Peircean Evidence‑Weighted RMST ‐ Methode Paper

Foundational principles

A Model‑Free, Assumption‑Free, Empirical Survival Method

Author: Inspired by C. S. Peirce and Hernán • Implementation by AI • Assisted by gitfrid

This page outlines the foundational principles of the AE) C.S. Pierce Evidence-Weighted RMST script. The method bridges the gap between clinical epidemiology and information theory, calculating not just "if" a result occurred, but the integrity of the signal itself.

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1. Mathematical Foundation (The Statistics)

The core of the analysis relies on Restricted Mean Survival Time (RMST) and the Hypergeometric Distribution.

• Assumption-Free: RMST is mathematically superior to the Hazard Ratio because it does not require "proportional hazards." It measures the area under the curve without assuming the vaccine's effect is constant over time.
• Information Influence ($I_t$): The script uses Hypergeometric logic to measure the probability of an outcome given a finite population. This is a robust frequentist approach to quantify daily statistical certainty.

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2. Causal Integrity (The Science)

The script follows the Target Trial Emulation framework (as championed by Miguel Hernán), ensuring observational data mimics a randomized controlled trial:

• Time Zero ($T_0$) Alignment: Everyone starts the analysis at the same relative moment (FIRST).
• Eliminating Immortal Time Bias: Vaccinated clones only "enter" the risk set at the actual moment of vaccination.
• Eliminating Healthy User Bias: The +14-day buffer ensures the analysis does not credit the "forced health" of individuals.

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3. The Peircean "Digital Sieve" (The Philosophy)

The integration of Peircean logic (Firstness, Secondness, Thirdness) serves as a mathematical "Stress Test" for the data. While standard $p$-values can be fragile, this script uses Information Weighting.

• Filtering Variance: It prevents a few outcomes at the end of a study (low-density data) from dictating the results for the majority at the start.
• Secondness vs. Thirdness: The script distinguishes between a "Second" (a brute, random accident/noise) and a "Third" (a law or persistent pattern).
• Abductive Flags: The script identifies "nodes of truth"—points where the survival benefit (or harm) is so statistically surprising that it demands a causal explanation.

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4. Application to High-Risk Groups (AG70+)

In older populations, data is often "noisy" due to competing risks. Standard models may see a random spike in deaths and misinterpret it as a change in vaccine efficacy.

The Peircean script asks: "Is this spike an accident, or is it a law?" By using the Weighted ΔRMST, the script prioritizes the "Thirdness" (the actual truth) and allows random noise to fall through the sieve.

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Final assessment

It is a High-Integrity Scientific Tool. By removing the standard avenues for "for distortions" in observational data (immortal time and healthy user bias) and adding an Information-Theoretic filter, the script provides a honest report of survival data.

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The Method

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1. Abstract

This method compares survival outcomes between two dynamic strategies by constructing counterfactual "clone" trajectories. It weights daily survival differences by their Directional Surprisal, effectively distinguishing between brute empirical observations and gaps supported by non-random statistical evidence.

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2. Empirical Survival Curves

For each day $t$, we define the empirical survival probabilities based on the count of individuals alive ($Y$) in the untreated ($u$) and treated ($v$) clone populations:

$$S_{u}(t) = \frac{1}{N} \sum_{i=1}^{N} Y_{i,u}(t)$$

$$S_{v}(t) = \frac{1}{N} \sum_{i=1}^{N} Y_{i,v}(t)$$

The daily survival difference is: $$\Delta S(t) = S_{v}(t) - S_{u}(t)$$

The total empirical RMST difference up to horizon $T$ is the cumulative sum of these differences:

$$\text{RMST}_{\text{emp}} = \sum_{t=0}^{T} \Delta S(t)$$

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3. Directional Peircean Weighting

Evidence is defined as the degree of "surprise" relative to chance. We calculate a daily p-value $p(t)$ via a rolling-window hypergeometric test.

The Directional Surprisal $I(t)$ is defined as: $$I(t) = \text{sgn}(\Delta S(t)) \cdot (-\ln(p(t)))$$

To ensure the weighted RMST remains on a comparable scale to the empirical RMST, we normalize the weight by the maximum observed surprisal: $$w(t) = \frac{|I(t)|}{\max |I(t)|}$$

The Peircean Evidence-Weighted RMST is then calculated as:

$$\text{RMST}_{\text{P}} = \sum_{t=0}^{T} (\Delta S(t) \cdot w(t))$$

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4. Interpretation of Metrics

Metric	Interpretation
$\text{RMST}_{\text{emp}}$	The raw life-days gained/lost as observed in the population (The Brute Fact).
$\text{RMST}_{\text{P}}$	The portion of life-days gained/lost supported by a statistically significant signal (The Purified Signal).
$I(t) > 0$	Periods where evidence statistically favors the treatment strategy.
$I(t) < 0$	Periods where evidence favors the control or suggests noise/bias.

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5. Summary of Advantages

• Assumption-Free: Requires no proportional hazards (PH) assumptions or regression models.
• Immortal-Time Correction: Uses clone-based target trial emulation to handle dynamic treatment timing.
• Transparency: High-surprise days are flagged as "Abductive Surprises," providing a traceable audit trail of which dates drove the statistical conclusion.