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

David L Carlson dcarlson at tamu.edu
Mon Nov 14 19:04:41 CET 2016

```Usually you want to use the geometric mean on variables measured on the same scale, but in your case, transforming weight didn't change much. Adding cube root transformation as another approach (since weight should increase as the cube of the linear measures), the correlations with the 3 linear measurements are about the same for the transformed values and the 3 transformations are very strongly correlated:

> wgt.log <- log(df\$weight)
> wgt.cube <- df\$weight^(1/3)
> cor(cbind(weight=df\$weight, wgt.log, wgt.cube), df[, -1])
interoc    cwidth   clength
weight   0.3048239 0.2545593 0.4884446
wgt.log  0.3096511 0.2807528 0.4841830
wgt.cube 0.3077714 0.2724312 0.4863528
> cor(cbind(weight=df\$weight, wgt.log, wgt.cube))
weight   wgt.log  wgt.cube
weight   1.0000000 0.9862879 0.9939102
wgt.log  0.9862879 1.0000000 0.9984574
wgt.cube 0.9939102 0.9984574 1.0000000

David C

-----Original Message-----
From: Sidoti, Salvatore A. [mailto:sidoti.23 at buckeyemail.osu.edu]
Sent: Monday, November 14, 2016 11:41 AM
To: David L Carlson; Jim Lemon; r-help mailing list
Subject: RE: [R] Principle Component Analysis: Ranking Animal Size Based On Combined Metrics

Fascinating! So it appears that I can simply take the geometric mean of all 4 metrics (unscaled), including weight, then designate that value as a relative measure of "size" within my sample population. The justification for using the geometric mean is shown by the high correlation between PC1 and the size values:

pc1         gm
pc1  1.0000000 -0.8458024
gm  -0.8458024  1.0000000

Pearson's product-moment correlation
data:  pc1 and gm
t = -10.869, df = 47, p-value = 2.032e-14
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.9104585 -0.7407939
sample estimates:
cor
-0.8458024

Salvatore A. Sidoti
PhD Student
Behavioral Ecology

-----Original Message-----
From: David L Carlson [mailto:dcarlson at tamu.edu]
Sent: Monday, November 14, 2016 11:07 AM
To: Sidoti, Salvatore A. <sidoti.23 at buckeyemail.osu.edu>; Jim Lemon <drjimlemon at gmail.com>; r-help mailing list <r-help at r-project.org>
Subject: RE: [R] Principle Component Analysis: Ranking Animal Size Based On Combined Metrics

The first principal component should be your estimate of "size" since it captures the correlations between all 4 variables. The second principle component must be orthogonal to the first so that if the first is "size", the second pc is independent of size, perhaps some measure of "shape". As would be expected, the first principal component is highly correlated with the geometric mean of the three linear measurements and moderately correlated with weight:

> gm <- apply(df[, -1], 1, prod)^(1/3)
> pc1 <- prcomp(df, scale.=TRUE)\$x[, 1]
> plot(pc1, gm)
> cor(cbind(pc1, gm, wgt=df\$weight))
pc1         gm        wgt
pc1  1.0000000 -0.9716317 -0.5943594
gm  -0.9716317  1.0000000  0.3967369
wgt -0.5943594  0.3967369  1.0000000

-------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77840-4352

-----Original Message-----
From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of Sidoti, Salvatore A.
Sent: Sunday, November 13, 2016 7:38 PM
To: Jim Lemon; r-help mailing list
Subject: Re: [R] Principle Component Analysis: Ranking Animal Size Based On Combined Metrics

Hi Jim,

Nice to see you again! First of all, apologies to all for bending the rules a bit with respect to the mailing list. I know this is a list for R programming specifically, and I have received some great advice in this regard in the past. I just thought this was an interesting applied problem that would generate some discussion about PCA in R.

Yes, that is an excellent question! Indeed, why not just volume? Since this is still a work in progress and we have not published as of yet, I would rather not be more specific about the type of animal at this time ;>}. Nonetheless, I can say that the animals I study change "size" depending on their feeding and hydration state. The abdomen in particular undergoes drastic size changes. That being said, there are key anatomical features that remain fixed in the adult.

Now, there *might* be a way to work volume into the PCA. Although volume is not a reliable metric since the abdomen size is so changeable while the animal is alive, but what about preserved specimens? I have many that have been marinating in ethanol for months. Wouldn't the tissues have equilibrated by now? Probably... I could measure volume by displacement or suspension, I suppose.

In the meantime, here's a few thoughts:

1) 	Use the contribution % (known as C% hereafter) of each variable on principle components 1 and 2.

2) 	The total contribution of a variable that explains the variations retained by PC1 an PC2 is calculated by:

sum(C%1 * eigenvalue1, C%2 * eigenvalue2)

3) Scale() to mean-center the columns of the data set.

4) Use these total contributions as the weights of an arithmetic mean.

For example, we have an animal with the following data (mean-centered):
weight:	1.334
interoc:	-0.225
clength:	0.046
cwidth:	-0.847

The contributions of these variables on PC1 and PC2 are (% changed to proportions):
weight:	0.556
interoc:	0.357
clength:	0.493
cwidth:	0.291

To calculate size:
1.334(0.556) - 0.225(0.357) + 0.046(0.493) - 0.847(0.291) = 0.43758 Then divide by the sum of the weights:
0.43758 / 1.697 = 0.257855 = "animal size"

This value can then be used to rank the animal according to its size for further analysis...

Does this sound like a reasonable application of my PCA data?

Salvatore A. Sidoti
PhD Student
Behavioral Ecology

-----Original Message-----
From: Jim Lemon [mailto:drjimlemon at gmail.com]
Sent: Sunday, November 13, 2016 3:53 PM
To: Sidoti, Salvatore A. <sidoti.23 at buckeyemail.osu.edu>; r-help mailing list <r-help at r-project.org>
Subject: Re: [R] Principle Component Analysis: Ranking Animal Size Based On Combined Metrics

Hi Salvatore,
If by "size" you mean volume, why not directly measure the volume of your animals? They appear to be fairly small. Sometimes working out what the critical value actually means can inform the way to measure it.

Jim

On Sun, Nov 13, 2016 at 4:46 PM, Sidoti, Salvatore A.
<sidoti.23 at buckeyemail.osu.edu> wrote:
> 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
>
>                         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
> 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
>
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