Assignment 10.7Falconer & Mackay, chapter 10

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

From last lecture:

- to estimate genotypic variance, we ask how much less alike are DZ than MZ twins

VEc = variance due to common environment

VEw = variance due to unique environment (i.e., within-pair environmental variance)

- twice the difference between the MZ and DZ correlation = (VA + 1.5VD)/VP

Between pairs, varB Within pairs, varW Correlation (r= varB/VP)

MZ VA + VD + VEc VEw (VA + VD + VEc)/VP

DZ ½ VA + ¼ VD + VEc ½ VA + ¾ VD + VEw (½ VA + ¼ VD + VEc)/VP

Difference ½ VA + ¾ VD ½ VA + ¾ VD (½ VA + ¾ VD)/VP

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

2 4 6 8 10 12 14

02

46

810

Offspring

Phen

otyp

ic v

alue

μN

μL

μLB

μ

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

VA/VP + VEc/VP = 1

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

VA/VP + VEc/VP = 1

.5 + VEc/VP = 1

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

VA/VP + VEc/VP = 1

.5 + VEc/VP = 1

VEc/VP = 1 - .5

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

VA/VP + VEc/VP = 1

.5 + VEc/VP = 1

VEc/VP = 1 - .5

VEc/VP = .5

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .75

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .75VP

VW = VP (1 – .75)

VW = .25VP

.25VP = ½VA + ¾VD + VEw

½VA = .25VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .25VP

VA = .5VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.75 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .5 + 1.5

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

VA/VP + VEc/VP = 1

.5 + VEc/VP = 1

VEc/VP = 1 - .5

VEc/VP = .5

So:

VA/VP = .5VEc/VP = .5VD/VP = 0VEw/VP = 0

10.7. Show that if the correlation of full sibs is .75, the heritability of the character cannot be greater than .5. The full-sib correlation of weaning weight in mice is .8; what is the maximum heritability compatible with this?

corFS = .8

VW = ½VA + ¾VD + VEw

VW = VP – covFS

VW = VP – .8VP

VW = VP (1 – .8)

VW = .2VP

.2VP = ½VA + ¾VD + VEw

½VA = .2VP - ¾VD – VEw

h2 is max here if VD = VEw= 0:

½VA = .2VP

VA = .4VP , if VD = VEc= 0.

2(corMZ - corDZ) = (VA + 1.5VD)/VP

2corMZ – 2*.8 = VA/VP (because corFS=corDZ, and VD=0)

2corMZ = .4 + 1.6

corMZ = 1

corMZ = (VA + VD + VEc)/VP

corMZ = (VA + VEc)/VP

(VA + VEc)/VP = 1

VA/VP + VEc/VP = 1

.4 + VEc/VP = 1

VEc/VP = 1 - .4

VEc/VP = .6

So:

VA/VP = .4VEc/VP = .6VD/VP = 0VEw/VP = 0