Hi everyone,

As part of my PhD research, I work with a lot of ordinal variables. As such, I was really pleased to see that @paul.buerkner and @matti’s recent tutorial paper on the matter. One of the key points they make is about modelling the variance of the latent Y distribution using the `disc`

parameter in brms.

I have a few of questions about interpreting unequal variances. If anyone would be kind enough to answer, I’d really appreciate it.

First, how is `disc`

/latent variance related to the thresholds? Using the stem cell example in the paper, if we expect liberals to have less variation than moderates, my impression would be that we would expect the thresholds for the liberals to be closer together than those for the moderates. Is this true?

Second, if so, what is the difference between a model with varying thresholds, e.g. `bf(rating ~ 1 + (1 | belief))`

, and a model with unequal variances, e.g. `bf(rating ~ 1 + belief) + lf(disc ~ 0 + belief, cmc = FALSE)`

?

Finally, in terms of notation, is there any convention around how we might write out the linear predictor components of the latent variance parameter? Imagine, for instance, that we have a simple experiment with a treatment T and a cumulative ordinal attitudinal outcome on a 1-5 scale, O. My impression is that the model assuming *equal* variance would be written something like:

O_{i} \sim \text{Cumulative}(\eta_{i}, \tau) \\ \eta_{i} = \beta T_{i} \\ \beta \sim Normal(0, 1) \\ \tau_t \sim \text{Normal}(0, 1.5) \text{, for } t=1..4

But this notation doesn’t make explicit the latent scale. As such, I’m not sure how one would extend it to including linear predictors of the latent variance parameter. Any advice? Sorry is this last point is banal – I have to teach myself this stuff in my own time and I’m aware that I need to up my notation game.