Re: [MORPHMET] Interpreting PCA results

[N. MacLeod]

I agree with Jim. However, this discussion does beg the question of what the 
status of landmark, semilandmark, or indeed pixel brightness configurations 
within multivariate spaces is? Very similar spaces have been used in the area 
of theoretical morphology to conduct various sorts of experiments dealing with 
the nature of morphological evolution, especially the development of patterns 
based on null models to which empirical patterns can be compared. Moreover, 
machine learning specialists are now using morphologies generated artificially, 
in ways that aren’t very different from the ways in which such visualisations 
can be created by morphometricians, to train their AI systems. McGhee 
distinguishes "theoretical morphospaces” derived from graphics equations (e.g., 
Raup’s coiling models) from (what he terms) the “empirical morphospaces” we 
deal with as morphometricians and that lie at the heart of this conversation. 
But are the two really that different? If so, why and in what cases? If not 
what does that mean for the ways in which we might use such spaces? I’ve long 
found this an interesting question to ponder. Any thoughts from the community?

Norm MacLeod


> On 15 May 2017, at 18:28, F. James Rohlf  wrote:
> 
> What is important is not the fact that one is going +/- one standard 
> deviation along each axis. When shape changes are subtle one may need to go 
> beyond the observed range to make it more obvious to the eye what the changes 
> are. Exactly how far one goes away from the mean is arbitrary. It is a 
> visualization – not statistics.
>  
> --
> F. James Rohlf New email: f.james.ro...@stonybrook.edu
> Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.
> & Research Professor. Dept. of Anthropology
> Stony Brook University 11794-4364
> WWW: http://life.bio.sunysb.edu/morph/rohlf
> P Please consider the environment before printing this email
>  


_

Professor Norman MacLeod
The Natural History Museum, Cromwell Road, London, SW7 5BD
(0)207 942-5204 (Office Landline)
(0)785 017-1787 (Mobile)
http://paleonet.org/MacLeod/

Department of Earth Sciences, University College
London, Gower Street, London WC1E 6BT, UK

Nanjing Institute of Geology & Palaeontology,
Chinese Academy of Sciences, 39 Beijing, Donglu, Nanjing, China
_







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RE: [MORPHMET] Interpreting PCA results

[F. James Rohlf]

What is important is not the fact that one is going +/- one standard deviation 
along each axis. When shape changes are subtle one may need to go beyond the 
observed range to make it more obvious to the eye what the changes are. Exactly 
how far one goes away from the mean is arbitrary. It is a visualization – not 
statistics.

 

--

F. James Rohlf New email: f.james.ro...@stonybrook.edu

Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.

& Research Professor. Dept. of Anthropology

Stony Brook University 11794-4364

WWW: http://life.bio.sunysb.edu/morph/rohlf

P Please consider the environment before printing this email 

 

From: mahendiran mylswamy [mailto:mahenr...@gmail.com] 
Sent: Monday, May 15, 2017 2:31 AM
To: f.james.ro...@stonybrook.edu
Cc: morphmet@morphometrics.org; dsbriss_dmd ; K. James 
Soda 
Subject: RE: [MORPHMET] Interpreting PCA results

 

Dear all,

I read interesting comments and the attached manuscript as well.

I find David question us interesting.

If any one could answer David question in a simple way?

I am quoting his question below?.

"What I do not quite understand is what exactly is the purpose of applying 
standard deviation(s) to the PCA and then warping the Procrustes average shape 
to these standard deviations? "

 

 

On 15 May 2017 6:08 a.m., "F. James Rohlf" mailto:f.james.ro...@stonybrook.edu> > wrote:

I agree with these comments but would like to add another point. I prefer to 
think that the purpose of the PCA is to produce a low-dimension space that 
captures as much of the overall variation (in a least-squares sense) as 
possible. Within that space there is no need to limit the visualizations to the 
extremes of each axis – one can investigate any direction within that space if 
there is a pattern in the data that suggests an interesting direction. The 
directions of the axes are mathematical constructs and not bases on any 
biological principles. Perhaps one sees some clusters in the PCA ordination but 
the variation within or between clusters need not be parallel to one of the PC 
axes. One can then look in other directions. That is why the tpsRelw program 
allows one to visualize any point in the ordination space – not just parallel 
to the axes. That means for publication one has to decide which directions are 
of interest – not just mechanically display the extremes of the axes.

 

--

F. James Rohlf New email: f.james.ro...@stonybrook.edu 
<mailto:f.james.ro...@stonybrook.edu> 

Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.

& Research Professor. Dept. of Anthropology

Stony Brook University 11794-4364

WWW: http://life.bio.sunysb.edu/morph/rohlf

P Please consider the environment before printing this email 

 

From: K. James Soda [mailto:k.jamess...@gmail.com 
<mailto:k.jamess...@gmail.com> ] 
Sent: Sunday, May 14, 2017 7:28 PM
To: dsbriss_dmd mailto:orthofl...@gmail.com> >
Cc: MORPHMET mailto:morphmet@morphometrics.org> >
Subject: Re: [MORPHMET] Interpreting PCA results

 

Dear David,

Great question!  I disagree with the statement that the samples' variance in 
shape space is not biologically real or, perhaps more accurately, is less real 
than the variance in any other space.  As far as I see it, the basic strategy 
in any biostatistical study, be it GM or otherwise, is that a researcher 
represents a real biological population as an abstract statistical population.  
They then use this abstract statistical population as a proxy for the real one 
so that inferences in the statistical space have implications for the real 
world.  

For example, a PCA finds a direction in the statistical space in which the 
statistical population tends to be spread out.  This is interesting to the 
researcher because this direction has a correspondence to certain real world 
variables.  As a result, the PCA tells the researcher in what ways the real 
population tends to vary.  The key point, though, is that the researcher 
transitioned from the statistical space to the real world.

Moving from shape space to the real world is no different in principle.  We 
have a real population of specimens, whose shape are of interest to us, and we 
represent them using vectors of shape variables.  The vectors are abstractions; 
it is not as if we can hold a vector in our hands.  However, this is irrelevant 
because they are just proxies, no less real than any other quantitative 
representation.  What matters is if we can tie them back to the real world.  
This is why morphometricians implement a visualization step.  In a PCA, the PCs 
describe how our proxies vary, and we visualize in order to see how this 
variation appears in the real world.  It is infeasible to visualize every point 
along this axis, so we instead visualize a handful.  Since the core goal in PCA 
(at least in this context) is to describe variance, we generally describe the 
loca

Re: [MORPHMET] Interpreting PCA results

[mahendiran mylswamy]

yeah, your inference makes absolute sense.Thanks for the quick response,
Carmelo.

On Mon, May 15, 2017 at 12:10 PM, Carmelo Fruciano 
wrote:

> Dear Mahediran,
>
> to my understanding from David's phrasing, it is just a way to visualize
> shape variation along a given PC axis (as the value at 0 is the mean). One
> could use some other criterion (for instance the maximum and minimum scores
> along that given PC).
>
> But, of course, all the other considerations of whether or not it makes
> sense to interpret (or at least explore) the patterns predicted along a
> given PC axis still apply.
>
> Best,
>
> Carmelo
>
> Il 15/05/2017 4:30 PM, mahendiran mylswamy ha scritto:
>
> Dear all,
> I read interesting comments and the attached manuscript as well.
> I find David question us interesting.
> If any one could answer David question in a simple way?
> I am quoting his question below?.
> "What I do not quite understand is what exactly is the purpose of applying
> standard deviation(s) to the PCA and then warping the Procrustes average
> shape to these standard deviations? "
>
>
> On 15 May 2017 6:08 a.m., "F. James Rohlf" 
> wrote:
>
>> I agree with these comments but would like to add another point. I prefer
>> to think that the purpose of the PCA is to produce a low-dimension space
>> that captures as much of the overall variation (in a least-squares sense)
>> as possible. Within that space there is no need to limit the visualizations
>> to the extremes of each axis – one can investigate any direction within
>> that space if there is a pattern in the data that suggests an interesting
>> direction. The directions of the axes are mathematical constructs and not
>> bases on any biological principles. Perhaps one sees some clusters in the
>> PCA ordination but the variation within or between clusters need not be
>> parallel to one of the PC axes. One can then look in other directions. That
>> is why the tpsRelw program allows one to visualize any point in the
>> ordination space – not just parallel to the axes. That means for
>> publication one has to decide which directions are of interest – not just
>> mechanically display the extremes of the axes.
>>
>>
>>
>> --
>>
>> F. James Rohlf *New email: f.james.ro...@stonybrook.edu
>> *
>>
>> Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.
>>
>> & Research Professor. Dept. of Anthropology
>>
>> Stony Brook University 11794-4364
>>
>> WWW: http://life.bio.sunysb.edu/morph/rohlf
>>
>> P Please consider the environment before printing this email
>>
>>
>>
>> *From:* K. James Soda [mailto:k.jamess...@gmail.com]
>> *Sent:* Sunday, May 14, 2017 7:28 PM
>> *To:* dsbriss_dmd 
>> *Cc:* MORPHMET 
>> *Subject:* Re: [MORPHMET] Interpreting PCA results
>>
>>
>>
>> Dear David,
>>
>> Great question!  I disagree with the statement that the samples' variance
>> in shape space is not biologically real or, perhaps more accurately, is
>> less real than the variance in any other space.  As far as I see it, the
>> basic strategy in any biostatistical study, be it GM or otherwise, is that
>> a researcher represents a real biological population as an abstract
>> statistical population.  They then use this abstract statistical population
>> as a proxy for the real one so that inferences in the statistical space
>> have implications for the real world.
>>
>> For example, a PCA finds a direction in the statistical space in which
>> the statistical population tends to be spread out.  This is interesting to
>> the researcher because this direction has a correspondence to certain real
>> world variables.  As a result, the PCA tells the researcher in what ways
>> the real population tends to vary.  The key point, though, is that the
>> researcher transitioned from the statistical space to the real world.
>>
>> Moving from shape space to the real world is no different in principle.
>> We have a real population of specimens, whose shape are of interest to us,
>> and we represent them using vectors of shape variables.  The vectors are
>> abstractions; it is not as if we can hold a vector in our hands.  However,
>> this is irrelevant because they are just proxies, no less real than any
>> other quantitative representation.  What matters is if we can tie them back
>> to the real world.  This is why morphometricians implement a visualization
>> step.  In a PCA, the PCs describe how our proxies vary, and we visualize in
>> order to see how thi

Re: [MORPHMET] Interpreting PCA results

[Carmelo Fruciano]

Dear Mahediran,

to my understanding from David's phrasing, it is just a way to visualize 
shape variation along a given PC axis (as the value at 0 is the mean). 
One could use some other criterion (for instance the maximum and minimum 
scores along that given PC).


But, of course, all the other considerations of whether or not it makes 
sense to interpret (or at least explore) the patterns predicted along a 
given PC axis still apply.


Best,

Carmelo


Il 15/05/2017 4:30 PM, mahendiran mylswamy ha scritto:

Dear all,
I read interesting comments and the attached manuscript as well.
I find David question us interesting.
If any one could answer David question in a simple way?
I am quoting his question below?.
"What I do not quite understand is what exactly is the purpose of 
applying standard deviation(s) to the PCA and then warping the 
Procrustes average shape to these standard deviations? "



On 15 May 2017 6:08 a.m., "F. James Rohlf" 
mailto:f.james.ro...@stonybrook.edu>> 
wrote:


I agree with these comments but would like to add another point. I
prefer to think that the purpose of the PCA is to produce a
low-dimension space that captures as much of the overall variation
(in a least-squares sense) as possible. Within that space there is
no need to limit the visualizations to the extremes of each axis –
one can investigate any direction within that space if there is a
pattern in the data that suggests an interesting direction. The
directions of the axes are mathematical constructs and not bases
on any biological principles. Perhaps one sees some clusters in
the PCA ordination but the variation within or between clusters
need not be parallel to one of the PC axes. One can then look in
other directions. That is why the tpsRelw program allows one to
visualize any point in the ordination space – not just parallel to
the axes. That means for publication one has to decide which
directions are of interest – not just mechanically display the
extremes of the axes.

--

F. James Rohlf *New email: f.james.ro...@stonybrook.edu
<mailto:f.james.ro...@stonybrook.edu>*

Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.

& Research Professor. Dept. of Anthropology

Stony Brook University 11794-4364

WWW: http://life.bio.sunysb.edu/morph/rohlf
<http://life.bio.sunysb.edu/morph/rohlf>

PPlease consider the environment before printing this email

*From:* K. James Soda [mailto:k.jamess...@gmail.com
<mailto:k.jamess...@gmail.com>]
*Sent:* Sunday, May 14, 2017 7:28 PM
*To:* dsbriss_dmd mailto:orthofl...@gmail.com>>
*Cc:* MORPHMET mailto:morphmet@morphometrics.org>>
*Subject:* Re: [MORPHMET] Interpreting PCA results

Dear David,

Great question!  I disagree with the statement that the samples'
variance in shape space is not biologically real or, perhaps more
accurately, is less real than the variance in any other space.  As
far as I see it, the basic strategy in any biostatistical study,
be it GM or otherwise, is that a researcher represents a real
biological population as an abstract statistical population.  They
then use this abstract statistical population as a proxy for the
real one so that inferences in the statistical space have
implications for the real world.

For example, a PCA finds a direction in the statistical space in
which the statistical population tends to be spread out.  This is
interesting to the researcher because this direction has a
correspondence to certain real world variables.  As a result, the
PCA tells the researcher in what ways the real population tends to
vary.  The key point, though, is that the researcher transitioned
from the statistical space to the real world.

Moving from shape space to the real world is no different in
principle.  We have a real population of specimens, whose shape
are of interest to us, and we represent them using vectors of
shape variables.  The vectors are abstractions; it is not as if we
can hold a vector in our hands. However, this is irrelevant
because they are just proxies, no less real than any other
quantitative representation.  What matters is if we can tie them
back to the real world.  This is why morphometricians implement a
visualization step. In a PCA, the PCs describe how our proxies
vary, and we visualize in order to see how this variation appears
in the real world.  It is infeasible to visualize every point
along this axis, so we instead visualize a handful.  Since the
core goal in PCA (at least in this context) is to describe
variance, we generally describe the locations where a
visualization occurs in units of standard deviations from the
mean.  We could use absolute distances along an axis, b

RE: [MORPHMET] Interpreting PCA results

[mahendiran mylswamy]

Dear all,
I read interesting comments and the attached manuscript as well.
I find David question us interesting.
If any one could answer David question in a simple way?
I am quoting his question below?.
"What I do not quite understand is what exactly is the purpose of applying
standard deviation(s) to the PCA and then warping the Procrustes average
shape to these standard deviations? "


On 15 May 2017 6:08 a.m., "F. James Rohlf" 
wrote:

> I agree with these comments but would like to add another point. I prefer
> to think that the purpose of the PCA is to produce a low-dimension space
> that captures as much of the overall variation (in a least-squares sense)
> as possible. Within that space there is no need to limit the visualizations
> to the extremes of each axis – one can investigate any direction within
> that space if there is a pattern in the data that suggests an interesting
> direction. The directions of the axes are mathematical constructs and not
> bases on any biological principles. Perhaps one sees some clusters in the
> PCA ordination but the variation within or between clusters need not be
> parallel to one of the PC axes. One can then look in other directions. That
> is why the tpsRelw program allows one to visualize any point in the
> ordination space – not just parallel to the axes. That means for
> publication one has to decide which directions are of interest – not just
> mechanically display the extremes of the axes.
>
>
>
> --
>
> F. James Rohlf *New email: f.james.ro...@stonybrook.edu
> *
>
> Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.
>
> & Research Professor. Dept. of Anthropology
>
> Stony Brook University 11794-4364
>
> WWW: http://life.bio.sunysb.edu/morph/rohlf
>
> P Please consider the environment before printing this email
>
>
>
> *From:* K. James Soda [mailto:k.jamess...@gmail.com]
> *Sent:* Sunday, May 14, 2017 7:28 PM
> *To:* dsbriss_dmd 
> *Cc:* MORPHMET 
> *Subject:* Re: [MORPHMET] Interpreting PCA results
>
>
>
> Dear David,
>
> Great question!  I disagree with the statement that the samples' variance
> in shape space is not biologically real or, perhaps more accurately, is
> less real than the variance in any other space.  As far as I see it, the
> basic strategy in any biostatistical study, be it GM or otherwise, is that
> a researcher represents a real biological population as an abstract
> statistical population.  They then use this abstract statistical population
> as a proxy for the real one so that inferences in the statistical space
> have implications for the real world.
>
> For example, a PCA finds a direction in the statistical space in which the
> statistical population tends to be spread out.  This is interesting to the
> researcher because this direction has a correspondence to certain real
> world variables.  As a result, the PCA tells the researcher in what ways
> the real population tends to vary.  The key point, though, is that the
> researcher transitioned from the statistical space to the real world.
>
> Moving from shape space to the real world is no different in principle.
> We have a real population of specimens, whose shape are of interest to us,
> and we represent them using vectors of shape variables.  The vectors are
> abstractions; it is not as if we can hold a vector in our hands.  However,
> this is irrelevant because they are just proxies, no less real than any
> other quantitative representation.  What matters is if we can tie them back
> to the real world.  This is why morphometricians implement a visualization
> step.  In a PCA, the PCs describe how our proxies vary, and we visualize in
> order to see how this variation appears in the real world.  It is
> infeasible to visualize every point along this axis, so we instead
> visualize a handful.  Since the core goal in PCA (at least in this context)
> is to describe variance, we generally describe the locations where a
> visualization occurs in units of standard deviations from the mean.  We
> could use absolute distances along an axis, but this is probably more
> arbitrary than standard deviation units.  The standard deviations come from
> the data's distribution, whereas the absolute distance is really only
> well-defined in the mathematical space.
>
> To summarize: i) Nearly all quantitative analyses involve an abstraction
> to a mathematical space.  ii) The description of points in a mathematical
> space is useful to the researcher because the researcher is able to
> translate the abstract mathematical space into a real world
> interpretation.  iii) In GM, the shape variables are traditionally
> translated into the real world via visualization.  Ergo, morphometri

RE: [MORPHMET] Interpreting PCA results

[F. James Rohlf]

I agree with these comments but would like to add another point. I prefer to 
think that the purpose of the PCA is to produce a low-dimension space that 
captures as much of the overall variation (in a least-squares sense) as 
possible. Within that space there is no need to limit the visualizations to the 
extremes of each axis – one can investigate any direction within that space if 
there is a pattern in the data that suggests an interesting direction. The 
directions of the axes are mathematical constructs and not bases on any 
biological principles. Perhaps one sees some clusters in the PCA ordination but 
the variation within or between clusters need not be parallel to one of the PC 
axes. One can then look in other directions. That is why the tpsRelw program 
allows one to visualize any point in the ordination space – not just parallel 
to the axes. That means for publication one has to decide which directions are 
of interest – not just mechanically display the extremes of the axes.

 

--

F. James Rohlf New email: f.james.ro...@stonybrook.edu

Distinguished Professor, Emeritus. Dept. of Ecol. & Evol.

& Research Professor. Dept. of Anthropology

Stony Brook University 11794-4364

WWW: http://life.bio.sunysb.edu/morph/rohlf

P Please consider the environment before printing this email 

 

From: K. James Soda [mailto:k.jamess...@gmail.com] 
Sent: Sunday, May 14, 2017 7:28 PM
To: dsbriss_dmd 
Cc: MORPHMET 
Subject: Re: [MORPHMET] Interpreting PCA results

 

Dear David,

Great question!  I disagree with the statement that the samples' variance in 
shape space is not biologically real or, perhaps more accurately, is less real 
than the variance in any other space.  As far as I see it, the basic strategy 
in any biostatistical study, be it GM or otherwise, is that a researcher 
represents a real biological population as an abstract statistical population.  
They then use this abstract statistical population as a proxy for the real one 
so that inferences in the statistical space have implications for the real 
world.  

For example, a PCA finds a direction in the statistical space in which the 
statistical population tends to be spread out.  This is interesting to the 
researcher because this direction has a correspondence to certain real world 
variables.  As a result, the PCA tells the researcher in what ways the real 
population tends to vary.  The key point, though, is that the researcher 
transitioned from the statistical space to the real world.

Moving from shape space to the real world is no different in principle.  We 
have a real population of specimens, whose shape are of interest to us, and we 
represent them using vectors of shape variables.  The vectors are abstractions; 
it is not as if we can hold a vector in our hands.  However, this is irrelevant 
because they are just proxies, no less real than any other quantitative 
representation.  What matters is if we can tie them back to the real world.  
This is why morphometricians implement a visualization step.  In a PCA, the PCs 
describe how our proxies vary, and we visualize in order to see how this 
variation appears in the real world.  It is infeasible to visualize every point 
along this axis, so we instead visualize a handful.  Since the core goal in PCA 
(at least in this context) is to describe variance, we generally describe the 
locations where a visualization occurs in units of standard deviations from the 
mean.  We could use absolute distances along an axis, but this is probably more 
arbitrary than standard deviation units.  The standard deviations come from the 
data's distribution, whereas the absolute distance is really only well-defined 
in the mathematical space.

To summarize: i) Nearly all quantitative analyses involve an abstraction to a 
mathematical space.  ii) The description of points in a mathematical space is 
useful to the researcher because the researcher is able to translate the 
abstract mathematical space into a real world interpretation.  iii) In GM, the 
shape variables are traditionally translated into the real world via 
visualization.  Ergo, morphometricians often interpret PCA results via 
visualizations along individual PCs.  To aid in interpretation, this tends to 
occur in standard deviation units because the standard deviation is more easily 
tied to the real world relative to arbitrary selecting a unit of distance.

Perhaps some of these points are up for debate, but remember that statistics is 
largely the study of VARIATION.  If the variation in shape space did not have 
any biological significance, almost no analysis after alignment would be 
possible.

Hope somewhere in this long commentary, you found something helpful,

James

 

On Tue, May 9, 2017 at 4:56 PM, dsbriss_dmd mailto:orthofl...@gmail.com> > wrote:

Good afternoon all, I have a question about interpretation of PCs.  I have come 
across several articles in orthodontic lite

Re: [MORPHMET] Interpreting PCA results

[K. James Soda]

Dear David,

Great question!  I disagree with the statement that the samples' variance
in shape space is not biologically real or, perhaps more accurately, is
less real than the variance in any other space.  As far as I see it, the
basic strategy in any biostatistical study, be it GM or otherwise, is that
a researcher represents a real biological population as an abstract
statistical population.  They then use this abstract statistical population
as a proxy for the real one so that inferences in the statistical space
have implications for the real world.

For example, a PCA finds a direction in the statistical space in which the
statistical population tends to be spread out.  This is interesting to the
researcher because this direction has a correspondence to certain real
world variables.  As a result, the PCA tells the researcher in what ways
the real population tends to vary.  The key point, though, is that the
researcher transitioned from the statistical space to the real world.

Moving from shape space to the real world is no different in principle.  We
have a real population of specimens, whose shape are of interest to us, and
we represent them using vectors of shape variables.  The vectors are
abstractions; it is not as if we can hold a vector in our hands.  However,
this is irrelevant because they are just proxies, no less real than any
other quantitative representation.  What matters is if we can tie them back
to the real world.  This is why morphometricians implement a visualization
step.  In a PCA, the PCs describe how our proxies vary, and we visualize in
order to see how this variation appears in the real world.  It is
infeasible to visualize every point along this axis, so we instead
visualize a handful.  Since the core goal in PCA (at least in this context)
is to describe variance, we generally describe the locations where a
visualization occurs in units of standard deviations from the mean.  We
could use absolute distances along an axis, but this is probably more
arbitrary than standard deviation units.  The standard deviations come from
the data's distribution, whereas the absolute distance is really only
well-defined in the mathematical space.

To summarize: i) Nearly all quantitative analyses involve an abstraction to
a mathematical space.  ii) The description of points in a mathematical
space is useful to the researcher because the researcher is able to
translate the abstract mathematical space into a real world
interpretation.  iii) In GM, the shape variables are traditionally
translated into the real world via visualization.  Ergo, morphometricians
often interpret PCA results via visualizations along individual PCs.  To
aid in interpretation, this tends to occur in standard deviation units
because the standard deviation is more easily tied to the real world
relative to arbitrary selecting a unit of distance.

Perhaps some of these points are up for debate, but remember that
statistics is largely the study of VARIATION.  If the variation in shape
space did not have any biological significance, almost no analysis after
alignment would be possible.

Hope somewhere in this long commentary, you found something helpful,

James

On Tue, May 9, 2017 at 4:56 PM, dsbriss_dmd  wrote:

> Good afternoon all, I have a question about interpretation of PCs.  I have
> come across several articles in orthodontic literature having to do with
> morphometric analysis of sagittal cephalograms that discuss warping a
> Procrustes analysis along a principal component axis.  Essentially the
> authors discuss finding whatever principal components represent shape
> variance, then determining the standard deviation(s) of those PC's, and
> applying the standard deviations to the Procrustes shape to warp the
> average shape plus or minus.  So if you have an average normodivergent
> Procrustes shape, one warp perhaps in the negative direction might give you
> a brachycephalic shape, while the opposite would give you a dolichocephalic
> shape.  But I don't know where this idea comes from.  I have been involved
> with 8 or 9 morphometrics projects over the last few years and I have never
> been able to figure this out or the rationale for performing such an
> application with the PC results.
>
> As an example of what I am talking about here is a passage from the
> Journal of Clinical & Diagnostic research, doi:  10.7860/JCDR/
> 2015/8971.5458 
>
> "Here, the first 2 PCs are shown & the Average shape (middle) was warped
> by applying each PC by amount equal to 3 standard deviations in negative
> (left) and positive (right) direction {[Table/Fig-10
> ]: PC1
> with standard deviation, [Table/Fig-11
> ] PC 2
> with standard deviation}."
>
> I did not include the graphs from the article but if it would help to
> answer this question I can su