I hope this is the right site to post this.
The example I have in my mind is a GLMM model, where we infer random effects, and a random effect caterpillar plot (with confidence intervals):
Now, suppose we start applying topological data analysis tools to the above plot, for example the Mapper algorithm, with the filter determined by: if two confidence intervals overlap by at least X%, they are then grouped into one node. For example if we pick a $X=40$% in the above picture, we should get three different components.
Question 1: Is there anything interesting, from a statistics point of view about the persistant components that arise from the above reasoning, as we tune $X$ from 100% to 0 (i.e. looking at the resulting barcodes)?
Question 2: The above reasoning for example, could be done with something simpler like an ANOVA study. Are there any other interesting examples of statistical models which would yield non-trivial topological structures from their confidence intervals? I picked the GLMM model in particular because it has nice non-trivial variance structure.
My main reason for interest in this is to better conceptualize the notion of assigning rankings to each random effect. This obviously depends both on whether or not two confidence intervals overlap and how we define cutoffs of percentiles, whether it's by fixing cutoff regions or by demanding some maximum confidence interval overlap. To this end, is there any reasonable argument that can be made for picking a particular value of $X$ above to declare that the resulting groupings (and therefore rankings) are optimal?