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Principal components analysis

Welcome! In this practical we will do principal component analysis (PCA), one of the fundamental tools of population genetics for identifying sample clustering and outliers. PCA is several things at the same time: a way of reducing the dimension of a giant matrix of genotypes by projecting it onto a smaller set of important directions; a way to evaluate and intepret relatedness and population structure between samples; a way to find data outliers and unexpected samples.

Because these features are often correlated to environmental factors in a large dataset, this makes it a great way to potentially control for environmental confounding in genome-wide association studies.

How PCA works​

Imagine we a giant matrix of numbers - say one with LL rows and NN columns, like this:

X=column 1column 2⋯column Nrow 1x11x12⋯x1Nrow 2x21x22⋯x2N⋮⋮⋱row LxL1xL2⋯xLNX = \begin{array}{c|cccc} & \text{column 1} & \text{column 2} & \cdots & \text{column $N$} \\ \hline \text{row $1$} & x_{11} & x_{12} & \cdots & x_{1N} \\ \text{row $2$} & x_{21} & x_{22} & \cdots & x_{2N} \\ \vdots & \vdots & & \ddots & \\ \text{row $L$} & x_{L1} & x_{L2} & \cdots & x_{LN} \end{array}

For example - it could be a matrix of genotypes in a genome-wide association study (rows are SNPs and columns are samples). Or it could ba a matrix of expression levels (rows are genes and columns are samples). Whatever it is, it it's a big multidimensional object, and it would be nice to have a simple way to visualise and analyse its structure.

Principal components analysis (PCA) does this by re-writing the matrix in terms of a small number of 'principal components' - directions in space. The directions are orthogonal to each other (i.e. at right angles) so, in a sense, they each represent a seperate part of the matrix. Together, the principal components capture the major structure of the matrix and often reveal the most important features.

Principal components for GWAS​

In this tutorial you will carry out principal components analysis on a set of human samples using the plink software package. Because this is for use in a GWAS study, we will carry out many of the steps that are usually done in those studies, including including pruning SNPs for local correlation patterns, identifying and excluding closely-related individuals, and comparing samples to a global reference panel. Before doing that, however, we'll also try a simple simulated PCA example in R.

When you're ready, go and get setup for the practical. The complete set of pages is as follows: