Hi all! I’m relatively new to rstan, and for an Epidemiology project I need to run a Bayesian Gamma regression, with inverse link. I have gotten it to work already within the rstanarm library, but I tried to recreate it myself to understand it better and to get used to the stan language; and I’ve had some issues.

Just to clarify what the inverse link entails, we have y \sim Gamma(\alpha, \beta), with \alpha being the shape parameter, and \beta the rate. The inverse link is \mu^{-1} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 = \eta. If we reparametrize with \mu = \alpha / \beta, we can write y \sim Gamma(\alpha, \eta\alpha).

My issue is as follows: because both shape and rate need to be strictly positive, the linear predictor \eta should also be positive. Despite running the below code (constraining it to be positive), I run into lots of divergent transitions (like, hundreds) and chains that are all over the place.

In r:

```
# Simulate some data
x1 <- 1:150
x2 <- x1^2
b <- c(0, 0.05, -0.00001)
eta <- (b[1] + b[2] * x1 + b[3] * x2)
y <- rgamma(150, shape = 0.7, rate = 0.7 * eta)
```

With stan:

```
data {
int<lower=0> N;
vector<lower=0>[N] y;
vector[N] x1;
vector[N] x2;
}
parameters {
real b0;
real b1;
real b2;
real<lower=0.001,upper=100 > shape;
}
transformed parameters{
vector<lower=0>[N] eta; // linear predictor
eta = b0 + b1 * x1 + b2 * x2;
}
model {
b0 ~ normal(0, 50);
b1 ~ normal(0, 50);
b2 ~ normal(0, 50);
shape ~ uniform(0.01, 50);
for(i in 1:N){
y[i] ~ gamma(shape, shape * eta[i]);
}
}
```

My viewer is full messages such as:

Rejecting initial value:

Error evaluating the log probability at the initial value.

Exception: validate transformed params: eta[1] is -3.50457, but must be greater than or equal to 0 (in ‘model865c2a234c78_gamm’ at line 18)

Any ideas on what I am doing wrong here? What does rstanarm do differently? (I have looked at some of the source code, but I am lost to be honest) Thanks in advance!

NB: Any code I have found so far deals with the log link, which is substantially nicer since \eta is by default positive!

(See: http://seananderson.ca/2014/04/08/gamma-glms/)