Robust and powerful

sibpair test for rare variant association

Sebastian ZöllnerUniversity of Michigan

Acknowledgements

Matthew Zawistowski

Keng-Han Lin

Mark Reppell

GWAS have been successful. Only some heritability is explained by common variants. Uncommon coding variants (maf 5%-0.5%) explain less. Rare variants could explain some ‘missing’ heritability.

◦ Better Risk prediction.◦ Rare variants may identify new genes.◦ Rare exonic variants may be easier to annotate functionally and

interpret.

Rare Variants –Why Do We Care?

Testing individual variants is unfeasible.◦Limited power due to small number of

observations.◦Multiple testing correction.

Alternative: Joint test.◦Burden test (CMAT, Collapsing, WSS)◦Dispersion test (SKAT, C-alpha)

Burden/Dispersion Tests

Gene-based tests have low power.◦ Nelson at al (2010) estimated that 10,000 cases &

10,000 controls are required for 80% power in half of the genes.

Large sample size required More heterogeneous sample =>Danger of

stratification Stratification may differ from common variants

in magnitude and pattern.

Challenges of Rare Variant Analysis

(202 genes, n=900/900, MAF < 1%,

Nonsense/nonsynonymous variants)

Stratification in European Populations

Variant Abundance across Populations

African-American

Southern AsiaSouth-Eastern Europe

Finland

South-Western Europe

Northern Europe

Central EuropeWestern Europe

Eastern EuropeNorth-Western Europe

A gradient in diversity from Southern to Northern Europe

Sample SizeExpe

cted

Num

ber o

f var

iant

s pe

r kb

Allele Sharing

Median EU-EU: 0.71 Median EU-EU: 0.86 Median EU-EU: 0.98

• Measure of rare variant diversity.• Probability of two carriers of the minor alleles being

from different populations (normalized).

1. Select 2 populations.2. Select mixing parameter

r.3. Sample 30 variants from

the 202 genes.4. Calculate inflation based

on observed frequency differences.

General Evaluation of Stratification

Inflation by Mixture Proportion

Zawistowski et al. 2014

Inflation across Comparisons

If multiple affected family members are collected, it may be more powerful to sequence all family members.

Family-based tests can be robust against stratification. TDT-Type tests are potentially inefficient. How to leverage low frequency?

◦ Low frequency risk variants should me more common in cases.◦ And even more common on chromosomes shared among

many cases.

Family-based Test against Stratification

• Consider affected sibpairs.• Estimate IBD sharing.• Compare the number of

rare variants on shared (solid) and non-shared chromosomes (blank).

Any aggregate test can be applied.

Family Test S=0

S=2

S=1

Twice as many non-shared as shared chromosomes.

Null hypothesis determines test:

Shared alleles : Non-shared alleles=1:2Test for linkage or association

Shared alleles : Non-shared alleles=Shared chromosomes : Non-shared chromosomes

Test for association only

Basic Properties

IBD sharing is known. Individuals don’t need phase to identify shared variants. Except one configuration: IBD 1 and both sibs are heterozygous

Under null, probability of configuration 2 is allele frequency. Under the alternative, we need to use multiple imputation.

Haplotypes not required

Configuration 1

+1 shared

Configuration 1

+2 non-shared

Assume chromosome sharing status is known for each sibpair.

Count rare variants; impute sharing status for double-heterozygotes.

Compare number of rare variants between shared and non-shared chromosomes with chi-squared test (Burden Style).

Evaluation of Internal Control

S=0

S=2

S=1

Classic Case-Control

Selected Cases

Enriching Based on Familial Risk

S=0

S=2

S=1

Internal Control

Consider 2 populations. p=0.01 in pop1, p=0.05 in pop2. 1000 sibpairs for internal control design. 1000 cases, 1000 controls for selected cases. 1000 cases and 1000 controls for case-control. Sample cases from pop1 with proportion . Test for association with α=0.05.

Stratification

Robust to Population Stratification

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.4

0.8

Proportion

Type

I E

rror

Rat

e

Internal ControlSelected CasesConventional

Realistic rare variant models are unknown◦ Typical allele frequency◦ Number of risk variants/gene◦ Typical effect size◦ Distribution of effect sizes◦ Identifiabillity of risk variants

Goal: Create a model that summarizes these unknowns into◦ Summed allele frequency◦ Mean effect size◦ Variance of effect size

Evaluating Study Designs

Assume many loci carrying risk variants. Risk alleles at multiple loci each increase

the risk by a factor independently. Frequency of risk variant:

◦ Independent cases

◦ On shared chromosome

Basic Genetic Model

)()|()|( RPRAAPAARP

A Affected AA Affected relative

pairR Risk locus

genotypeP(A|R)P(R)ARP )|(

Relative risk is sampled from distribution f with mean , variance σ2.

Simplifications: ◦ Each risk variant occurs only once

in the population.◦ Each risk variant on its own

haplotype. Then the risk in a random case

is

Effect Size Model

2121 )()(),|( 212121rrrr mfmfmmrrAP

A Affected r1,r2

Carrier status of chromosome 1,2

m1,m2

Relative risk of risk variants on 1,2

Mean effect sizeσ2 Variance of effect

size

To calculate the probability of having an affected sib-pair we condition on sharing S.

For S>0, the probability depends on σ2. E.g. (S=2):

Effect in Sib-pairsAA Affected rel pairri Carrier stat

chrom imi Relative risk of

variant on if Distribution of

RR Mean RRσ2 Variance of RRS Sharing status

2121 )()()( 2222 rrrr fEfE

)()(

)2,,|(

2122

21

21

21 mfmfmm

SrrAAPrr

Select μ, σ2 and cumulative frequency f Calculate allele frequency in

cases/controls P(R|A). Calculate allele frequency in shared/non-

shared chromosomes.=> Non-centrality parameter of χ2 distribution.

Analytic Power Analysis

Minor Allele Frequency

1 2 3 4 5

0.0

0.2

0.4

0.6

f=0.2f=0.01

sMA

F

1 2 3 4 5Mean Relative Risk

1 2 3 4 5

Conventional Case-Control Internal

ControlSelected Cases

Power Comparison by Mean Effect Size

1.0 2.5 4.0

0.0

0.4

0.8

Pow

er

f=0.01

1.0 2.5 4.0

sapp

ly(x

, fun

ctio

n(x)

pow

er.s

as(m

u =

x, s

igm

a2 =

sig

ma2

, f =

0.0

5,

n

_sb

= n1

))

f=0.05

Mean Relative Risk1.0 2.5 4.0

sapp

ly(x

, fun

ctio

n(x)

pow

er.s

as(m

u =

x, s

igm

a2 =

sig

ma2

, f =

0.2

,

n_s

b =

n1))

f=0.2

Internal ControlSelected CasesConventional

Power Comparison by Variance

0 1 2 3 4

0.0

0.4

0.8

Pow

er

f=0.01

0 1 2 3 4

sapp

ly(x

, fun

ctio

n(x)

pow

er.s

as(m

u =

mu,

sig

ma2

= x

, f =

0.0

5,

n

_sb

= n1

))

f=0.05

Variance of Relative Risk0 1 2 3 4

sapp

ly(x

, fun

ctio

n(x)

pow

er.s

as(m

u =

mu,

sig

ma2

= x

, f =

0.2

,

n_s

b =

n1))

f=0.2

Internal ControlSelected CasesConventional

Gene-gene interaction affects power in families. For broad range of interaction models, consider

two-locus model. G now has alleles g1,g2. The joint effect is

We compare the effect of while adjusting L and G to maintain marginal risk.

Gene-Gene Interaction

))((2121

21212121),,,|( ggrrggG

rrLggrrAP

Power for Antagonistic Interaction

0.2 0.4 0.6 0.8 1.0

0.0

0.4

0.8

Pow

er

Interaction Coefficient

IC SRR=2IC SRR=8Conventional

Power for Positive Interaction

1.0 1.2 1.4 1.6 1.8 2.0

0.0

0.4

0.8

Pow

er

Interaction Coefficient

IC SRR=2IC SRR=8Conventional

Stratification is a strong confounder for rare variant tests.

Family-based association methods are robust to stratification.

Comparing rare variants between shared and non-shared chromosomes is substantially more powerful than case-control designs.

All family based methods/samples depend on the model of gene-gene interaction. Under antagonistic interaction power can be lower than a population sample.

Conclusions

Questions?Thank you for your attention