Rescale posterior samples for identification

Applies post-hoc rescaling to MCMC posterior samples to resolve identification indeterminacy. For unconstrained models, this centers item difficulties and (for 2PL/3PL) normalizes discriminations.

Usage

dpmirt_rescale(samples_obj, rescale = TRUE)

Arguments

samples_obj: A dpmirt_samples object.
rescale: Logical. If TRUE (default), apply rescaling. If FALSE, wrap untransformed posterior samples in a dpmirt_fit object using the same downstream fit schema.

Value

A dpmirt_fit S3 object containing rescaled posterior samples, chain/run metadata, diagnostics (ESS, WAIC, cluster info), and model configuration. This object can be passed directly to dpmirt_estimates, dpmirt_resume, and other downstream functions.

Details

Post-hoc rescaling resolves the identification indeterminacy inherent in unconstrained IRT models:

Rasch (location only): For each MCMC iteration $s$: $$\beta^*_i = \beta_i - \bar{\beta}$$ $$\theta^*_j = \theta_j - \bar{\beta}$$

2PL/3PL IRT parameterization (location + scale): Let $c_s = \bar{\beta}_s$ and $d_s = (\prod_i \lambda_i)^{-1/I}$: $$\beta^*_i = (\beta_i - c_s) / d_s$$ $$\lambda^*_i = \lambda_i \cdot d_s$$ $$\theta^*_j = (\theta_j - c_s) / d_s$$

2PL/3PL slope-intercept parameterization: Let $c_s = \sum_i \gamma_i / \sum_i \lambda_i$ and $d_s = (\prod_i \lambda_i)^{-1/I}$: $$\gamma^*_i = \gamma_i - \lambda_i c_s$$ $$\lambda^*_i = \lambda_i \cdot d_s$$ $$\theta^*_j = (\theta_j + c_s) / d_s$$

Constrained models and calls with rescale = FALSE use the same dpmirt_fit wrapper but preserve the raw sampled scale.

After rescaling: $\bar{\beta}^* = 0$ and $(\prod_i \lambda^*_i)^{1/I} = 1$.

References

Paganin, S., Paciorek, C. J., Wehrhahn, C., Rodriguez, A., Rabe-Hesketh, S., & de Valpine, P. (2023). Computational strategies and estimation performance with Bayesian semiparametric item response theory models. Journal of Educational and Behavioral Statistics, 48(2), 147–188.

Examples

if (FALSE) { # \dontrun{
sim <- dpmirt_simulate(200, 20, model = "rasch", seed = 42)
spec <- dpmirt_spec(sim$response, model = "rasch", prior = "normal",
                    identification = "unconstrained")
compiled <- dpmirt_compile(spec)
samples <- dpmirt_sample(compiled, niter = 5000, nburnin = 1000)

# Apply rescaling
rescaled <- dpmirt_rescale(samples)
} # }