BSMS205 · Genetics

Partitioned
Variance

Chapter 17 · Part III · Complex Traits

A question to start with

If two things move
together — does one
cause the other?

From theory to the world

Last lecture · Ch16

Fisher's VP = A + D + E
Heritability h²
Additive vs dominant variance

Today · Ch17

Apply to real traits
Real biomedical studies
Causation vs confounding

The simple, powerful idea

Variation comes from
many sources.
Pull them apart
before drawing conclusions.

Real traits · real numbers

Trait	Approx. h²	Notes
Height	~80%	Among most heritable traits in humans
Schizophrenia	~65%	Polygenic, twin-study based
BMI	~30-50%	Strong gene–environment interaction
Depression	~40%	Heterogeneous · context-dependent
Educational attainment	~40%	Mostly environmental + social

Roadmap for today

The framework — VP partitioned across layers
Case 1 · the microbiome and autism
Case 2 · Tylenol and autism — sibling controls
Why h² differs across populations
Gene × environment interactions
Lessons for causal inference
Summary & bridge to GWAS

§ 1

The Framework,
Revisited

Fisher's equation · the universal template

V_P = V_A + V_D + V_E

V_A · additive genetic variance
V_D · dominance variance
V_E · environmental variance
Plus V_GE for gene–environment interaction

Shared C vs unique E environment

Shared environment · C

Same household, parents, neighborhood
Diet, schools, language
Makes siblings resemble each other

Unique environment · E

Differs between siblings
Specific experiences, accidents, friends
Makes siblings differ

Twin and family designs

Design	What it estimates	Logic
MZ vs DZ twins	h² (additive)	MZ share 100% genes; DZ share 50%
Adoption studies	Genes vs C	Adoptees share C, not genes
Sibling controls	Removes A & C	Within-family comparison
SNP-based h²	Common-variant h²	Uses genome-wide SNPs directly

Variance can be partitioned across layers

Not just genes vs environment
Genome · transcriptome · microbiome · behavior
Each layer contributes some fraction of variance
The same logic — applied to any source of variation

Fisher's framework is a template,
not just a genetics formula.

§ 2

Case 1
The Microbiome
and Autism

The popular hypothesis

Autistic children show different gut bacteria
Some studies → "microbiome → autism"
Spawned commercial products claiming to treat ASD
Real correlation — but is it causal?

Yap et al., 2021 · the design

247

children · ASD & controls

Measured 3 things:
Autism diagnosis
Dietary patterns
Gut microbiome composition

A new partitioning metric

b² · the microbiome
analog of h².

Fraction of variance in a trait explained by microbiome composition.

What the b² values showed

Trait	Microbiome b²	Interpretation
ASD diagnosis	0 – 9%	Essentially zero
Dietary patterns	40 – 64%	Microbiome reflects diet
Stool consistency	~50%	Direct gut effect

The microbiome explains almost none of ASD variance.

The headline finding

≈ 0%

of ASD variance explained by the microbiome

Knowing a child's gut bacteria → tells you nothing about ASD
The popular story is backwards

The hidden chain · behavior mediates

autism → selective eating → altered diet
→ different microbiome

Many autistic children have selective eating patterns
Texture, flavor, routine preferences
Limited diet → less diverse microbiome
The microbiome is a mirror, not a cause

Layered variance · what's really there

Layer	Role in ASD
Genetic (A, D)	Neurodevelopment, sensory processing
Behavioral	Selective eating, routines
Environmental	Diet, microbiome (responds to behavior)
G × E	Possible — but not driving ASD

Why this matters

Real-world claims about microbiome → autism collapsed
Commercial "microbiome treatments" lose their basis
Public health → don't treat the mirror
The framework protects us from bad inference

Treat the cause, not the reflection.

§ 3

Case 2
Tylenol & Autism
Sibling Controls

The setup

Tylenol · acetaminophen — common in pregnancy
Generally considered safe for fever & pain
Past decade · several studies → small association with ASD
Pregnant people worried; clinicians unsure

The dataset

2,480,000

children · Sweden · 24 years

National medical registries
Detailed medication records during pregnancy
Childhood diagnoses of autism & ADHD

Step 1 · the population analysis

Outcome	Hazard ratio	Looks like
Autism (ASD)	~1.05	5% higher risk
ADHD	~1.07	7% higher risk

A small but real-looking association. Same as past studies.

The clever pivot · sibling controls

Compare siblings within the same family
One pregnancy with acetaminophen, the next without
Holds constant: genes, parents, SES, home, lifestyle
What's left → only the within-mother difference

Sibling control · the result

1.0

hazard ratio · all outcomes

Autism · no association
ADHD · no association
Other outcomes · same flat result

What was the confound?

Familial confounding — shared family factors
More health problems · more stress · different care behaviors
Same factors → both medication use and child outcomes
Genes + shared environment → A + C

Hidden chain · shared family factors → both Tylenol use
and child neurodevelopment.

Connecting back to Fisher

V_P = V_A + V_C + V_E

Population analysis · drug confounded with A + C
Sibling analysis · subtracts A & C
Leaves only E — the unique pregnancy effect
Drug effect on E · zero

The clinical impact

Acetaminophen during pregnancy → no causal effect on ASD/ADHD
Confirms safety for fever & pain management
Removes a major source of parental anxiety
Variance partitioning → actual medical guidance

§ 4

Why h²
Differs

h² is a ratio · context-dependent

h² = V_A / V_P

Same V_A, different V_P → different h²
Bigger environmental variation → smaller h²
Smaller environmental variation → larger h²
h² is a population statistic, not an individual one

Height · the textbook example

Population	h² for height	Why
Modern wealthy nation	~80%	Nutrition uniformly good
1900s Western Europe	~65%	Variable nutrition
Famine-affected region	much lower	Environment dominates

Same genes — different heritability.

Educational attainment · environment dominates

h² for years of schooling · ~40%
~60% of variance is environmental + social
Schools, family resources, peer networks
Policy can change the outcome dramatically

Schizophrenia & depression · context matters

Trait	Approx. h²	Sensitivity
Schizophrenia	~65%	Stable across populations
Major depression	~40%	Highly context-dependent
Anxiety disorders	~30–40%	Stress & trauma sensitive

§ 5

Gene ×
Environment

What G × E means

Same genotype, different environments → different phenotype
Same environment, different genotypes → different phenotype
The two effects don't simply add — they multiply
Captured as V_GE in the variance equation

Real-world G × E examples

Trait	Gene	Environment
Lung cancer	CYP1A1, GSTM1	Smoking
Obesity	FTO	High-calorie diet, exercise
Depression	5-HTT, CRHR1	Childhood stress, trauma
PKU	PAH	Dietary phenylalanine
Lactose intolerance	LCT	Dairy intake

FTO and BMI · the cleanest case

FTO risk allele · ~3 kg heavier on average
Effect much smaller in physically active people
Effect much larger in sedentary, high-calorie diets
Same gene, different environment → different impact

Genes load the gun.
Environment pulls the trigger.

PKU · environment as cure

Mutations in PAH · cannot break down phenylalanine
Normal diet → severe intellectual disability
Phenylalanine-restricted diet → normal development
100% genetic disease — and environmentally cured

Why G × E matters for h²

If V_GE is large → h² estimates depend on environment
Different populations → different environmental mixtures
Same gene effect can be amplified or silenced
One reason heritability is not portable across populations

§ 6

Lessons for
Causal Inference

The two-study pattern

	Yap (microbiome)	Ahlqvist (Tylenol)
Question	Microbiome → ASD?	Tylenol → ASD?
Method	b² across layers	Sibling controls
Confound	Behavior (eating)	Family factors (A + C)
Real effect	≈ 0%	HR ≈ 1.0

Lesson 1 · partition before concluding

Apparent associations often vanish after partitioning
Microbiome → ASD · vanished
Tylenol → ASD · vanished
If you skip the partition step → wrong answer

Lesson 2 · A + C must be controlled

Family members share A and C
Studies that ignore them → spurious causation
Sibling controls · twin studies · adoption studies
Polygenic scores · molecular replacements for these designs

Lesson 3 · the framework is universal

Fisher's V_P = V_A + V_D + V_E · over 100 years old
Works for genome, microbiome, behavior, environment
Any biological or social layer
Still the gold standard for complex trait analysis

Lesson 4 · don't trust raw correlations

"X causes Y" headlines
without partitioning are not science.

Did they account for shared family factors?
Did they consider behavioral pathways?
Did they partition variance into A, C, E?
If not — grain of salt

§ 7

Summary

What to take away

VP = A + D + E + GE · the universal partitioning template
h² varies by population · height ~80% · BMI ~30-50% · depression ~40%
Microbiome → ASD · b² ≈ 0 · behavior was the confound
Tylenol → ASD · HR = 1.0 in siblings · family factors were the confound
G × E (FTO, PKU, smoking) · same gene, different impact

Five things to take away. One — V P equals A plus D plus E, possibly plus G E, is the universal partitioning template. It works on genes, on the microbiome, on family-level confounds. Two — heritability varies across populations because the environment varies; height around eighty percent, B M I around fifty percent, depression around forty percent. Three — the Yap microbiome study showed the apparent microbiome-autism link is actually a behavioral pathway, with b squared near zero. Four — the Ahlqvist Tylenol study showed the apparent drug-autism link is actually shared family factors, with hazard ratio one in siblings. Five — gene by environment interactions, like F T O and obesity, P K U and diet, smoking and lung cancer genes, mean that the same gene can have very different effects in different environments. Hold those five points.

The big picture

Heritability is real.
Confounding is also real.
Partition before concluding.

Theory protected us from bad inference twice today
Same template — applied to two completely different questions
That is why we learned the math

Next lecture

If heritability is real —
can we find the
actual variants behind it?

Chapter 18 · Genome-Wide Association Studies (GWAS)