How do I fit multilevel IRT models?
||Fitting multilevel IRT models in gllamm
Minjeong Jeon, University of California, Berkeley
Here we consider multilevel IRT models for binary responses.
See also the FAQ on single-level item response models for binary responses.
Multilevel two parameter IRT models in gllamm
Now I consider a multilevel extension of the IRT models.
Suppose students are nested within schools and the school identifier is
To take into account dependence among the students within the same school, we introduce a
school-level random intercept and the resulting model becomes
a 3-level multilevel model with the item responses at level 1, students at level 2,
and schools at level 3.
The syntax for fitting the multilevel two-parameter IRT model can be written as
eq load: i1-i15
constraint def 1 [pid1_1l]i2 = [sch2_1l]i2
constraint def 2 [pid1_1l]i3 = [sch2_1l]i3
constraint def 3 [pid1_1l]i4 = [sch2_1l]i4
constraint def 4 [pid1_1l]i5 = [sch2_1l]i5
gllamm y i1-i15, nocons link(logit) family(binomial) i(pid sch) ///
eqs(load load) constraints(1/4) adapt
This syntax is similar to the one for a simple two-parameter IRT model.
Here for the multilevel two parameter IRT model, we need to specify loadings for levels 2 and 3
In addition, note that
the discrimination parameters are assumed to be the same at levels 2 and 3 in the model.
I therefore defined
four constraints for the factor loadings (or item discrimnation parameters) using
Since the discrimination parameter for the first item (item 1) is constrained to 1 at both levels,
we constrain the factor loadings for items 2 to 4 to be the same at level 2 and level 3.
We then specify these constraints with the
1/4 means 1 to 4.
Instead of using parameter constraints, the model can also be estimated using the
bmatrix() option as demonstrated in the first example below.
- Multilevel two-parameter model
- Multilevel one-parameter model
Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2004).
multilevel structural equation modelling. Psychometrika
69 (2), 167-190.Local
Rabe-Hesketh, S. and Skrondal, A. (forthcoming).
In van der Linden, W. J. and Hambleton, R. K.
Handbook of Item Response Theory: Models,
Statistical Tools, and Applications.
Boca Raton, FL: Chapman & Hall/CRC Press, volume 3, chapter 30.