Simon A Larsson

Stockholm University | SU · Department of Physical Geography

I'm not really a "stats guy", but while working on my doctoral thesis and final paper of my PhD project, I became aware that statistical analysis cannot be performed on compositional data (such as tephra oxide concentrations) in a mathematically sound way due to the constant-sum constraint - yet the average tephrochronologist, when doing a PCA or similar, uses the raw or normalised compositional data anyway (at least in the majority of cases, as far as I've seen in my reading). I've learned that log-ratio transformations are "the" solution to the constant-sum constraint and have seen that this has indeed been done in at least a handful of tephra studies (and that it's becoming more routinely applied for e.g. XRF data), but there's not really any widespread acknowledging of this issue and even less any commonly accepted procedure as to the data curation with log-ratio transformations.

I commented on this in my thesis, and during my defence I was encouraged to publish something about this, aimed specifically at the tephra community. So, I've started working on this "something", but I need to hear some opinions on the specifics of the log-ratio transformation part of what could be a formal suggestion for a common tephra compositional data curation procedure. What I want to discuss is what kind of log-ratio transformation should be used if we were to agree upon one particular way of doing it - which is something that would be preferable in order to allow straight-forward comparisons of results between different studies, kind of in the same way that biplots of particular elements have become more or less routine to display in tephra papers.

A log-ratio transformation is basically lrt(x) = log10(x/_) where the divisor of the fraction (i.e. "_") can vary. I've seen single element oxides being used as the divisor, but without real motivation as to why that particular oxide was selected. In my thesis I argued for a centred lrt, where the divisor is the geometric mean of all oxides, and it makes sense to use something like that because it may be less ambiguous than just selecting one oxide without explaining why. However, it has the issue that no oxide concentration can be 0 (because then the divisor will be 0, and you can't divide by 0) and in many cases at least one oxide will be below EPMA detection levels and register as 0 - particularly P2O5 but also MnO and MgO in some analyses. Furthermore, P2O5 is not always reported (especially in older publications) so using a centered lrt would probably require omitting P2O5 (and then still not work with some few analyses where MnO or MgO were 0).

My main question then is, what log-ratio transformation would YOU suggest to be the norm, and why? Maybe using a single oxide is the better option, but if so, which one? To my understanding, which variation of the transformation is used doesn't really matter as long as the same one is used for all data. So let's just decide how to do it! I would really appreciate your input and suggestions. Thanks in advance!

PS. I'm not here to argue for or against the necessity of log-ratio transformations (mathematically speaking, it is required if statistical analyses are to be applied to EPMA data), although the floor is open and thoughts on any aspect of this issue are welcome. DM me if you're interested in more details about the paper I'm preparing, too. :)