PCA on genotype matrix with multiple alleles

Question

Consider an m x n genotype matrix of m haploid samples and n SNPs where each value is an allele encoded by an integer (0,1,2,3).

Is there a good/standard way to encode the alleles in order to perform a PCA on this matrix to investigate population structure ?

I have seen this with matrices of diploid samples where each SNP is encoded as 0/1/2 to represent the number of non-reference allele, but is there a way to consider more than 2 alleles ?

e.g. difference between alleles 1 and 3 should be equal to difference between alleles 1 and 2

Answer 1

You can make a 'dummy variable' for each allele. That means that you don't have info per SNP, but for SNPs with more alleles, the allele is present (1) or not (0).

Answer 2

I recommend the Hamming distance and doing a multidimensional scaling (a procedure similar to PCA but for distances), that way you don't create new variables for the same position.

The distance function can be defined as

hamming_dist <- function(x, y) {
    if (x != y) {
        1
    } else {
        0
    }

Or you can use a package:

library(e1071)
H <- hamming.distance(as.matrix(X))

Or as you yourself pointed other implementations.

You can use them to calculate the distance between all samples and then use it for the multidimensional analysis (cmdscale in R)

plot(cmdscale(H))