Abstract
We generalize the power divergence (PD) family of statistics to the two-parameter logistic IRT model for the purpose of constructing hypothesis tests and confidence intervals of the person parameter. The well-known score test statistic is a special case of the proposed PD family. We also prove the proposed PD statistics are asymptotically equivalent and converge in distribution to \(\chi _{1}^2\). In addition, a moment matching method is introduced to compare statistics and choose the optimal one within the PD family. Simulation results suggest that the coverage rate of the associated confidence interval is well controlled even under small sample sizes for some PD statistics. Compared to some other approaches, the associated confidence intervals exhibit smaller lengths while maintaining adequate coverage rates. The utilities of the proposed method are demonstrated by analyzing a real data set.
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Notes
\(2nI^\lambda \) in its entirety is generally referred to as the PD statistic and has a Chi-squared asymptotic distribution.
This result is perhaps well-known. But, for self-containess and rigor, a formal proof is provided here.
One reviewer remarked correctly that it is straightforward to mathematically show the monotonicity assuming \(a_j = 1 \text { and } b_j = b \,\forall j\); However, it is non trivial to prove under the general case.
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Liu, X., Yang, J., Chae, H.S. et al. Power Divergence Family of Statistics for Person Parameters in IRT Models. Psychometrika 85, 502–525 (2020). https://doi.org/10.1007/s11336-020-09712-7
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DOI: https://doi.org/10.1007/s11336-020-09712-7