Principal Components Analysis - how to get the contribution (%) of each parameter to a Prin.Comp.?

Question

I want to know to what degree a measurement/parameter contributes to one of the calculated principal components.

A real-world description:

1. i\'ve got fi

Answer 1

You want the $loadings component of the returned object:

R> class(pca$loadings)
[1] "loadings"
R> pca$loadings

Loadings:
  Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
a -0.198  0.713        -0.671       
b  0.600         0.334 -0.170  0.707
c -0.600        -0.334  0.170  0.707
d  0.439        -0.880 -0.180       
e  0.221  0.701         0.678       

               Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
SS loadings       1.0    1.0    1.0    1.0    1.0
Proportion Var    0.2    0.2    0.2    0.2    0.2
Cumulative Var    0.2    0.4    0.6    0.8    1.0

Note that this has a special print() method which suppresses printing of small loadings.

If you want this as a relative contribution then sum up the loadings per column and express each loading as a proportion of the column (loading) sum, taking care to use the absolute values to account for negative loadings.

R> load <- with(pca, unclass(loadings))
R> load
      Comp.1       Comp.2      Comp.3     Comp.4        Comp.5
a -0.1980087  0.712680378  0.04606100 -0.6713848  0.000000e+00
b  0.5997346 -0.014945831  0.33353047 -0.1698602  7.071068e-01
c -0.5997346  0.014945831 -0.33353047  0.1698602  7.071068e-01
d  0.4389388  0.009625746 -0.88032515 -0.1796321  5.273559e-16
e  0.2208215  0.701104321 -0.02051507  0.6776944 -1.110223e-16

This final step then yields the proportional contribution to the each principal component

R> aload <- abs(load) ## save absolute values
R> sweep(aload, 2, colSums(aload), "/")
      Comp.1      Comp.2     Comp.3     Comp.4       Comp.5
a 0.09624979 0.490386943 0.02853908 0.35933068 0.000000e+00
b 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
c 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
d 0.21336314 0.006623362 0.54544349 0.09614059 3.728970e-16
e 0.10733880 0.482421595 0.01271100 0.36270762 7.850462e-17

R> colSums(sweep(aload, 2, colSums(aload), "/"))
Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 
     1      1      1      1      1

---

If using the preferred prcomp() then the relevant loadings are in the $rotation component:

R> pca2 <- prcomp(my_table, scale = TRUE)
R> pca2$rotation
         PC1          PC2         PC3        PC4           PC5
a -0.1980087  0.712680378 -0.04606100 -0.6713848  0.000000e+00
b  0.5997346 -0.014945831 -0.33353047 -0.1698602 -7.071068e-01
c -0.5997346  0.014945831  0.33353047  0.1698602 -7.071068e-01
d  0.4389388  0.009625746  0.88032515 -0.1796321 -3.386180e-15
e  0.2208215  0.701104321  0.02051507  0.6776944  5.551115e-17

And the relevant incantation is now:

R> aload <- abs(pca2$rotation)
R> sweep(aload, 2, colSums(aload), "/")
         PC1         PC2        PC3        PC4          PC5
a 0.09624979 0.490386943 0.02853908 0.35933068 0.000000e+00
b 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
c 0.29152414 0.010284050 0.20665322 0.09091055 5.000000e-01
d 0.21336314 0.006623362 0.54544349 0.09614059 2.394391e-15
e 0.10733880 0.482421595 0.01271100 0.36270762 3.925231e-17