Multidimensional IRT in brms

Dear Paul,
in your recent excellent paper “Bayesian Item Response Modeling with brms and Stan”, you said that “the non-linear multilevel formula syntax of brms allows for a flexible yet concise specification of multidimensional IRT models”. Could you please give a short example of brms code how to specify a multidimensional model for polytomous data with estimated discrimination? Eg. 10 items in total, 5 of them in one dimension and another 5 in othoer dimension. Thanks a lot. Martin

Please also provide the following information in addition to your question:

  • Operating System: Windows
  • brms Version: 2.11.1

Hi Martin,

welcome to the Stan forums. Under your circumstances, the 1PL model could look as follows:

bf(y ~ (1 | item) + (0 + itemtype | person),
   family = bernoulli())

where itemtype is a factor assigning the items to the two dimensions. The corresponding 2PL model, could look as follows:

bf(y ~ disc * eta,
   eta ~ (1 | item) + (0 + itemtype | person),
   disc ~ (1 | item),
   nl = TRUE,
   family = bernoulli()
)
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Dear Paul,
thanks a lot for your reply. Unfortunately, my items are polytomous, as I said previously, so the family will be “cumulative” (or graded response model in IRT terminology). As you have said: “We have to use slightly different formula syntax, though, as the non-linear syntax of brms cannot handle the ordinal thresholds in the way that is required when adding discrimination parameters.” Your example: “formula_va_ord_2pl <- bf(resp ~ 1 + (1 |i| item) + (1 | id), disc ~ 1 + (1 |i| item))”. Should I add (0+itemtype) to (1|id) to obtain (0 + itemtype | id)?

Yes, exactly. The key is to replace (1 | id) by (0 + itemtype | id).