Guide to reducing data dimensions

In a world where collecting enormous amounts of complex and co-linear, data is increasingly the norm, techniques that reduce data dimensions to something that can be used in statistical models is essential . However, in ecology at least the means of doing this are unclear and the info out there is confusing. Earlier this year Meng et al provided a nice overview to what’s out there (see link below) specifically for reducing omics data sets, but is equally relevant for ecologists. One weakness of the paper is they provided only small amounts of practical advice particularly on how to interpret the resultant dimension reduced data.  Overall though this is an excellent guide and I aim to give a bit extra practical advice on dimension reduction using the techniques that i use.

Anyway, before going forward – what do we mean by dimension reduction? Paraphrasing from Meng et al – Dimension reduction is the mapping of data to a lower dimensional space such that redundant variance in the data is reduced , allowing for a lower-dimensional representation (say 2-3 dimensions, but sometime many more) without significant loss of information. My philosophy is to try to use the data in a raw form wherever possible, but where this is problematic due to problems with co-linearity etc and where machine learning algorithms such as Random Forests are not appropriate (eg., your response variable is a matrix….) this is my brief practical guide to three common ones:

PCA : dependable principal components analysis – excellent  if you have lots of correlated continuous predictor variables with few missing values and 0’s. There are a large number of PCA based analyses that may be useful (e.g., penalized PCA for feature selection, see Meng et al), but I’ve never used them. Choosing  the right number of PCAs is subjective and is a problem for lots of these techniques -an arbitrary cutoff of selecting PCAs that account for ~70% of the original variation seems reasonable. However, if your PCs only explain a small amount of variation you have a problem as fitting a large number of PCs to a regression model is usually not an option (and if PC25 is a significant predictor what does that even mean biologically?). Furthermore,and if there are non-linear trends this technique  won’t be useful.

PCoA :  Principal co-ordinate analysis or classical scaling is similar to PCA but used on  matrices. Has the same limitations of PCA.

NMDS: Non-metric multidimensional scaling is a much more flexible method that can cope much better with non-linear trends in the data. This method is trying to best preserve distances between objects (using a ranking system), rather than finding the axes that best represent variation in the data as PCA and PCoA do. This means that NMDS also captures variation often in a few dimensions (often 2-3), though it is important to assess NMDS fit by assessing ‘stress’ (values below 0.1 are usually OK). There is debate how useful these axis scores are (see here: https://stat.ethz.ch/pipermail/r-sig-ecology/2016-January/005246.html) as they are rank based and the axis 1 doesn’t explain the largest amount of variation and so fourth as is the case with PCA/PCoA. However I still think this is a useful strategy (see the Beale 2006 link below).

I stress (no pun intended!) the biggest problem with this techniques is interpretation of new variables. Knowing the raw data inside and out and how they are mapped onto the new latent variables is important. For example, high loading’s on PCA1 reflect  high soil moisture and high pH. If you don’t know this  interpreting regressions coefficients in a meaningful way is going to be impossible. It also leads to annoying statements like ‘PcOA 10 was the most significant predictor’ without any further biological reasoning for what the axes actually represents. Tread with caution and data dimension reduction can be a really useful thing to do.

Meng et al: http://bib.oxfordjournals.org/content/early/2016/03/10/bib.bbv108.full

Beale 2006: http://link.springer.com/article/10.1007/s00442-006-0551-8

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