*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 = XNote 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)._{father}+ X_{mother}+ E.

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(XSo what is Var(P), the variance of height as we observe it in the population?_{father}) = V_{additive}/2 -- each parent provides half theadditive genetic'input'

Var(X_{mother}) = V_{additive}/2 -- each parent provides half theadditive genetic'input'

Var(E) = V_{environment}.

Var(P) = Var(XMessy, but if we assume X_{father}) + Var(X_{mother}) + Var(E) +

2Cov(X_{father}, X_{mother}) + 2Cov(X_{father}, E) + 2Cov(X_{mother}, E).

_{father}, X

_{mother}and E are independent, their covariances are zero, so

Var(P) = Var(XThe fraction of the population phenotypic variation due to genetic, additive effects is then simply_{father}) + Var(X_{mother}) + Var(E),

V_{phenotype}= V_{additive}+ V_{environment}

h= V^{2}= V_{additive}/V_{phenotype}_{additive}/(V_{additive}+ V_{environment}).

**This is the definition of heritability, h**

^{2}.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:

This is the standard definition.ρ = Cov(A,B)/√(Var(A) * Var(B)).

In the case of one parent and their offspring, under some simplifying assumptions,

Cov(parent,offspring) = V- this takes a few lines to work out, setting most of the X_{additive}/2

_{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) = VSo using the formula for ρ above,_{phenotype},

ρ = (VThis shows that heritability_{additive}/2) / V_{phenotype}= h^{2}/2.

*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 = rh**.

^{2}---

**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-heightwith 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 = h**. *

^{2}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?

Yes, the child has regressed towards the mean.Answer: predicted-offspring-height = β * (3 + 1)/2 = 2h^{2}= 1.35 inches.

---

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**where x is the independent variable.

_{y}/σ_{x})
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