Full Q matrices by default

[ScottClaessens]

This is a copy of the issue opened by Erik, since his GitHub account is currently playing up. He said:

Currently, off-diagonals of the drift matrix (Q) are set to 0. I prescribed this based on some early results I had with some latent variable models where the cross-selection could not be identified simultaneously with correlated drift. So I assumed independent drift terms. But this has always bothered me somewhat, and revisiting the issue I think what I had found previously was a corner-case.

I have done some experiments with the two vignette datasets (authority and primates) and both fit just fine with the full Q matrices being estimated (p.s. the cross-selection results from the Austronesian example are qualitatively the same, just a bit more uncertain I guess owing to the greater model complexity).

In any case, I think this should probably be the default to estimate going forward. It is only ~4 lines of code difference in the Stan models, and should be easy to generalize. I think the only other change would be in the output of summary(object). @ScottClaessens I'll send you the example Stan files if you have time to implement, otherwise I can open a branch and do it.

[ScottClaessens]

We agreed to label this "drift covariance" and represent it in the summary output using double tildes (~~) as in the lavaan package. We will also include an argument in coev_fit() to allow users to set these to zero if they want to.

[ScottClaessens]

I'm working on this now. I'm thinking that, to be consistent with the rest of the output, we should probably use the form cor(X,Y) to present correlated drift in the summary output.

[ErikRingen]

cor(x,y) or cov(x,y)? If the former, we can construct the correlation matrix in generated quantities (R = LL'). In fact, it might be good to do this regardless.

[ScottClaessens]

Yep, the former. I’ve gone for the generated quantities approach. Should be done tinkering by tomorrow :)