# Using principal components analysis vs correspondence analysis

I am analyzing a data set concerning intertidal communities. The data are percent cover (of seaweed, barnacles, mussels, etc) in quadrats. I am used to thinking about correspondence analysis (CA) in terms of species counts, and principle component analysis (PCA) as something more useful for linear environmental (not species) trends. I haven't really had any luck figuring out if PCA or CA would be a better fit for percent cover (can't find any papers), and I'm not even sure how something that is capped up to 100% would be distributed?

I am familiar with the rough guideline that if the length of the first detrended correspondence analysis (DCA) axis is greater than 2, then you can safely assume that CA should be used. The length of DCA axis 1 was 2.17, which I don't find helpful.

• Both PCA and CA are related and both can be based on SVD algorithm. The fundamental formal difference (not mentioned in @Gavin's otherwise deep answer) is that PCA decomposes relations between columns only (e.g. by decomposing their covariance matrix), treating rows as "cases"; while CA decomposes columns and rows simultaneously, treating them symmetrically, as cross-tabulation "categories". Hence the biplot left by CA and the quasi-biplot (loadings + scores) that could be plotted after PCA give conceptually quite different information. Sep 20, 2013 at 0:58

PCA works on the values where as CA works on the relative values. Both are fine for relative abundance data of the sort you mention (with one major caveat, see later). With % data you already have a relative measure, but there will still be differences. Ask yourself

• do you want to emphasise the pattern in the abundant species/taxa (i.e. the ones with large %cover), or
• do you want to focus on the patterns of relative composition?

If the former, use PCA. If the latter use CA. What I mean by the two questions is would you want

A = {50, 20, 10}
B = { 5,  2,  1}


to be considered different or the the same? A and B are two samples and the values are the %cover of three taxa shown. (This example turned out poorly, assume there is bare ground! ;-) PCA would consider these very different because of the Euclidean distance used, but CA would consider these two samples as being very similar because the have the same relative profile.

The big caveat here is the closed compositional nature of the data. If you have a few groups (Sand, Silt, Clay, for example) that sum to 1 (100%) then neither approach is correct and you could move to a more appropriate analysis via Aitchison's Log-ratio PCA which was designed for closed compositional data. (IIRC to do this you need to centre by rows and columns, and log transform the data.) There are other approaches too. If you use R, then one book that would be useful is Analyzing Compositional Data with R.

• As always, a really excellent answer Gavin. Thank you! That clarifies things a lot, and I'll be using PCA then. Given that the intertidal community is 3 dimensional, the percent cover actually went 100% in some cases when the the organisms grew over each other. This is not the closed compositional form that you are talking about though, right? Sep 20, 2013 at 0:06
• No it's not what he's talking about. By closed I believe he means a system in which with three species A, B, C, you have %C = 100% - %B - %A May 13, 2017 at 16:21
• and what about DCA? Jun 11, 2019 at 4:34
• DCA is a messed-up version of CA so the same general principles apply to it. DCA is doing some weird torturing of the data and I don't think we need to bother with it as a method in our toolbox today, but others' opinions will vary on that. Jun 11, 2019 at 16:48

Sorry to revive an oldie, but a student asked me about this. I might be looking at the wrong end of the stick, but the answer by @GavinSimpson seems to me a bit on a tangent, not really aiming at a key distinction between the two methods. A PCA can be run either on a correlation matrix or a covariance matrix. If you run it on the correlation matrix (instead of the covariance matrix), then the absolute values are not being considered. Any multiplication of a row by a constant would not change the results. So the main difference between PCA and a CA must be something else. Underneath the hood, the PCA considers Euclidean distances between points, the CA a qui-squared distances between points. In practice the CA seems to work better with non-continous data (as in nominal, categorical, a considerable amout of 0's, etc). If someone can explain what is the difference in methods regarding the linear algebra underneath the hood in a way that is diggestable to humans, then hats off to them and please post here as I'd love to read it.