Should I reword the Factor if it shows all negative loadings?

Question

I've seen very similar questions but none really address my issue. I ran a PCA on 29 items to identify different factors of stress and worry. All items were on a scale from 1 to 3 and none of these were reversed coded (Higher scores mean higher stress and worry). When I run the PCA the factors I obtain make sense in the grouping of the items; I've called these Depressive stress, Impulsive and Physiological. The fourth one incorporates items that reflect "Low energy" and they all have negative loadings. I have created factor scores for all 4 and run a cluster analysis. The cluster analysis only seems to make sense if I refer to the last factor as "High energy" rather than low, and I'm guessing this is because of the negative factor loadings. What should I do? I would like to use the name "Low Energy" so that it similarly indicates high stress as the other factors. Can I some how reverse the factor score?

cdalitz · Answer

Your "factors" are the eigen vectors of the covariance matrix. They are only defined up to length (usually normalized to one or to the sqaure root of the corresponding eigenvalue) and sign: the eigen value equation
$$Avec{v} = lambda vec{v}$$
still holds if $vec{v}$ is replaced with $-vec{v}$. You can thus simply multiply your "energy factor" with minus one.

Should I reword the Factor if it shows all negative loadings?

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