Interpreting PCA scores

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

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.

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

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