add new functions for group means and correlations

[Nathaniel-Haines]

I discovered an issue with how group mean parameters were being calculated (post hoc in the generated quantities block) that traces back to hBayesDM. As described in the issue I opened here, the method where we simply do, e.g., mu_Arew = Phi_approx(mu_p[1]) in Stan actually gives us an estimate of the group median rather than group mean. This is a rather subtle bug, so I don't expect it to change results much (if at all).

I opened this PR with a new custom function in the Stan codes called Phi_approx_group_mean_rng which computes the group mean as intended. It involves a simulation step as described in the issue I linked. I also added a new function termed Phi_approx_corr_rng that will simulate the test-retest correlations on the scale of interest, which avoids the need for us to do the pop_correct_MCMC step post-hoc. Now, we can just do:

pars = rstan::extract(orl_fit)

pars$R_Arew

and the resulting correlation we get from Stan will be of the correlation on the parameters with the correct transformation. So, no more need to do the iterative correlations part outside of Stan!

@hollysully @jeremy-haynes I know this is a strange change—let me know if you have any questions! Going forward, this should be more robust given that we have had some trouble with the pop_correct_MCMC code before.

NOTE: I still need to update the R analysis code that computes the iterative correlations/runs the pop_correct_MCMC step, as it is no longer necessary. That said, I was not sure what all we want to save in that analysis file. If anyone wants to update that before we merge these changes, feel free to! (and push changes to this branch)

[jeremy-haynes]

@Nathaniel-Haines @hollysully this sounds good to me. I was looking at the new code. I was wondering if these were kind of like a bootstrap-type of approach to calculating the group means and the correlations (but bootstrapped for each MCMC sample)? In any case, I can work on updating the RMDs that have the iterative correlations and whatnot.

[Nathaniel-Haines]

@Nathaniel-Haines @hollysully this sounds good to me. I was looking at the new code. I was wondering if these were kind of like a bootstrap-type of approach to calculating the group means and the correlations (but bootstrapped for each MCMC sample)? In any case, I can work on updating the RMDs that have the iterative correlations and whatnot.

It is similar in that simulation is used, but this method is really more like a posterior predictive simulation. Since we have the full posteriors on group-level parameters, we can simulate results forward from them and make inference on the resulting values 🤓