EXP-001: The Cost of Linearity

How much statistical power do ALS clinical trials lose by assuming linear ALSFRS-R decline when the true progression is nonlinear with latent subgroups?

The ALSFRS-R (ALS Functional Rating Scale — Revised) is the primary outcome measure in virtually every ALS trial. It's a 48-point scale tracking physical function across 12 domains. The standard analysis assumes each patient's score declines at a roughly constant rate over time. But a growing body of literature — Gomeni et al. (2014), van Eijk et al. (2025), Gordon et al. (2010) — shows this assumption is wrong.

Data-Generating Process. We simulated patient trajectories using three latent classes derived from the literature:

Slow progressors follow a decelerating curve (slope = 0.3 pts/month, quadratic term = 0.01). Fast progressors decline rapidly with slight acceleration (slope = 1.2 pts/month). The stable-then-crash group holds near baseline for ~9 months then drops precipitously (2.0 pts/month post-crash). These parameters draw from Gomeni et al.'s 2-cluster PRO-ACT analysis, van Eijk et al.'s nonlinear decline findings (N=7,030), and Gordon et al.'s demonstration that quadratic fits outperform linear.

Each simulated patient also has random intercept (SD = 3.0) and random slope (SD = 0.15) effects, plus residual noise (SD = 2.0). We modeled informative dropout — patients with lower scores are more likely to drop out — using a logistic function, reflecting real-world trial attrition in ALS.

Fig. 1 — The three latent trajectory classes used in the data-generating process. Each thin line is a simulated patient; bold line is the class mean.

Treatment Scenarios. We tested four scenarios:

• Null: No treatment effect (validates Type I error control)
• Uniform 25% slowing: Drug slows decline by 25% in all classes
• Class-specific 50% slowing: Drug halves decline rate in slow progressors only
• Trajectory modification: Drug delays the crash in stable-then-crash patients from 9 to 15 months

Sample sizes: 100, 200, 400, and 800 patients per arm. Each configuration was simulated 500 times.

Fig. 2 — Power curves across all four scenarios. The oracle method (green) reaches 80% power at dramatically smaller sample sizes.

Click each scenario below to see the detailed power tables.

Fig. 3 — Mean estimated treatment effect by method and scenario. Note: ANCOVA coefficient (change score) and LMM coefficient (per-month slope) are on different scales and should not be directly compared as a ratio. ANCOVA estimates appear larger due to this scale difference; LMM and Oracle track the true effect on the per-month slope scale.

Fig. 4 — Type I error rates under the null scenario. All methods stay within the acceptable range around α = 0.05.

Finding 1: Type I error is well-controlled. Under the null scenario, all three methods produce false positive rates near 5%, confirming the simulation is valid and the methods are calibrated.

Finding 2: The oracle dominates everywhere. Class-aware analysis reaches >98% power at N=100/arm across all treatment scenarios. It's not just better — it's in a different league.

Finding 3: ANCOVA consistently underperforms LMM. By collapsing to a single timepoint comparison, ANCOVA loses information. In the uniform scenario at N=100, LMM has 71% power while ANCOVA has just 37%.

Finding 4: The penalty is worst for realistic scenarios. The class-specific and trajectory-modification scenarios — arguably the most biologically plausible — show the largest gaps between standard and oracle methods.

ALS clinical trials are expensive, slow, and heartbreaking when they fail. Over 50 drugs have shown promise in preclinical models and failed in Phase II/III trials. The standard explanation is that the drugs didn't work. But this simulation suggests another possibility: some of those drugs might have worked, in some patients, and we couldn't see it.

If even a fraction of failed ALS trials suffered from the power loss demonstrated here, the implications are significant. Drugs that were abandoned might deserve re-examination. Future trials could be designed with stratification approaches that account for trajectory heterogeneity.

The oracle method is hypothetical — you can't perfectly classify patients in a real trial. But you don't need perfection. Growth mixture models, machine learning classifiers, and even simple baseline slope estimates could recover a substantial portion of this lost power. The gap between "perfect classification" and "no classification" is so enormous that even imperfect classification should help.

This isn't about one statistical method being better than another. It's about recognizing that ALS patients are not all the same, and designing trials that account for that heterogeneity rather than averaging over it.

The full simulation code is open and will be published alongside our PRO-ACT analysis. The study is fully reproducible: 500 simulations per cell, fixed random seed (42), Python with NumPy, statsmodels, and SciPy.

PRO-ACT validation. This simulation used literature-derived parameters. The next step is to fit these trajectory classes to real patient data from the PRO-ACT database (~10,000 ALS patient records) and verify whether the heterogeneity patterns hold.

Growth mixture modeling. We're developing a practical classification approach that could be deployed in real trials — not an omniscient oracle, but a feasible approximation that recovers meaningful statistical power.

Retrospective re-analysis. If the PRO-ACT validation confirms trajectory heterogeneity, the logical next question becomes: were any of the 50+ failed ALS drugs actually effective in a subgroup?