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**Association Tests for Rare Variants Using Sequence Data**

Guimin Gao, Wenan Chen, & Xi Gao Department of Biostatistics, VCU

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**Introduction to Association tests: two hypotheses**

Common variant-common disease Common variant: Minor allele frequencies (MAF) >= 5% Using linkage disequilibrium(LD) Rare variant-common disease Rare variant: MAF < 1% (or 5%) High allelic heterogeneity: collectively by multiple rare variants with moderate to high penetrances Associations through LD would not be suitable

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**Association tests for Common variants**

Test a single marker each time Cochran-Armitage’s trend test (CATT) (assuming additive (ADD)) Power: High for additive (ADD) or Multiplicative (MUL); low recessive (REC) or Dominant (DOM) Genotype association test (GAT) using chi-square statistic Power: a little lower for ADD, higher for REC MAX3 = maximum of three trend test statistics across the REC, ADD, and DOM models (Freidlin et al Hum Hered.) Power: lower than CATT under ADD higher than CATT & CAT under REC

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**Association tests for Common variants**

Test for single marker (CATT, GAT, & MAX3) Low power when MAF <10% No power for rare variants with MAF<1% Multivariate test Considering a group of variants (ex. SNPs in a gene) each time Multiple logistic regression (or Hotelling test, Fisher’s product) Xij = 0, 1, 2, the count. of the minor alleles of indiv i at locus j Power: higher than single-marker test; still very low due to large d.f = No. of SNPS = k Need new methods for rare variants Collapsing SNPs into a single marker to reduce d.f. testing a single SNP doesn’t work for rare variants; Need new methods for rare variants Collapsing SNPs into a single marker to reduce d.f.

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**Outline Introduction to association tests**

Three well-known collapsing methods for rare variants: CAST, CMC, & Weighted Sum methods An evaluation using GAW 17 data Extension to the three collapsing methods Future research Combined multivariate and collapsing (CMC) method (Li & Leal 08) Weighted sum method (Madsen & Browning 09)

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**Three association tests for Rare variants**

Collapsing a set of rare variants (into a single marker) A cohort allelic sums test (CAST) (Morgenthaler & Thilly 2007, Mutat. Res.) Combined Multivariate and Collapsing (CMC) (Li & Leal, 2007, AJHG) Division into subgroups, collapsing in each subgroup Weighted Sum statistic (Madsen & Browning, 2009; PloS Genet. Price et al. 2010, AJHG)

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**A cohort allelic sums test (CAST)**

A group of n variants (SNPs) in a unit (ex. one gene, LD block) Collapsing the genotypes across the variants Indicator coding for individual j xj = 1, if rare alleles present at any of the n variants; xj = 0, otherwise Testing if the proportions of individuals with rare variants (xj = 1) in cases and controls differ Higher power than method testing single variant each time Only for rare variants Only for rare variants

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**Combined Multivariate and Collapsing (CMC) Method (Li & Leal 08)**

Consider SNPs in a unit with MAF< a threshold (0.01 or 0.05) Division and Collapsing Divided into several sub-groups based on the MAF Ex. Subgroups : (0, 0.001], [0.001, 0.005), [0.005, 0.01) SNPs are collapsed in each sub-group xij = 1, if indiv j has rare alleles present in the i-th subgroup; xij = 0, otherwise Multivariate analysis of variance

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**Combined Multivariate and Collapsing (CMC) Method (Li & Leal 08)**

Multivariate test of collapsed sub-groups Hotelling T2 test, MANOVA, Fisher’s product method Power: often higher than CAST Different threshold may have different power

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**Weighted Sum Method (Madsen & Browning 09)**

A group of variants (SNPs) in a unit A weight for SNP i by the S.t.d of No. of minor alleles in the sample qi is the minor allele freq in controls Calculate a weighted genetic score for indiv j Iij = 0, 1, 2, the count of the minor allele of indiv i at locus j Obtain the Rank (Vj); Sum of the ranks of affected indivs cases and controls under H0, no threshold

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**Permutation for p-value estimation**

From observed data: Permutation to estimate p-value: Phenotype labels are permuted 1000 times, x1, …x1000 Calculate the mean (μ) and standard deviation (σ) of 1000 xs Assume z ~ N(0, 1) under null hypothesis Obtain the p-value from N(0, 1) Fast, p-value ~U[0,1]

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**Weighted Sum Method (Madsen & Browning 09)**

Power comparison: Simulations assuming genotypic relative risk is proportion to MAF at disease loci (Madsen & Browning 09) Weighted Sum Method (WSM) > CMC > CAST (WSM) > CMC may not be true in other situations Can be applied to rare variants & common variants Disadvantage: Give very high weights to very rare alleles (singleton), very low weights to common variants.

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**An evaluation of the CMC method and Weighted sum method by using GAW 17 data**

Both methods are powerful (based on the authors’ simulation) Our evaluation based on simulated datasets from GAW 17 GAW 17 data: a subset of genes with real sequence data available in the genome project Simulated phenotypes Unrelated individuals, families Dataset of 697 unrelated individuals 24487 SNPs in 3205 genes from 22 autosomal chromosomes Only test for the 2196 genes with non-synonymous SNPs In favor of their method

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**GAW 17 dataset of unrelated individuals**

Four phenotypes: Q1, Q2, Q4 and disease status. Q1, Q2, and Q4 are quantitative traits Q1 associated with 39 SNP in 9 genes, Q2 associated with 72 SNPs in 13 genes Q4: not related to any genes Disease status is a binary trait: affected or unaffected, associated with 37 genes 200 simulated phenotype replicates Only one replicate of genotype data (original data)

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**Transforming Phenotypes**

Methods: case-control design Transform Q1, Q2, Q4 into binary traits Splitting at the top 30% percentile of the distributions

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**Criteria for evaluation of Tests**

Familywise error rate (FWER) 2196 genes with non-synonymous SNPs, 2196 tests 2196 null hypotheses Hj0: gene not associated with the trait Q1 associated in 9 genes, 9 null hypotheses are not true. (2196-9) null hypotheses are true FWER = Pr(reject at least one true null hypothesis) = Nf/200 Nf : No. of replicates, at least one true hypothesis are rejected Average Power Mean of power for all the 9 genes that affect the phenotypes Evaluating power: Q1, Q2, Disease Evaluate FWER: Q4 the corresponding

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**Distribution of MAF in the GAW 17 dataset**

Figure 1. Distribution of MAF of SNPs in GAW 17

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**Figure 1. Group SNPs based on MAFs for CMC**

>=0.1 Similar to Madsen & Browning (2009)

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**Table 1: Average power Traits CMC method Weighted sum method Q1 0.144**

0.112 Q2 Disease

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**Table 2: FWER (nominal α = 0.05)**

Trait CMC method Weighted sum method Q4 0.115 0.0100 CMC has FWER inflation Population stratification or admixture, Samples from Asian, Europe,… Relatedness among samples Similar results in Power and FWER were reported at GAW 17

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**Variable-Threshold Approach (Price et al 2010)**

Given a threshold T, calculate a score for indiv j Iij = 0, 1, 2, the count of the minor allele of indiv i at locus j Calculate the sum of score for cases: Calculate Z(T) = V(T)/Var(V(T)) Find T to maximize Z(T), Zmax = max (Z(T)) Permutation to estimate p-value for Zmax Power: >CMC; Extended to quantitative traits

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**A weighted approach (Price et al 2010)**

Calculate a weighted score for indiv j Iij = 0, 1, 2 Calculate the sum of score for cases Possible weight Power: similar to the weighted sum method (Madsen & Browning 09)

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**A weighted approach (Price et al 2010)**

Calculate the sum of score for cases Iij = 0, 1, 2 Calculate weight by the prediction of functional effects PolyPhen-2 is used to predict damaging effects of missense mutations with probabilistic scores. Probabilistic scores as weights may reduce the noise of non- functional variants. Higher Power than other methods

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**A data-adaptive sum test (Han & Pan 2010, Hum Hered)**

Logistic model xij = 0, 1, 2, the count of the minor allele of indiv i at locus j Effect on opposite directions If j <0, with p-value < threshold (0.1), change xij into 2- xij Permutation to estimate p-value

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Conclusion Collapsing methods have higher power than single-marker test For genome-wide data analysis, collapsing methods don’t have much power after multiple testing adjusting Weighted sum methods are promising, need prior information from biological data

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**Modifying the weighted sum method (in progress)**

Future research Modifying the weighted sum method (in progress) Very high weights to very rare variants Smoothing weights w’ = 0.5w +0.5 (average of all w)

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