Whole Genome Regression : Lasso, Ridge, 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

2. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Contents
1 Limitation of GWAS or Linkage analysis
2 Introduction of WGP
©
3 Lasso estimation
4 Bayesian inference of Lasso
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

3. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Limitation
1 GWAS & Linkage : No consistent result ! poor prediction.
2 Complex traits : Overall eect (e.g:cardiovascular, cancer,
etc..).
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

4. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Example of GWAS
GWASÐ ìì genetic informationD ¨Ð ìhܤ”
ƒ@ ˆ¥.
1 1 trait VS 1 locus ! SNP /Ì| µÄÉ lä.
2 Ș ´ SNP informationD à$t| XÀ multiple
comparison p-value| t©XŒ ä. (ex: p-value cuto-
5 108)
3 Signi

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

11. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
©
Core context of WGP

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

13. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
©
1 Lasso@ Bayesian inferenceX uì Ь@ Dt´Ì Lt
ä.
2 Lasso (¤À| t©X0 pt0 ¬| ` ˆä.
3 „@ ÙT ! Ltü ø¼ Ý1.
4 Data@ phenotype …% ! |8Ð ]` Ltü ø¼!!
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

15. )T (~y X

16. ) s.t
Xp
j=1

17. 2
j t
$ minimize (~y X

18. )T (~y X

19. ) +
Xp
j=1

20. 2
j
(4)
Lasso
minimize (~y X

21. )T (~y X

22. ) s.t
Xp
j=1
j

23. j j t
$ minimize (~y X

24. )T (~y X

25. ) +
Xp
j=1
j

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(

28. 2) VS Abs(j

29. j)
3 0.04 VS 0.2 : ñt ôä ‘@

30. D 0 T ˜
ô¸ä.
4 t T pt, ‰ T Î@

31. äD 0 ô¸ä.
5 Lasso ridgeôä T Î@

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

34. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Choosing
K-fold cross validation pt0Ð k € n käD Àà
Modeling Ä tƒD kX sampleÐ ©Xì error
l Ä øƒäD ä Éà ƒD CV error| ä.
CV erroräX ÉàD ŒT X” lä.
(CV error)() = E((CV error)()
k ) (6)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

35. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
10 fold CV
Figure : 10 fold CV
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

36. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Bayesian inference
Introduction ! ThinkBayes X] gogo!!
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

37. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Lasso ! Bayesian Lasso

38. i j2
2 ej

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;
;

44. ; 2) =
Y
N(yi ji +
XJ
j=1
xij
j +
XL
l=1
zij

45. j ; 2) (9)
y = fyig;
= f
jg;

46. = f

47. lg
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

48. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Prior construction: hierarchial model
1 Intercept() sex, smoking, BMI(
) : vague
prior(non-informative)
2 Residual variance - standard assumption of bayesian regression
: scaled-inverse Chi-square density 2(2jdf ; S)
3 marker eect - bayesian Lasso
p(

49. ; Q 2; 2jH; 2) = p(

50. j 22)p( 2j2)p(2jr ; s)
= f
L
l=1 N(

51. l j0; 2
l 2)Exp( 2
l j2)gG(2jr ; s)
Jinseob Kim, MD, MPH Whole Genome Prediction Using Penalized Regression

52. Limitation of GWAS or Linkage analysis
Introduction of WGP
Lasso estimation
Bayesian inference of Lasso
Prior
p(;
; 2;

53. ; 2; 2jH) /
2(2jdf ; S)f
YL
l=1
N(

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

57. j ; 2)
YL
2(2jdf ; S)f
l=1
N(

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