Weighted Sequential Construction of Goodness-of-Fit Statistics from Pairwise Concordance Marginal Information in Sparse Binary IRT Models with Symmetric Intercepts and Slopes

When many binary items are analyzed, most cells in the complete response table are empty. Full-table statistics may then be unreliable, and an omnibus result does not locate the misfit. This paper develops a local goodness-of-fit testing method for binary item response models by extending the weighted sequential sums-of-squares construction to pairwise concordance margins. The construction gives ordered one-degree-of-freedom components for individual item pairs. They are compared with Cholesky components and four local diagnostics under zero and wide symmetric intercepts. Omnibus statistics are examined separately under symmetric slopes. The sequential components have an empirical Type I error close to the nominal levels. With zero intercepts, all methods show substantial power for the target pairs, although pair order moves some shared discrepancy into non-target components. Cholesky gives higher target-pair power and broader moderate elevations across non-target pairs, whereas the sequential components show more concentrated, order-dependent peaks. With wide symmetric intercepts, power differs greatly among target pairs, and the Lagrange multiplier statistic is particularly sensitive to intercept distance. The main qualitative patterns persist across three nominal levels. Lower-order omnibus statistics generally maintain an empirical Type I error close to the nominal levels and detect the slope alternatives, whereas Pearson–Fisher is liberal in the sparse full table. A symmetric parameter vector therefore need not give uniform local diagnostic results.