Yes, the PC axes are “comparable”. I think the best way to think about what a 
PCA does is to interpret it as a projection of a multidimensional space down to 
a low dimensional space that captures as much of the overall variation as 
possible. The first axis is somewhat special because it represents the best 
1-dimensional space. Past that one should think of 1 and 2 giving the best 
2-dimensional space and 1, 2, and 3 giving the best 3-dimensional space, etc. 
The axes themselves are not of a priori interest in an application – it is the 
space that is of interest. A consequence is that plots showing projections of 
points relative to PC1, PC2,etc. must be plotted to the same scale (i.e., 
consistent with the fact that the eigenvalues give the variances along each 
axis). If, as unfortunately often the case, the axes are plotted using 
different scales then the space has been distorted and is no longer the space 
that best accounts for the overall variation in the data. That also distorts 
the impressions one gets in looking at the plot as using different scales 
changes the relative distances between points.


Within that reduced space one may find that particular axes can seem to be 
interpretable but one should really look at the space and decide which 
directions within the space are most interesting based on the patterns of the 
data. That is, the data need to suggest interesting direction unless one has 
some a priori groups one wishes to compares. Often the first PC is of special 
interest but that is often due to allometry and the relatively large impact of 
size variation. That is, by now, a rather boring result! The individual PC axes 
are defined based on convenient mathematical properties – not based on any 
biological models so each one should not be considered separately as things of 
special interest.


The above also means that one need not just visualize variation along each axis 
separately. One can, as in tpsRelw software, visualize any specified point 
within the PC space or in any direction of interest within the PC space.


_ _ _ _ _ _ _ _ _

F. James Rohlf, Distinguished Prof. Emeritus

Depts. of Anthropology and of Ecology & Evolution



From: Yinan Hu <> 
Sent: Friday, May 18, 2018 2:19 PM
Subject: Re: [MORPHMET] How to project shape difference onto different PC


Dear James,


Thanks for the reply. Yes I have completed a PCA on a GM dataset with 11 
landmarks, and you got it exactly right that I'm trying to decompose shape 
differences onto individual PCs.  


The reason I was hesitating to do the vector projection is that I'm not sure if 
PC scores on different PCs are directly comparable to each other. For 
simplicity, let's say I'm only considering PC1 and PC2, which explains 80% of 
shape variation in total (60% + 20%). Group A has a mean PC1 score of 0.5, and 
PC2 score of 0.1; where as Group B has a mean PC1 score of 0.4 and PC2 score of 
0.3.  Then I'm looking at a 0.1 difference along PC1 and a 0.2 difference along 
PC2 between these two groups. 


Would this mean they differ twice as much along PC2 than PC1, such that in the 
80% of shape variation explained by these two PCs, 1/3 is along PC1 and 2/3 is 
along PC2?


But considering that PC1 explains three times more variation than PC2 (60% vs 
20%), would this mean I should weigh the PC score difference (distance along 
each PC)? i.e. although the absolute difference in mean PC1 score is 0.1, it 
should be weighed three times more than the difference along PC2 so in the 80% 
of shape variation explained by these two PCs, 3/5 is along PC1 and 2/5 is 
along PC2?



On the other hand, I agree visualizing the shape difference along each PC can 
be helpful, and I'm pretty sure the plotRefToTarget function from the R package 
geomorph can achieve this.


Thanks again.





On Friday, May 18, 2018 at 12:53:32 PM UTC-4, K. James Soda wrote:

Dear Dr. Hu,

Let me begin by restating how I understand the question: You have completed a 
PCA on a morphological data set in which there are two subsets of interest. Now 
you would like to decompose the difference between the two subsets into 
differences along individual PCs. Here is my two cents on the issue:

I would say that the literal solution to this problem would probably be 
something along the lines of what you proposed. For simplicity, say that you 
summarized each subset using its mean position in the PC space. This would be 
expressed as a vector where each element is a position along a single PC. The 
difference between these two vectors would then be a decomposition of how far 
you would need to move along each PC axis to move from one mean to the other. 
You could then standardize the elements so that their absolute values sum to 
one. This would be an expression of what percentage of the distance is along 
each PC.

What I perceive as the subtext of your question, though, is whether this sort 
of decomposition has a reasonable interpretation, and the answer to this 
question is somewhat trickier. Assuming this is a GM data set, the more 
relevant point might be how you convert the difference into visualizations. A 
nice feature of GM data is that each PC will correspond to a "type" of 
deformation. This feature can be used to decompose the difference between two 
shapes in a shape-PC space as well. For example, imagine you moved from one 
mean shape in the PC space to the other by only moving parallel to PC axes. If 
you are interested in two PCs, this could be accomplished in two ways. You 
could then visualize the shape at the points where you make a turn; that is, 
you would visualize how mean shape 1 would need to be deformed to have the same 
PC1 or PC2 score as mean shape 2 if all other PCs were held constant. The 
degree of deformation would then provide a qualitative measure of how radical 
each PC's contribution is to the shape difference. Of course, this is not a 
quantitative measure, as you requested, but I would argue it is a more helpful 
assessment b/c it directly corresponds to observable phenomena. How helpful, 
though, will depend on your research question.

Hope something in there helps a little,



On Thu, May 17, 2018 at 10:15 AM, Yinan Hu < <javascript:> > 

Dear colleagues,


I'm trying to figure out how to break down shape differences onto individual PC 
axes.  I have a morphospace where PC1 explains 60% of shape variation and PC2 
explains 20% of variation. Two subsets of samples of particular interest do not 
differ much along PC1, but differs significantly along PC2. How should I 
project the shape difference between these subsets onto seperate PC axes, such 
that I can quantitatively show X% of shape difference between them are along 
PC1 and Y% is along PC2?


A simple vector projection (i.e. using the mean difference of PC1 score and PC2 
score) doesn't feel right to me as I don't think PC scores are directly 
comparable between different PCs. Or am I wrong?

Any suggestions would be greatly appreciated.


Many thanks for your time.

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