My first point is that the over-parameterization that I noted is due to overlap/redundancy in parameterization of the RANDOM and REPEATED statements in Proc MIXED (and equivalently, in the RANDOM and RANDOM/RESIDUAL statements in Proc GLIMMIX) for certain--actually most--covariance structures. This issue is addressed in the appendix of the link to the Littell, Henry and Ammerman (1998) that @SteveDenham provides. It is also addressed in the Littell, Pendergast & Natajaran (Statistics in Medicine, 2000) paper I referenced in the code I posted previously. In this latter paper, the authors note the distinction between an AR(1) structure and an AR(1)+RE structure, as does Walt Stroup (Generalized Linear Mixed Models (2013)) in Ch 14 (Fig 14.2 does not correctly illustrate the different structures, IMO).
My second point about whether we can model R-side covariance structures is addressed by Stroup in his text on p 435, where he writes
"The primary ambiguity in repeated measures model building for non-Gaussian data occurs when we have a member of the two-parameter exponential family [e.g., gamma]. How do we model repeated measures for beta or negative binomial or other distributions in this family? .... For G-side models, it is not clear how the random ts(a)_ijk [i.e., time x subject(treatment) random effects] effect coexists with f(y|b)'s scale parameters."
In the 10 years or so since Walt wrote his text, he has continued to revisit issues. For example, he and Elizabeth Claassen published a paper in the Journal of Agricultural, Biological, and Environmental Statistics just last month (https://doi.org/10.1007/s13253-020-00402-6) entitled "Pseudo-Likelihood or Quadrature? What We Thought We Knew, What We Think We Know, and What We Are Still Trying to Figure Out". So he may currently hold a different view about R-side covariance structure modelling with non-Gaussian distributions. Then again, he just retired and could be on to other things 🙂
My third observation is that a random coefficients model (RCM) with time continuous will be quite different from a model with time categorical.
My fourth observation is that although UN, CHOL, and ARH(1) are identical when you are modeling a RCM with random intercepts, random slopes, and one covariance (i.e., a 2 x 2 matrix), you would not want to apply an AR constraint on covariances in a covariance matrix that was larger than 2 x 2 because that would be silly. I'd stick with UN or CHOL, or in my practice VC (or UN(1)) because my datasets are always small, and I have never been able to fit a decent intercept-slope covariance.
Mathematical statistics is not my superpower; my explorations tend to be empirical. My practical experience is consistent with the point that Walt makes in his text, and with the Littell et al. papers.
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