We're told that intelligence is 60-80% heritable, and that personality is 40-60% heritable. In some hand-wavy way, we know that heritability captures the
nature side of the
nature-nurture contribution to traits.
But what does
heritability really mean? It's a rather slippery concept. We'll get there by stages.
1. The contribution of genes to a phenotype
Let's take height as our running example (pretty much the same heritability as intelligence). Let's take a person with height P (P stands for phenotype - the measured trait). P is measured in inches away from the population mean height.
How did a person get to be that height? Nature and nurture, right?
We assume that the alleles the person got from their father contributes X
father inches of height, X
mother counts the inches they received from their mother's alleles they inherited, and then there is a nurture - or environmental - term E inches. So their total height,
P = Xfather + Xmother + E.
Note these are genetic additive effects: each additional allele is plausibly assumed to make its independent contribution into raising or lowering X a fraction. Dominance and
epistatic effects are neglected in this simplified conceptual model (in a polygenic trait, they tend not to be large).
Since we're measuring deviations from the mean, the average values across the population of X
father, X
mother and E must all be zero. And so, therefore, must be the average value of P.
So without loss of generality, we assume X
father, X
mother and E are normally distributed random variables with mean zero and variances as follows:
Var(Xfather) = Vadditive/2 -- each parent provides half the additive genetic 'input'
Var(Xmother) = Vadditive/2 -- each parent provides half the additive genetic 'input'
Var(E) = Venvironment.
So what is Var(P), the variance of height as we observe it in the population?
Var(P) = Var(Xfather) + Var(Xmother) + Var(E) +
2Cov(Xfather, Xmother) + 2Cov(Xfather, E) + 2Cov(Xmother, E).
Messy, but if we assume X
father, X
mother and E are independent, their covariances are zero, so
Var(P) = Var(Xfather) + Var(Xmother) + Var(E),
Vphenotype = Vadditive + Venvironment
The fraction of the population phenotypic variation due to genetic, additive effects is then simply
h2 = Vadditive/Vphenotype = Vadditive/(Vadditive + Venvironment).
This is the definition of heritability, h2.
So if h
2 is 0.5, then 50% of the variance in the phenotype is genetic in origin (additive-genetic, that is) and 50% is environmental (everything else).
Note that the more you reduce environmental variance, for example making sure that everyone's well-fed, properly educated and not knocked about, the more genetic differences predominate .. and heritability goes up. Not what the SJWs really want to hear!
---
2. Correlations
What is the correlation, ρ, between a parent and child for height?
If we have two random variables, A and B, the correlation between them is defined as follows:
ρ = Cov(A,B)/√(Var(A) * Var(B)).
This is the standard definition.
In the case of one parent and their offspring, under some simplifying assumptions,
Cov(parent,offspring) = Vadditive/2
- this takes a few lines to work out, setting most of the X
father, X
mother and E cross-terms to zero. It reflects the 50% of genetic material they have in common.
More obviously,
Var(parent) = Var(offspring) = Vphenotype,
So using the formula for ρ above,
ρ = (Vadditive/2) / Vphenotype = h2/2.
This shows that heritability
is not the same as the correlation between a child and one of its parents.
In general, the correlation, ρ, on a trait between relatives is equal to the
coefficient of relatedness times the heritability, ie
ρ = rh2.
---
3. Predictions
If we know the height of both the parents, what's our best prediction of the height of their offspring? In our mind, we draw the best-fit regression line on the scatter-plot of parental-midpoint and offspring heights measured across the population.
If we centre the graph-axes at the mean values of the two populations (parental mid-point heights and offspring heights) then the regression line goes through the origin, with slope β. Then the equation of the regression line takes this simple form:
predicted-offspring-height = β * parental-midpoint-height
with both heights measured as inches in deviation from the respective means.
How do we compute β?
In this special case it turns out that β equals the heritability, so
β = h2. *
This should remind you of the
Breeder's Equation.
---
Example: suppose the heritability of height is 0.673 and we know that one parent is 3 inches above the population mean while the other parent is 1 inch above the mean, what's the predicted (expected) height deviation from the mean for their child?
Answer: predicted-offspring-height = β * (3 + 1)/2 = 2h2 = 1.35 inches.
Yes, the child has regressed towards the mean.
---
This is problem 6.3 (p. 149) from '
Population Genetics: a concise guide' by John H. Gillespie, from which all the material above has been summarised.
---
* In general,
β = ρ * (σy/σx) where x is the independent variable.