Analytically Corrected Bayesian Modularization for Local Item Calibration

Abstract

The continuous calibration of pilot items embedded in operational assessments is challenging when pilot samples are small and adaptively routed. We formalize a Bayesian modularization framework for local item calibration that blocks feedback from pilot responses to the operational latent scale: Plausible Values are drawn from the operational posterior, and each pilot item is calibrated by its own local logistic regression. This construction is computationally scalable, protects operational trait estimates from malfunctioning pilot items, and admits Firth's penalized likelihood for sparse routed samples. Because treating imputed traits as fixed predictors induces attenuation, we derive closed-form disattenuation mappings that recover the generating item parameters under normal-ogive, posterior-normality, homoscedasticity, and joint-normality approximations. The resulting Modular Local Calibration (MLC) estimator reaches the target of marginal-likelihood calibration without per-item numerical integration. A multivariate delta-method covariance combined with Rubin's-rules pooling propagates operational item-parameter uncertainty into the focal item standard errors. Monte Carlo simulations for the unidimensional 2PL show that corrected MLC substantially reduces attenuation bias and yields nominal-to-conservative interval coverage in the studied conditions, including under restricted-range MAR routing.