Chen, H., Li, Z. and Lin,X., 2016. Test for rare variants by environment interactions insequencing association studies. Biometrics, 72, 156-164.Lin, X, 1997. Variance component testing in generalised linearmodels with random effects. Biometrika, 84, 309-326.Oliveira, C., Pereira, A., de Andrade, M., Soler, J. and Krieger,J., 2008. Heritability of cardiovascular risk factors in a brazilianpopulation: Baependi heart study. BMC Med. Genet., 9, 32.Shen, X., Alam, M., Fikse, F. and Ronnegard, L., 2013. A novelgeneralized ridge regression method for quantitative genetics.Genetics, 193, 1255-1268.Zhang, D. and Lin, X, 2003. Hyphotesis testing in semiparame-tric additive mixed models. Biostatistics 2003, 4, 7-74.


References

Satterthwaite, F., 1941. Synthesis of variance. Psychometrika,6, 309-316.Fu, W.J., 2005.Nonlinear GCV and quasi-GCV for shrinkagemodels. Journal of Statistical Planning and Inference, 131, 333-347.Mazo, M. A., Coombes, B. and de Andrade, M., 2017. AnEfficient Test for Gene-Environment Interaction in GeneralizedLinear Mixed Models with Family Data. International JournalOf Environmental Research And Public Health, 14, 1134-1146.


References

Chen, H., Wang, C., Conomos, M., Stilp, A., Li, Z., Sofer,T., Szpiro, A., Chen, W., Brehm, J., Celedon, J., Redline, S.,Papanicolaou, G., Thornton, T., Laurie, C., Rice, K. and Lin,X., 2016. Control for population structure and relatedness forbinary traits in genetic association studies via logistic mixedmodels. The American journal of human genetics, 98, 653-666.Leal, S., Yan, K. and Muller-Myhsokb, B, 2005. SimPed: ASimulation Program to Generate Haplotype and Genotype 308Data for Pedigree Structures. Hum Hered., 119-122.


Family data and kinship matrix

Return


Ridge regression estimation

g(µd2) = Xβ + Gθ + d2 (d2 = Kb)

ql(β, θ, φR , σR) = −12 log

∣∣∣∣σ2RφR

(KKT )WR + In

∣∣∣∣+

N∑i=1

ni∑j=1

qlij(β, θ, φR ; d2) − 12 dT

2(σ2

RKKT)−1 d2,

where d2 is choosen to maximize the sum of the last two terms,WR = diag

{ωij/

[ν(µd2

ij )g ′(µd2ij )2

]}and

qlij(β,θ, φR ; d2) =∫ µ

d2ij

Yij

ωij(Yij − µ)φRν(µ) dµ.



g(µd2) = Xβ + Gθ + d2 (d2 = Kb)

ql(β, θ, φR , σR) = −12 log

∣∣∣∣σ2RφR

(KKT )WR + In

∣∣∣∣+

N∑i=1

ni∑j=1

qlij(β, θ, φR ; d2) − 12 dT

2(σ2

RKKT)−1 d2,

where d2 is choosen to maximize the sum of the last two terms,WR = diag

{ωij/

[ν(µd2

ij )g ′(µd2ij )2

]}and

qlij(β,θ, φR ; d2) =∫ µ

d2ij

Yij

ωij(Yij − µ)φRν(µ) dµ.



Ridge regression estimator of θ, is obtained by minimazing the fun-ction

[ql(β,θ, φR , σR)× φR ]− 12λθ

Tθ.

where λ is a penalizing factor.

XT WRX XT WR XT WRWRX WR + λ(GGT )−1 WR

WRX WR WR + φRσ2

R(KKT )−1

( βd1d2

)=

XT WRYWRYWRY

where d1 = Gθ. Return

Documents

Seminario - Testing gene-environment interaction in generalized … · Breslow, N. and Clayton, D, 1993. Approximate inference in generalizedlinearmixedmodels.J. Am. Stat. Assoc.,88,9-25