Test significance of clusters on a PCA plot

Question

Is it possible to test the significance of clustering between 2 known groups on a PCA plot? To test how close they are or the amount of spread (variance) and the amount of overlap between clusters etc.

Answer 1

Here is a qualitative method that uses ggplot(...) to draw 95% confidence ellipses around clusters. Note that stat_ellipse(...) uses the bivariate t-distribution.

library(ggplot2)

df     <- data.frame(iris)                   # iris dataset
pca    <- prcomp(df[,1:4], retx=T, scale.=T) # scaled pca [exclude species col]
scores <- pca$x[,1:3]                        # scores for first three PC's

# k-means clustering [assume 3 clusters]
km     <- kmeans(scores, centers=3, nstart=5)
ggdata <- data.frame(scores, Cluster=km$cluster, Species=df$Species)

# stat_ellipse is not part of the base ggplot package
source("https://raw.github.com/low-decarie/FAAV/master/r/stat-ellipse.R") 

ggplot(ggdata) +
  geom_point(aes(x=PC1, y=PC2, color=factor(Cluster)), size=5, shape=20) +
  stat_ellipse(aes(x=PC1,y=PC2,fill=factor(Cluster)),
               geom="polygon", level=0.95, alpha=0.2) +
  guides(color=guide_legend("Cluster"),fill=guide_legend("Cluster"))

Produces this:

Comparison of ggdata$Clusters and ggdata$Species shows that setosa maps perfectly to cluster 1, while versicolor dominates cluster 2 and virginica dominates cluster 3. However, there is significant overlap between clusters 2 and 3.

Thanks to Etienne Low-Decarie for posting this very useful addition to ggplot on github.

Answer 2

You could use a PERMANOVA to partition the euclidean distance by your groups:

data(iris)
require(vegan)

# PCA
iris_c <- scale(iris[ ,1:4])
pca <- rda(iris_c)

# plot
plot(pca, type = 'n', display = 'sites')
cols <- c('red', 'blue', 'green')
points(pca, display='sites', col = cols[iris$Species], pch = 16)
ordihull(pca, groups=iris$Species)
ordispider(pca, groups = iris$Species, label = TRUE)

# PerMANOVA - partitioning the euclidean distance matrix by species
adonis(iris_c ~ Species, data = iris, method='eu')

Answer 3

Hy, after seeing that prcomp plotting can be highly time-consuming, based on the work of Etienne Low-Decarie posted by jlhoward, and adding vector plotting from envfit{vegan} objects (Thanks to Gavin Simpson). I designed a function to create ggplots.

## -> Function for plotting Clustered PCA objects.
### Plotting scores with cluster ellipses and environmental factors
## After: https://stackoverflow.com/questions/20260434/test-significance-of-clusters-on-a-pca-plot
#         https://stackoverflow.com/questions/22915337/if-else-condition-in-ggplot-to-add-an-extra-layer
#         https://stackoverflow.com/questions/17468082/shiny-app-ggplot-cant-find-data
#         https://stackoverflow.com/questions/15624656/labeling-points-in-geom-point-graph-in-ggplot2
#         https://stackoverflow.com/questions/14711470/plotting-envfit-vectors-vegan-package-in-ggplot2
#         http://docs.ggplot2.org/0.9.2.1/ggsave.html

plot.cluster <- function(scores,hclust,k,alpha=0.1,comp="A",lab=TRUE,envfit=NULL,
                         save=FALSE,folder="",img.size=c(20,15,"cm")) {

  ## scores = prcomp-like object
  ## hclust = hclust{stats} object or a grouping factor with rownames
  ## k = number of clusters
  ## alpha = minimum significance needed to plot ellipse and/or environmental factors
  ## comp = which components are plotted ("A": x=PC1, y=PC2| "B": x=PC2, y=PC3 | "C": x=PC1, y=PC3)
  ## lab = logical, add label -rownames(scores)- layer
  ## envfit = envfit{vegan} object
  ## save = logical, save plot as jpeg
  ## folder = path inside working directory where plot will be saved
  ## img.size = c(width,height,units); dimensions of jpeg file

  require(ggplot2)
  require(vegan)
  if ((class(envfit)=="envfit")==TRUE) {
    env <- data.frame(scores(envfit,display="vectors"))
    env$p <- envfit$vectors$pvals
    env <- env[which((env$p<=alpha)==TRUE),]
    env <<- env
  }
  if ((class(hclust)=="hclust")==TRUE) {
    cut <- cutree(hclust,k=k)
    ggdata <- data.frame(scores, Cluster=cut)
    rownames(ggdata) <- hclust$labels
  }
  else {
    cut <- hclust
    ggdata <- data.frame(scores, Cluster=cut)
    rownames(ggdata) <- rownames(hclust)
  }
  ggdata <<- ggdata
  p <- ggplot(ggdata) +
    stat_ellipse(if(comp=="A"){aes(x=PC1, y=PC2,fill=factor(Cluster))}
                 else if(comp=="B"){aes(x=PC2, y=PC3,fill=factor(Cluster))}
                 else if(comp=="C"){aes(x=PC1, y=PC3,fill=factor(Cluster))},
                 geom="polygon", level=0.95, alpha=alpha) +
    geom_point(if(comp=="A"){aes(x=PC1, y=PC2,color=factor(Cluster))}
               else if(comp=="B"){aes(x=PC2, y=PC3,color=factor(Cluster))}
               else if(comp=="C"){aes(x=PC1, y=PC3,color=factor(Cluster))},
               size=5, shape=20)
  if (lab==TRUE) {
    p <- p + geom_text(if(comp=="A"){mapping=aes(x=PC1, y=PC2,color=factor(Cluster),label=rownames(ggdata))}
                       else if(comp=="B"){mapping=aes(x=PC2, y=PC3,color=factor(Cluster),label=rownames(ggdata))}
                       else if(comp=="C"){mapping=aes(x=PC1, y=PC3,color=factor(Cluster),label=rownames(ggdata))},
                       hjust=0, vjust=0)
  }
  if ((class(envfit)=="envfit")==TRUE) {
    p <- p + geom_segment(data=env,aes(x=0,xend=env[[1]],y=0,yend=env[[2]]),
                          colour="grey",arrow=arrow(angle=15,length=unit(0.5,units="cm"),
                                                    type="closed"),label=TRUE) +
      geom_text(data=env,aes(x=env[[1]],y=env[[2]]),label=rownames(env))
  }
  p <- p + guides(color=guide_legend("Cluster"),fill=guide_legend("Cluster")) +
    labs(title=paste("Clustered PCA",paste(hclust$call[1],hclust$call[2],hclust$call[3],sep=" | "),
                     hclust$dist.method,sep="\n"))
  if (save==TRUE & is.character(folder)==TRUE) {
    mainDir <- getwd ( )
    subDir <- folder
    if(file.exists(subDir)==FALSE) {
      dir.create(file.path(mainDir,subDir),recursive=TRUE)
    }
    ggsave(filename=paste(file.path(mainDir,subDir),"/PCA_Cluster_",hclust$call[2],"_",comp,".jpeg",sep=""),
           plot=p,dpi=600,width=as.numeric(img.size[1]),height=as.numeric(img.size[2]),units=img.size[3])
  }
  p
}

And as an example, using data(varespec) and data(varechem), note that varespec is transposed to show distance between species:

data(varespec);data(varechem)
require(vegan)
vare.euc <- vegdist(t(varespec),"euc")
vare.ord <- rda(varespec)
vare.env <- envfit(vare.ord,env=varechem,perm=1000)
vare.ward <- hclust(vare.euc,method="ward.D")

plot.cluster(scores=vare.ord$CA$v[,1:3],alpha=0.5,hclust=vare.ward, k=5,envfit=vare.env,save=TRUE)

Answer 4

I have found two distances to represent what you see on a PCA plot into numbers.

Mahalanobis distance:

require(HDMD)
md<-pairwise.mahalanobis(iris[,1:4],grouping=iris$Species)
md$distance 

         [,1]     [,2]      [,3]
[1,]   0.0000 91.65640 178.01916
[2,]  91.6564  0.00000  14.52879
[3,] 178.0192 14.52879   0.00000

Bhattacharyya distance:

require(fpc)
require(gtools)
lst<-split(iris[,1:4],iris$Species)

mat1<-lst[[1]]
mat2<-lst[[2]]
bd1<-bhattacharyya.dist(colMeans(mat1),colMeans(mat2),cov(mat1),cov(mat2))

mat1<-lst[[1]]
mat2<-lst[[3]]
bd2<-bhattacharyya.dist(colMeans(mat1),colMeans(mat2),cov(mat1),cov(mat2))

mat1<-lst[[2]]
mat2<-lst[[3]]
bd3<-bhattacharyya.dist(colMeans(mat1),colMeans(mat2),cov(mat1),cov(mat2))

dat<-as.data.frame(combinations(length(names(lst)),2,names(lst)))
dat$bd<-c(bd1,bd2,bd3)
dat

          V1         V2        bd
1     setosa versicolor 13.260705
2     setosa  virginica 24.981436
3 versicolor  virginica  1.964323

To measure significance between clusters

require(ICSNP)
HotellingsT2(mat1,mat2,test="f")