I have 400 samples chemically analysed for 48 elements. Some elements are correlated. My goal is to identify outliers in a multivariate space, samples that are unique in their composition. I compute Mahalanobis distance after auto-scaling and normalizing the data to find anomalous samples. Now, I like to know what variables (elements) or variable combinations contribute most to extreme Mahalanobis distance. I like to display the results in a bar diagram.
I use factor analysis with oblique rotation to determine factor loadings and sample scoring on the components. I retain all 48 components. Now, I investigate high scores on the components and high residuals ( sample being off the line of components). I expect high scores on the components contribute most to MAHA-distance. The same should apply to high residuals but I don't know how to describe it mathematically. So far I haven't considered the scaling along the component vectors (shrinking or enlarging) to account for correlation. It has to do with the eigenvalue of the component. Do I take care of the scaling by multiplying the sample score on the component by 1/eigenvalue? I'm not sure if all of it is a viable approach or if I should rather look into multivariate regression to solve my goal?