Howdy,

I do not believe that `brms`

can do this. You would need to do it in Stan. Also, as I thought more about what you are trying to do here, I am wondering if it even makes sense…

What you are calling `tau`

is the standard deviation of group-level effects in `brms`

, which would be a single parameter. This is the standard deviation of the varying intercepts of a theoretical population of groups. If I understand correctly, you are wanting to be able to predict this parameter also based on some, say, continuous predictor variable. But what this implies is a vector of `tau`

’s for every group, not a single `tau`

, because for every (infinite) value of the predictor, then you would have a different value for `tau`

. Unless multiple groups had the exact same value for `tau`

… I tried to think about this by writing a simulation in R and model in Stan. Here, your `tau`

’s are called `sd_1`

in the model, and they are a vector of parameters that are estimated as the outcome variable in an exponential model predicted by the predictor `v`

. They are also used in the varying intercepts for the binomial model using a non-centered parameterization (the default in `brms`

and usually works better unless you have a ton of data in each group).

The model runs fine and appears to work, although b_sd is underestimated (perhaps due to the partial pooling effect from the binomial model??). I think there is still partial pooling going on, as the `sd_1`

still are part of the binomial model. But now you get a different `sd_1`

, or `tau`

as you call it, for every `id`

, rather than a singe one for the entire group.

Honestly, even after writing this thing up, I am not entirely sure what to think about it from a conceptual or theoretical view…so take this with a grain of salt. Maybe @jsocolar would have some thoughts.

```
library(rstan)
#data simulation
n_id <- 100 #number of id's
n_len <- 10 #number of observations for each id
n <- n_id * n_len #total number of observations
id <- rep(1:n_id, each=n_len) #id label
trials <- rpois(n, lambda=50) #random number of trials
x <- rnorm(n) #predictor for binomial outcome model
v <- rnorm(n_id) #predictor for vector of sd's for varying intercepts
a_sd <- 0 #intercept for exponential model of sd's
b_sd <- -1.5 #coefficient for v
mu_sd_z <- a_sd + b_sd*v #linear model for sd's
sd_z <- rexp(n_id, rate = exp(-(mu_sd_z))) #generative model for the vector of sd's of varying intercepts
z <- rnorm(n_id, 0, sd_z) #generate varying offsets (varying intercepts) for the binomial outcome model
a <- (-2) #intercept of the binomial outcome model
b <- 1 #coeffient for x
p <- plogis((a + z[id]) + b*x) #probability p for linear model
y <- rbinom(n=n, size=trials, prob=p) #generative model for outcome y's
#stan data
d1 <- list(N = n, #total number of observations
Y = y, #binary response
x = x, #predictor for binomial outcome y
v = v, #predictor for vector of sd's of varying intercepts
J = n_id, #number of id's
id = id, #id
trials = trials #trials
)
#stan code
stan_code1 <- "
data {
int<lower=1> N;
int Y[N];
vector[N] x;
int<lower=1> J;
int<lower=1> id[N];
vector[J] v;
int trials[N];
}
parameters {
real a; // intercept for binomial outcome model
vector<lower=0>[J] sd_1; // vector of group-level standard deviations for varying intercepts
vector[J] z_1; // standardized group-level effect, needed for non-centered parameterization
real b; // coefficient for x for binomial outcome model
real a_sd; // intercept for exponential model for vector of sd's of varying intercepts
real b_sd; // coeffient for v for exponential model for vector of sd's of varying intercepts
}
transformed parameters {
vector[J] r_1_1; // actual varying intercepts (offsets)
r_1_1 = (sd_1 .* z_1); // non-centered parameterization
}
model {
for (j in 1:J) {
target += exponential_lpdf(sd_1[j] | exp(-(a_sd + b_sd * v[j]) )); // exponential model for vector of sd's for varying intercepts
}
for (n in 1:N) {
target += binomial_logit_lpmf(Y[n] | trials[n], a + r_1_1[id[n]] + b * x[n]); // binomial outcome model
}
// priors
target += normal_lpdf(a | 0, 2.5);
target += normal_lpdf(sd_1 | 0, 2.5)
- 1 * normal_lccdf(0 | 0, 2.5);
target += std_normal_lpdf(z_1);
target += normal_lpdf(b | 0, 2.5);
target += normal_lpdf(a_sd | 0, 2.5);
target += normal_lpdf(b_sd | 0, 2.5);
}
generated quantities {
}
"
#fit model
fit1 <- stan(model_code = stan_code1, data = d1,
chains = 4, iter = 2000, warmup = 1000,
thin = 1, cores = 4)
#print selected parameters
print(fit1, pars=c("a", "b", "a_sd", "b_sd"))
```