用于有界特质测量与增长的锚定逻辑斯蒂族：先确定原点再确定单位
An Anchored Logistic Family for Bounded Trait Measurement and Growth: Origin Before Unit
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中文总结 AI 辅助
该研究提出锚定逻辑斯蒂族，将认知特质模型推广至任意精通锚点，采用贝叶斯模态EM估计参数，速度远超MCMC采样器，可用于有界特质测量与增长建模，其贡献在解释与计算而非测量本身。
中文摘要 AI 辅助
潜在特质模型在测量内容上的差异小于其设定的约束差异。项目反应理论将特质从所施测的项目中解放出来，但在此过程中，将量表的原点与单位让渡给了惯例。认知特质模型（CTM；Choi，2022）通过将特质限定在[0,1]区间内，同时恢复了原点与单位：其中0代表无知水平，1代表任务领域定义的精通水平。我们将CTM背后的推导推广为由精通锚点L索引的族：已发表的模型对应L=1的情况，而当L→∞时则得到半截断链接，其具有绝对原点但无上限。先确定原点再确定单位。有界支撑允许使用高斯-勒让德求积法而无截断误差，因此项目参数与被试参数可通过贝叶斯模态期望最大化（Bayes modal EM）而非马尔可夫链蒙特卡洛（MCMC）得到。在相同数据与先验下，该方法重现了已发表的估计值（θ相关系数为0.9999），同时比收敛的随机游走采样器快35倍，比NUTS快73倍；单次自适应测试更新耗时约5微秒。将该链接应用于时间轴时，它成为一个四参数增长曲线，涵盖加速、减速、平台期和下降轨迹，但不包含非单调轨迹。仅当所有项目具有相同斜率时，该族才退化为逻辑斯蒂模型的重新参数化，因此原则上可能具有测量优势；三次模拟未发现此类优势，其贡献在于解释与计算而非测量本身。在LSAT再分析中，满分考生的精通程度在67%、75%或100%之间，具体取决于估计量——参照标准的主张仅在附带其不确定性时才有意义，且该坐标通常并非个体能在领域中表现的比例。
英文摘要
Latent trait models differ less in what they measure than in what they fix. Item response theory frees the trait from the items administered but, in doing so, surrenders the origin and unit of the scale to convention. The Cognitive Trait Model (CTM; Choi, 2022) restores both by bounding the trait on [0, 1], where 0 denotes an ignorance level and 1 a mastery level defined by the task domain. We generalize the derivation behind CTM into a family indexed by the mastery anchor $L$: the published model is the case $L = 1$, while the limit $L \to \infty$ gives a half-truncated link with an absolute origin but no ceiling. Origin before unit. Bounded support admits Gauss-Legendre quadrature without truncation error, so item and person parameters follow from Bayes modal EM rather than MCMC. On identical data and priors this reproduces the published estimates ($θ$ correlation 0.9999) while running 35 times faster than a converged random-walk sampler and 73 times faster than NUTS; a single adaptive-testing update costs about five microseconds. Applied to the time axis, the same link becomes a four-parameter growth curve spanning accelerating, decelerating, plateauing and declining trajectories, though not non-monotone ones. The family reduces to a reparameterization of the logistic model only when all items share a slope, so a measurement advantage is possible in principle; three simulations find none, locating the contribution in interpretation and computation rather than in measurement itself. In the LSAT reanalysis, examinees with perfect scores read 67%, 75% or 100% of mastery depending on the estimator -- the criterion-referenced claim is meaningful only with its uncertainty attached, and the coordinate is not in general the proportion of the domain a person can perform.
