implement SDT models

[GidonFrischkorn]

Because of the Master thesis I am supervising, I had a look into options for implementing SDT models and found out that brms already supports basically all link functions that are relevant for this: mostly probit for SDT with normal noise and cloglog for SDT with Gumbel noise. Additional options would be cauchit or logit for SDT models with Cauchy or Logistic noise. But I don't think these are really relevant. These links are implement for both bernoulli or binomial distributions for single threshold SDT models, as well as for the cumulative distribution needed for the multiple threshold/confidence SDT models.

I was thinking about implementing two SDT models for 1) the single threshold case and 2) the multiple threshold case and in addition to the response variable let the users specify which noise distribution they want to assume (i.e. "normal" or "gumbel"). For the multiple threshold models, there would be an additional option for unequal variance models, providing the option to predict a "discrimination" parameter.

Would you mind, if I take a shot at these implementations? Or do you want to do this? Just because originally, we discussed that you would take the lead in this.

Apart from that I think these models should support all of the brms functionality, given that the distributions and link functions are natively supported by brms.

[venpopov]

Sure, go ahead. Less work for me 😄 but also less fun for me 😢 I can focus on trying to get the full sdm working

[GidonFrischkorn]

The simple SDT models without confidence ratings are mostly implemented. When I had a look into the confidence rating SDT models I noticed that it is more complicated to have the thresholds vary according to a function. This is mainly because there are several thresholds (specifically one less that rating points on the rating scale). This issue is discussed here: https://discourse.mc-stan.org/t/estimating-two-sets-of-thresholds-in-an-ordinal-model/20380

So, we have to make a decision, how to deal with the need to have thresholds vary over a set of predictor variables and how to include that in the bmmformula. If we want to accommodate these predictions in the bmmformula then we need to transform the predictions into a grouping variable telling brms to estimate separate sets of thresholds for certain conditions. Something like: resp | thres(gr = thres_predictor) ~ dprime*isold

As of now, I am not sure how we would deal with random effects in this case. Intuitively, the random effects of the thresholds should then also be grouped according to the same predictors. What do you think?

Maybe, we can have a chat about that when you are back at the office. For now, I will finalize the simple SDT models and make a commit for them by tomorrow.

[venpopov]

Yes, let's chat next week when I'm back