Should ordinal variables be normalized for PCA? [duplicate]

Question

I need to analyze my (ecological) data with PCA, but the data don't seem to meet the assumption of normality very well. The problem is, that out of my 9 variables only two are continuous and the others are ordinal. So I was wondering if it's possible / recommended to apply any transformations to these ordinal data in order to normalize them, or should I only transform the two continuous variables and leave the rest as they are (and are there any references available on the matter)? I am aware that PCA may not be the most ideal option for analyzing this kind of data, but I must run it anyway.

As to interpreting analyses made with normalized / transformed data, I'm also a bit confused if there should be some kind of 'back-transformations' or can the outcome be interpreted just as it is (regarding PCA in particular)?

Answer 1

It isn't really possible to turn ordinal data into normally distributed data. However, PCA typically decomposes the correlation matrix (you can also decompose the covariance matrix, but this is less common). So you can create your own correlation matrix by using Pearson product-moment correlations between the continuous variables, and other types of correlations, as appropriate, between the ordinal variables. Two possibilities might be Spearman's rank correlation, or the polychoric correlation, if you believe that your ordinal data come from a categorization of an underlying normal distribution.

As far as the interpretation of your PCA goes, I don't see any immediate issues. There wouldn't be any need for back-transformation, for example. PCA will give you a set of eigenvectors that tell you how your original variables are related to the principal components (your new variables), and a set of eigenvalues that tells you how much your data spread out along each of the new variables / how elongated (as opposed to spherical) your data cloud is. To understand PCA better, it may help you to read this excellent CV thread: Making sense of principal component analysis, eigenvectors & eigenvalues.