Hi!

I’m fairly new to Bayesian modeling in general and Stan in particular so I apologize if I even use incorrect phrasing of what I’m trying to ask…!

I’m working on a hierarchical model of reinforcement learning in a repeated measures design. I’ve posted the model code (close to my latest version) in this post.

In the given experiment a number of subjects `nS`

performed a learning task under `nC`

different (drug) conditions (i.e. each subject repeats the learning paradigm in `nC`

times). The reinforcement model then uses a set of parameters to model the subjects behavior in the learning task. Of interest for my current question is the parameter `Arew`

that models the learning rate. It needs to be constrained to a range of `[0,1]`

.

To account for the hierarchical structure (subject repeated in condition) each parameter has a non-centered parameterization. The baseline condition is parameterized as follows

```
Arew_normal[s,] = Arew_m + Arew_cond_s * Arew_cond_raw[s,] + Arew_vars[s,1];
```

That is, there is a group mean `Arew_m`

, a subject specific offset `Arew_cond_raw[s,]`

with sd `Arew_cond_s`

and a random part for each subject that is correlated across conditions.

Each non-baseline condition (that is, each drug condition) has an additional offset `Arew_cond_grp_m`

plus a subject specific random part:

```
Arew_normal[s,v] += cond_vars[s,v,kk] * (Arew_cond_grp_m[kk] + Arew_vars[s,kk+1]);
```

where `cond_vars[s,v,kk]`

is a dummy ccoding the condition.

Now for the reinforcement model the parameter `Arew`

needs to be in the range `[0,1]`

. This is achieved by an `inv_logit()`

transform:

```
Arew[s,] = inv_logit(Arew_normal[s,]);
```

That means, that I can interpret the subject-wise parameters `Arew`

on my `[0,1]`

scale. However, the estimates for the group-level parameters for the overall mean of the parameter (`Arew_m`

) and the group-level parameter for the condition effect (`Arew_cond_grp_m`

) are only available on the unconstrained space before the transform happens.

What would be the way to interpret these parameters?

That is, how can I tell (and report) what effect my drug manipulation has on my parameter `Arew`

?

I’m extending here a model that is used in the hBayesDM package. This simple model, that does not implement repeated measures, uses a non-centered parameterization for the learning rate parameter as follows (using `Phi_approx()`

instead of `inv_logit()`

to transform the parameter to a range of `[0,1]`

):

```
A[s] = Phi_approx(A_m + sigma * A_raw[s])
```

To get the group-level parameter transformed back to an interpretable scale it then uses

```
mu_A = Phi_approx(A_m);
```

in the `generated quantities`

block.

Is there a way to obtain interpretable values for `Arew_m`

and `Arew_cond_grp_m`

using such kind of back-transformation? I’d assume

```
mu_Arew = inv_logit(Arew_m)
mu_Arew_cond_grp = inv_logit(Arew_cond_grp_m)
```

to be misleading here since the parameter `Arew`

comprises of a sum of the 2 parameters.

Any idea on this or a suggestion how to better deal with such a case is highly appreciated!

Thanks a lot!