The aim of PCA is to replace a large number of correlated variables with a smaller number of uncorrelated variables. These variables, called PC, are linear combinations of the observed variables. I really appreciated the explanation of PCA here Making sense of principal component analysis, eigenvectors & eigenvalues.
Would it be possible to include the results of PCA (loadings) as a grouping factor in the model? Here I mean the random effect part in mixed-effect models.
For example, you have different sites with some data. You know that the relationship between response and the explanatory variable is different between those sites and due to this I would like to include the "site" as a random effect. However, there are too many sites compared to the data and I would like to groups them based on the same characteristics. Could I use the PC1 loadings for this?