BSMS205 · Genetics

Heritability

Chapter 16 · Part III · Complex Traits

A question to start with

Is height genetic?
Yes — but how much?

Where we are in the story

Last chapter

Polygenic model — many small effects
~12,000 SNPs for height
Almost all additive

This chapter

How much of the variance is genetic?
One number: h²
What it does — and does not — mean

The answer in advance

h²

heritability · the proportion of variance explained by genes

A number between 0 and 1
A property of a population, not a person
Depends on the environment as much as the genes

Roadmap for today

Partitioning the variance · V_P = V_G + V_E
Defining h² and H²
Estimating it · twins, relatives, SNPs
Why h² changes across environments
What heritability is not
Summary & bridge to Chapter 17

§ 1

Partitioning
the Variance

Fisher's central equation

V_P = V_G + V_E

V_P · phenotypic variance · what we measure
V_G · genetic variance · differences in DNA
V_E · environmental variance · everything else

Total variation = genes + environment. That's the move.

Breaking V_G down further

V_G = V_A + V_D + V_I

V_A · additive · each allele adds a fixed effect
V_D · dominance · heterozygote not the average
V_I · epistasis · gene-by-gene interaction

For most human traits, V_A dominates.

The full Fisher decomposition

V_P = V_A + V_D + V_I + V_E

Component	What it is	Typical share
V_A	Additive allele effects	Most of V_G
V_D	Dominance deviations	Small (< 1% additive)
V_I	Gene × gene interaction	Tiny at population level
V_E	Environment, chance, noise	Trait-dependent

Additive · the picture

Allele B adds 0.5 cm

Genotype	Height
AA	170.0 cm
AB	170.5 cm
BB	171.0 cm

Each B copy: +0.5 cm
AB sits exactly halfway
No dominance · linear
This is the model that works

§ 2

What h²
Actually Measures

Narrow-sense heritability

h² = V_A / V_P

Fraction of variance from additive effects only
Predicts parent-offspring resemblance
The number that matters for response to selection
The default in most modern studies

Broad-sense heritability

H² = (V_A + V_D + V_I) / V_P = V_G / V_P

Includes all genetic effects
Captures total genetic influence
Less useful for prediction · dominance & epistasis scramble in offspring
Common in twin studies

h² vs H² · side by side

	h² (narrow)	H² (broad)
Numerator	V_A	V_A + V_D + V_I
Predicts inheritance?	Yes	No
Predicts evolution?	Yes	No
Captures all genes?	No	Yes
Default in human GWAS	This one	Twin-study estimates

A worked example

If V_P = 100, V_A = 68, V_E = 32 …

h² = 68 / 100 = 0.68

That's the height answer in modern populations
68% of the variation between people is genetic
32% is environment — nutrition, illness, chance

§ 3

How We
Estimate h²

The core idea · shared DNA, shared traits

Relationship	Shared DNA	Trait similarity
Identical (MZ) twins	~100%	Very high
Fraternal (DZ) twins	~50%	Moderate
Full siblings	~50%	Moderate
First cousins	~12.5%	Low
Strangers	~0%	None expected

More shared DNA → more covariance in the trait.

Twin studies · the classic design

MZ twins

~100% shared DNA
Same household
Same age

DZ twins

~50% shared DNA
Same household
Same age

If MZ > DZ in similarity → genetic influence.

Twin study · Falconer's formula

h² ≈ 2 × (r_MZ − r_DZ)

r_MZ = correlation between MZ twins
r_DZ = correlation between DZ twins
Doubling reflects the 2× difference in shared DNA
For height: r_MZ ≈ 0.9, r_DZ ≈ 0.5 → h² ≈ 0.8

The shared environment caveat

MZ twins often share more environment than DZ twins
Same room · same friends · dressed alike
That inflates r_MZ beyond pure genetics
Twin studies probably overestimate h² slightly

Twins reared apart = the cleaner test — but rare.

Adoption studies · disentangling genes & home

Compare adopted child vs biological parents
Compare adopted child vs adoptive parents
Genes go with biological · environment with adoptive
If trait correlates with biological parents → genetic

SNP-based heritability · the modern way

Take ~50,000 unrelated individuals
Genotype them at millions of SNPs
Compare genetic similarity to trait similarity
Methods: GREML, LDSC

No pedigree needed — the genome is the pedigree.

SNP-h² for height · 0.68

0.68

SNP-based h² for height (common SNPs only)

Twin estimate: ~0.80
SNP estimate: ~0.68
Gap = the missing heritability problem
Add rare variants → climbs back toward 0.80

Three methods · three angles

Method	What it captures	h² for height
Twin studies	All shared genetics (≈ H²)	~0.80
SNP-h² (common)	Common additive variants	~0.68
WGS + rare variants	Common + rare additive	~0.79

§ 4

Why h²
Varies by Trait & Place

It's a ratio · both sides matter

h² = V_A / V_P = V_A / (V_A + V_E)

Shrink V_E → h² goes up
Grow V_E → h² goes down
Genes haven't changed — the denominator changed

Heritability across human traits

Trait	h²	Comment
Height	~0.68	Strong polygenic, modest V_E
Schizophrenia	~0.65	Highly heritable disease
BMI	~0.3-0.5	Diet & lifestyle dominate V_E
Blood pressure	~0.25	Stress, diet, medication
Educational attainment	~0.20	Mostly environment

The two-fields thought experiment

Same seeds · different fields

Field A · rich soil

Plants 150–180 cm
Within-field h² = 0.80

Field B · poor soil

Plants 100–130 cm
Within-field h² = 0.80

Difference between fields = 100% environment.

Here is the famous thought experiment that should burn into your memory. Take one hundred genetically identical seeds and plant fifty in field A — rich soil, lots of water. Plant the other fifty in field B — poor soil, dry. Within field A, plants vary from one hundred fifty to one hundred eighty centimeters. Within field B, they vary from one hundred to one hundred thirty. In each field, all the variation within that field is genetic, because the environment within the field is uniform. So h squared within each field is high — say zero point eight. But the difference between the two fields — sixty centimeters on average — is one hundred percent environmental. The fields differ, the seeds do not. High within-population heritability tells you nothing about between-population differences.

Why this matters · environment can be invisible

If everyone has good nutrition → V_E shrinks → h² rises
If everyone has poor nutrition → similar story
The bigger the environmental contrast, the bigger V_E
Heritability tells you about within-group variation only

§ 5

What h²
Is Not

Mistake 1 · "h² = 0.68 means your height is 68% genes"

Wrong. Your height is 100% genes and 100% environment.

h² describes variation between people, not one person
You do not have a "68% genetic" phenotype
Genes and environment interact in every cell, every day

Mistake 2 · "High h² means the trait is fixed"

Wrong. High h² is compatible with large environmental shifts.

20th century: average height increased ~10 cm
Genes did not change in 100 years
Heritability stayed ~0.7 the whole time
All of the change = environment (nutrition)

Mistake 3 · "Low h² means genes don't matter"

Wrong. Low h² may just mean V_E is large.

BMI h² ~ 0.3-0.5 — but obesity has clear genetic risk
Most people have similar genes for hair count → h² ≈ 0 for "having hair"
Heritability requires genetic variation in the population

Mistake 4 · "h² explains group differences"

Wrong. Within-group h² says nothing about between-group differences.

Two-fields experiment makes this concrete
Group A taller than Group B → could be 100% environment
Even if h² = 0.99 within each group

Five things h² is

Hold these straight

A population statistic, not a personal one
A ratio · depends on V_E too
A snapshot · changes with environment
About variation, not causation in individuals
A useful summary, not a destiny

§ 6

The Missing
Heritability Problem

Three numbers · one trait

Source	h² for height
Twin studies	~0.80
SNP-h² (common variants, GREML/LDSC)	~0.68
Sum of GWAS-significant SNPs (~12,000)	~0.40

Where did the difference go?

Where it went

Sub-threshold SNPs — too small to reach genome-wide significance, but real
Rare variants — missed by SNP arrays
Structural variants — CNVs, large deletions
Imperfect tagging — common SNPs don't capture all common variation
Some twin estimate inflation from shared environment

§ 7

Summary

What to take away

V_P = V_G + V_E · Fisher's central decomposition
h² = V_A / V_P · narrow-sense, the workhorse
Estimated by twins, relatives, SNPs · three angles
Height ~0.68 · BMI ~0.3-0.5 · varies hugely
A population statistic — not personal, not destiny

Next lecture

Now let's see this with
real data examples.

Chapter 17 · Partitioned Variance in Practice