バイオメトリクス授業資料 iwatawiki lec06 s

(1)

060310391 0560565

6 2015/11/6 13:00-14:45

@1 - 4

1

(associa4on analysis)

(associa4on study)

……T……C……A……G……T………A…………A………

• 

DNA

……T……C……A……A……T………A…………A………

……T……A……G……G……T………A…………A………

……G……C……A……A……T………C…………T………

……T……A……A……G……T………A…………A………

……T……C……A……A……T………A…………A………

……T……C……A……G……T………A…………T………

……T……A……G……G……T………C…………A………

……G……C……A……G……C………A…………T………

……T……A……A……G……C………A…………A………

……T……C……A……G……C………C…………T………

……T……A……A……A……C………C…………A………

……T……C……G……G……C………A…………T………

……G……A……A……G……C………A…………A………

……T……C……A……A……C………C…………A………

A

B

C

:

: :

DNA

200

70/80

23/120

200

2

AD2 T( Ab N 1 73 7 0

0

T

0

C

• •  ^QTL

• •  ^GWAS

•  χ

²

•  Fisher

• 

• •  ¹ ²

• 

3

•  ³

AA, Aa, aa 3

^30,000

• 

2005

4

(2)

QTL

• DNA

QTL

• 

×

6

• DNA

SNP

• 

ATCGAG TAGACT TATACG

ATCGAG TAGACA TATACG

7

(single nucleo4de polymorphisms: SNPs)

GeneChip Rice 44K SNP Genotyping Array

•  44,100 SNPs (10kb 1 SNP)

Tung et al. (2010) Rice 3:205-217 ⁸

QTL

DNA

GWAS

DNA

Morrell et al. (2012) Nature Review Gene4cs 13:85

QTL vs.

(3)

10

QTL

_Yes _No

linkage disequilibrium: LD

11

QTL

(Modiﬁed from Balding 2006)

SNP

( )

SNP

( )

SNP

13

(4)

(Rafalski 2002

●^B ●^b

●^A ^pAB ^pAb ^pA

●^a ^paB ^pab ^pa

p_B p_b

r²= ^D

2

p_Ap_ap_Bp_b

(●● 1,●● 0

)

D = p

_AB

− p

_A

p

_B

= p

_AB

p

_ab

− p

_Ab

p

_aB

r²⁼ ^0.25

2

0.5× 0.5 × 0.5 × 0.5

= 1

AB

r²^{= 0.1024} r²^{= 0}

A B

14

• 

(Rafalski 2002 )

15

LD

Gupta et al. (2005)

• 

16

LD

(Zhu et al. 2007)

LD

→ LD

↑

↓

17

(5)

AA

aa

AA aa

χ² Fisher Fisher’s exact test

18

χ

²

(R) (S)

AA f11 (11) f12 (4) f1. (15) aa f21 (3) f22 (7) f2. (10) f.1 (14) f.2 (11) n (25)

p(R) = f.1 / n = 14 / 25 = 0.56 p(S) = f.2 / n = 11 / 25 = 0.44 AA p(AA) = f1. / n = 15 / 25 = 0.60 aa p(aa) = f2. / n = 10 / 25 = 0.40

↓ R AA n × p(R) × p(AA) = 25 × 0.56 × 0.60 = 8.4 R aa n × p(R) × p(aa) = 25 × 0.56 × 0.40 = 5.6 S AA n × p(S) × p(AA) = 25 × 0.44 × 0.60 = 6.6 S aa n × p(S) × p(aa) = 25 × 0.44 × 0.40 = 4.4

χ²= ^(obs^{− exp)}

2

∑

exp ⁼^{(11− 8.4)}²

8.4 ⁺

(3− 5.6)²

5.6 ⁺

(4− 6.6)²

6.6 ⁺

(7− 4.4)² 4.4 ^{= 4.57}

(r-1)(c-1) χ² r c

5%

χ_0.01² (1) = 6.63 >χ²= 4.57 >χ_0.05² (1) = 3.84

5

19

-4 -2 0 2 4 6 8

mQQ mqq

QTL 㑇ఏᏊᆺ

QQ qq

࣐ 勖

࢝ 勖㑇 ఏ Ꮚ ᆺ

AA

aa

40 10

10 40

-4 -2 0 2 4 6 8

₯ᅾⓗ࡞ΰྜศᕸ

-4 -2 0 2 4 6 8

࣐࣮࣮࢝

⾲⌧ᆺศᕸ

mAA

-4 -2 0 2 4 6 8

maa

y _i = u + β _j x _ij + e _i

x y

-0.2 0.2 0.6 1.0

-20246

Marker genotype

Phenotype

i AA x_i=2

aa x_i=0

x_i yi

N0 0

( ( CA

0 2 20

-3 _-2 _-1 0 ₁ ₂ 3

455055

x

y

_i

= α + β x

_i

+ ε

_i

= y _ˆ

_i

+ ε

_i

y

dependent (response) variable

independent (explanatory) variable

SNP

β

regression coeﬃcient

ε

residuals y_i

ˆ

y _i=α+βx_i

x_i εi

α

intercept, constant term y

21

(6)

The method of least squares

ε

_i

= y

_i

− (α + βx

_i

)

( 2

SSE ⁼ ^εi 2

i n

∑

⁼ ^(yⁱ⁻^{α − βx}ⁱ⁾²

i n

∑

:

∂SS_E

∂β ⁼⁻² _i ^(yⁱ⁻^{α − βx}ⁱ^)xⁱ

n

∑

^{= 0}

∂SS_E

∂α ⁼⁻² _i ^(yⁱ⁻^{α − βx}ⁱ⁾

n

∑

^{= 0}

a = y_i

i

∑

n ⁿ^{− b} _i^xi

∑

n n = y − bx b =

x_iy_i−

i

∑

n _i^xi

∑

n _i^yi

∑

n ⁿ

x_i²− x_i

i

∑

n

( )

² ⁿ

i

∑

n ⁼

(x_i− x )(y_i− y )

i

∑

n

(x_i− x )²

i

∑

n

n x_i

i

∑

n

x_i

i

∑

n ^xi 2 i

∑

n

#

$

%

&

' ( ( a b

#

$ %

& ' ( = ⁱ^yⁱ

∑

n

x_iy_i

i

∑

n

#

$

%

&

' (

-3 _-2 _-1 0 ₁ ₂ 3 (

455055

x

y

y_i ˆ

y _i=α+βx_i

x_i εi

22

(a b α β

y _i = u + β _j x _ij + e _i

1311

SNPs

All materials can be downloaded from hkp://ricediversity.org/

SNPs …

0e+00 1e+08 2e+08 3e+08

05102030

position (bp)

−log10(p)

LD

Rafalski and Morgante 2004 Oraguzie et al. 2007) 25

(7)

p 

… suppose that a would-be gene7cist set out to

study the “trait” of ability to eat with chops7cks in the

San Francisco popula7on by performing an

associa7on study with the HLA complex. The allele

HLA-A1 would turn out to be posi7vely associated

with ability to use chops7cks … because the allele

HLA-A1 is more common among Asians than

Caucasians.”

Lander and Schork (1994)

HLA Human Leukocyte Antigen

26

?

1

2

(Modiﬁed from Balding 2006)²⁷

•  Indica

Japonica

y _i = u + β _j x _ij + v _k q _ik

k=1

K

∑ ^{+ e} ⁱ

1.0

0.0 0.5

Bayesian

Structure Pritchard et al. (2000) Gene4cs 155:945– 959

4

6 K=6

2

q_i1 = 0.78, q_i3 = 0.32, 0

(8)

0e+00 1e+08 2e+08 3e+08

0510152025

position (bp)

−log10(p)

…

→ ³¹

A

•  Yu et al. (2006) Nat. Genet. 38: 203

y _i = u + β _j x _ij + v _k q _ik

k=1

K

∑ ⁺ ^α ⁱ ^{+ e} ⁱ

a ~ N (0, Aσ _α ² )

var(α

_i

) = a(i, i) σ

_α²

cov( α

i

, α

i_'

) = a(i, i ') σ

_α²

→

0e+00 1e+08 2e+08 3e+08

01234

position (bp)

−log10(p)

(9)

1SNP

SNP

A A

B B

B A

34

B A

SNPs

SNP

A

A B

B ^A

35

B

Bayesian

QTL

y _i = u + β _j γ _j x _ij

j=1

J

∑ ⁺ ^v ^k ^q ^ik

k=1

K

∑ ⁺ ^α ⁱ ^{+ e} ⁱ

•  Iwata et al. (2007) Theor Appl Genet 114:1437–1449

γ

_j

= ¹

0 ! "

#

j SNP

γ

_j

~ B(1, p

_j

)

0e+00 1e+08 2e+08 3e+08

0.00.20.40.60.81.0

position (bp)

Posterior prob. of QTL

Bayesian

→

γ j

(10)

…

Chr. 3

GS3

63 kb

id3008333

546 kb

id3008127

Chr. 5

qSW5

id5002699

0.95 1.0

66 kb 0.88

0e+00 1e+08 2e+08 3e+08

0.00.20.40.60.81.0

position (bp)

Posterior prob. of QTL

SNPs

GWAS & GS

GWAS

GS

• 

•  NA

• 

•  ^NA

(11)

•  AC 0

Q, K

•  Q

–  Structure (Pritchard et al. 2000)

–  Price et al. 2006)

–  Zhu and Yu 2009)

•  K:

–  Loiselle et al. (1995),

Ritland et al. (1996))

–  Zhao et al. 2007

– 

43

GWAS

Atwell et al. (2010) Nature 465: 627

→ a

⁴⁵

(12)

A B (

C ( D

false posi4ve rate: FPR = B / (A + B)

false nega4ve rate: FNR = C / (C + D)

false discovery rate: FDR = B / (B + D)

⁴⁶

•  FDR: false discovery rate

1.  K ^P

2.  P

3.  ^FDR ^q* ^5%

i

4.  1 ⁱ

Benjamini and Hochberg (1995)

) ( )

2 ( ) 1

(

P P

_K

P ≤ ≤ ^! ≤

K

P

_i

iq

* )

(

≤

•  ^p

•  ^K ^P

p K × P

• 

P

_(i)

^≤ ^iq

*

K ^{⇔ q}

*

_≥ ^KP

(i)

i

•  q^* ⁱ

K × P

•  ^q^* ^5%

GWAS

49

•  GWAS Hd6

Yano et al. (2016) Nature Gene4cs 48: 927

(13)

Yano et al. (2016) Nature Gene4cs 48: 927 51 ⁵³

Yang et al. (2003) Am. J. Hum. Genet 73: 627

•  ACTN3 ^α-c4nin-3

•  ^X ^α-c4nin-3

•  a-ac4nin-3 ^a-ac4nin-2 ^ACTN3

(14)

126,559 GWAS

•  Fig. 1 3 ^SNPs

Rietveld et al. 2014, PNAS 13790, Ward et al. 2014 PLoS ONE e100248)

•  ^R² ^0.02%

•  ^Fig. 2

54

Rietveld et al. (2013) Science 340: 1467

• 

• •  ^DNA

QTL

55

• 

• QTL

• •  ^:

(2000/01)

•  ISBN-10: 4130602063

•  ISBN-13: 978-4130602068

56

hkp://www.amazon.co.jp/

6

1.  ¹⁰⁰ ²

A, B AABB,

aaBB, AAbb, aabb 40, 10, 10, 40

AB D r²

2.  ^AB

AB 1

57

χ

_0.01²

(1) = 6.63, ^χ

0.05

2

(1) = 3.84

(15)

PDF MS-Word PDF

e

[email protected]

2016 11 14

58

Fisher

Fisher’s exact test

• “ ”

• 

59

R S

AA a c g

aa b d h

e f n

e, f, g, h P(e, f ,g,h) = ⁿ

e

"

# $ ^%

& ' p^e^q^f× ⁿ

g

"

# $ ^%

&

' r^g^s^h⁼ ^(n!)

2

e! f !g!h!^p

eq^fr^gs^h

p = e/n, q = f/n, r = g/n, s = h/n

a, b, c, d e, f, g, h

P(a,b,c, d e, f ,g,h) = p(a,b,c,d,e, f ,g,h) / p(e, f ,g,h) =^{e! f !g!h!} n!

1 a!b!c!d!

e, f, g, h

a, b, c, d P(a,b,c, d,e, f ,g,h) = ⁿ

e

"

# $ ^%

& ' p^e^q^f × ^e

a

"

# $ ^%

& ' r^a^s^b× ^f

c

"

# $ ^%

& ' r^c^s^d ⁼ ^n!

e! f ! e! a!b!

f ! c!d!^p

eq^fr^gs^h

60

P(B)

P(A ∩ B)

P(A | B) = P(A ∩ B) /P(B)

a,b,c,d

11 4

3 7

R S

AA 11 4 15

aa 3 7 10

14 11 25

p_x

6

p_x=14!11!15!10! 25!

1 11!3!4!7!

"

# $ ^%

& ' = 0.0367

Fisher

χ²

5%

12 3

2 8

13 2

1 9

14 1

0 10

10 5

4 6

9 6

5 5

8 7

6 4

7 8

7 3

6 9

8 2

5 10

9 1

4 11

10 0

p =14!11!15!10! 25!

1 11!3!4!7!⁺

1 12!2!3!8!⁺

1 13!1!2!9!⁺

1 14!0!1!10!⁺

1 5!9!10!1!⁺

1 4!10!11!0

"

# $ ^%

& '

= 0.0486

p_x

61

(16)

-3 _-2 _-1 0 ₁ ₂ 3

455055

-3 _-2 _-1 0 ₁ ₂ 3

455055

-3 _-2 _-1 0 ₁ ₂ 3

455055

y

_i

ˆ

y

_i

y

(y_i− y )²

i

∑

n ⁼ ^{( ˆ}^yi^{− y )} 2 i

∑

n ⁺ ^(yi^{− ˆ}^yi⁾ 2 i

∑

n

(a)

SS_Y

(a)

(b)

(c)

(b)

SS_R

(c)

SS_E F

(b)

SS_R 1 MS_R =

SS_R

MS_R/MS_E

(c)

SS_E n - 2 MS_E = SS_E /(n-2)

(a)

SS_Y n - 1

F

y

_i

y

ˆ

y

_i

(

62

J = (yi−α − βxi)²

i=1

∑

N

y

_i

= α + β x

_i

+ ε

_i

^ε

ⁱ

^{~ N(0,} ^σ

²

⁾

0 σ²

y

_i

~ N( α + β x

_i

, σ

²

)

^y ^α+βx ^σ²

L = p(y₁, y₂,..., yN) = ^exp{−(yⁱ^{−α − βx}ⁱ⁾

2_/2σ2

} 2πσ²

i=1 N

∏

⁼^exp{− ^(yⁱ^{−α − βx}ⁱ⁾

2_/2σ2 i=1

∑

N ^}

(2πσ²)^N

0.00.10.20.30.4

α + βxi y_i

p(y_i) = ¹ 2πσ²^exp⁻

(yi−α − βxi)² 2_σ² ' ( )

* + ,

_{ln L = −} ¹

2σ² ^(yⁱ^{−α − βx}ⁱ⁾

2 i=1

∑

N ⁻^N₂^log2πσ²

∂ln L

∂β ⁼⁻ 1

σ² _i^(yⁱ⁻^α⁻^β^xⁱ^)xⁱ

n

∑ ^{= 0}

∂ ln L

∂α ⁼⁻ 1

σ² _i^(yⁱ^{−α − βx}ⁱ⁾

n

∑

^{= 0}

0 σ²

₆₃

8a

バイオメトリクス授業資料 iwatawiki lec06 s

060310391 0560565

6

2015/11/6 13:00-14:45

@1 - 4

(associa4on analysis)

(associa4on study)

70/80

23/120

•

• QTL

•

• GWAS

• χ

• Fisher

•

•

• 1 2

•

• 3

AA, Aa, aa 3

•

•

QTL

×

(single nucleo4de polymorphisms: SNPs)

QTL vs.

linkage disequilibrium: LD

D = p

− p

p

= p

p

− p

p

•

•

LD

LD

χ

∑

y i = u + β j x ij + e i

y

= α + β x

+ ε

= y ˆ

+ ε

The method of least squares

ε

= y

− (α + βx

)

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

( )

∑

∑

∑

∑

∑

∑

∑

∑

y i = u + β j x ij + e i

1311

SNPs

SNPs …

LD

… suppose that a would-be gene7cist set out to

study the “trait” of ability to eat with chops7cks in the

San Francisco popula7on by performing an

associa7on study with the HLA complex. The allele

• 

•  ^QTL

• 

•  ^GWAS

•  χ

•  Fisher

• 

• 

•  ¹ ²

• 

•  ³

• 

• 

• 

• 

y _i = u + β _j x _ij + e _i

= y _ˆ

y _i = u + β _j x _ij + e _i

•  Indica

y _i = u + β _j x _ij + v _k q _ik

∑ ^{+ e} ⁱ

•  Yu et al. (2006) Nat. Genet. 38: 203

y _i = u + β _j x _ij + v _k q _ik

∑ ⁺ ^α ⁱ ^{+ e} ⁱ

a ~ N (0, Aσ _α ² )

y _i = u + β _j γ _j x _ij