How do I fit multilevel IRT models?

Title   Fitting multilevel IRT models in gllamm
Author Minjeong Jeon, University of California, Berkeley
Date July 2012

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 schid. 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 the eqs() option. 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 constraint def. 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 constraints(1/4) option, where 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.



  • Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2004). Generalized multilevel structural equation modelling. Psychometrika 69 (2), 167-190.Local
  • Rabe-Hesketh, S. and Skrondal, A. (forthcoming). GLLAMM software. 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.