Basically, this allows covariates for the mean value, the variance of the between-subject random effects, and the variance of the within-subject error terms. There is example code by Williams et al 2019 on how to do this with a non-linear version of this model (https://link.springer.com/article/10.3758/s13428-019-01255-9), but I cannot figure out how to implement the original linear version in brms. Is this even possible?

Please also provide the following information in addition to your question:

but it doesnâ€™t say how to model the â€śbetween-subjectâ€ť variance (i.e., the group-level intercepts) with covariates. In effect, Iâ€™m not sure how to model the \sigma_{u_{ij}} portion of the model above, since the parameters are not part of the syntax, but is required as both the coefficient for \theta_{1i} and as a non-linear outcome within the model in the form of \log\left(\sigma^2_{\upsilon_{ij}}\right).

I am not sure I understand. You can predict sigma by whatever covariate you want as you do with X1 above. What is it that you donâ€™t know how to model then?

The sigma that brms allows to model as you have pointed out is the standard deviation of the residuals. However, as in Hedeker and Nordgren 2013 (https://www.ncbi.nlm.nih.gov/pubmed/23761062), I also wish to model the standard deviation of the group effects:

In terms of the variance structure, covariates can be specified to have effects on both the between-subject and within-subject variances. Another use is for clustered data in which subjects are nested within clusters (e.g., clinics, hospitals, schools, etc.) and interest is in modeling the between-cluster and within-cluster variances in terms of covariates.

This means, in a panel data set with individuals as level-2 units, modeling both the within-group variability (that is, within-person variability over time), and the between-group variability (that is, variability between persons). sigma as it is allows me to model the within-group variability, but not the between-group variability. Hope this makes more sense?

To my knowledge, brms does not let you do this [yet?]. Itâ€™s fairly simple to do via stan itself, however. The Williams/Rast/Zimprich paper does not model the between-group SD afaik.

So you can either code it up yourself in Stan, or you might be able to use a package of mine (https://github.com/stephensrmmartin/ICCier Use Split_bp_wp branch!) to fit the model. Itâ€™s intended for modeling ICCs, but we ultimately use a MELSM. Due to the focus on ICCs, the summary output, by default, hides some parameters (if I recall correctly), but can show the full parameter matrix of interest.
The basic specification may be: within ~ level1ResidSDPredictors|level2ResidSDPredictors, between ~ RandomEffectSDPredictors, mean ~ level1locationPredictors|level2locationPredictors. So, e.g.: ICCier(y, subj, within ~ 1|X1, between ~ X1, mean ~ 1|X1, data = ds) would model the residSD and between-group SDs from a level-2 variable: X2. Again, by default, the focus is on ICCs, so the summary(output) may hide some of the parameters, but thereâ€™s an option I believe for showing the full thing.

Did you ever succeed in modeling the BS variance in Stan or brms? If so, Iâ€™d love to hear/see how you did it because Iâ€™m working on the same problem.

Hi, unfortunately not. This paper (https://doi.org/10.1080/00273171.2018.1477577) seems to have some code to do it in Stan, but I havenâ€™t yet looked at it closely and modified it for my own use. Let if know if you make any progress on it!

Hi Madeline! Did you ever happen to get this working? I want to compute ICCs for each subject using brms in stan and Iâ€™m a little stuck. Curious to know if youâ€™ve figured this out for the group-level?

Iâ€™m working on validating the solution proposed in the latter now and hope to have that plus code implementing/validating the former done in the next month or so.

I checked with @paul.buerkner a while back to determine whether it is possible to model the between subject variance in addition to the within subject variance using brms. He said that itâ€™s not currently possible and that it isnâ€™t high on the priority list.

To my knowledge, youâ€™d have to write the model youâ€™re interested in in Stan. I tried this and gave up after a sincere attempt. I found it to be pretty complicated.