Shannon Entropy Trajectories Reveal Between-Arm Distributional Structure Invisible to Standard Endpoint Analysis in Pooled ALS Clinical Trials

Standard analysis of amyotrophic lateral sclerosis (ALS) clinical trials evaluates therapeutic efficacy by comparing linear slopes of total ALS Functional Rating Scale (ALSFRS) scores between treatment arms. This approach compresses multidomain ordinal data into a single scalar trajectory, discarding distributional structure. When subgroup-level trends differ in timing or direction, such aggregation can attenuate or eliminate them, a phenomenon known as Simpsons paradox. Here we apply Shannon entropy, computed from item-level score distributions within each ALSFRS functional domain following the framework established in [8], to the PRO-ACT database, stratified by treatment arm (Active: n = 4,581; Placebo: n = 2,931; 19 monthly time points). The entropy trajectories of drug-treated and placebo populations diverge visibly and systematically across all four functional domains (Bulbar, Fine Motor, Gross Motor, Respiratory). In the Fine Motor domain, the placebo population reaches peak entropy at month 8 and reverses, while the active population does not peak until month 13, a five-month delay in the populations transit toward functional loss. This divergence is model-independent: it is present in the raw Shannon entropy trajectories before any dynamical model is applied. A permutation test shuffling patient-level arm labels (n = 1,000 permutations) confirms that the total integrated absolute divergence across all four domains exceeds the null distribution at p < 0.001 (observed: 4.48; null: 2.03 {+/-} 0.33; 7.5 standard deviations above the null mean), with Fine Motor (p = 0.001) and Respiratory (p < 0.001) individually significant. The quantity that differs between arms, the shape and timing of the populations distributional evolution, does not exist as a measurable quantity in the total-score linear-slope framework used to evaluate these trials. Whether this signal reflects genuine treatment effects, compositional artifacts from pooling heterogeneous trials, or both cannot be determined from the anonymized public database alone. What can be determined is that the standard ALS clinical trial endpoint makes an implicit assumption, that the distributional information it discards is uninformative, and the present results demonstrate empirically that this assumption is false.