Watanabe's Learning Coefficients in Item Response Theory
We study the joint-parameter two-parameter logistic (2PL) model in item response theory, in which both item parameters and respondent abilities are treated as unknown fixed parameters. We first characterize its singular structure through the Fisher information matrix. The parameterization has an inherent scale indeterminacy; when all discrimination parameters are nonzero and the ability parameters are not all equal, its tangent direction is the only null direction and the quotient model is locally regular. We then classify the additional rank deficiencies caused by zero discrimination parameters and degenerate ability configurations. This classification separates the strata governed by a nondegenerate quadratic approximation from those requiring an analysis of higher-order product singularities. Under an asymptotic regime in which the numbers of items and respondents are fixed and the number of independent response matrices tends to infinity, we derive explicit local and global learning coefficients and multiplicities for every realizable true response distribution. The global formulas depend only on the true response probabilities, including cases in which some probabilities equal \(1/2\) or are constant across respondents. These results extend previous calculations restricted to nonzero true discrimination parameters and provide the quantities required for applying sBIC to the joint-parameter 2PL model. Key Words: item response theory; two-parameter logistic model; singular learning theory; learning coefficient; real log canonical threshold; Fisher information matrix; sBIC.