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gwasBackground.Rmd
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---
title: "GWAS background"
author: "Lieven Clement"
date: "statOmics, Ghent University (https://statomics.github.io)"
output:
bookdown::html_document2:
code_download: true
df_print: paged
theme: flatly
highlight: tango
toc: true
toc_float: true
number_sections: true
code_folding: show
---
# Background
## DNA
- 6 Billion base pairs: 3 billion from father and 3 billion from mother
- Organised in $2 \times 23$ chromosomes of length 50 - 250 milion bp
</br>
## Transcription / translation
## Variation in DNA
- 90% of variation in DNA are SNP: single nucleotide polymorphism. Different base at a single position in the DNA
- Humans: $\pm$ 5 million SNPs
- Most of them are neutral: high redundancy in the genomic code
- Sometimes they are not neutral:
{width=25%}{width=75%}
- Genomic recombination of parental chromosomes when producing germ cells.
- Linkage disequilibrium: SNPs often occur together because of genomic recombination!
## GWAS in University of Bergen
- GWAS: Genome Wide Association Studies
- Studies in large cohorts
- Use SNPs to identify genes associated with a particular trait: e.g. birth weight, pacenta weight, BMI, ... .
# Linear models for GWAS
In GWAS one often corrects for population stratification using the following linear model.
$$
\tag{1}
\mathbf{y} = \mathbf{x}_\text{test}\beta_\text{test} + \mathbf{X}_c\boldsymbol{\beta}_c + \mathbf{X}_\text{PCA} \boldsymbol{\beta}_\text{PCA} +\boldsymbol{\epsilon}
$$
with
- $\mathbf{y}$ an $N\times1$ vector of the phenotype
- $\mathbf{x}_\text{test}$ an $N\times1$ vector with the genotype for the candidate SNP
- $\beta_\text{test}$ the association of candidate SNP and the phenotype
- $\mathbf{X}_c$ an $N\times C$ matrix with the covariate pattern for $C$ known covariates (vector of ones (intercept), age, gender, batch,...)
- $\boldsymbol{\beta}_c$ the $p\times 1$ vector of parameters modeling the association of the p covariates and the phenotype.
- $\mathbf{X}_\text{PCA}$ an $N\times p$ matrix with p PCs used to correct for population stratification
- $\boldsymbol{\epsilon}$ an $N\times 1$ vector with environmental residuals that are assumed to be i.i.d. $\epsilon_i \sim N(0,\sigma_u^2)$ with $i = 1\ldots N$
Let $\mathbf{Z}$ be an $N\times M$ genetic relationship matrix with all $M$ normalised genotypes.
Then with the SVD we can decompose $\mathbf{Z}$
$$
\mathbf{Z} = \mathbf{U}\boldsymbol{\Delta}\mathbf{V}^T
$$
Note, that the $N\times M$ matrix $\mathbf{V}$ are also the M PCs of an PCA.
So we can approximate $\mathbf{Z}$ using a truncated PCA, e.g. by using the first $p$ PCs.
$$
\mathbf{Z}_p = \mathbf{U}_{p} \boldsymbol{\Delta}_p\mathbf{V}^T_p
$$
with
$$
\mathbf{X}_\text{PCA} = \mathbf{U}_{p} \boldsymbol{\Delta}_p
$$
the scores on the first p PCs that can be used to correct for population stratification.
# Linear mixed model for GWAS
## Specification
$$
\tag{1}
\mathbf{Y} = \mathbf{x}_\text{test}\beta_\text{test} + \mathbf{X}_c\boldsymbol{\beta}_c + \mathbf{Z}_{GRM}\mathbf{u} +\boldsymbol{\epsilon}
$$
with
- $\mathbf{Y}$ an $N\times1$ vector of the phenotype
- $\mathbf{x}_\text{test}$ an $N\times1$ vector with the genotype for the candidate SNP
- $\beta_\text{test}$ the association of candidate SNP and the phenotype
- $\mathbf{X}_c$ an $N\times p$ matrix with the covariate pattern for $C$ known covariates (vector of ones (intercept), age, gender, batch,...)
- $\boldsymbol{\beta}_c$ the $p\times 1$ vector of parameters modeling the association of the p covariates and the phenotype.
- $\mathbf{Z}$ an $N\times M$ genetic relationship matrix (GRM) with all normalised genotypes
- $\mathbf{u}$ an $M\times 1$ vector with i.i.d. random effect for each SNP $\mathbf{u}\sim \text{MVN}(0,\mathbf{I}\sigma_u^2)$
- $\boldsymbol{\epsilon}$ an $N\times 1$ vector with environmental residuals that are assumed to be independent of $\mathbf{u}$ and i.i.d. $\boldsymbol{\epsilon}\sim \text{MVN}(0,\mathbf{I}\sigma_\epsilon^2)$
Random effects are used to model the correlation structure in the data. They imply a certain covariance structure of $\mathbf{y}$
## Covariance structure
Covariance structure of $\mathbf{y}$ implied by GWAS mixed model:
$$
\begin{array}{ccl}
\text{var}\left[\mathbf{Y}\right] &=& \text{var}\left[\mathbf{x}_\text{test}\beta_\text{test} + \mathbf{X}_c\boldsymbol{\beta}_c + \mathbf{Z}_\text{GRM}\mathbf{u} +\boldsymbol{\epsilon}\right]\\\\
&\updownarrow& \mathbf{u} \perp \boldsymbol{\epsilon}\\\\
&=& \text{var}[\mathbf{Z}_\text{GRM}\mathbf{u}] + \text{var}[\boldsymbol{\epsilon}]\\\\
&=&\mathbf{Z}_\text{GRM}\text{var}[\mathbf{u}]\mathbf{Z}_\text{GRM}^T + \mathbf{I} \sigma^2\\\\
&=&\mathbf{Z}_\text{GRM}\mathbf{I}\sigma^2_u\mathbf{Z}_\text{GRM}^T + \mathbf{I} \sigma^2_\epsilon \\\\
&=&\mathbf{Z}_\text{GRM}\mathbf{Z}_\text{GRM}^T \sigma^2_u+ \mathbf{I} \sigma^2_\epsilon
\end{array}
$$
Note that the model is often also written in another way:
$$
\tag{1}
\mathbf{Y} = \mathbf{x}_\text{test}\beta_\text{test} + \mathbf{X}_c\boldsymbol{\beta}_c + \mathbf{g} +\boldsymbol{\epsilon}
$$
- with $\mathbf{g} \sim \text{MVN}(\mathbf{0},\mathbf{K}\sigma^2_g)$
- $\mathbf{K}$ the $N \times N$ empirical kinship matrix
$$
\mathbf{K} = \frac{\mathbf{Z}_\text{GRM}\mathbf{Z}^T_\text{GRM}}{M}
$$
- $\sigma_g^2$ the polygenic variance $\sigma_g^2=M\sigma_u^2$
## Main advantages of LMM method
1. Better control of false positive associations by correcting for population or relatedness structure
2. An increase in power:
- Through the correction for this structure.
- by conditioning on associated loci other than the candidate locus.
## Pitfalls of LMM
1. Computational complexity:
- $M > 500.000$, $N > 70000$
- Building the GRM ($M \times M$ matrix)
- Estimating the mean and variance components for each of the $M$ candidate SNP!
- Association statistics for each variant (for each SNP!)
2. Loss in power when the candidate marker is included in the GRM
3. Using a small subset of markers in the GRM can compromise correction for stratification