In clinical trials and other areas of applied statistics, we often need to model longitudinal data.
In most introductions to modelling longitudinal data with mixed effects models that I have seen, there seems to be very little focus on modelling the residual covariance structure. In R the most popular package for fitting mixed models seems to be lme4 and yet, as far as I know, it uses a compound symmetric (CS) structure with no option to specify anyything else. I know there are other packages in R (glmmTMB, mmrm and nlme for example) and other software entirely that allows other structures.
There is an expectation that not only will within-person repeated measures be correlated, but the correlations will not be equal accross all time points, but rather dimishish, the further apart the measurements are. Such a structure can be modelled by an autoregressive AR(1) structure. Another popular choice is of an unstructured (US) covariance matrix, where all the individual correlations are estimated seperately. I can see that with an US covariance matrix the software will have to estimate many more parameters that with AR(1) or CS structure, so it is far more flexble at the cost of being less efficient.
In many discussions on here and elsewhere, I have seen the recommendation to use AR(1), such as this answer by Frank Harrell:
It is important to explicitly model the autocorrelation. You might model cluster effects as random intercepts, but within-cluster effects need to be handled with something like a continuous-time AR(1) correlation structure.
But then I read today this answer by Björn, who recommends an unstructured covariance matrix, and argued against AR(1):
you flexibly estimate how correlated different timepoints (and the eyes in the same person) are from the data without imposing something like a AR(1) structure (that tends to be very wrong for any real data, but if you assumed a more structured covariance structure that is appropriate you might have a gain in efficiency)
So how should we choose between, say AR(1) and US (or any other the other structures that some software supports such as Toeplitz, Heterogeneous AR(1) and Heterogeneous Toeplitz) ? Is it simply a matter of fitting several structures and using a significance test to choose between them ? If so then I assume there should be an adjustment for multiple testing ?