Bayesian Modal Estimation of the Four-Parameter Item Response Model in Real, Realistic, and Idealized Data Sets

In this study, we explored item and person parameter recovery of the four-parameter model (4PM) in over 24,000 real, realistic, and idealized data sets. In the first analyses, we fit the 4PM and three alternative models to data from three Minnesota Multiphasic Personality Inventory-Adolescent form factor scales using Bayesian modal estimation (BME). Our results indicated that the 4PM fits these scales better than simpler item Response Theory (IRT) models. Next, using the parameter estimates from these real data analyses, we estimated 4PM item parameters in 6,000 realistic data sets to establish minimum sample size requirements for accurate item and person parameter recovery. Using a factorial design that crossed discrete levels of item parameters, sample size, and test length, we also fit the 4PM to an additional 18,000 idealized data sets to extend our parameter recovery findings. Our combined results demonstrated that 4PM item parameters and parameter functions (e.g., item response functions) can be accurately estimated using BME in moderate to large samples (N ⩾ 5, 000) and person parameters can be accurately estimated in smaller samples (N ⩾ 1, 000). In the supplemental files, we report annotated code that shows how to estimate 4PM item and person parameters in (Chalmers, 2012).

本研究针对24000余个真实、模拟与理想化数据集，探究了四参数模型（four-parameter model, 4PM）的项目与被试参数恢复性能。在首轮分析中，我们采用贝叶斯众数估计（Bayesian modal estimation, BME），将4PM与三种备选模型拟合至明尼苏达多相人格调查表-青少年版（Minnesota Multiphasic Personality Inventory-Adolescent）的3个因素分量表数据。研究结果显示，相较于更为简洁的项目反应理论（Item Response Theory, IRT）模型，4PM对上述分量表的拟合效果更优。随后，我们基于上述真实数据分析得到的参数估计结果，在6000个模拟数据集内估算4PM的项目参数，以确立可实现准确项目与被试参数恢复的最小样本量要求。我们还采用了涵盖项目参数、样本量与测验长度离散水平的析因设计，将4PM拟合至额外18000个理想化数据集，以拓展参数恢复相关的研究结论。综合所有研究结果可知，在中等至大样本量（N≥5000）条件下，采用BME可准确估算4PM的项目参数与参数函数（如项目反应函数（item response functions)）；而在较小样本量（N≥1000）条件下，即可准确估算被试参数。补充材料中提供了带注释的代码，展示了如何在（Chalmers, 2012）中实现4PM的项目与被试参数估算。