I have an intracluster correlation matrix, symmetric matrix capturing correlation between clusters (based on k means). What is a nice way to represent these clusters graphically using this matrix and the post intra-cluster correlation of each variable?

To graphically represent clusters based on an intracluster correlation matrix, especially when you have performed clustering using methods like K-means, you can use several visualization techniques. Here are some suggestions:

Correlogram

You can use the corrplot package in R to create a correlogram, which is a graphical display of a correlation matrix. This can help visualize the correlations between different variables within and across clusters.

library(corrplot)

# Assuming 'corr_matrix' is your intracluster correlation matrix
corrplot(corr_matrix, type = "upper", order = "hclust",
         tl.col = "black", tl.srt = 45)

This will create a colored matrix where the color intensity and the size of the circle are proportional to the correlation coefficients. Positive correlations are typically displayed in blue and negative correlations in red2.

Heatmap

A heatmap is another effective way to visualize the correlation matrix. You can use the heatmap function in R.

# Get some colors
col <- colorRampPalette(c("blue", "white", "red"))(20)

# Assuming 'corr_matrix' is your intracluster correlation matrix
heatmap(corr_matrix, col = col, symm = TRUE)

This will produce a heatmap where the colors represent the correlation coefficients, making it easy to identify high and low correlations2.

Cluster Dendrogram with Correlogram

You can combine a dendrogram (showing the clustering structure) with a correlogram to provide a more comprehensive view.

library(corrplot)

# Assuming 'corr_matrix' is your intracluster correlation matrix
corrplot(corr_matrix, type = "upper", order = "hclust",
         tl.col = "black", tl.srt = 45, addrect = 3)

The addrect argument can be used to add rectangles around clusters based on the dendrogram2.

Scatter Plots and Box Plots

For a more detailed look at the distribution of variables within each cluster, you can use scatter plots or box plots. These can help visualize the spread and central tendency of the variables within each cluster.

# Example using ggplot2 for box plots
library(ggplot2)

# Assuming 'data' is your dataset with a 'cluster' column
ggplot(data, aes(x = cluster, y = variable_name)) +
  geom_boxplot()

This will create box plots for each cluster, showing the median, quartiles, and outliers for the specified variable4.

Combining Correlogram with Significance Test

If you want to include the significance of the correlations, you can use the rcorr function from the Hmisc package to compute the correlation matrix and the corresponding p-values, and then visualize it using corrplot.

library(Hmisc)
library(corrplot)

# Compute correlation matrix and p-values
res.cor2 <- rcorr(data)

# Visualize with significance levels
corrplot(res.cor2$r, type = "upper", order = "hclust",
         tl.col = "black", tl.srt = 45, p.mat = res.cor2$P)

This will display the correlation coefficients along with their significance levels (indicated by stars)2.

By using these methods, you can effectively visualize the intracluster correlations and the structure of your clusters.