I have data on several respondents placement of parties on a left-right political scale, and the idea is to estimate the latent party placements by a model that looks like this (This is also called an Aldrich-McKelvey scaling model):
y* = \alpha_i + \beta_i \theta_j
where y* is the placement on the latent scale (let’s say I am assuming a latent normal distribution), \alpha_i is the individual respondent’s shift of the scale, \beta_i the respondent’s scaling of the scale and \theta_j the latent party placements. The model is not identifiable if \beta_i is not restricted by sign.
By following @paul.buerkner s paper on item response theory in brms I realized this is identical (at least in concept) to an ordinal item response theory model, so in brms I set up:
bf(value ~ 1+ (1|id|rowid)+ (1|party),
disc ~ 1+ (1|id|rowid))
with cumulative(link=“probit”) as family.
The problem is that some (albeit a few) respondents have answered as if the scale were flipped, put left-wing parties to the right and vice versa. In this parametrization those get very high scaling values, which of course is reasonable, since the variance/scaling would be seen as high when the scaling is restricted to positive. It could be interpreted as if the model regards them as very unprecise.
See this plot of correlation of each respondent’s placements’ correlation with the mean party placement of the latent scale versus the estimated mean scaling (I think I inversed the link correctly to make it interpretable as variance).
One alternative would be to formulate the model as an non-linear one (which was where I began before realizing the conceptual equivalence to ordinal IRT models), but I cannot figure out on how to add some kind of term that would model this scale flip.
This is how I formulated the model as a non-linear formula initially:
bf(value ~ 1 + exp(scalepar) * partylat + shiftpar,
shiftpar ~ 0+(1|id|rowid),
scalepar ~ 0+(1|id|rowid),
partylat ~ 0+(1|party),
nl = TRUE)
(exp(scalepar) is to make the model identifiable. This model also took ridiculously long time to estimate in comparison with the one above.)
The literature says that one fixed scaling parameter will make the model identifiable, but since brms does not support priors (also not constant priors) for individual random effects it would seem I am out of options. The pooling would also make this difficult to estimate properly I guess.
Do you have any input on how to incorporate the scale flip for some respondents?