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" <f.james.ro...@stonybrook.edu <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 <orthofl...@gmail.com <mailto:orthofl...@gmail.com>>
    *Cc:* MORPHMET <morphmet@morphometrics.org
    <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, 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 <orthofl...@gmail.com
    <mailto:orthofl...@gmail.com>> 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
        <https://dx.doi.org/10.7860%2FJCDR%2F2015%2F8971.5458>

        "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
        <https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4347171/figure/F10/>]:
        PC1 with standard deviation, [Table/Fig-11
        <https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4347171/figure/F11/>]
        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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