1. Population Stratification
Qunyuan Zhang
Division of Statistical Genomics
GEMS Course M21-621
Computational Statistical Genetics
Mar. 24, 2011

2. What is Population Stratification (PS) ?
In narrow sense PS is the presence of a systematic difference in allele frequencies between subpopulations in a population, possibly due to different ancestry or origins, especially in the context of genetic association studies. Population stratification is also referred to as population structure.
In broad sense PS can be regarded as the presence of a difference in relatedness between individuals in a population, due to different subpopulations, family/pedigree structure and/or cryptic relation.

3. PS & False Positives
False Positives (inflation) Association could be due to the underlying structure of the population, even there is no disease-locus association.

4. An Example of PS-caused False Positive
Sub-population 1
case
control
total
risk A 72 8 80
9/1 a 18 2 20 90 10
100
Sub-population 2 3 27 30
1/9 7 63 70
Mixed population 75 35
110
2.14 25 65
0.38
200
1.00
No disease-locus association.
Risk difference between sub-populations.
Allele Frequency difference between sub-populations.
False disease-locus association in mixed population. (any allele with higher frequency in higher-risk sub-population seems to be risk allele)

5. Mantel-Haenszel Test for Stratification
(1)
Adjusted RR
(2)
Standard error
An Example
Chi-square test
(3)

6. Linear Model Usually Q is unknown, needs to be estimated Marker data
Population structure variable
Genetic background variable
Membership variable
Subgroup/sub-population variable
Ancestry/admixture proportion variable
Usually Q is unknown, needs to be estimated

7. Estimating Q by Eigen-analysis
singular values
X = U S VT
idv1
idv2
idv3
snp1 2 1
snp2
snp3
snp4
snp5
3.81
0.00
2.05
1.13 T S2
eigenvalues
-0.28
-0.95
0.11
-0.75
0.29
0.59
-0.60
0.08
-0.80
-0.55
0.33
0.34
-0.78
-0.10
-0.27
-0.16
0.04
-0.71
-0.20
0.14
0.52
-0.15
-0.93
0.20
14.51
0.00
4.21
1.28
Q Q Q3
Eigenvector of COV(X)
References: Patterson et al. 2006, Price et al (software EIGENSTRAT)
Or SAS Proc PRINCOM; R svd() and eigen()

8. Eigen-analysis of HapMap PopulationsQ2 Q1

9. (for admixed population)
Estimating Q by MLE
(for admixed population)
G: Observed genotypes of admixed [and parental populations]
Q: Allelic frequencies in parental populations
P : Individual membership to be estimated
Goal: obtain P that maximizes Pr(G|P,Q)
Assign prior values for Q (randomly or estimated from parental population genotype data) & P (randomly)
Compute P(i) by solving
Compute Q(i) by solving
Iterate Steps 1 and 2 until convergence.
Tang et al. Genetic Epidemiology, 2005(28): 289–301

10. (for admixed population)
Estimating Q by MCMC
(for admixed population)
Observed G : genotypes of admixed [and parental populations]
Unknown Z : admixed individuals’ membership from ancestral populations
Problem: How to estimate Z ?
Bayesian and Markov Chain Monte Carlo (MCMC) methods
Assume ancestral population number K (see next slide)
Define prior distribution Pr(Z) under K
Use MCMC to sample from posterior distribution Pr(Z|G) = Pr(Z)∙ Pr(G|Z)
Average over large number of MCMC samples to obtain estimate of Z
Falush et al. Genetics, 2003(164):1567–1587
Software : STRUCTURE

11. Infer Population Number (K)

12. Linear Model (an example including m Q-variables)
SAS Proc REG, Proc GENMOD; R lm(), glm()
Generalized, can fit binary/categorical y

13. Unified Mixed Model (more general)
Inferred population membership
SNP(s)
Covariate(s)
ID matrix
Modeling the resemblance
among individuals
V = Z G Z ' + R

14. Multi-Variate Normal Distribution (MVN) & Likelihood of Mixed Model
Based on MVN, the likelihood of trait (y) in a matrix form is:
no. of individuals (in a pedigree)
mean phenotype vector
nn variance-covariance matrix
phenotype vector
Kinship (IBD) matrix (nn )
V = Z G Z ' + R

15. Kinship Inbreeding Coefficient
The inbreeding coefficient of an individual is the probability that the pair of alleles carried by the gametes that produced it are Identical By Descent (IBD).
Identical By Descent (IBD)
Two alleles come from the same ancestry.
Kinship/Coancestry
The inbreeding coefficient of an individual is equal to the coancestry between its parents. For example if parents X and Y have a child Z, then
inbreeding coefficient of Z = coancestry between X and Y
Software: SAS (PROC INBREED), MERLIN, SPAGedi , R(kinship, emma) et al. (need pedigree and/or marker data)

16. Kinship Matrix (expected probability of allele sharing among relatives)

17. Resources for Mixed Model with Kinship Matrix
Software
Kinship
Mixed Model
Data
SAS
Proc INBREED
Proc MIXED
Quantitative trait
Pedigree data
Proc GLIMMIX
Quantitative/qualitative trait, Pedigree data
R : kinship
makekinship()
lmekin()
R: emma
emma.kinship()
emma.REML.t()
Using maker data to calculate kinship
EMMAX
emmax-kin
emmax

18. Diagnosis of Inflation of False Positives
Inflation: more false positives than expected under the null
In GWAS, usually due to PS
Can be caused by inappropriate statistical methods even with no PS
May (not necessarily) indicate PS

19. Theoretical Basis of Diagnosis
Uniform distribution [0,1] of p-values under the null
no inflation
inflation
Histogram
-log10(p)
Q-Q plot

20. Inflation Rate (IR) For Binary Trait For Continuous Trait
Devlin et al. 2004
For Binary Trait
For Continuous Trait
Amin , Duijn, Aulchenko, 2007

21. Genomic Control (by IR)
For Binary Trait
For Continuous Trait
Or based on p-value

22. Practice Download and unzip the data from
dsgweb.wustl.edu/qunyuan/data/ popstra2011hw.zip
Ignore pedigree.csv, test each SNP in snp.csv for association (with trait in trait.csv);
Investigate p-values to see if there is any inflation;
Try to explain why;
List some possible methods to reduce or control the inflation;
Choose one method, apply it to the data;
Does it work?
Try to explain why.
Clearly document each step of you analysis.
The is no standard answer, feel free to try anything you like !
Report back to and in one week. Thanks !