Doing a meta-analysis with regression on three covariates x_1, x_2, x_3, I want to include the prior information we have about the order of the covariates coefficients.

**Model**

My model is a regression model with a random intercept:

k_i \sim Bin(n_i,p_i) (data)

p_i = \text{logit}^{-1}(\alpha_i + \beta_1 \cdot x_{1,i}+\beta_2 \cdot x_{2,i}+\beta_3 \cdot x_{3,i})

with \alpha_i \sim \mathcal{N}(\alpha,\sigma^2) (random intercept)

We know that \beta_1>\beta_3 and \beta_2>\beta_3. As it is partial ordering, I cannot declare the vector \beta :=(\beta_1,\beta_2,\beta_3) as an `ordered`

vector. I was therefore thinking of declaring the coefficients \beta as

```
parameters {
real beta_1;
real beta_2;
real < upper = min_beta_1_2 > beta_3;
}
transformed parameters {
vector[2] beta_1_2;
beta_1_2[1] = beta_1;
beta_1_2[2] = beta_2;
real min_beta_1_2 = min(beta_1_2);
}
```

**Q1**: Is it a correct formulation?

Then, I thought I could write it in another way, putting the constraint on \beta_1 and \beta_2:

```
parameters {
real beta_3;
real < lower = beta_3 > beta_1;
real < lower = beta_3 >beta_2;
}
```

**Q2** : Is there any difference between the two models in terms of results, computation time and â€śconvergenceâ€ť? If yes, which one should I use?