1. Robust and powerful sibpair test for rare variant association
Sebastian Zöllner
University of Michigan

2. Acknowledgements
Keng-Han Lin
Matthew Zawistowski
Mark Reppell

3. Rare Variants –Why Do We Care?
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.

4. Burden/Dispersion Tests
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)

5. Challenges of Rare Variant Analysis
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.

6. Stratification in European Populations
Test is symmetric…
Add in lambda values…
but note that genomic control lambda not enough to correct – b/c effect depends on total # of variants per gene… so per-gene correction factors needed. Likely to overcorrect genes that have small variation.
(202 genes, n=900/900,
MAF < 1%, Nonsense/nonsynonymous variants)

7. Variant Abundance across Populations
African-American
Southern Asia
South-Eastern Europe
Finland
South-Western Europe
Northern Europe
Central Europe
Western Europe
Eastern Europe
North-Western Europe
Expected Number of variants per kb
To-do:
Add text box to mark up y-axis and fix typo
A gradient in diversity from Southern to Northern Europe
Sample Size

8. Allele Sharing Measure of rare variant diversity.
Probability of two carriers of the minor alleles being from different populations (normalized).
To-do:
Sharing nubmers in middle panel are for (1,2.5]% but plot is for 1-5%
Move to world map…
Median EU-EU: 0.71
Median EU-EU: 0.86
Median EU-EU: 0.98

9. General Evaluation of Stratification
Select 2 populations.
Select mixing parameter r.
Sample 30 variants from the 202 genes.
Calculate inflation based on observed frequency differences.

10. Inflation by Mixture Proportion
Zawistowski et al. 2014

11. Inflation across Comparisons

12. Family-based Test against Stratification
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.

13. Family Test 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.
S=1
S=2

14. Basic Properties Twice as many non-shared as shared chromosomes.
Null hypothesis determines test:
Shared alleles : Non-shared alleles=1:2
Test for linkage or association
Shared alleles : Non-shared alleles=
Shared chromosomes : Non-shared chromosomes
Test for association only

15. Haplotypes not required
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.
Configuration 1
+1 shared
Configuration 1
+2 non-shared

16. Evaluation of Internal Control
S=0
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).
S=1
S=2

17. Enriching Based on Familial Risk
Classic Case- Control
Internal Control
Selected Cases
S=0
S=2
S=1

18. Stratification 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.

19. Robust to Population Stratification

20. Evaluating Study Designs
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

21. Basic Genetic Model 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 A
Affected AA
Affected relative pair R
Risk locus genotype

22. Effect Size Model A Affected r1,r2 Carrier status of chromosome 1,2
m1,m2
Relative risk of risk variants on 1,2 
Mean effect size σ2
Variance of effect size
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

23. Effect in Sib-pairs AA Affected rel pair ri Carrier stat chrom i mi
Relative risk of variant on i f
Distribution of RR 
Mean RR σ2
Variance of RR S
Sharing status
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):

24. Analytic Power Analysis
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 χ distribution.

25. Minor Allele Frequency
Conventional Case-Control
Internal Control
Selected Cases

26. Power Comparison by Mean Effect Size

27. Power Comparison by Variance

28. Gene-Gene Interaction
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.

29. Power for Antagonistic Interaction

30. Power for Positive Interaction

31. Conclusions
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.

32. Thank you for your attention
Questions?
Thank you for your attention