Population genetics as a case study

Two is better than one

• Population genetics is interested in identifying and describing clusters of genetically related individuals

  • Population structure within a species’ range

  • Multivariate approaches will also produce synthetic variables built as linear combination of alleles

    $$ \text{new variable}=a_1\text{allele}_1+a_2\text{allele}_2+a_3\text{allele}_3+...+a_n\text{allele}_n $$

• Interested in among-individual diversity but also among-group ones

• Genetic variability can be decomposed using a standard multivariate ANOVA model of total variance = (variance between groups) + (variance within groups)

  $$ \text{var}(\bold{X})=B(\bold{X})+W(\bold{X}) $$

• PCA focuses on $\text{var}(\bold{X})$ to describe the global diversity, on the contrary DAPC optimises $B(\bold{X})$ while minimising $W(\bold{X})$ by producing discriminant functions

Identifying clusters

• Like LDA, DAPC requires a priori knowledge of groups, which is usually unknown or uncertain
• Achieved using *k-*means
  • For every given number of k, maximise $B(\bold{X})$
  • https://en.wikipedia.org/wiki/K-means_clustering#/media/File:K-means_convergence.gif
  • k-means is run sequentially with increasing k, and clustering solutions are compared using Bayesian Information Criterion (BIC)
  • Optimal clustering solution corresponds to the lowest BIC
• k-means is run after transforming that data using PCA, which reduces number of variables to speed up the clustering algorithm

Describing clusters

• It provides an efficient description of genetic clusters using a few synthetic variables constructed as linear combinations of the original variables (alleles)