Association Analysis of Rare Genetic Variants

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Association Analysis of Rare Genetic Variants

• Qunyuan Zhang
• Division of Statistical Genomics
• Course M21-621
• Computational Statistical Genetics

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Rare Variants

• Low allele frequency usually less than 1
• Low power for most analyses, due to less
  variation of observations
• High false positive rate for some model-based
  analyses, due to sparse distribution of data,
  unstable/biased parameter estimation and inflated
  p-value.

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An Example of Low Power
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Jonathan C. Cohen, et al.
Science 305, 869 (2004)
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An Example of High False Positive Rate(Q-Q plots
from GWAS data, unpublished)
N2500 MAFgt0.03
N2500 MAFlt0.03
N50000 MAFlt0.03 Bootstrapped
N2500 MAFlt0.03 Permuted
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Three Levels of Rare Variant Data

• Level 1 Individual-level
• Level 2 Summarized over subjects
• Level 3 Summarized over both subjects and
  variants

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Level 1 Individual-level

Subject V1 V2 V3 V4 Trait-1 Trait-2
1 1 0 0 0 90.1 1
2 0 1 0 . 99.2 1
3 0 0 0 0 105.9 0
4 0 0 0 0 89.5 0
5 0 . 0 0 97.6 0
6 0 0 0 0 110.5 0
7 0 0 1 0 88.8 0
8 0 0 0 1 95.4 1
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Level 2 Summarized over subjects (by group)
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Jonathan C. Cohen, et al.
Science 305, 869 (2004)
Jonathan C. Cohen, et al.
Science 305, 869 (2004)
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Level 3 Summarized over subjects (by group) and
variants (usually by gene)
Variant allele number Reference allele number Total
Low-HDL group 20 236 256
High-HDL group 2 254 256
Total 22 490 512
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Methods For Level 3 Data
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Single-variant Test vs Total Freq.Test (TFT)
Jonathan C. Cohen, et al.
Science 305, 869 (2004)
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What we have learned

• Single-variant test of rare variants has very low
  power for detecting association, due to extremely
  low frequency (usually lt 0.01)
• Testing collective effect of a set of rare
  variants may increase the power (sum test,
  collective test, group test, collapsing test,
  burden test)

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Methods For Level 2 Data

• Allowing different samples sizes for different
  variants
• Different variants can be weighted differently

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CAST A cohort allelic sums test
Morgenthaler and Thilly, Mutation Research 615
(2007) 2856
Under H0 S(cases)/2N(cases)-S(controls)/2N(contro
ls) 0 S variant number N sample size T
S(cases) - S(controls)N(cases)/N(controls)
S(cases) - S(controls) (S can be calculated
variant by variant and can be weighted
differently, the final Tsum(WiSi)
) ZT/SQRT(Var(T)) N (0,1) Var(T) Var
(S(cases) - S (controls) ) Var(S(cases))
Var(S (controls)) Var(S(cases))
Var(S(controls)) X N(cases)/N(controls)2
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C-alpha
PLOS Genetics, 2011 Volume 7 Issue 3
e1001322
Effect direction problem
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C-alpha
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QQ Plots of Existing Methods (under the null)

• EFT and C-alpha
• inflated with false positives
• TFT and CAST
• no inflation, but assuming single
• effect-direction
• Objective
• More general, powerful methods

EFT TFT
CAST C-alpha
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More Generalized Methods For Level 2 Data
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Structure of Level 2 data
variant 1
variant 2


variant k
variant 3
variant i
Strategy Instead of testing total freq./number,
we test the randomness of all tables.
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Exact Probability Test (EPT)
1.Calculating the probability of each table based
on hypergeometric distribution
2. Calculating the logarized joint
probability (L) for all k tables
3. Enumerating all possible tables and L
scores
4. Calculating p-value P Prob.( )
ASHG Meeting 1212, Zhang
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Likelihood Ratio Test (LRT)
Binomial distribution
ASHG Meeting 1212, Zhang
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Q-Q Plots of EPT and LRT(under the null)
EPT N500
LRT N500
LRT N3000
EPT N3000
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Power Comparison significance level0.00001
Variant proportion Positive causal 80 Neutral
20 Negative Causal 0
Power
Power
Power
Sample size
Sample size
Sample size
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Power Comparison significance level0.00001
Variant proportion Positive causal 60 Neutral
20 Negative Causal 20
Power
Sample size
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Power Comparison significance level0.00001
Variant proportion Positive causal 40 Neutral
20 Negative Causal 40
Power
Sample size
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Methods For Level 1 Data

• Including covariates
• Extended to quantitative trait
• Better control for population structure
• More sophisticate model

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Collapsing (C) test
Li and Leal,The American Journal of Human
Genetics 2008(83) 311321
Step 1
Step 2 logit(y)a b X e (logistic
regression)
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Variant Collapsing
() () (.) (.)
Subject V1 V2 V3 V4 Collapsed Trait
1 1 0 0 0 1 1
2 0 1 0 0 1 1
3 0 0 0 0 0 0
4 0 0 0 0 0 0
5 0 0 0 0 0 0
6 0 0 0 0 0 0
7 0 0 1 0 1 0
8 0 0 0 1 1 1
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WSS
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WSS
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WSS
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Weighted Sum Test
Collapsing test (Li Leal, 2008), wi 1 and
s1 if sgt1 Weighted-sum test (Madsen Browning
,2009), wi calculated based-on allele freq. in
control group aSum Adaptive sum test (Han Pan
,2010), wi -1 if blt0 and plt0.1, otherwise
wj1 KBAC (Liu and Leal, 2010), wi left tail
p value RBT (Ionita-Laza et al, 2011), wi log
scaled probability PWST p-value weighted sum
test (Zhang et al., 2011) , wi rescaled left
tail p value, incorporating both significance and
directions EREC( Lin et al, 2011), wi
estimated effect size
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() ()
Subject V1 V2 Collapsed Trait
1 1 0 1 3.00
2 0 1 1 3.10
3 0 0 0 1.95
4 0 0 0 2.00
5 0 0 0 2.05
6 0 0 0 2.10
When there are only causal() variants
Collapsing (Li Leal,2008) works well, power
increased
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() () (.) (.)
Subject V1 V2 V3 V4 Collapsed Trait
1 1 0 0 0 1 3.00
2 0 1 0 0 1 3.10
3 0 0 0 0 0 1.95
4 0 0 0 0 0 2.00
5 0 0 0 0 0 2.05
6 0 0 0 0 0 2.10
7 0 0 1 0 1 2.00
8 0 0 0 1 1 2.10
When there are causal() and non-causal(.)
variants
Collapsing still works, power reduced
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() () (.) (.) (-) (-)
Subject V1 V2 V3 V4 V5 V6 Collapsed Trait
1 1 0 0 0 0 0 1 3.00
2 0 1 0 0 0 0 1 3.10
3 0 0 0 0 0 0 0 1.95
4 0 0 0 0 0 0 0 2.00
5 0 0 0 0 0 0 0 2.05
6 0 0 0 0 0 0 0 2.10
7 0 0 1 0 0 0 1 2.00
8 0 0 0 1 0 0 1 2.10
9 0 0 0 0 1 0 1 0.95
10 0 0 0 0 0 1 1 1.00
When there are causal() non-causal(.) and
causal (-) variants
Power of collapsing test significantly down
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P-value Weighted Sum Test (PWST)
() () (.) (.) (-) (-)
Subject V1 V2 V3 V4 V5 V6 Collapsed pSum Trait
1 1 0 0 0 0 0 1 0.86 3.00
2 0 1 0 0 0 0 1 0.90 3.10
3 0 0 0 0 0 0 0 0.00 1.95
4 0 0 0 0 0 0 0 0.00 2.00
5 0 0 0 0 0 0 0 0.00 2.05
6 0 0 0 0 0 0 0 0.00 2.10
7 0 0 1 0 0 0 1 -0.02 2.00
8 0 0 0 1 0 0 1 0.08 2.10
9 0 0 0 0 1 0 1 -0.90 0.95
10 0 0 0 0 0 1 1 -0.88 1.00
t 1.61 1.84 -0.04 0.11 -1.84 -1.72
p(xt) 0.93 0.95 0.49 0.54 0.05 0.06
2(p-0.5) 0.86 0.90 -0.02 0.08 -0.90 -0.88
Rescaled left-tail p-value -1,1 is used as
weight
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P-value Weighted Sum Test (PWST)
Power of collapsing test is retained even there
are bidirectional effects
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PWSTQ-Q Plots Under the Null
Direct test Inflation of type I error
Corrected by permutation test (permutation of
phenotype)
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Generalized Linear Mixed Model (GLMM) Weighted
Sum Test (WST)
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GLMM WST
Y quantitative trait or logit(binary trait) a
intercept ß regression coefficient of weighted
sum m number of RVs to be collapsed wi
weight of variant i gi genotype (recoded) of
variant i Swigi weighted sum (WS) X
covariate(s), such as population structure
variable(s) t fixed effect(s) of X Z design
matrix corresponding to ? ? random polygene
effects for individual subjects, N(0, G),
G2s2K, K is the kinship matrix and s2 the
additive ploygene genetic variance
e residual
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Weight

• Base on allele frequency, binary(0,1) or
  continuous, fixed or variable threshold
• Based on function annotation/prediction SIFT,
  PolyPhen etc.
• Based on sequencing quality (coverage, mapping
  quality, genotyping quality etc.)
• Data-driven, using both genotype and phenotype
  data, learning weight from data or adaptive
  selection, permutation test
• Any combination

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Application 1 Family Data

• Adjusting relatedness in family data for
  non-data-driven test of rare variants.

Unadjusted
Adjusted
? N(0,2s2K)
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• Q-Q Plots of log10(P) under the Null

Li Leals collapsing test, ignoring family
structure, inflation of type-1 error
Li Leals collapsing test, modeling family
structure via GLMM, inflation is corrected
(From Zhang et al, 2011, BMC Proc.)
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Application 2 Permuting Family Data
MMPT Mixed Model-based Permutation
Test Adjusting relatedness in family data for
data-driven permutation test of rare variants.
? N(0,2s2K)
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• Q-Q Plots under the Null

WSS
Permutation test, ignoring family structure,
inflation of type-1 error
aSum
SPWST
PWST
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(From Zhang et al, 2011, IGES Meeting)
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• Q-Q Plots under the Null

WSS
Mixed model-based permutation test (MMPT),
modeling family structure, inflation corrected
aSum
SPWST
PWST
(From Zhang et al, 2011, IGES Meeting)
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Burden Test vs. Non-burden Test
Burden test
Non-burden test
T-test, Likelihood Ratio Test, F-test, score
test,
SKAT sequence kernel association test
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SKAT sequence kernel association test
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Extension of SKAT to Family Data
kinship matrix
Polygenic heritability of the trait
Residual
Han Chen et al., 2012, Genetic Epidemiology
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Other problems

• Missing genotypes imputation
• Genotyping errors QC (family consistency,
  sequence review)
• Population Stratification
• Inherited variants and de novo mutation
• Family data linkage infomation
• Variant validation and association validation
• Public databases
• And more

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