Hae Kyung Im, PhD

Principal Component Analysis and Population Structure

April 11, 2022

Genotype Matrix, a Treasure Trove

2

Principal Components Reveals Demographic History

3

J. Novembre, et al “Genes mirror geography within Europe,” Nature, vol. 456, no. 7218, pp. 98–101, Aug. 2008.

Could Population Structure Bias GWAS Results?

Spurious Association Due to Population Structure

5

Case

Control

Spurious Association Due to Population Structure

6

Control

maf=50%

maf=25%

maf=50%

maf=25%

maf=40%

maf=35%

Case

How to Correct for Population Structure?

1. Correcting with genomic control (Devlin and Roeder 1999)

​

2. Inferring the latent sub-populations (Pritchard et al 2000)

Fit association in each population separately and combine

​

3. Adjusting for principal components

(Patterson 2006, Novembre 2008, Price et al 2010)

​

4. Mixed effects modeling (EMMAX, Kang et al 2010)

7

Principal Component Analysis

Principal Component Analysis (SVD)

9

DATA

n x M

=

u₁

u₂

d₁

d₂

v'₁

v'₂

Geometric Interpretation of Singular Value Decomposition

10

https://en.wikipedia.org/wiki/Singular_value_decomposition

https://en.wikipedia.org/wiki/Singular_value_decomposition#/media/File:Singular_value_decomposition.gif

X

D

Example Population Structure

HapMap Project

12

An international project to create a haplotype map of the human genome

1000 Genomes Project

13

Auton, A., Altshuler, D. M., Durbin, R. M., Chakravarti, A., Clark, A. G., Donnelly, P., et al. (2015). A global reference for human genetic variation. Nature, 526(7571), 68–74. http://doi.org/10.1038/nature15393

HapMap Phase 3 Populations

14

HapMap Phase 3 Populations

15

https://hakyimlab.github.io/hgen471/L6-population-structure.html

Population Structure in HapMap

16

https://hakyimlab.github.io/hgen471/L6-population-structure.html

Population Structure in HapMap

17

Population Structure in HapMap

18

Population Structure in HapMap

19

PCA in UK Biobank

20

GWAS in Multi-ancestry Samples

Example: Growth Phenotype by Population

22

H. K. Im et al, “Mixed effects modeling of proliferation rates in cell-based models: consequence for pharmacogenomics and cancer.,” PLoS Genetics, 2012.

Example: Growth Phenotype by Population

23

https://hakyimlab.github.io/hgen471/L6-population-structure.html

Populations Differences Lead to Inflation of Small P-values

24

https://hakyimlab.github.io/hgen471/L6-population-structure.html

Populations Differences Lead to Inflation of Small P-values

25

https://hakyimlab.github.io/hgen471/L6-population-structure.html

Populations Differences Lead to Inflation of Small P-values

26

https://hakyimlab.github.io/hgen471/L6-population-structure.html

what happens if we add principal components as covariates in the regression?

Growth GWAS Adjusted with PCs

28

Heritability

Types of Heritability

• Broad sense heritability: var()/var()
• Narrow sense heritability: var()/var()
• Chip heritability: additive component captured by the chip/imputation

30

Review

Matrix Algebra

Matrix Algebra

32

Addition

Scalar

Multiplication

Transposition

Matrix Multiplication

33

By File:Matrix multiplication diagram.svg:User:BilouSee below. - This file was derived from: Matrix multiplication diagram.svg,

CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=15175268

Matrix Multiplication

34

Matrix Form of System of Linear Equations

35

https://en.wikipedia.org/wiki/Matrix_(mathematics)

Derive Linear Regression Solution with Matrix Notation

36

Hardy Weinberg Equilibrium

Hardy Weinberg Equilibrium

• Random mating
• Genes inherited from the father is independent of the ones inherited from the mother
• If p = minor allele frequency of a biallelic SNP
• Suppose the variants seen in the population are A and C
• Allele frequency of A is 20% and C is 80%
• Minor allele frequency p = ?
• p(AA) = p^2
• AC = 2*p*(1-p)
• CC = (1-p)^2
• Tested using a chi2 test

38

Title Text

• The Hardy-Weinberg equilibrium is a principle stating that the genetic variation in a population will remain constant from one generation to the next in the absence of disturbing factors. When mating is random in a large population with no disruptive circumstances, the law predicts that both genotype and allele frequencies will remain constant because they are in equilibrium.
• The Hardy-Weinberg equilibrium can be disturbed by a number of forces, including mutations, natural selection, nonrandom mating, genetic drift, and gene flow. For instance, mutations disrupt the equilibrium of allele frequencies by introducing new alleles into a population. Similarly, natural selection and nonrandom mating disrupt the Hardy-Weinberg equilibrium because they result in changes in gene frequencies. This occurs because certain alleles help or harm the reproductive success of the organisms that carry them. Another factor that can upset this equilibrium is genetic drift, which occurs when allele frequencies grow higher or lower by chance and typically takes place in small populations. Gene flow, which occurs when breeding between two populations transfers new alleles into a population, can also alter the Hardy-Weinberg equilibrium.
• Because all of these disruptive forces commonly occur in nature, the Hardy-Weinberg equilibrium rarely applies in reality. Therefore, the Hardy-Weinberg equilibrium describes an idealized state, and genetic variations in nature can be measured as changes from this equilibrium state.

39

https://www.nature.com/scitable/definition/hardy-weinberg-equilibrium-122/