🤖 AI Summary
Traditional long assessments exhibit low statistical efficiency in estimating population-level latent trait means. This work proposes a closed-form approximation method within the item response theory (IRT) framework, wherein item information functions are approximated via Gaussian convolution. By integrating marginal maximum likelihood estimation with a stochastic item sampling mechanism, the approach explicitly models the trade-off between test length and sample size and incorporates a variance inflation factor accounting for item pool mismatch. The method enables unbiased inference of population means even with extremely short tests—including single-item assessments. Validation through exact benchmarks, Monte Carlo simulations, and resampling experiments on NAEP mathematics data demonstrates that estimates remain approximately unbiased even when based on a single item, substantially enhancing measurement efficiency.
📝 Abstract
When the target of inference is a group mean rather than an individual score, long assessments can be statistically inefficient. From the marginal score of the likelihood, we derive the asymptotic standard error of the marginal maximum likelihood estimator of a latent IRT mean under random item sampling, and obtain a closed-form approximation via a Gaussian-convolution treatment of the item information. The formula makes the test-length--sample-size trade-off explicit and includes an inflation factor for jointly estimating the population variance when the item pool is mistargeted. We validate it against exact benchmarks, Monte Carlo simulation under pool--population mismatch, and a NAEP mathematics resampling study; group means remain approximately unbiased even with single-item forms.