Interpreting PCA scores

Question

Can anyone help me in interpreting PCA scores? My data come from a questionnaire on attitudes toward bears. According to the loadings, I have interpreted one of my principal components as "fear of bears". Would the scores of that principal component be related to how each respondent measures up to that principal component (whether he/she scores positively/negatively on it)?

Answer 1

Basically, the factor scores are computed as the raw responses weighted by the factor loadings. So, you need to look at the factor loadings of your first dimension to see how each variable relate to the principal component. Observing high positive (resp. negative) loadings associated to specific variables means that these variables contribute positively (resp. negatively) to this component; hence, people scoring high on these variables will tend to have higher (resp. lower) factor scores on this particular dimension.

Drawing the correlation circle is useful to have a general idea of the variables that contribute "positively" vs. "negatively" (if any) to the first principal axis, but if you are using R you may have a look at the FactoMineR package and the dimdesc() function.

Here is an example with the USArrests data:

> data(USArrests)
> library(FactoMineR)
> res <- PCA(USArrests)
> dimdesc(res, axes=1)  # show correlation of variables with 1st axis
$Dim.1
$Dim.1$quanti
         correlation  p.value
Assault        0.918 5.76e-21
Rape           0.856 2.40e-15
Murder         0.844 1.39e-14
UrbanPop       0.438 1.46e-03
> res$var$coord  # show loadings associated to each axis
         Dim.1  Dim.2  Dim.3   Dim.4
Murder   0.844 -0.416  0.204  0.2704
Assault  0.918 -0.187  0.160 -0.3096
UrbanPop 0.438  0.868  0.226  0.0558
Rape     0.856  0.166 -0.488  0.0371

As can be seen from the latest result, the first dimension mainly reflects violent acts (of any kind). If we look at the individual map, it is clear that states located on the right are those where such acts are most frequent.

You may also be interested in this related question: What are principal component scores?

Answer 2

For me, PCA scores are just re-arrangements of the data in a form that allows me to explain the data set with less variables. The scores represent how much each item relates to the component. You can name them as per factor analysis, but its important to remember that they are not latent variables, as PCA analyses all variance in the data set, not just the elements held in common (as factor analysis does).

Answer 3

PCA results (the different dimensions or commponents) generally can't be translated into a real concept I think is wrong to assume that one of the components is "fear of bears" what lead you to think that was what the component meant? Principal components procedure transforms your data matrix to a new data matrix with the same or less amount of dimensions, and the resulting dimensions range from the one that better explains the variance to the one that explains it the less. This components are calculated based on a combination of the original variables with the calculated eigenvectors. Overal PCA procedure does convert the original variables to orthogonal ones (linearly independent). Hope this helps you clarify a little about pca procedure