Hello!

I need help writing the algebraic notation of complex nonlinear `brms`

models. I have already fitted the models with success. I now need to “translate” the `brms`

notation I wrote to algebraic notation, to better understand the difference between the models I have fitted.

The database to which I fitted the models is structured as follows:

```
sample phase SC gamma tau
1 A020102 2 60 70.030 3164
2 A020102 2 60 38.490 2810
3 A020102 2 60 20.480 2522
```

In which “phase” and “SC” are categorized grouping factors, “gamma” is the independent variable and "tau"is the response variable.

I am fitting the following nonlinear equation to my data:

\tau = a + b\gamma^c

With a, b and c as the model parameters.

Here are the models set up in brms syntax:

```
model_A <- brms::bf(tau ~ a + b * (gamma^c),
a ~ 1 + (1|SC|phase),
b ~ 1 + (1|SC|phase),
c ~ 1 + (1|SC|phase),
nl = TRUE)
model_B <- brms::bf(tau ~ a + b * (gamma^c),
a ~ 1 + (1|SC) + (1|SC:phase),
b ~ 1 + (1|SC) + (1|SC:phase),
c ~ 1 + (1|SC) + (1|SC:phase),
nl = TRUE)
model_C <- brms::bf(tau ~ a + b * (gamma^c),
a ~ 1 + (1|SC) + (1|phase),
b ~ 1 + (1|SC) + (1|phase),
c ~ 1 + (1|SC) + (1|phase),
nl = TRUE)
```

I now need to write these models in algebraic notation, using the correct index notation. I’m specially confused about how to describe nested random effects, the meaning of the `:`

operator and the difference between `(1|SC) + (1|phase)`

and `(1|SC|phase)`

.

This StackExchange question helped a lot, but does not answer all I need. There are also a few similar topics (1, 2, 3) already on Stan Forums, but none of them deal with the `:`

operator. Also, the `equatiomatic`

package was removed from the CRAN repository, thus I’m assuming it’s not entirely reliable to use.

Can anyone please help me?

Any help is deeply appreciated.