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I have a total of $N$ units that are assigned to $D$ classes based on certain combinations of characteristics. If each unit is assigned to only one class, then the problem reduces to the usual compositional data with $D$ compositions, which are non-overlapping. For such a scenario, we may apply centred log-ratio transformation (clr) prior to, for example, comparing groups. However, I have a problem where one unit might be assigned to 1 or more classes and hence the classes are overlapping. In such cases, the traditional clr seems to be not useful. Therefore, I would appreciate it if someone suggests to me a better way to handle such overlapping compositions to make a meaningful comparison between groups.

Alemu
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  • Could you explain why the CLR is no longer useful in your situation? – whuber May 10 '22 at 20:59
  • I always assumed CLR is applicable only for non-overlaping compositions. Think of it as a multinomial experiment where the total size is partitioned into K disjoint classes each with their probability pj, and pj sum to 1. Here the classes are no longer disjoint and the pj do not sum to 1 (because the intersection has a non zero probability). Therefore, CLR seems to me not the right choice here. – Alemu May 10 '22 at 22:21
  • The issue isn't whether the classes overlap, but whether a transformation can be helpful in the analysis of the data. It's quite possible it will help even with overlapping classes. See my post at https://stats.stackexchange.com/a/259223/919 for some related discussion. – whuber May 10 '22 at 22:33
  • Thanks for the comment and the reference. Indeed, our question is whether the transformation helps to meaningfully compare compositions across groups. – Alemu May 11 '22 at 19:56
  • That comes down to what "meaningful" might be. For instance, if it requires unbiased estimation of class means, then nonlinear transformations like the CLR might not be meaningful. But if you are looking for insight or straightforward interpretation, finding a transformation that simultaneously makes many of the group distributions approximately isotropic could be helpful. – whuber May 11 '22 at 20:50

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