Nearly all of the papers and posts I've seen focus on the equivalence of confirmatory factor analysis (CFA) or structural equation modeling (SEM) with item response theory (IRT) models when the data are binary or ordered. Here's a related, example post that doesn't address this Item Response Theory vs Confirmatory Factor Analysis How, then, are the two related in the case of continuous responses?
More specifically, suppose I have multiple, continuous item responses per person in a long data structure, such that each item response is in its own row. I want to fit a measurement model of the underlying latent variables for the items. Rather than model the covariance structure as in CFA/SEM, I want to employ an IRT-style hierarchical model -- my data are already in the long IRT structure!
Unlike the typical IRT logistic or cumulative link functions and non-normal (e.g., Bernoulli) likelihoods, I could use an identity link and Gaussian likelihood. This produces a hierarchical linear model with some combination of fixed and/or random effects for the person and item parameters: IRT models are essentially hierarchical models with item and person parameters.
For example, I could fit a model $y_{ik} \sim Normal(\tau_i - \lambda_i \theta_j, \sigma_y)$ where $\tau$ is a fixed item-specific intercept (difficulty/item intercept), $\theta$ is a random person-specific intercept (ability/factor score), and $\lambda$ (discrimination/factor loading) is a coefficient relating the latent variable to the item response. I can let these parameters vary across items, persons, etc. I could also include observed or latent predictors (as in explanatory item response models) and allow for correlated random effects with multivariate distributions for varying intercepts and slopes.
In light of my distributional choices:
- Can I still interpret the parameters as ability, discrimination, etc.? Or does my "IRT version of CFA with continuous variables" change the substantive interpretation of the parameters?
- Relatedly, since the location and scale of the Gaussian are not inherently correlated (unlike in the Binomial case), is $\lambda$ no longer a scale parameter?
