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Lecture 4: Heritability. Heritability. Narrow vs. board sense. Narrow sense: h 2 = V A /V P. Slope of midparent-offspring regression (sexual reproduction). Board sense: H 2 = V G /V P. Slope of a parent - cloned offspring regression (asexual reproduction).

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slide2

Heritability

Narrow vs. board sense

Narrow sense: h2 = VA/VP

Slope of midparent-offspring regression

(sexual reproduction)

Board sense: H2 = VG/VP

Slope of a parent - cloned offspring regression

(asexual reproduction)

When one refers to heritability, the default is narrow-sense, h2

h2 is the measure of (easily) usable genetic variation under

sexual reproduction

slide3

Why h2 instead of h?

Blame Sewall Wright, who used h to denote the correlation

between phenotype and breeding value. Hence, h2 is the

total fraction of phenotypic variance due to breeding values

Heritabilities are functions of populations

Heritability values only make sense in the content of the population

for which it was measured.

Heritability measures the standing genetic variation of a population,

A zero heritability DOES NOT imply that the trait is not genetically

determined

slide4

Heritabilities are functions of the distribution of

environmental values (i.e., the universe of E values)

Decreasing VP increases h2.

Heritability values measured one one environment

(or distribution of environments) may not be valid

under another

Measures of heritability for lab-reared individuals

may be very different from heritability in nature

slide5

Heritability and the prediction of breeding values

If P denotes an individual's phenotype, then best linear

predictor of their breeding value A is

The residual variance is also a function of h2:

The larger the heritability, the tighter the distribution of true

breeding values around the value h2(P - mP) predicted by

an individual’s phenotype.

slide6

Heritability and population divergence

Heritability is a completely unreliable predictor of

long-term response

Measuring heritability values in two populations that

show a difference in their means provides no information

on whether the underlying difference is genetic

sample heritabilities

Traits more closely

associated with fitness

tend to have lower

heritabilities

Sample heritabilities
estimation one way anova

Trait value in sib j from family i

Deviation of sib j from the family mean

Common Mean

Effect for family i =

deviation of mean of i from

The common mean

Estimation: One-way ANOVA

Simple (balanced) full-sib design: N full-sib families,

each with n offspring

One-way ANOVA model:

zij = m + fi + wij

s2f = between-family variance = variance among family means

s2w = within-family variance

s2P = Total phenotypic variance = s2f + s2w

slide9

Covariance between members of the same group

equals the variance among (between) groups

The variance among family effects equals the

covariance between full sibs

The within-family variance s2w = s2P - s2f,

slide11

Since

Estimating the variance components:

2Var(f) is an upper bound for the additive variance

slide12

Nice fact: Under normality, the (large-sample) variance

for a mean-square is given by

)

(

)

(

Assigning standard errors ( = square root of Var)

slide13

-

-

-

Hence, h2 < 2 tFS

Estimating heritability

An approximate large-sample standard error

for h2 is given by

worked example

VA < 10

h2 < 2 (5/25) = 0.4

Worked example

10 full-sib families, each with 5 offspring are measured

estimation nested anova

Value of the kth offspring

from the kth dam for sire i

Effect of sire i = deviation

of mean of i’s family from

overall mean

Within-family deviation of kth

offspring from the mean of the

ij-th family

Effect of dam j of sire i = deviation

of mean of dam j from sire and overall

mean

Overall mean

Estimation: Nested ANOVA

Balanced full-sib / half-sib design: N males (sires)

are crossed to M dams each of which has n offspring

Nested ANOVA model:

zijk = m + si + dij+ wijk

s2s = between-sire variance = variance in sire family means

s2d = variance among dams within sires =

variance of dam means for the same sire

s2w = within-family variance

s2T = s2s + s2d + s2w

slide18

s2w = s2z - Cov(FS)

Estimation of sire, dam, and family variances:

Translating these into the desired variance components

• Var(Total) = Var(between FS families) + Var(Within FS)

• Var(Sires) = Cov(Paternal half-sibs)

slide19

Summarizing,

Expressing these in terms of the genetic and environmental variances,

slide20

4tPHS = h2

h2 < 2tFS

Intraclass correlations and estimating heritability

Note that 4tPHS = 2tFS implies no dominance

or shared family environmental effects

parent offspring regression

Single parent - offspring regression

The expected slope of this regression is:

Residual error variance (spread around expected values)

(

)

Shared environmental values

To avoid this term, typically regressions are male-offspring, as

female-offspring more likely to share environmental values

Parent-offspring regression
slide23

Midparent - offspring regression

Key: Var(MP) = (1/2)Var(P)

Residual error variance (spread around expected values)

)

(

(

)

The expected slope of this regression is h2, as

Hence, even when heritability is one, there is considerable spread

of the true offspring values about their midparent values,

with Var = VA/2, the segregation variance

standard errors

Single parent-offspring regression,

N parents, each with n offspring

Squared regression slope

Sib correlation

Total number of offspring

• Midparent-offspring variance half that of single parent-offspring variance

Standard errors

Midparent-offspring regression,

N sets of parents, each with n offspring

estimating heritability in natural populations

A lower bound can be placed of heritability using

parents from nature and their lab-reared offspring,

)

(

where

Covariance between breeding value in nature and BV in lab

is the additive genetic covariance between

environments and hence g2 < 1

Estimating Heritability in Natural Populations

Often, sibs are reared in a laboratory environment,

making parent-offspring regressions and sib ANOVA

problematic for estimating heritability

Why is this a lower bound?

defining h 2 for plant populations

Genotype i

Environment j

Effect of plot k for Genotype I

in environment j

deviations of individual

Plants within this plot

Interaction between Genotype I

and environment j

Number of Environments

Number of plots/environment

Number of individuals/plot

Defining H2 for Plant Populations

Plant breeders often do not measure individual plants

(especially with pure lines), but instead measure a plot or a block of individuals.

This can result in inconsistent measures of H2

even for otherwise identical populations.

If we set our unit of measurement as the average over all plots,

the phenotypic variance becomes

Hence, VP, and hence H2, depends on our choice of e, r, and n