Genetic background and
population stratification

Shaun Purcell1,2 & Pak Sham1

1Social, Genetic & Developmental Psychiatry Research Centre, IoP, KCL, London.

2Whitehead Institute, MIT, Cambrdige, MA, USA.

Association & stratification

Sewall Wright (1951)

concepts of population structure & impact on the evolutionary process

C. C. Li (1972)

impact of population structure on disease-gene association studies

increase in type I errors

decrease in power

Signatures of stratification

At a single locus

non-independence of paternal and maternal alleles

Across loci

non-independence of alleles across loci

linkage disequilibrium, LD

use LD to map genes

spuriously infer indirect association

At a single locus

Allele frequencies

A1	p

A2	q

Genotype frequencies

expected under “Hardy-Weinberg equilibrium”

A1A1	p2

A1A2	2pq

A2A2	q2

At a single locus

Sub-population

1	2

A1	0.1	0.9

A2	0.9	0.1

A1A1	0.01	0.81

A1A2	0.18	0.18

A2A2	0.81	0.01

1+2

0.5

0.5

0.41 (0.25)

0.18 (0.50)

0.41 (0.25)

Quantifying population structure

Expected average heterozygosity

in random mating subpopulations (HS)

in combined total population (HT)

from the previous example,

HS = 0.18 , HT = 0.5

Wright’s fixation index

FST = ( HT - HS ) / HT

FST = 0.64

0.01 - 0.05 for European populations

0.1 - 0.3 for most divergent populations

Across loci

200 Scandinavians

B1	B2

A1	160	160	2 = 0

A2	40	40

200 Spaniards

B1	B2

A1	160	40	2 = 0

A2	160	40

Across loci

400 Scandinavians and Spaniards combined

B1	B2

A1	320	200	2 = 7.81

A2	200	80

Spurious association

not reflective of genetic distance

A and B might be on different chromosomes

Solutions

Family controls

related individuals share same sub-population

e.g. TDT test, between-within model

Index of membership

self-reported ethnicity

not always accurate / effects may be subtle

infer from an individual’s genetic background

detection

look for signatures of population stratification

correction

correct tests for inferred substructure

Key references

Pritchard et al (2000) Genetics

STRUCTURE method

Thomas & Witte (2002) Can Epi, Biomarkers & Prev

Wacholder et al (2002) Can Epi, Biomarkers & Prev

Point/Counterpoint: should we worry about stratification

Cardon & Palmer (2003) The Lancet

review article

Rosenberg et al (2002) Science (2003) AJHG

Diversity panel; informativeness of markers

Freedman et al (2004) Nat Genet

empirical assessment of stratification in case/control studies

Genetic background approaches

Genomic Control

Structured Association

Method: multilocus genotype data to detect and correct for stratification

Premise: stratification operates globally – on whole genome, whereas LD operates locally at short scales

Genomic control

2 statistics not distributed as 2 under PS “overdispersion”

Pritchard & Rosenberg (1999)

assess whether 2 statistics for unlinked markers are okay

Devlin & Roeder (1999)

null locus test statistic TN distributed 21

in presence of stratification, TN /  ~ 21

estimate 

statistic at test locus T /  ~ 21

Genomic control

Test locus

Unlinked ‘null’ markers

2

No stratification

2

Stratification  adjust test statistic

Genomic control

Simple estimate of inflation factor

using the median protects from outliers

i.e. if some of the null markers are also QTL

bounded at minimum of 1

i.e. should never increase test statistic

principle extended to multiple alleles, haplotpes, quantitative traits

Must formulate all tests as 1 df tests, however

Genomic control

	Inflation factor

R	number of cases (controls)

F	Wright’s FST coefficient of inbreeding

gk (fk)	Proportion of cases (controls) from subpopulation k

Example

2 equifrequent subpopulations, FST = 0.01

Disease twice as common in one subpopulation

R = 1000

  1.5

Structured association

Assignment of individuals to subpopulations

Test for association conditional on subpopulation

Distance-based approaches

Model-based approaches

Pritchard et al (2000) Genetics

Bayesian framework (STRUCTURE / STRAT)

Satten et al (2001) AJHG

Latent class analysis model

Purcell & Sham (2005) Hum Hered

Latent class analysis model (L-POP / L-ASSOC)

IBS distance-based metric

A

G/J K/U H/H W/W S/S ...

G/G K/K H/H U/B S/S ...

IBS	1 1 2 0 2 ...

G/J K/U H/H W/W S/S ...

K/J L/A G/D U/B S/L ...

IBS	1 0 0 0 1 ...

Pair from same population

Pair from different population

1. Distance matrix (applying clustering or multidimensional scaling)

2. Permutation test: do these two individuals come

from different populations?

Rosenberg et al (2002) Science

Structured association

Unlinked ‘null’ markers

LD observed under stratification

Subpopulation A

Subpopulation B

Advantages of SA

Structure of intrinsic interest

Any test of association can be used

Allows allelic heterogeneity between subpopulations

Does not assume constant FST across the genome

Structured association

Genotype a number of loci across the genome

Loci must be unlinked

in a non-stratified sample, would not expect to observe correlations between these loci

in a stratified sample, would not expect to observe correlations between these loci within sub-population

Latent Class Analysis

K sub-populations, latent classes

Sub-populations vary in allele frequencies

Random mating within subpopulation

Within each subpopulation

Hardy-Weinberg and linkage equilibrium

For population as a whole

Hardy-Weinberg and linkage disequilibrium

Latent Class Analysis

Goal : assign each individual to class C of K

Key : conditional independence of genotypes, G within classes

P(C | G)	posterior probabilities

P(C)	prior probabilities

P(G | C)	class-specific allele frequencies

E-M algorithm

P(C | G)

P(G |C)

P(C)

E step:

counting individuals and alleles in classes

M step:

Bayes theorem, assume conditional independence

-2LL

Converged?

Sum over j = 1 to K classes

Assumes conditional independence

Product over l = 1 to L loci

M-step

For each individual, posterior probabilities

Likelihood

Likelihood of an individual

Use AIC to select optimal K solution

Allowing for admixture

Stratification within a sample

we have assumed sub-populations are distinct

Admixture within an individual

an individual’s genome has descended from 2 or more pure sub-populations

Example #1

Example #1

Example #1

Example #2

3 subpopulations, 1000 individuals, 30 SNPs

70% : 20% : 10%

allele frequency U[0.001 - 0.999] + N(0, 0.2)

AIC

K= 3

Sub-population

B

C

A

Class 2

Class 3

Class 1

Latent class

690

212

88

4

4

1

1

Factors affecting power

Number of markers

Selection of most informative markers vs random

ancestry informative markers (AIMs)

Extreme

substructure

Very mild

substructure

Big impact on

type I error

Small impact on

type I error rates

Easy to detect

Hard to detect

Notes on L-POP

Example parameter file (http://statgen.iop.kcl.ac.uk/lpop/)

Example parameter file

DATAFILE mydata.raw

STRUCTURE

PHENO 4

CLASS 2

TAG cl2

RAND 0

REPEAT 10

VERBOSE2

1st line is title

required

file format

model specification

Name tag for results

Random # seed

# attempts

at convergence

Verbosity of output

(1-3)

# cols to skip

Results format for L-POP

grep P: results	get prior class probabilities

grep K: results	get likelihood, AIC

grep k: results	get likelihood, AIC from all	E-M convergences

grep I: results	get posterior class probabilties

grep D: results	get genetic distance matrix

grep I:cl3: results	get P(C|G) for solution	with TAG cl3 only

Correction

Satten et al

Test of association combined with detection of structure

Binary disease traits

P(C|G) as covariates

K-1 covariates

Alternatively, assign to class with highest P(C|G)

Applicable to any type of analysis / trait

Can allow for interactions (i.e. different effects between subpopulations)

Testing for association

Weighted likelihood

Model probability of genotype conditional on trait

Individual’s class probabilities

(estimated using L-POP)

Class-specific likelihood

of genotype conditional

on trait

Parameters p, a, d

(potentially class-specific)

Notes on L-ASSOC

Data :

Individuals only, quantitative trait

.ped file and .dat file

weights as covariates (C in .dat file)

Parameters:

used to build alt and null models

Universal	Class-specific

Allele frequency:	p	P

Additive genetic value:	a	A

Dominance deviation:	d	D

Notes on L-ASSOC

Standard test of association

lassoc ––file data ––alt pa ––null p

Test of association allowing for stratification

lassoc ––file data ––alt Pa ––null P

Test of allele frequency differences between strata

lassoc ––file data ––alt P ––null p

Test of QTL by strata interaction

lassoc ––file data ––alt PA ––null Pa

Test of all effects

lassoc ––file data ––alt PAD ––null P

lassoc --file data --alt pa --null p

Model SP p a d va vd

--------------------------------------------------------

H1 1 0.498 0.020 0.005

2 0.498 0.020 0.005

3 0.498 0.020 0.005

HO 1 0.498

2 0.498

3 0.498

----------------------------

-2LL(H1) 209.839

-2LL(H0) 216.029

LRT 6.190

df 1

p-value 0.013

----------------------------

lassoc --file data --alt Pa --null P

Model SP p a d va vd

--------------------------------------------------------

H1 1 0.624 0.017 0.004

2 0.443 0.017 0.004

3 0.502 0.017 0.004

HO 1 0.622

2 0.446

3 0.508

----------------------------

-2LL(H1) 209.839

-2LL(H0) 216.029

LRT 1.190

df 1

p-value 0.734

----------------------------

Practical session

Goal

using QTDT, LPOP and LASSOC, analyse the data under the pshaun/strat/ directory

1. For the two SNP test markers, what does standard association analysis reveal?

2. Is there evidence for population substructure?

3. What is the effect of testing for association conditional on any substructure, using family-based tests?

(I) Individuals

QTDT

dind.ped, dind.dat

Type 50 null loci

Collect siblings

QTDT

dfam.ped, dfam.dat

LPOP

dnull.ped

QTDT

dcov.ped, dcov.dat

LASSOC

dcov.ped, dcov.dat

LASSOC

dnull.ped, dnull.dat

(II) Family-based analysis

(III) GC / SA analysis

(IV) SA

(V) GC

Generate weights

dind.ped

1 1 0 0 1 1 1 1 2 1.576

2 1 0 0 1 1 2 1 1 0.368

3 1 0 0 1 2 1 1 1 -0.423

dnull.ped

1 1 0 0 1 1 1 1 2 1.576 1 1 1 2 2 1 2 2 1 2 2 1 …

2 1 0 0 1 1 2 1 1 0.368 1 2 1 1 2 1 2 2 2 1 2 1 …

3 1 0 0 1 2 1 1 1 -0.423 2 1 1 1 1 2 1 1 2 1 1 1 …

Null markers

PED details, QTL & trait

dcov.ped

1 1 0 0 1 1 1 1 2 1.576 0.000 0.000 1.000

2 1 0 0 1 1 2 1 1 0.368 0.000 0.150 0.850

3 1 0 0 1 2 1 1 1 -0.423 0.998 0.001 0.001

Posterior probabilities (LPOP)

dfam.ped

1 3 0 0 1 -9 -9 -9 -9 -9

1 4 0 0 1 -9 -9 -9 -9 -9

1 1 3 4 1 1 1 1 2 1.576

1 2 3 4 1 1 2 1 2 1.576

“Parents”

Siblings

Trait

QTL

PED details

Standard QTDT analysis (not controlling for stratification)

qtdt –p dind.ped –d dind.dat -at -weg

Family-based QTDT analysis (not controlling for stratification)

qtdt –p dfam.ped –d dfam.dat -at –weg

Family-based QTDT analysis (within test, controlling for stratification)

qtdt –p dfam.ped –d dfam.dat -ao –weg

Family-based QTDT analysis (test of stratification)

qtdt –p dfam.ped –d dfam.dat -ap –weg

L-POP stratification analysis

lpop < param1 > results

lpop < param2 >> results

lpop < param3 >> results

lpop < param4 >> results

Get lowest AIC

grep AIC results

Get prior class probabilities for 3 class solution (TAG cl3)

grep P:cl3: results

Get posterior probabilities from the 3 class solution

grep I:cl3: results

grep I:cl3: results | gawk ‘{print $4,$5,$6}’ > postprob

QTDT analysis, using covariates

qtdt –p dcov.ped –d dcov.dat -at -weg

LASSOC analysis, not controlling

lassoc --file dcov --alt pa --null p

LASSOC analysis, controlling stratification

lassoc --file dcov --alt Pa --null P

LASSOC analysis, testing for stratification

lassoc --file dcov --alt P --null p

LASSOC analysis, allowing for QTL x strata interaction

lassoc --file dcov --alt PA --null P

LASSOC analysis of all null loci

lassoc --file dnull --alt pa --null p

Get median test statistic, divide by 0.456, use to correct QTL tests

e.g. using grep to extract test statistics efficiently

lassoc --file dnull --alt pa --null p > gcresults

grep LRT gcresults

L-POP

Shaun Purcell

2002-01-14T17:54:38