Diagnosing Block-Position Effects on the Item Difficulty Scale: A Stage-Specific Localized Difficulty-Shift Model

Abstract

Block-position effects in large-scale assessments are often modeled as common difficulty shifts, although variation may differ across administration stages and across examinee-item ability-difficulty relationships. This study develops a staged two-parameter logistic framework: M0 has no position effect, M1 adds a common linear shift, M2 permits stage-specific nonlocalized shifts, and M3 allows those shifts to vary with ability-difficulty proximity.

The framework was evaluated in controlled balanced and TIMSS-structured sparse simulations and applied to TIMSS 2019 Chinese Taipei Grade 8 mathematics data. Across both simulations, M1 was sufficient for linear effects, M2 for nonlinear but nonlocalized stage effects, and M3 added information when localization was genuinely present. M3 primarily improved recovery of position-effect and probability-effect functions and threshold-proximal calibration, whereas sparse administration reduced precision and weakened identification of the nonlocalized boundary.

In the TIMSS application, the common linear trend was small, whereas the empirical administration design separated the items into two disconnected position families, Positions 1 and 4 versus Positions 2 and 3. M3 yielded a proximity-varying fitted pattern aligned with these two families and concentrated near the baseline response threshold. Because no items connected the two families, this between-family pattern cannot be attributed uniquely to block position; the directly connected within-family contrasts did not show clear differences. Global held-out predictive differences were very small. The fitted family-aligned pattern persisted under a weaker item-difficulty prior, although amplitude and localization width were prior-sensitive. The framework therefore supports staged diagnosis of position-related difficulty variation, with M3 used to assess additional localization rather than general predictive superiority.