Good morning
 I understand this is a bit late to come to this discussion, but there are 
some points I would like to add. First, there is an excellent discussion of 
PCA after Procrustes superimposition in Dryden and Mardia (1998) 
Statistical shape analysis, Wiley (pages 96-108). They present steps to 
visualize the effects of each PC, and that is probably where the idea of 
using a range of -3 to 3 standard deviations came from. They also mention 
(as Rohlf did) that this is arbitrary and there are instances where the 
shape changes are subtle and a larger range needs to be used to enhance 
visualization. The graphics produced (Dryden and Mardia call them icons) 
are not related to specific individuals, but that does not make them less 
real. A PC is a latent variable (not directly observed) and our hope is 
that the ordination produced is the result of a real biological process. 

The icons depict shape changes along the PCs and can be interpreted as a 
"contribution" of these changes to the ordination of points. That is to 
say, if a PC icon depicts an increase in the cranial base angle in the 
positive direction, individuals with greater angles will have higher scores 
along this PC. Because the PCs are orthogonal from each other, one needs to 
keep in mind that the icons are extremes in a single PC axis with zero 
scores in all other PCs, so they are unlikely to be equal to an individual 
shape (this is not their purpose anyway). In order to get an actual shape 
for an individual, we would need to sum all PC vectors weighted by the 
scores for that individual, but this defeats the purpose of data 
summarizing and finding a latent variable.

Going back to the example presented by David in his last message about 
using PCA as a predictive tool, if you have a sample of individuals 
measured in different times during a treatment and the shape changes are 
linear, and the treatment is the main factor generating shape variation, 
the first PC will probably depict the shape changes associated with the 
treatment. Of course, there are many other issues to be considered as 
pointed out in the other messages, but it should fulfil the "predictive" 
role at least in a general sense.

The statements about covariance between cranial base and mandible have to 
do with methods for the study of integration. This is a topic with a huge 
amount of literature using geometric morphometric methods and you can 
certainly measure the amount of covariation between cranial base and 
mandible using geometric morphometrics.

Best regards,

Leandro R. Monteiro
Laboratorio de Ciencias Ambientais
Universidade Estadual do Norte Fluminense
Av. Alberto Lamego 2000.
Campos dos Goytacazes, RJ, cep 28013-602
CV Lattes:

Em terça-feira, 16 de maio de 2017 09:55:20 UTC-3, dsbriss_dmd escreveu:
> Good morning all,
> I would like to thank everyone so far for generously taking the time to 
> reply to my questions:  I didn't think I would generated such an 
> interesting discussion!  As an amateur morphometrician I am trying to keep 
> up and have started reading some of the literature you all have cited.
> From what I understand so far, the PCA is a statistical result that 
> describes the variance in a shape, and the warp visualization that is 
> extrapolated from the PCs is one method to describe the statistical 
> variance.  I think this is what I was getting at by saying that the PCA 
> didn't have a "real" biological basis (sorry for my inaccurate language).   
> James you identified my main problem, in that how does one move from this 
> virtuality into the real world?  Or more to the point, how does the reader, 
> who is not necessarily well versed in geometric morphometrics, interpret 
>  PCA results in real-world shape spaces, perhaps without this 
> visualization?  
> The replies from Profs. MacLeod and Rohlf also get a bit at what I was 
> after, in that how does one decide which axes in PCA are of interest in the 
> first place, or indeed which landmarks are of interest, and avoid the trap 
> of mechanically displaying a warp (of whatever extreme) simply to provide a 
> visualization?  That question about the way we use these spaces is also 
> important to know, as one of the questions I usually get from my residents 
> or faculty colleagues is what clinical application the PCA has; I usually 
> find that I have to explain that it doesn't have a clinical significance or 
> application, as by itself it is not a description of a real clinical 
> situation.  
> What they seem to want me to say is, can the PCs derived from GPA be used 
> as a predictive tool to describe how an individual shape will change over 
> time.  My usual answer is no, it cannot be used that way.  I think that the 
> warped PCA, whatever criteria are selected, might help to visually explain 
> how an individual differs from the Procrustes shape, but in the average 
> orthodontic reader I am not sure it is interpreted this way.  This may be a 
> quirk of our specialty, since we have been using landmark-based linear and 
> angular analyses as growth predictive tools since the 1940's.  
> I don't want to say that we are wrong to do this, but the issue comes in 
> trying to apply those long-used clinical tools to geometric morphometrics, 
> and I don't think they mesh very well.  And as we get closer to 3 
> dimensional analysis those older tools won't be able to apply anymore. 
>  From a standard cephalometric approach I might be able to claim that 
> cranial base angle (Nasion-Sella-Basion) has some correlation with 
> mandibular prognathism, but I am not sure that this is true from a 
> geometric morphometric perspective, as I can't (yet) answer what the 
> covariance is between the cranial base and the mandible, for example.
> Anyway thank you all again, this is a very interesting thread and I 
> appreciate all the input so far.  I have been sharing it with my residents 
> who are in the midst of working on their research, I hope it will also be 
> able to help them.
> Best,
> David
> On Sunday, May 14, 2017 at 2:22:10 PM UTC-4, 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 supply them.
>> 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?  Maybe my understanding of PCA is 
>> limited, but I was under the impression that in GPA the principal 
>> components are only statistical variance, and don't represent something 
>> biologically real.  So to see how an individual varies from the shape 
>> average you have to go back and look at whatever landmark(s) represent that 
>> specific individual and compare that shape to the Procrustes average. 
>>  Maybe this is not correct?  
>> Thanks in advance, I appreciate any help you can give me.
>> David

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