I have a question about conducting a PCA between variables that are measured in different units. I understand the importance of using a correlation matrix versus a covariance matrix to minimize variance. The data I'm working with is not normally distributed and has not been transformed in other tests.
For example, there are three variables A, B, and C, and 20 observations, where 10 observations are measured using 1 set of units, and the other 10 observations are measured using another set of units*. The values between the units are quite different in in value and variance (expected). The data is not normal in either units and has not been transformed.
The measurements using the first set of units is 2 to 3 orders of magnitude higher than those measured using the other units (expected). I have conducted a PCA using a correlation matrix and interpreted results. However a non-statistician recommended I `standardize' the measured data, such that I'm using ratio or fractions for all the observations for each of the variables: Variable A/(sum of all 3 variables) and so on and so forth for Variables B and C.
However, a PCA using the contributing fraction of each variable is different from PCA using the measured value in different units leading to different eigenvalues and eigenvectors, thus leading to two different scientific interpretations.
Beyond using the appropriate association matrix, Is this "standardization" step valid and or necessary from a statistical perspective? Update: Should PCA be done on compositional data?