📊 statistics

A critical evaluation of longitudinal proportional effect models

This paper critically evaluates nonlinear longitudinal proportional effect models, demonstrating that their assumption of a fixed treatment effect over time leads to bias, inflated Type I error rates, and safety concerns regarding the detection of treatment harm, even when the assumption holds.

Original authors: Michael C. Donohue, Philip S. Insel, Oliver Langford

Published 2026-01-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael C. Donohue, Philip S. Insel, Oliver Langford

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A "Magic" Formula That Isn't Magic

Imagine you are running a race to see if a new running shoe (the Active Treatment) helps people run faster than their old shoes (the Control Group).

Scientists have been excited about a new mathematical tool called the "Longitudinal Proportional Effect Model." The pitch is that this tool is a "super-charger." It claims to be more powerful than standard methods and gives a direct answer to a very specific question: "By what percentage did the new shoes improve the runner's speed?"

However, this paper by Donohue, Insel, and Langford is a critical warning label. They argue that this "super-charger" is actually broken. It doesn't just give better answers; it gives biased answers that trick you into thinking a treatment works better than it actually does, while hiding the fact that it might be harmful.

The Core Problem: The "Zero Point" Trap

To understand the bias, imagine a thermometer.

Standard Method (The Ruler): Measures the difference in degrees. If the room goes from 70° to 75°, the change is +5°. If it goes from 0° to 5°, the change is still +5°. It doesn't matter where you start.
The "Proportional" Method (The Percentage): Measures the change relative to the starting point.
- If you start at 70° and go to 75°, that's a small percentage gain.
- If you start at 0° and go to 5°, the math breaks. You can't calculate a percentage of zero.
- If you start at -10° (below zero) and go to -5°, the math gets weird and unstable.

The paper argues that in medical trials (especially for diseases like Alzheimer's), the "starting point" (the control group's average score) often hovers near zero or even goes negative (representing a decline in health). When the math tries to calculate a "percentage improvement" from a number near zero, it becomes unstable and starts hallucinating results.

The "Favoritism" Bias: A One-Way Street

The authors discovered that this model has a built-in favoritism.

Imagine a judge who is secretly rooting for the new shoe brand.

If the new shoes help: The model exaggerates how much they helped. It makes the improvement look huge.
If the new shoes hurt: The model struggles to see the damage. It acts like a blind spot, making it very hard to detect if the treatment is actually making things worse.

The paper shows that this bias happens even when the rules of the model are followed perfectly. It's not a mistake in how the scientists used the tool; the tool itself is flawed.

The "Group Labeling" Trick

The paper points out a strange quirk: The model's answer changes depending on which group you call "Group A" and which you call "Group B."

If you label the New Treatment as the main group, the model biases the result to make the treatment look good.
If you swapped the labels and made the Control Group the main group, the model would bias the result to make the control look good.

This is like a scale that tips to the left if you put the object on the left side, and tips to the right if you put it on the right side. A good scale should give the same weight regardless of where you put the object. Because this model changes its mind based on labeling, its results are unreliable.

The Simulation Results: The "93% Error"

The authors ran computer simulations (virtual clinical trials) to test this. They found terrifying results:

When the control group's average score was close to zero, the model started rejecting the idea that "nothing happened" 93% of the time, even when there was no treatment effect at all.
In a standard test, you expect to be wrong about 5% of the time (this is called Type I error). This model was wrong nearly 20 times more often than it should be.
In many cases, the model acted like a one-sided test: It would only shout "It works!" but rarely shout "It hurts!"

The Safety Concern: Hiding Harm

The most dangerous part of this bias is safety.
In diseases like Alzheimer's, there is a real fear that a drug could make memory worse (treatment harm).

Because this model is biased toward seeing improvement, it is very bad at seeing decline.
The authors warn that using this model could lead to approving a drug that actually harms patients, simply because the math was too busy trying to find a "percentage improvement" to notice the damage.

The Conclusion: Don't Use It

The authors conclude that despite the promise of higher statistical power (the ability to find a signal in the noise), this model is not worth the risk.

The "power" it claims to have is actually just bias masquerading as success.
Standard, linear models (the "rulers") are safer, more honest, and don't change their answers based on how you label the groups.
They recommend that scientists avoid using these proportional effect models, especially for safety reasons.

In short: The paper says this new mathematical tool is like a broken speedometer that always reads "fast" when you are driving a new car, but fails to warn you when you are driving off a cliff. It's better to use an old, reliable speedometer than a fancy one that lies to you.

Technical Summary: A Critical Evaluation of Longitudinal Proportional Effect Models

Problem Statement
Nonlinear longitudinal proportional effect models have recently been proposed (e.g., by Wang et al. [2024]) to estimate proportional treatment effects in randomized clinical trials, with claims of increased statistical power and direct clinical interpretability. These models assume that the mean trend of the active intervention group is a fixed proportion of the control group mean over time, parameterized as $f_A(t) = f_C(t)(1 - \theta)$ .

However, the authors identify critical flaws in this approach. The primary issue is that these models are inherently biased, even when the proportional effect assumption holds true. This bias is sensitive to the labeling of treatment groups; the specific parameterization suggested in existing literature induces a bias favoring the active treatment group. Consequently, this leads to inflated Type I error rates, potential one-sided testing behavior, and a reduced ability to detect treatment harm (safety signals). Furthermore, the models become undefined or unstable when the control group mean approaches zero, a scenario plausible in presymptomatic disease stages or specific cognitive outcomes.

Methodology
The paper employs a combination of theoretical analysis and simulation studies to evaluate the performance of proportional effect models against standard linear models.

Cross-Sectional Analysis: The authors first examine a special case of the proportional model at a single time point. They compare the nonlinear least squares estimator for the proportional effect ( $\hat{\theta}$ ) against the familiar two-sample t-test estimator for the additive difference ( $\hat{\delta}$ ). Simulations are conducted across varying control group means ( $\beta_C$ ) and treatment effects ( $\delta$ ).
Longitudinal Simulation: The authors simulate 10,000 clinical trials under four distinct scenarios (A–D) representing different mean trends over time:
- Scenarios A–C: Linear trends where the control group mean is non-zero, satisfying the proportional effect assumption.
- Scenario D: A linear trend where the control group mean is fixed at zero, violating the finite definition of the proportional effect.
- Models Compared: For each dataset, two models are fitted:
  - A linear Mixed Model Repeated Measures (MMRM) with an unstructured mean and additive treatment effect ("slope").
  - A nonlinear Proportional MMRM (pMMRM) with an unstructured control mean and a fixed proportional treatment effect.
- Both models include participant-specific random intercepts and constrain baseline group means to be equivalent (constrained Longitudinal Data Analysis, or cLDA).
Evaluation Metrics: The study assesses statistical power, Type I error rates, and the proportion of rejections favoring the active treatment under the null hypothesis. "Zipper plots" are used to visualize the distribution of estimates and confidence intervals, specifically looking for asymmetry and bias.

Key Contributions and Results
The simulations reveal that the purported efficiency gains of proportional effect models are largely artifacts of bias rather than true statistical power.

Bias and Sensitivity to Labeling: The proportional estimator is sensitive to the location of the control group mean. When $\beta_C$ is near zero, the model exhibits significant bias. The specific parameterization used in the literature favors the active group, leading to asymmetric confidence intervals and a high proportion of rejections in favor of the active treatment even when no treatment effect exists.
Type I Error Inflation: In longitudinal simulations, as the control group mean approaches zero (moving from Scenario A to D), the Type I error for the proportional model increases dramatically. In Scenario A (control mean near zero), the Type I error reached 5.3% with 93.2% of rejections favoring the active group. In Scenario D (control mean = 0), the Type I error soared to 34.0%, with 100% of rejections favoring the active group.
Power vs. Safety: While the proportional model appeared to show higher power (e.g., 93.8% vs. 88.2% in Scenario A) when the treatment effect was positive, this "improvement" was driven by the bias. Conversely, the model demonstrated diminished power to detect treatment harm ( $\delta < 0$ ). In cross-sectional simulations with $\delta < 0$ , the standard t-test generally dominated the proportional test.
Comparison to Linear Models: The linear "slope" model demonstrated consistent operating characteristics across all scenarios, maintaining the nominal 5% Type I error and symmetric rejection rates under the null hypothesis. The proportional model's performance degraded as the control mean approached zero, whereas the linear model remained robust.

Significance and Claims
The paper concludes that the longitudinal proportional effect models promoted in recent tutorials are fundamentally flawed for clinical trial analysis. The authors assert that:

The models are biased: The bias exists even when model assumptions are correct and is exacerbated when the control group mean is near zero.
Safety concerns: The bias makes it difficult to detect treatment-emergent worsening (harm), posing a significant safety risk. This is particularly relevant for therapies where cognitive worsening is a known risk (e.g., BACE inhibitors).
Recommendation: Biostatisticians and clinical trialists should avoid using these models. The apparent power advantages are erroneously inflated by bias.
Alternatives: The authors suggest that uncertainty intervals for proportional effects can be derived from flexible linear models using the delta method, bootstrap, or Markov chain Monte Carlo, without incurring the bias inherent in the nonlinear parameterization.

The paper does not propose a new model but serves as a critical warning against the adoption of the specific nonlinear proportional effect parameterization currently being discussed in the literature, citing its potential to mislead trial outcomes and compromise patient safety.