First, is there a test to be able to say an ellipse in a PCA plot is "smaller" than another?
Second, what is the significance/meaning of a smaller ellipse? (Greater constriction/Less variability/etc?)
I'm attaching a plot as an example. Here, I see that the ellipses for my three groups overlap, but the red one looks smaller than the other two.
Thanks ahead of time.
Characteristic of the PCA is the absence of an associated probability density or generative model.
Your answer should be in the link:
https://www.researchgate.net/post/Ellipse-on-PCA-analysis
Otherwise, you could check the Probabilistic Principal Component Analysis (PPCA); however, I do not know how (and if) it works.
It seems that these are the error ellipse (The error ellipse represents an iso-contour of the Gaussian distribution, and allows you to visualize a 2D confidence interval. Each ellipse shows a 95% confidence ellipse for a set of 2D normally distributed data samples. This confidence ellipse defines the region that contains 95% of all samples that can be drawn from the underlying Gaussian distribution). In this case, it seems that the plots are 95% interval for PCA with one principal component (the PC1 or the PC2) and two principal components.
It looks to me as though the ellipses are fitted to the PCA plane so as to include 95% of the points in each of the 3 groups, rather than fitted to the full multivariate data cloud. If so, then the size of each ellipse tells you how similar/dissimilar samples are within groups in the plane of the ordination. This ignores several facts. PCA is a projection, so samples can be far apart in the full multivariate space but close together in the chosen plane. The plane chosen for an ordination (PC1 v PC2) explains 67% of the total variance so a third of the variance is not explained by the plot. Without the associated results (eigenvectors, eigenvalues etc.) it is difficult to say more. If you really want to ask questions like 'are groups of samples separated in multivariate space' or 'are samples in group A more dispersed than in group B' then I'd recommend using tests that are designed to address such questions, like Marti Anderson's PERMANOVA and PERMDISP.
The size is directly rely to the variance and you can compare assuming your using normalized data.
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