An Anchored Logistic Family for Bounded Trait Measurement and Growth: Origin Before Unit

Abstract: 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.