[R] Principle Component Analysis: Ranking Animal Size Based On Combined Metrics

[Sidoti, Salvatore A.]

Let's say I perform 4 measurements on an animal: three are linear measurements in millimeters and the fourth is its weight in milligrams. So, we have a data set with mixed units.

Based on these four correlated measurements, I would like to obtain one "score" or value that describes an individual animal's size. I considered simply taking the geometric mean of these 4 measurements, and that would give me a "score" - larger values would be for larger animals, etc.

However, this assumes that all 4 of these measurements contribute equally to an animal's size. Of course, more than likely this is not the case. I then performed a PCA to discover how much influence each variable had on the overall data set. I was hoping to use this analysis to refine my original approach.

I honestly do not know how to apply the information from the PCA to this particular problem...

I do know, however, that principle components 1 and 2 capture enough of the variation to reduce the number of dimensions down to 2 (see analysis below with the original data set).

Note: animal weights were ln() transformed to increase correlation with the 3 other variables.

df <- data.frame(
  weight = log(1000*c(0.0980, 0.0622, 0.0600, 0.1098, 0.0538, 0.0701, 0.1138, 0.0540, 0.0629, 0.0930,
             0.0443, 0.1115, 0.1157, 0.0734, 0.0616, 0.0640, 0.0480, 0.1339, 0.0547, 0.0844,
             0.0431, 0.0472, 0.0752, 0.0604, 0.0713, 0.0658, 0.0538, 0.0585, 0.0645, 0.0529,
             0.0448, 0.0574, 0.0577, 0.0514, 0.0758, 0.0424, 0.0997, 0.0758, 0.0649, 0.0465,
             0.0748, 0.0540, 0.0819, 0.0732, 0.0725, 0.0730, 0.0777, 0.0630, 0.0466)),
  interoc = c(0.853, 0.865, 0.811, 0.840, 0.783, 0.868, 0.818, 0.847, 0.838, 0.799,
              0.737, 0.788, 0.731, 0.777, 0.863, 0.877, 0.814, 0.926, 0.767, 0.746,
              0.700, 0.768, 0.807, 0.753, 0.809, 0.788, 0.750, 0.815, 0.757, 0.737,
              0.759, 0.863, 0.747, 0.838, 0.790, 0.676, 0.857, 0.728, 0.743, 0.870,
              0.787, 0.773, 0.829, 0.785, 0.746, 0.834, 0.829, 0.750, 0.842),
  cwidth = c(3.152, 3.046, 3.139, 3.181, 3.023, 3.452, 2.803, 3.050, 3.160, 3.186,
             2.801, 2.862, 3.183, 2.770, 3.207, 3.188, 2.969, 3.033, 2.972, 3.291,
             2.772, 2.875, 2.978, 3.094, 2.956, 2.966, 2.896, 3.149, 2.813, 2.935,
             2.839, 3.152, 2.984, 3.037, 2.888, 2.723, 3.342, 2.562, 2.827, 2.909,
             3.093, 2.990, 3.097, 2.751, 2.877, 2.901, 2.895, 2.721, 2.942),
  clength = c(3.889, 3.733, 3.762, 4.059, 3.911, 3.822, 3.768, 3.814, 3.721, 3.794,
              3.483, 3.863, 3.856, 3.457, 3.996, 3.876, 3.642, 3.978, 3.534, 3.967,
              3.429, 3.518, 3.766, 3.755, 3.706, 3.785, 3.607, 3.922, 3.453, 3.589,
              3.508, 3.861, 3.706, 3.593, 3.570, 3.341, 3.916, 3.336, 3.504, 3.688,
              3.735, 3.724, 3.860, 3.405, 3.493, 3.586, 3.545, 3.443, 3.640))

pca_morpho <- princomp(df, cor = TRUE)

summary(pca_morpho)

Importance of components:
                         		Comp.1    	Comp.2    	Comp.3    	Comp.4
Standard deviation     	1.604107 	0.8827323 	0.7061206 	0.3860275
Proportion of Variance 	0.643290 	0.1948041 	0.1246516 	0.0372543
Cumulative Proportion  	0.643290 	0.8380941 	0.9627457 	1.0000000

Loadings:
        		Comp.1 	Comp.2	Comp.3 	Comp.4
weight  	-0.371  	0.907        			-0.201
interoc 	-0.486 	-0.227 	-0.840       
cwidth  	-0.537 	-0.349  	0.466 		-0.611
clength 	-0.582         			0.278  	0.761

               		Comp.1	Comp.2	Comp.3	Comp.4
SS loadings      	1.00   		1.00   		1.00   		1.00
Proportion Var   	0.25   		0.25   		0.25 		0.25
Cumulative Var   	0.25   		0.50   		0.75   		1.00

Any guidance will be greatly appreciated!

Salvatore A. Sidoti
PhD Student
The Ohio State University
Behavioral Ecology