Appropriateness of PCA to visualize clusters in genetic data

Question

I've seen PCA improperly applied in genetic research quite often. I wanted to clarify : when is it appropriate to use PCA as a visualization tool in your analysis?

Some examples:

1) Rarely is the % variance reported for the components. With human data it's been my experience that the first three components (which are often plotted) tend to contain very little % of the variance. How meaningful are you visual results (i.e. clustering) when the first three components cumulatively account for only (say) 10% of the variance?

2) Once you've actively performed feature selection, let's say simple a t-test, and you've widdeled your large data set down to a small set of features, should you perform PCA to visualize clusters? I have heard it argued that since you're so actively doing feature selection that PCA clustering, after the fact, is not really relevant. Is that true?

3) If you are going perform PCA, what are important parameters to report? I expect the %Variance each component covers, but is there something else?

Answer 1

I do share your concern about the matter of visual rendering of clusters by means of PCA. If the 1st 2 or 3 principal components account for only small portion of multidimensional variability they are likely to miss most of the directions along which the clusters differentiate. Moreover, even when components are strong they can fail: just consider two oblong parallel clusters in 2 dimensions which are separated from each other by a slit along their length. The 1st pc will lie also along their length and won't show the presence of clusters.

Your second question implies now that you know your clusters in advance (because you mention using t-test or similar comparison method). If you succeeded in selecting features (dimensions) which differentiate them most strongly, then using PCA on those selected features may turn meaningless - if the features don't correlate, or may turn valuable - if they all or some of them are pretty well correlated.

If you speak of PCA and not of factor analysis in strict meaning and you do neither rotations nor interpretations of the extracted PCs then %Variance is the only important statistic to report, for me. You could also show the scree-plot.

Answer 2

As your data likely is very noisy, you can try to improve the PCA performance using a robust variation of it, see Wikipedia for details.

But in general I do share your concern. Because in complex data sets such as genetic data, different clusters may show different correlations that cannot be adequately represented by global PCA.

The quality of using PCA for dimensionality reduction (e.g. to 2D or 3D for visualization) does depend a lot on the amount of variance captured. But you can't go by the direct relative shares. If we have 1000 dimensions, and the first two explain $10\%$ this can (I did not test this) be quite significant. In 10 dimensions, it is completely meaningless, for uniform i.i.d. data the first single eigenvector will necessarily already explain more than this. A better control is the value $$\frac{\text{explained variance}}{\text{expected explained variance}}$$.

Just a few days ago I posted a question here about the expected distribution of eigenvalues. If we find some distribution for this, we can test whether the result is significant:

Estimated distribution of eigenvalues for i.i.d. (uniform or normal) data

If you look at the example I posted, it is not unusual to see eigenvalues range from $0.119$ to $0.006$ (a difference of factor 20!) in a uniform i.i.d. dataset with just 20 dimensions, at least when the sample is small. So the eigenvalues seem to be rather unreliable when it comes to indicating whether the projection is actually capturing something.

Feature selection will not cover rotations. Which is when PCA gets interesting: did it actually rotate the data much, or did it just select a number of features (i.e. low angle between one of the axes and one of the eigenvectors)? Try plotting the axes of the original attributes in your visualization to show the relationship to the original data and the attributes used by PCA.

Answer 3

One theory. For human, the proportion of variation within groups (i.e. same ethnic group) account for ~85% of the genetic variation within individuals. Conversely, the variation within populations (i.e. continental scale) account for only ~15% of the genetic variation within individuals.

Even though the first three components (i.e. PC1, PC2, PC3) contained only a small fraction of the total variance, most often the magnitude of associated eigenvalues can be 50 to 70 times that of higher components. In other words, the first three components can explain substantially more (50x-70x) variance than any other component when comparing by individual basis.

While sometimes these higher components do explain hidden substructure within groups, bear in mind that individuals from the same group has ~85% genetic variation among themselves. Hence, most of the higher components might just be explaining this within groups variation. For analysis of genetic clusters, this is of no interest to geneticists. These higher components can thus be treated as background noises. Geneticists are mainly interested in variation within populations, which often is very ancient and strongly separated. Thus, when population clusters are formed in the first three components, it can be argued that they form mainly due to the variation within populations.

Summary: The low variance (<10%) cumulatively accounted by the first three components can be justified by the fact that variation within populations is only ~15% of the genetic variation within individuals.