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!

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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.

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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.

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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.

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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.

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* In general, β = ρ * (σ_y/σ_x) where x is the independent variable.

Emergent Spacetime

An extract from this book - (publisher, you have no idea how naff that front cover is).

"“Spacetime can’t be fundamental,” says the theorist Nima Arkani-Hamed. “It has to come out of something more basic.”

This thinking completely inverts physics. Nonlocality is no longer the mystery; it’s the way things really are, and locality becomes the puzzle. When we can no longer take space for granted, we have to explain what it is and how it arises, either on its own or in union with time.

Clearly, constructing space isn’t going to be as straightforward as melding molecules into a fluid. What could its building blocks possibly be? Normally we assume that building blocks must be smaller than the things you build out of them. A friend of mine and his daughter once erected a detailed model of the Eiffel Tower out of popsicle sticks; they hardly needed to explain that the sticks were smaller than the tower.

When it comes to space, though, there can be no “smaller,” because size itself is a spatial concept. The building blocks cannot presume space if they are to explain it. They must have neither size nor location; they are everywhere, spanning the entire universe, and nowhere, impossible to point to. What would it mean for things not to have positions? Where would they be? “When we talk about emergent space-time, it must come out of some framework that is very far from what we’re familiar with,” Arkani-Hamed says."

Emergent spacetime is a hot topic. Relativity assumes spacetime as a prior manifold and imposes geometry upon it. Quantum theory knows nothing - ab initio - about spacetime; its setting is a high/infinite dimensional complex vector space known as Hilbert space.

How are the two reconciled? Physical observables such as spatial position (momentum, energy, spin state are others) define coordinate systems (sets of basis vectors) within Hilbert space. It seems that Hilbert space is more fundamental than the spacetime we find ourselves in, but how do we get our perceived universe out of quantum theory? A unified theory needs to tell us but no compelling narrative has yet emerged.

I'm hoping this book can bring me up to date (in a sort of, a bit like, resembling kind of way).

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A review from Backreaction (which de-risked this purchase for me).