I am trying to set some informed priors (from previous studies) on a “correlation” between clinical symptoms. Symptoms (in this case) are evaluated on a likert scale 0-5. So, both outcome and predictor are on an ordinal scale.
The posterior estimate of the relation is (per brms docs) the average change (in log-odds) in the outcome (across the levels) of a change of 1 in the predictor. Which is great per se. But I now want to set a prior on that inspired by previous studies, which are either reporting pearson R, or spearman Rho. Let’s say 0.2 (se: 0.05). This is where it gets tricky: how do I convert that to the scale of the estimate I’m interested in?
I ran a few rough simulations and if the outcome were gaussian and scaled 0-1, the posterior estimate of the relation corresponds very roughly to spearman rho / (levels-1). But I haven’t cracked what happens when the ordinal outcome yet.
The interpreation of the ordinal coefficients depend on the specific ordinal family. Which of them have you used? I like to use the probit link and the cumulative family. Then, the latent continous response variable can be though of being standard normal, which helps when interpresting coefficient scale.
sorry, forgot to specify. I used cumulative w the default logit (which is why I mentioned log-odds). So if I use probit (z scale), the beta would be on a sd/(levels-1) scale? and therefore could be turned into a -1/1 approximate scale by multiplying by (levels-1) and dividing by 3 (very roughly, being the normal roughly within 3 sd from the mean)?
I am not sure what you mean by the level - 1 scaling. With the cumulative family, the scaling of the latent variable is independent of the number of levels. More levels mean just a more fine grained resoultion of the observed y to inform the latent y.
yes, that makes sense. Back to the simulation machine. Thanks!