In his article on item response theory in `brms`

(https://arxiv.org/pdf/1905.09501.pdf?), Paul shows how to use his package for a so-called *two-parameter graded response model*. These are cumulative logit models with two item-specific parameters. An *easiness* parameter, say b_i, and a *discrimination* parameter, say a_i, for each item i.

A simple version of my current code to fit such a model would be

```
family <- brmsfamily("cumulative", "logit")
formula <- bf(score ~ 1 + (1 | item) + (1 | rater), disc ~ 1 + (1 | item))
model <- brm(formula = formula, data = df, family = family)
```

However, mathematically, I can see two possible models that could be defined by this syntax. Let’s assume score has three ordinal categories Y_{i,j} \in \{1,2,3\} and that raters are enumerated by j = 1, ..., J and their ability is \theta_j. The probability to score a 1 (lowest category) could be defined in two ways now.

**Version 1**

Pr(Y_{i,j} = 1) = \text{expit}(\beta_1 - ((a_i) * (b_i + \theta_j)))

**Version 2**

Pr(Y_{i,j} = 1) = \text{expit}(a_i (\beta_1 - (b_i + \theta_j)))

In the first, the threshold parameter \beta_1 is not multiplied by the discrimination parameter, in the second one, it is. Of course, it’s the same for the definition of the probability of the second category, but this is enough to make the point.

To me, it is not clear which of the two models `brms`

is fitting here, even after carefully studying the associated paper. It seems they would refer to different underlying model definitions?