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

Acknowledgements Keng-Han Lin Matthew Zawistowski Mark Reppell

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.

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)

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.

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)

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

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

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.

Inflation by Mixture Proportion Zawistowski et al. 2014

Inflation across Comparisons

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.

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

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

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

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

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

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.

Robust to Population Stratification

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

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

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

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):

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 χ2 distribution.

Minor Allele Frequency Conventional Case-Control Internal Control Selected Cases

Power Comparison by Mean Effect Size

Power Comparison by Variance

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.

Power for Antagonistic Interaction

Power for Positive Interaction

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.

Thank you for your attention Questions? Thank you for your attention