Formula notation of nested varying effects

Interfaces brms
,
1

In the documentation to brmsformula stands the following:

If levels of the grouping factor belong to different sub-populations, it may be reasonable to assume a different covariance matrix for each of the sub-populations. For instance, the variation within the treatment group and within the control group in a randomized control trial might differ. Suppose that y is the outcome, and x is the factor indicating the treatment and control group. Then, we could estimate different hyper-parameters of the varying effects (in this case a varying intercept) for treatment and control group via y ~ x + (1 | gr(subject, by = x)).

I am wondering whether y ~ x + (1 | gr(subject, by = x)) is equivalent to y ~ x + (1 | x/subject)?

2

Welcome to the Stan Forums, @stfnrpplngr!

I don’t think y ~ x + (1 | gr(subject, by = x)) is equivalent to y ~ x + (1 | x/subject). For example, y ~ x + (1 | x/subject) also includes varying intercepts for x. This is not the case for y ~ x + (1 | gr(subject, by = x)) (at least in my understanding; I haven’t needed it yet).

Some extra info (perhaps you already know that): For the nesting syntax (y ~ x + (1 | x/subject)), I think brms follows lme4. And this lme4 vignette says on page 7 that (1 | g1/g2) is equivalent to (1 | g1) + (1 | g1:g2).