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> 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 > <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 <orthofl...@gmail.com> > *Cc:* MORPHMET <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> 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 > > > > > > > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+unsubscr...@morphometrics.org. > > > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+unsubscr...@morphometrics.org. > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > --- > You received this message because you are subscribed to the Google Groups > "MORPHMET" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to morphmet+unsubscr...@morphometrics.org. > -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- You received this message because you are subscribed to the Google Groups "MORPHMET" group. 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