Correlations Between CharactersIn “Genetics and Analysis of Quantitative traits”

byLynch, M. and Walsh, B.

PresentedSansak Nakavisut

Topics

•Covariance and Correlation

•Genetic Covariance

•Estimation of the Genetic Correlation

•Pairwise Comparison of Relatives

•Nested Analysis of Variance and

Covariance

•Regression of Family Means

Covariance

•Covariance measures how much 2 variables

vary together

• wt & age, age & grey hair; ADG & NBA

• If 2 variables vary in opposite direction, Cov

can be –ve eg. ADG & FCR

•Cov of a variable with itself = Variance

1

( )( )

1

n

i ii

XY

X X Y Y

n

2 1

( )( )

1

n

i ii

y

Y Y Y Y

n

Covariance & Variance

Animal ADG FCRTT2H0453 497.75 2.48 Cov(adg,fcr) -9.46TT2H0456 515.17 2.52 Var(adg) 3711.55TT2H0483 490.48 2.82 Var(fcr) 0.11TT2H0484 417.43 3.27TK1H0246 561.45 2.41TK1H0239 538.46 2.42TK1H0228 476.68 2.50TT2H0495 456.94 2.94TT1H7156 528.25 2.79TT1H7157 494.59 2.86TK1H0226 505.18 2.37TK1H0224 466.32 2.35TK3H0031 548.85 3.03TK3H0030 548.85 2.83TK1H0283 584.80 2.56TK3H0027 589.08 2.46TK3H0028 522.99 2.63TK1H0284 523.39 2.71TK1H0287 497.21 2.92TK1H0290 565.48 2.67

Example

y = -0.0025x + 3.9805R2 = 0.2103

0

1

2

3

4

5

0 200 400 600 800 1000

Correlation

2 2

XY XY

X Y X Y

r

A measure of the strength of a bivariate linear relationship–1 < r < +1

Correlation & Regression

2,

XY XY

X Y X

X

Y

r b

r b

Properties of covariance

• "The expected value of the cross product"

•Cov(a,Y) = 0

•Cov(aX,Y) = aCov(X,Y)

•Cov(X+W,Y) = Cov(X,Y) + Cov(W,Y)

•Cov(X,X) = Var(x)

•Cov(a+X,Y) = Cov(a,Y) + Cov(X,Y) = Cov(X,Y)

•Cov(X,Y) = Cov(Y,X)

Correlations between Characters

•Phenotypic correlations ie height & feet size

•Environmental correlations

•Genetic correlations pleiotropy gametic

phase disequilibrium

Genetic covariancei1α

j1αl1α

k1α

G1

i2α

j2αl2α

k2α

G21 ,1 ,1 ,1 ,1 ,1 ,1[ ] [ ]......

(21.1 ) 630

G i j k l ij kl

a page

2 ,2 ,2 ,2 ,2 ,2 ,2[ ] [ ]......

(21.1 ) 630

G i j k l ij kl

b page

2 2 2

2 2 2

(1) (1) (1)

(1) (1) (1) ...(21.2 )

G A D

AA AD DD a

2 2 2

2 2 2

(2) (2) (2)

(2) (2) (2) ...(21.2 )

G A D

AA AD DD b

G A D AA AD DD

A i,1 i,2 j,1 j,2 k,1 k,2 l,1 l,2

σ (1,2)=σ (1,2)+σ (1,2)+σ (1,2)+σ (1,2)+σ (1,2).....(21.3)

where

σ (1,2)=σ(α ,α )+σ(α ,α )+σ(α ,α )+σ(α ,α ).....

Estimation of the genetic correlation

•Three methods

•Pairwise Comparison of Relatives

•Nested Analysis of Variance and Covariance

•Regression of Family Means

•Extra method not in the book

Pairwise Comparison of Relatives

•Data from pairs of relatives

• ie mid-parent (x) values for Trait1 and Trait2

•And progeny means (y) for Trait1 and Trait2

•Four phenotypic Cov. can be computed

• Cov(x1,y1); Cov(x2,y2) >>> heritabilities T1&T2

• Cov(x1,y2); Cov(x2,y1) >>> rg(1,2)

Pairwise Comparison of Relatives2 2A AA

1x 1y

2 2A AA

2x 2y

A AA1x 2y

A AA2x 1y

σ (1) σ (1)σ(z ,z )= + +... (21.5a)

2 4

σ (2) σ (2)σ(z ,z )= + +... (21.5b)

2 4σ (1,2) σ (1,2)

σ(z ,z )= + +... (21.5c)2 4

σ (1,2) σ (1,2)σ(z ,z )= + +... (21.5d)

2 4

2(1 , 2 ) 2 (1,2) (1,2) (2 ) (1,2) 2 (1,2)... (21.4)G x y xy A xy D xy AA xy xy AD

Genetic correlation2 2A AA

1x 1y

2 2A AA

2x 2y

A AA1x 2y

A AA2x 1y

σ (1) σ (1)σ(z ,z )= + +... (21.5a)

2 4

σ (2) σ (2)σ(z ,z )= + +... (21.5b)

2 4σ (1,2) σ (1,2)

σ(z ,z )= + +... (21.5c)2 4

σ (1,2) σ (1,2)σ(z ,z )= + +... (21.5d)

2 4

1x 2y 2x 1y

1x 1y 2x 2y

σ(z ,z )+σ(z ,z )

2 σ(z ,z ).σ(z ,z )A

Example from my real datasiredam mid-parent-ADG mid-parent-FCR progeny-mean-ADG progeny-mean-FCR

CH1I1047TP1I1742 567.07 2.75 653.57 2.60CH1I1047TT1H7308 561.17 2.70 639.29 2.67CH1I1075TP1I1705 570.33 2.73 570.51 2.20

CH1J3131CH1J2419 542.89 2.58 540.86 2.59CH1J3131MK1J1338 536.42 2.99 542.19 2.28CH1J3131TT1K1236 585.11 2.98 623.38 2.41

CH1K4469CH1K3705 660.03 2.72 627.03 2.27CH1L4988CH1K3702 568.64 2.20 580.44 2.55CH1L4988CH1K3705 638.92 2.54 599.51 2.32CH1L4988CH1L5035 650.58 2.25 661.91 2.04CH1L4988CH1L5044 598.89 2.17 605.61 2.21CH1L4988CH1L5174 621.30 2.18 577.73 2.32CH1L4988CH1M5562 659.35 2.22 607.63 2.36

ADG-midP FCR-progM FCR-midP ADG-progM

ADG-midP ADG-progM FCR-midP FCR-progM

σ(z ,z )+σ(z ,z )

2 σ(z ,z ).σ(z ,z )Ar

cov(1x,2y) = -4.07cov(2x,1y) = -3.38

cov(1x,1y) = 1347.60cov(2x,2y) = 0.02

rA = -0.65

Estimate of additive genetic correlation between ADG & FCR

Nested Analysis of Var and Cov

•Nested full-sib and half-sib designs (Ch 18)

•Provide nested analysis of genetic variance

•Mean squared deviations of individual traits

•A parallel analysis > add. genetic covariance

•Mean cross-products of the deviations of

traits 1 and 2 rather than MS

Full-sib designT1 T2

T1 T2

T1 T2

T1 T2

T1 T2

T1 T2

1

2

11

12

T1 T2

T1 T2

T1 T221

Half-sib design

T1 T2

T1 T2

1

2

11

12

T1 T221

Analysis of Variance (half-sib)

Factor df SS MS E(MS)

Sire N-1 SSs/(N-1)

Within sire T-N SSs/(T-N)

Total T-1 SSt(T-1)2

1 1

( )N n

iji j

z z

2

1 1

( )N n

ij ii j

z z

2

1

( )N

i ii

n z z

2z

2e

2 20e sn

Analysis of Covariance (half-sib)

Factor df Sum cross-prod. MCP E(MCP)

Sire N-1 SCPs/(N-1)

Within sire T-N SCPe/(T-N)

Total T-1 SCPt(T-1)

1 1 2 21 1

( )( )N n

ij iji j

z z z z

1 1 2 21 1

( )( )N n

ij i iji j

z z z z

1 1 2 21

( )( )N

i i ii

n z z z z

(1,2)z

(1,2)e

(1,2) 0 (1,2)e sn

ANOVA (half-sib) ADG & FCR

Factor df SS MS E(MS)Sire 650 4210198 6477Within sire 1094 2387933 2182Total 1744 6598132 2

z

2e

2 20e sn

Factor df SS MS E(MS)Sire 650 116 0.180Within sire 1094 94 0.087Total 1744 211 2

z

2e

2 20e sn

ADG

FCR 0( 2.68)n

Analysis of Cov(ADG,FCR) (half-sib)

Factor df S cross-prod. MCP E(MCP)

Sire 650 -10715.1 -16.48Within sire 1094 -6605.7 -6.04Total 1744 -17320.8 -9.93 (1,2)z

(1,2)e(1,2) 0 (1,2)e sn

21 1603s

22 0.035s

(1,2) 3.88s

(1,2)

2 21 2

0.52sa

s s

r

Regression of Family means

1 2

1, 2,

( , )

( ) ( ).i i

i i

z za z

z z

r r

•Correlation between family mean phenotypes

•The Family size , the sampling errors

•Family mean phenotype Family mean genotype value

Regression of family means in practice

1, 2

1, 2,

( , )

( ) ( )

-7.140.47

. 52.78 0.29i i

i i

z z

a zz z

r r

siredam family-mean-ADG family-mean-FCRCH1I1047TP1I1742 653.57 2.60CH1I1047TT1H7308 639.29 2.67CH1I1075TP1I1705 570.51 2.20CH1J3131CH1J2419 540.86 2.59CH1J3131MK1J1338 542.19 2.28CH1J3131TT1K1236 623.38 2.41CH1K4469CH1K3705 627.03 2.27

This is how we do it now (REML)correlation between ADG & FCR Anim !P Sire !P Dam !P ADG FCR

chapter21.ped !ALPHA

data.dat !MAXIT 30

ADG FCR ~ Trait !r Tr.Anim

1 2 10Tr 0 US1 0.1 1Tr.Anim 2Tr 0 US1 0.1 1Anim

h1 = 0.5998 0.0198 h2 = 0.5109 0.0229 rp = -0.4391 0.0111

rg = -0.4001 0.0302

THE END