Hearchical model with varying sigma


My data points can be grouped in J groups and thus I choose a bayesian hiearchical random-intercept-slope model to model my data as follows:

\begin{aligned} \mathbf{y} &\sim \text{Normal}\left( \alpha + \alpha_j + \mathbf{X} \cdot \beta_j \cdot \tau_{\beta}, \sigma \right) \\ \alpha &\sim \text{Normal}(\mu_\alpha, \sigma_\alpha) \\ \alpha_j &\sim \text{Normal}(0, \tau_{\alpha}) \\ \beta_j &\sim \text{Normal}(0, 1) \\ \tau_{\alpha} &\sim \text{Gamma}(2, 0.1)\\ \tau_{\beta} &\sim \text{Gamma}(2, 0.1)\\ \sigma &\sim \text{Exponential}(\lambda_\sigma) \end{aligned}

Now, I also suspect that there is still some unexplained variability in my data that can be different across the J groups.
Therefore, I would also like to insert in the model above for each group a different \sigma_j and then a hyper prior \tau_{\sigma}.
I’m not sure what the best way is to go about this and which distributions on \sigma are recommended.
I hope somebody here can me some guidance on this problem or pointers to relevant literature?

Many thanks!

One approach that is used in psychology, searchable with the term “location-scale models”, is to a model the log variance in the same way as the mean is being modeled. I see here for code.

Thanks for the tip on the term location-scale models (and the link to your code)! Now I can search more pointedly. :)

There are also some packages out there that can fit these for you.

LMMELSM: (Disclosure: My package) - Handles latent, multivariate mixed effects location scale models. It can also do univariate, observed MELSMs of course. Benefits: Can also model random effect variances, not just residual variance. Cons: Only handles one random grouping effect (e.g., no nested or crossed random effects currently; maybe when I have some time and motivation…)

Brms: Obviously, an amazing package. It can handle MELSMs and LSMs. Benefits: Handles nested/crossed random effects. Cons: Does not handle latent variances, or random effect variances.

Some others: ICCier (Also my package, but I’d recommend LMMELSM/brms over this). Nlme can do /some/ variance models, but the support isn’t great. Hedeker has a couple of non-Bayesian MELSM packages too.

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