When fitting a beta likelihood model with individual random effect for matched pairs before-after measurement, the individuals’ effect estimated skewed in the same way as the outcome.

In particular, the model is of the form

```
logit(y) = intercept + treatment x time + respondent_id
```

where `respondent_id ~ N(0, 1)`

.

However, `y`

skewed toward 1, and the posterior estimated of `repsondent_id`

is also skewed in the same way as below.

So, there are few issues:

- when interpreting the treatment effect,
`respondent_id = 0`

is not an average individual. - effect size could vary depending the individual starting point
- non-normal random-effect could be a problem with the estimation?

I was thinking to

- consider
`respondent_id`

as fixed-effect (like econometrics’ approach), and report the estimated effect size for the all`respondent_id`

(~8000 in my data) - or transforming the random effect somehow

Could you recommend how would you aprroach this case?