# Re: [MORPHMET] How to project shape difference onto different PC

```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

Hope something in there helps a little,

James

On Thu, May 17, 2018 at 10:15 AM, Yinan Hu <yinanhu...@gmail.com> wrote:

> 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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