Imagine I've the following matrix, which gives the grades of students in the subjects German, Philosophy, Math and Physics:
ger = c(2,4,1,3,2,4,4,1,2,3) phi = c(3,4,1,2,2,3,3,2,2,2) mat = c(1,3,2,4,1,2,2,4,3,1) phy = c(2,2,2,5,2,2,3,4,3,3) A = cbind(ger,phi,mat,phy)
I combine everything to a matrix and scale the data:
As = scale(A)
Now, I perform a
summary on the PCA:
summary(princomp(As), loadings = TRUE)
Which returns the following output:
Importance of components: Comp.1 Comp.2 Comp.3 Comp.4 Standard deviation 1.3257523 1.1657791 0.59600603 0.35793402 Proportion of Variance 0.4882275 0.3775114 0.09867311 0.03558799 Cumulative Proportion 0.4882275 0.8657389 0.96441201 1.00000000 Loadings: Comp.1 Comp.2 Comp.3 Comp.4 ger 0.496 -0.502 0.519 0.482 phi 0.548 -0.443 -0.423 -0.570 mat -0.430 -0.572 -0.546 0.435 phy -0.518 -0.474 0.503 -0.503
I have a few hints for the first component (based on the loadings):
- There is a high positive correlation between german and philosophy and there is also a high positive correlation between math and physics.
- Who is good in language (german and philosophy) is often worse in MINT (math and physics) and the other way around.
And an idea about the second one, which I cannot interpret:
- It's a weighted arithmetic mean over all four variables.
But I have no idea how to interpret the
Comp. 3 and
Comp. 4 based on the loadings. Especially because all values of
Comp. 2 are all negative, or have the same orientation. Can someone help me? Thanks in advance!