Explanatory item response models add predictors of the $\theta$ "ability" parameter. However, these predictors are always observed covariates. Sometimes, one may want to include latent variables as predictors, such as a different dimension representing a different type of ability. This is the item response theory equivalent to a structural equation model.

What is the name for this model? Any resources suggested for learning more?

  • $\begingroup$ in educational assessment context this is often done using plausible value representations of said constructs. $\endgroup$
    – Tom
    Commented Oct 1, 2022 at 11:16

2 Answers 2


See Boeck and Wilson (2004): https://link.springer.com/book/10.1007/978-1-4757-3990-9

There is no special name for this. It is just an (possibly double) explanatory item response model.


To clarify, you are asking for information regarding an IRT model where a latent variable $\theta_y$ can be regressed onto one or more latent predictor variables $\theta_x$, correct? I am unaware of any formal presentation of such a model, though it should be pretty straightforward in a Bayesian framework using probabilistic programming languages (e.g., jags, stan).

One article (Lu, Thomas, & Zumbo, 2005) does come close to what you are looking for. First, they estimate the SEM's measurement model parameters using IRT. Second, they convert IRT parameters to their CFA/SEM parameterization$^1$; and third, they keep these parameters fixed during the estimation of structural parameters using SEM software.

$^1$ See Takane & De Leeuw (1987) for more information on the relationship between IRT and factor analysis of categorical data.


Lu, I. R., Thomas, D. R., & Zumbo, B. D. (2005). Embedding IRT in structural equation models: A comparison with regression based on IRT scores. Structural Equation Modeling, 12(2), 263-277.

Takane, Y., & De Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52(3), 393-408.

  • $\begingroup$ Come across the first paper quite a bit but nowhere subscribes to that journal! $\endgroup$ Commented Oct 1, 2022 at 1:12
  • $\begingroup$ Note: this appears do-able within the context of latent-variable linear mixed models. See page 383 here routledge.com/Longitudinal-Data-Analysis/… $\endgroup$ Commented Oct 1, 2022 at 2:44
  • $\begingroup$ Glad it helped! I don’t have access to the book you referenced, but I will take a look when I get the chance. $\endgroup$ Commented Oct 3, 2022 at 16:53

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