Hello Patric,

There's nothing about PCA or common factor analysis that forbids instances of cross-loadings, especially if you are trying to make sense out of an oblique rotation of the structure. Some folks just don't like them, however.

Yes, structures are easier to interpret (as is variable-component or variable-factor salience) when the "model" solution is that of Thurstone's simple structure. However, real data, manifest variables, method variance, and participant inconsistency all contrive to make results messy rather than in pristine alignment with some idealized standard.

From the sound of your query, however, I would urge you to consider using a different analytic method than PCA. Common factor analysis, using maximum likelihood or principal axis extraction, would make much more sense given the small hints you've dropped about your measure/variable set. As well, when it comes to rotation schemes, I'd start first with oblique rather than presume that orthogonal methods would work best.

Good luck with your work.

Patric Kavanagh, David Morse has given you some really good advice. I would add that different oblique rotations can produce different outcomes, so maybe check that out. In some of our own research, promax rotations yielded slightly more interpretable solutions than did direct oblimin rotations - and both were noticeably better than varimax rotations (which wasn't unexpected given that we thought our factors would be, and were, correlated).

In addition, I'd consider the extent of cross loadings. I confess to preferring them not to exist, but I don't worry if they are different by > .20. Cross loadings that differ by

I also agree with David Morse in recommending either maximum likelihood or principal axis extraction with oblique rotation. In particular, as a counter argument to using orthogonal rotation (i.e., zero correlation among the factors), I often ask my students to develop conceptual names for those factors that would indeed correspond to zero overlap in their content. Instead, the result is that is almost always some underlying relationship between what the factors appear to measure.

Many thanks for your comments. An oblique rotation using promax provided better results

Patric Kavanagh, thanks for conveying the good news. I'm pleased for you.

There is one residual thing that troubles me: the "definition" of a cross loading.

Some researchers say that if an item loads at greater than a certain satisfactory level (say, .40) on two or more factors, it should be regarded as cross-loading. Others say that there is no problem if an item loads on two or more factors above a specified satisfactory level (again, say, .40) as long as it loads to a GREATER extent (say .10 more, but preferably .20 more) on one factor relative to the other factor(s).

These are two slightly different perspectives. I wonder what you might have done, Patric, and what seems to be the recommended practice according to others.

If you want to use PCA, but do not want a lot of the small values in the eigenvectors, you should use a sparse PCA. See for example https://web.stanford.edu/~hastie/Papers/spc_jcgs.pdf.

ps. This is implemented in a few R packages, see for example https://cran.r-project.org/web/packages/sparsepca/sparsepca.pdf