# Are random-intercepts possible for delta in an equidistant, cumulative ordinal model?

Hey Everyone (particularly @paul.buerkner and @matti ),

I am implementing an ordinal regression model. In particular, it is a special sort of signal detection model. The specifics are not so important (check my post history if interested), because I have a basic question about the brms implementation. I could simulate and post data if needed. Also, for the sake of argument, let’s assume thresholds are set to ‘equidistant’ (which they are in my case). That means the distance between the thresholds is modelled by a combination of the parameter delta and the first threshold (i.e., the first threshold is estimated directly and all others are estimated by adding multiples of delta to that threshold).

In fitting a multilevel ordinal regression, I can apply a random-effect to the intercept. This allows the model to shift the thresholds to the left or right for each participant. However, I believe that it assumes the thresholds are otherwise identical from one participant to the next. That is to say, if the thresholds (for a 6 item scale) were [-1.2, -0.6, 0, 0.6 and 1.2], these could be shifted to [-1.8, -1.2, -0.6, 0, and 0.6] or to [0, 0.6, 1.2, 1.8, and 2.4] for different participants, but it would not be possible to account for a participant with [-1.8, -0.8, 0.2, 1.2, and 2.2], because delta for that participant would be 1 rather than 0.6.

In many cases, I imagine it feasible that participants would employ not only thresholds that differ in absolute position (i.e., the aforementioned shift), but also that some participants might have thresholds that are more spread out than others. This could naturally be handled by applying a random intercept to delta, allowing the spread across thresholds to vary from one subject to the next. However, it does not appear that predictors can be added for delta. Likewise, I believe from the tutorial paper that a category-specific random intercept would do the trick, but this is disallowed for cumulative models (e.g., here). Is there some other way to model variation in the relative spread of thresholds across participants? I can sort of fudge it by modelling a random intercept on disc (since allowing the SD of the underlying latent variable to vary by subject is equivalent to allowing the thresholds to vary…I think), but in my application I would like to fix disc = 2**.5.

Any advice appreciated, including arguments to the effect that I am completely off-base for asking.

• Operating System: R 3.6.1 on MacOS 10.14.6
• brms Version: brms 2.13.0
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Unfortunately, delta cannot be predicted at the moment and, as you say, category specific effects are not sensible in the current form for ordinal models. You may want to use another ordinal family to allow for category specific effects though.

Thanks, @paul.buerkner!
I very much appreciate your time, as always.

A few follow-up questions.

1.) The inability to allow thresholds to vary as a function of some grouping variable throws a wet-blanket on signal-detection models implemented in brms (for which the thresholds will almost always vary from one subject to the next). That is a real shame! Are there future plans to permit random-effects for the individual thresholds? Or is this a firm limitation?

2.) I am unsure how familiar you are with SDT models, specifically, so this one may be more for @matti , but do you recommend any of the other models you have implemented as suitable for SDT style models? I am only familiar with cumulative models (beyond what I read in your tutorial paper). Or are multilevel SDT models using ordinal ratings simply not advisable using brms at this time?

I actually was wrong on (1) you can allow thresholds to vary using the resp_thres function. Cumulative family is the standard one for SDT. Matti will be able to tell you more.

I’m not sure what the `resp_thres` function does, but it sounds like it might be useful. I might look into it at some point. The other ordinal families imply a different data generating model and so are inconsistent with SDT. That doesn’t mean that they aren’t good for the analysis that you have in mind though, they are just not sdt models.