statistics - PCA and Hotelling's T^2 for confidence intervall in R

Question

I made a principal component analysis and took the 2 first principal components. I made a chart of my points based on the score of the 2 PC. I would like to add on this graph a 95% confidence region corresponding to the Hotelling's T^2 test in order to detect the points that are out of the ellipse (outliers) How is it possible in R? Do you have any example?

I would do something like this and detect the points out of the ellipse:

Answer 1

We can plot the confidence ellipse for PCA with vegan or ggbiplot as below:

set.seed(1)
data <- matrix(rnorm(500), ncol=5) # some random data
data <- setNames(as.data.frame(rbind(data, matrix(runif(25, 5, 10), ncol=5))), LETTERS[1:5]) # add some outliers
class <- sample(c(0,3,6,8), 105, replace=TRUE) # 4 groups

library(vegan)
PC <- rda(data, scale=TRUE)
pca_scores <- scores(PC, choices=c(1,2))
plot(pca_scores$sites[,1], pca_scores$sites[,2],
     pch=class, col=class, xlim=c(-2,2), ylim=c(-2,2))
arrows(0,0,pca_scores$species[,1],pca_scores$species[,2],lwd=1,length=0.2)
ordiellipse(PC,class,conf=0.95)

library(ggbiplot)
PC <- prcomp(data, scale = TRUE)
ggbiplot(PC, obs.scale = 1, var.scale = 1, groups = as.factor(class), ellipse = TRUE, 
                                                    ellipse.prob = 0.95)