Heritability, correlation and prediction

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 = X_father + X_mother + 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(X_father) = V_additive/2    -- each parent provides half the additive genetic 'input'

Var(X_mother) = V_additive/2   -- each parent provides half the additive genetic 'input'

Var(E) = V_environment.

So what is Var(P), the variance of height as we observe it in the population?

Var(P) = Var(X_father) + Var(X_mother) + Var(E) +

        2Cov(X_father, X_mother) + 2Cov(X_father, E) + 2Cov(X_mother, E).

Messy, but if we assume X_father, X_mother and E are independent, their covariances are zero, so

Var(P) = Var(X_father) + Var(X_mother) + Var(E),

V_phenotype  = V_additive + V_environment

The fraction of the population phenotypic variation due to genetic, additive effects is then simply

h² = V_additive/V_phenotype = V_additive/(V_additive + V_environment).

This is the definition of heritability, h².

So if h² 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) = V_additive/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) = V_phenotype,

So using the formula for ρ above,

ρ = (V_additive/2) / V_phenotype = 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 * (σ_y/σ_x) where x is the independent variable.