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Whole Genome Regression using Bayesian Lasso
1. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Whole Genome Prediction Using Penalized
Regression
Bayesian Lasso
Jinseob Kim, MD, MPH
GSPH, SNU
February 27, 2014
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
5. cant SNPÌD Á ¨D l1 or combine
information via Genetic Risk Score
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
6. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Problem
1 Multiple comparison ! Power...
2 SNP X˜) trait@ „ ! LD information...
3 What is Genetic Risk Score??? €U À..
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
7. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Why this problem?
øå ä #à ŒÀ„Xt H L???
1 Multicolinearity issue!!! ! LD: similar allele information
2 n p issue: ‰, ¬Œôä À(SNP)/ Ît
ŒÀÄ ”t H(.
ŒÀÄX „°(variance)t 4 äÄä..... ”ˆ..
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
8. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Why?
9. ”ÉX unbiaseness| ì0XÀ JX0 L8tä.
Variance-bias trade-o!!
(a) (b)
Figure : Summary of variance-bias tradeo
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
10. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Variance-bias tradeo
Y = f (x) + , N(0; e ), ^f : estimate of f | L
Err (x) = E[(Y ^f (x))2] (1)
Err (x) = (E[^f (x) f (x)])2 + E[^f (x) E[^f (x)]]2 + e (2)
Err (x) = Bias2 + Variance + Irreducible error (3)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
12. unbiased estimator „D ì0ä!!!
1 WGP can use all available markers to regress phenotype onto
genomic information.
Ridge regression
Lasso (Least absolute shrinkage and selection operator)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
14. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Ridge VS Lasso
Ridge regression
minimize (~y X
26. j j
(5)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
27. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Ridge VS Lasso(2)
1 P )• ¨P Î@ betaäD 0 ô¸ä. äõ 1
t°, LD information .
2 Square(
32. äD 0 ô¸ä.
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
33. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Ridge VS Lasso(3)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
39. i j= : Laplace prior
1 The Laplacian prior assigns more weight to regions near zero
than the normal prior.
2 Interpretated as mixture of the hierarchical priors (Normal +
exponential)
a2
eajzj =
R 1
0
p1
2s
ez2=2s a2
2 eas2=2ds, a 0
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
40. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Laplace prior
Figure : Normal VS Laplace prior
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
41. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Example: Continuous case
Whole model
i = +
XJ
j=1
xij
j +
XL
l=1
zlj
42. j (7)
Likelihood
p(yi ji ; 2) = (22)1
2 expf
(yi i )2
22
g (8)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
43. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Likelihood
p(yj;
;
54. l j0; 2
l 2)Exp( 2
l j2)gG(2jr ; s)
(10)
H = fdf = 5; S = 170; = 1 104; s = 2g : For priors with small
in
uences on predictions
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
55. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Posterior
p(;
; 2;
56. ; 2; 2jy) /
Y
N(yi ji +
XJ
j=1
xij
j +
XL
l=1
zij
58. l j0; 2
l 2)Exp( 2
l j2)gG(2jr ; s)
(11)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
59. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Implementation
1 BLR(Bayesian Linear Regression) package in R
2 bayesm, splines and SuppDists for sampler
! BGLR(Bayesian Generalized Linear Regression) package in R
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
60. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Goodness of
61. t, DIC
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
62. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
äµ
BGLR package äµ : continuous trait (TG) binomial traint
(hyperTG)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
63. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
üX¬m
1 ø¬ À1È(conti VS categorial) À.
2 Lasso ø À(genotype)@ øå À(age)| l„
3 t` Lasso Ð ä´ xä@ ¨P T´|
ä. Àt õÉXŒ !´| X0 L8tä. Ș
allele count” 4pt 0,1,2tÀ ÁÆL.
4 Missingt Æ´| ä. GWAS” Missing |à LD
Ä°tüÀÌ BGLR@ øÀ Jä. Œä prediction
modeltÀ TT± xÐ missing Æ´| h: Imputation or
mean allele count.
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
64. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
üX¬m2
Validation ` ƒt|t
1 P SetX õµ SNPÌ !¨ l1Xì| ä.
2 P SetX allele count reference Ù|Xì| ä.
3 P SetÐ ¨P tù traitt ˆ´| ä.
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression
65. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
]
HP: 010-9192-5385
E-mail: secondmath85@gmail.com
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression