### A Method for Detecting Pleiotropy - Division of Statistical Genomics

```A Method for Detecting
Pleiotropy
Ingrid Borecki, Qunyuan Zhang, Michael Province
Division of Statistical Genomics
Washington University School of Medicine
Pleiotropy
Biological question:
Does a genetic variant have independent effects on
both of two traits?
Statistical question:
Can the correlation or a portion of the correlation
between two traits be explained by a genetic
variant?
Hypotheses & Models
Compound null:
no pleiotropy
Y1
Alternative:
pleiotropy
Y2
X
Y1
Y1
Y2
X
Y1
Y2
X
Y2
X
Statistical Parameter (δ) of Pleiotropy
& Hypotheses to Be Tested
Y1  1  1 X  1
Y2   2   2 X   2
Cov(Y1 , Y2 )  Cov(1 ,  2 )  12Var( X )
  Cov(Y1 , Y2 )  Cov(1 ,  2 )  1 2Var( X )
H0 :   0 vs. H A :   0
1  0 and(or) 2  0
Compound null:
no pleiotropy
1  0 and 2  0
Alternative:
pleiotropy
Estimating δ
Two traits are simultaneously fit into a mixed model
 Y1 
     X X  TT  T  X  XT  
 Y2 
 ~ N (0, R)
T is the trait indicating variable; R is block diagonal covariance matrix
(after re-ordering by individuals), with blocks corresponding to the
individuals and each block having the compound-symmetry structure
 2   12
 12 


2




12 
 12
When excluding X from the model
When including X in the model
Cˆov(Y1, Y2 )  ˆ12MLE
Cˆov(1,  2 )  ˆ12MLE
ˆ  Cˆov(Y1, Y2 )  Cˆov(1,  2 )
Testing δ
Pleiotropy Estimation Test
(PET)
z
ˆ
SDˆ
Q-Q Plot under the null
~ N (0,1)
Estimated by bootstrap
re-sampling 100 times
with replacement
-LOG10(P)
Other Methods for Comparison
•MANOVA (Wilks' test, wrong null)
•FCP: Fisher’s combined p-value test (meta-analysis
ignoring correlations, wrong null)
2
  2 log( pi )
2
i 1
•RCM: Reverse compound model (two tests)
logit( X )    1Y1  2Y2  
•SUM: Simple univariate model (two tests)
Y1*  1  1 X   1
Y2*   2   2 X   2
Testing if β1≠0 and β2≠0
Y1*
=Residual of Y1 adjusted by Y2
Y2*
=Residual of Y2 adjusted by Y1
Power Comparison
h12  h22  3%, r12  0.5, N  300, Simu.N  1000
PET
FCP
MANOVA
RCM
SUM
Power Comparison
h12  h22  1%, r12  0.5, N  300, Simu.N  1000
PET
FCP
MANOVA
RCM
SUM
Application
Correlation (WC, HOMA)= 0.542
SNP
1
2
3
4
5
6
7
WC
3.33E-06
1.77E-04
2.25E-03
2.29E-04
2.28E-04
1.92E-04
1.42E-02
HOMA
8.35E-06
8.96E-06
8.06E-06
4.04E-06
4.15E-06
9.84E-06
3.02E-05
PET
2.87E-09
8.33E-07
1.93E-06
5.68E-06
4.91E-05
7.18E-05
7.68E-05
Cov(%)
1.74
1.29
1.39
1.15
1.15
1.28
1.04
TG
1.83E-18
7.47E-01
2.61E-02
CAC
5.95E-01
3.28E-09
1.85E-05
PET
2.57E-01
6.76E-01
1.30E-04
Cov(%)
3.63
0.42
5.36
Correlation (TG, CAC)= 0.089
SNP
1
2
3
Conclusions
The PET Method
•Tests proper compound null for pleiotropy;
•Gives estimation of covariance due to pleiotropy;
• Has greater power other alternatives;
•Under mixed model framework, can easily be expanded to
other data (covariates, family data etc.) ;
•Practical to GWAS data (with 300 blades, R version takes
less than 1 day for the analysis of 2M SNPs and ~3000
subjects) ;
• Must be fit to primary phenotype and (typed or imputed)
genotype data.
Acknowledgement
Ling-Yun Chang (programming & testing)
Mary Feitosa (GWAS data and application)
```