# Multidimensional IRT in brms

**URL:** https://discourse.mc-stan.org/t/multidimensional-irt-in-brms/13075
**Category:** brms
**Created:** [February 11, 2020, 2:26pm UTC](https://discourse.mc-stan.org/t/multidimensional-irt-in-brms/13075 "2020-02-11T14:26:31Z")
**Posts on this page:** 1
**Showing post:** 14

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### Author: ![Famondir](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/famondir/32/11235_2.png) [@Famondir](https://discourse.mc-stan.org/u/Famondir)
#### Post date: [December 5, 2020, 11:03am UTC](https://discourse.mc-stan.org/t/multidimensional-irt-in-brms/13075/14 "2020-12-05T11:03:31Z")

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Dear multidim IRT modelers,  
I have a question about two competing possibilties to spedify a 2PL MIRT model. The first one sfollows the code you showed in your paper about “Analysing Standard Progressive Matrices”, @paul.buerkner:

```
formula <- bf(
  response ~ beta + exp(logalpha1) * theta1,
  nl = TRUE,
  logalpha1 ~ 1 + (1 | item),
  theta1 ~ 0 + (0 + dimension | person),
  beta ~ 0 + (1 | item),
  family = brmsfamily("bernoulli", link = "logit")
)

```

I just alterered the `(1 | person)` to `(0 + dimension | person)` and dropped the item correlations. The second specification looks like this:

```
formula <- bf(
  response ~ beta + exp(logalpha1) * map_dim1 * theta1 +
    exp(logalpha2) * map_dim2 * theta2 +
    exp(logalpha3) * map_dim3 * theta3,
  nl = TRUE,
  theta1 ~ 0 + (1 |p| person),
  theta2 ~ 0 + (1 |p| person),
  theta3 ~ 0 + (1 |p| person),
  beta ~ dimension + (1 | item),
  logalpha1 ~ 1 + (1 | item),
  logalpha2 ~ 1 + (1 | item),
  logalpha3 ~ 1 + (1 | item),
  family = brmsfamily("bernoulli", link = "logit")
)

```

The `map_dim` terms refer to integer columns in the data and are 1 if the item is related to the dimension and 0 else. I came up with this approach for models where items can have multiple dimensions and where the number of dimensions a item is related to can differ. It works.

When I came back an began to make some comparisms with simulated data on a model that could be described by both models I found the following points: The second approach…

- is slower. It takes almoast twice as long.
- gives some wide spread estimators for the logalphas that are multiplied by 0 in the main formula (\*).
- gets a little lower loo elpd.
- BUT better recovers the values for the logalpha SDs used to simulate the data.

The first model pools al `logalphas` so the SD over all `logalphas` corresponding to one dimension get shrunken to much and the difference in the SD between the dimensions gets shrunken as well. I think this is a drawback and I would like to hear your opion about this.

One possible solution I found after several tries to was:

```
formula <- bf(
  response ~ beta + exp(logalpha1) * theta1,
  nl = TRUE,
  logalpha1 ~ 1 + (1 | dimension/item),
  theta1 ~ 0 + (0 + dimension | person),
  beta ~ 0 + (1 | item),
  family = brmsfamily("bernoulli", link = "logit")
)

```

It should be noted that the estimation time for this model was the same as for the model with the mapping vectors. It models the underlying parameters better but the elpd\_loo does not improve more than one se\_diff compared to the model that does not model different logalpha SDs.

(\*): You can pass stan code via stan vars to set the ranefs of these to 0 manually. However these parameters don’t have an effect on the modelling process at all, because they are multiplied by 0 anyway. So to set these to zero is more cosmetic:

```
stan_code <- 'r_5_logalpha1_1 = r_5_logalpha1_1 .* map_alpha1;
r_6_logalpha2_1 = r_6_logalpha2_1 .* map_alpha2;
r_7_logalpha3_1 = r_7_logalpha3_1 .* map_alpha3;'

```

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