A Quarter-Power Law for Translating Human Development intoEquivalent Mouse and Rat Ages

Translating developmental time between laboratory rodents and humans is a central problem in preclinical research, yet no consensus mathematical framework exists. Linear models are known to be inadequate, and the logarithmic transformations conventionally used in the field do not fully linearize the relationship. Here we evaluated seven candidate functional forms relating human developmental age to rodent age, using a curated dataset of homologous developmental milestones in mouse, rat and human. Models were fitted by weighted nonlinear least squares and compared by leave-one-out cross-validation under a single model-independent weighting scheme derived from a LOESS variance estimate. Model specification was assessed using heteroscedasticity-robust RESET tests, which revealed significant residual curvature for the raw, logarithmic and log--log transformations, whereas the quarter-power transformation (X = Human1/4) was the only one leaving no detectable curvature in either species (mouse F = 0.90, p = 0.407; rat F = 1.83, p = 0.163). A quarter-power model with an additive constant (y = bx1/4 + c) achieved the lowest cross-validated error in mice and performed nearly as well as the best-performing model in rats, a free-power model with an additive constant, while using one fewer parameter. A third-power formulation, evaluated in light of the ongoing debate over metabolic scaling exponents, never ranked first in either species. Joint models incorporating a species dummy variable further supported a common quarter-power structure across both species, with species-specific coefficients. The additive constant was required in every formulation, indicating a systematic offset between developmental timescales that a purely scale-invariant power law does not capture. The resulting model provides a parsimonious, empirically supported framework for estimating equivalent developmental ages across humans, mice and rats.