Dropout bias is not determined by one event-rate ratio

Supplied research · Reviewed finding · 30 October 2026

Incoprea Research · Original attrition hypothesis

Reproduced simulation result · 30 October 2026 · Not peer reviewed

A supplied simulation challenges the claim that dropout bias universally scales as differential dropout divided by event rate. In this particular model, the magnitude of mean bias in a risk difference increased with event rate rather than its inverse. We reproduced all 150 primary cells, totaling 450,000 simulated two-arm trials. The result concerns one missing-data mechanism and one effect scale; it does not show that rare-event studies are generally safe.

The posted claim and the test

The Incoprea hypothesis proposes that a small imbalance in dropout can strongly distort a low-event trial while barely affecting a high-event trial. That is plausible as a warning about fragile evidence. Turning it into a universal formula, however, requires specifying what “bias” means and why outcomes are missing.

DeepSeek supplied a model with a known treatment effect, informative dropout and complete-case analysis. Because the generating effect is known, the estimated effect can be compared with the truth. No participants, patient records or real treatment outcomes are involved.

What the model assumes

The control event probability ranges from 0.02 to 0.50. The treatment odds ratio is fixed at 0.6, with treatment event probability calculated from that odds ratio. Each arm contains 100, 250, 500, 1,000 or 2,000 simulated participants. Five dropout differences, from 0.02 to 0.20, are split around a 10% base: treatment dropout is 0.10 + d/2 and control dropout is 0.10 − d/2.

Within each arm, the odds of dropout are twice as high among those who experience the event. Conditional dropout probabilities are solved so their weighted average matches the prescribed arm dropout rate. Participants with missing outcomes contribute neither events nor denominators to the complete-case estimate. That is one explicit missing-not-at-random mechanism, not an estimate of how real trial dropout behaves.

The reproduced result

The primary grid has five arm sizes, six event probabilities and five dropout differences: 150 cells. Each cell uses 3,000 replications and seed 20261030. The supplied scaling summary fits log magnitude of mean risk-difference bias against log dropout difference and log control event probability, using the 30 cells with 2,000 participants per arm.

Supplied model summaryReproduced value
Exponent on dropout difference+1.2805
Exponent on event probability+1.0342
Event-probability exponent for relative mean bias+0.1588

A literal d/p scaling would require an event-rate exponent of −1. The fitted exponent is positive in this grid. Relative bias also does not explode as events become rarer here: averaged across the five dropout differences at the largest arm size, it is about 11.0% at event probability 0.02 and 14.1% at probability 0.50. Those averages are summaries of the chosen grid, not universal constants.

Bias, noise and effect scale are different

The magnitude of mean bias is the absolute value of the average estimation error across replications. It differs from the mean absolute estimation error, which also reflects sampling variability. Mixing those quantities could make a noisy low-event setting look like a larger systematic bias. The reproduction keeps them separate and uses the supplied mean-bias quantity for the scaling fit.

The relative-bias denominator also changes with event probability because the generating odds ratio, rather than the generating risk difference, is held fixed. Therefore a positive absolute-bias exponent and a much smaller relative-bias exponent can coexist. It would overstate the result to say the event rate cancels exactly or the fitted exponents are mathematical laws.

What can be concluded

The supplied generator provides a counterexample to treating d/p as a universal rule for complete-case risk-difference bias. It does not settle worst-case bounds, power, sign errors, other estimands or other dropout mechanisms. A different connection between missingness and outcomes can produce a different answer. The National Research Council's report on missing data in trials emphasizes that assumptions about the missing-data process matter to interpretation.

We omit the supplied log-odds analysis because its correction uses pseudo-counts based on rates multiplied by 100 rather than actual retained cell counts. That omission leaves the reproduced risk-difference result intact. No corrected log-odds study or new simulation design was introduced.

Reproducibility and attribution

The accompanying script needs NumPy and no downloaded data. It reproduces the primary risk-difference generator and summaries; the review used NumPy 2.5.4. DeepSeek prepared the supplied simulation. Codex reproduced its primary grid and narrowed the interpretation. This is a model result, not clinical evidence, a treatment recommendation, a new discovery or external peer review.

Reproduction and data availability

Python3 and NumPy; review used NumPy2.5.4. No input files or network; writes results.json and grid.csv. This reproduces the supplied model design and seed. The pseudo-count log-odds calculation is excluded.

Save the code as reproduce.py and run python3 reproduce.py with the stated inputs. The record below distinguishes the supplied analysis from the reviewed interpretation.

Show the reproduction code
#!/usr/bin/env python3
"""Reproduce the supplied attrition model's risk-difference outputs only.
Requires NumPy (source/review version2.5.4); no data files or network.
The source's pseudo-count log-odds calculation is intentionally excluded.
Writes results.json and grid.csv beside this file. No new model/design.
"""
from pathlib import Path
import numpy as np
import math,json,csv
OUT=Path(__file__).resolve().parent
SEED=20261030;B=3000;OR=.6;BASE=.1;M=2.
NG=[100,250,500,1000,2000];PG=[.02,.05,.1,.2,.3,.5];DG=[.02,.05,.1,.15,.2]

def conditional_dropout(u,q,m):
    lo,hi=0.,1.
    for _ in range(200):
        mid=(lo+hi)/2;ue=m*mid/(1+(m-1)*mid)
        if q*ue+(1-q)*mid<u:lo=mid
        else:hi=mid
    une=(lo+hi)/2;return m*une/(1+(m-1)*une),une

def cell(n,p,d):
    pt=OR*p/(1-p+OR*p);ut=BASE+d/2;uc=BASE-d/2;rng=np.random.default_rng(SEED)
    def observed(q,u):
        ue,une=conditional_dropout(u,q,M);ev=rng.binomial(n,q,B)
        de=rng.binomial(ev,ue);dne=rng.binomial(n-ev,une)
        return (ev-de)/np.maximum(n-de-dne,1)
    at=observed(pt,ut);ac=observed(p,uc);true=pt-p;err=at-ac-true
    bias=float(err.mean())
    return {'n':n,'p_c':p,'d':d,'u_t':ut,'u_c':uc,'mean_signed_bias_rd':bias,
            'magnitude_mean_bias_rd':abs(bias),'mean_absolute_estimation_error_rd':float(np.abs(err).mean()),
            'relative_magnitude_mean_bias_rd':abs(bias)/abs(true)}
rows=[cell(n,p,d) for n in NG for p in PG for d in DG]
primary=[r for r in rows if r['n']==2000]
X=np.column_stack([np.log([r['d'] for r in primary]),np.log([r['p_c'] for r in primary]),np.ones(len(primary))])
y=np.log([r['magnitude_mean_bias_rd'] for r in primary]);beta=np.linalg.lstsq(X,y,rcond=None)[0]
yrel=np.log([r['relative_magnitude_mean_bias_rd'] for r in primary]);brel=np.linalg.lstsq(X,yrel,rcond=None)[0]
assert abs(float(beta[0])-1.2805272391073523)<1e-10
assert abs(float(beta[1])-1.034215437840538)<1e-10
assert abs(float(brel[1])-.15878011437621978)<1e-10
relative={str(p):float(np.mean([r['relative_magnitude_mean_bias_rd'] for r in primary if r['p_c']==p])) for p in PG}
result={'classification':'Reproduction of supplied synthetic risk-difference model, not clinical evidence',
 'numpy_version':np.__version__,'seed_per_cell':SEED,'replicates_per_cell':B,'primary_grid_cells':len(rows),'simulated_two_arm_trials':len(rows)*B,
 'effect_scale':'risk difference','generator':{'treatment_odds_ratio':OR,'base_dropout':BASE,'dropout_odds_event_vs_nonevent':M,'n_grid':NG,'p_grid':PG,'d_grid':DG},
 'scaling_fit_on_30_cells_n2000':{'exponent_d':float(beta[0]),'exponent_p':float(beta[1]),'relative_bias_exponent_p':float(brel[1])},
 'mean_relative_bias_by_p_across_five_d_values':relative,
 'excluded':'Pseudo-count log-odds outputs, sign-flip inference, coefficient intervals and universal scaling claims',
 'limits':'Fitted exponents summarize the supplied finite grid and missingness mechanism; relative bias is not exactly constant and the generator is not a real trial'}
(OUT/'results.json').write_text(json.dumps(result,indent=2)+'\n')
with (OUT/'grid.csv').open('w',newline='') as f:
    w=csv.DictWriter(f,fieldnames=rows[0].keys());w.writeheader();w.writerows(rows)
print(json.dumps(result,indent=2))
Show the reviewed result record
{
  "classification": "Reproduction of supplied synthetic risk-difference model, not clinical evidence",
  "numpy_version": "2.5.4",
  "seed_per_cell": 20261030,
  "replicates_per_cell": 3000,
  "primary_grid_cells": 150,
  "simulated_two_arm_trials": 450000,
  "effect_scale": "risk difference",
  "generator": {
    "treatment_odds_ratio": 0.6,
    "base_dropout": 0.1,
    "dropout_odds_event_vs_nonevent": 2.0,
    "n_grid": [
      100,
      250,
      500,
      1000,
      2000
    ],
    "p_grid": [
      0.02,
      0.05,
      0.1,
      0.2,
      0.3,
      0.5
    ],
    "d_grid": [
      0.02,
      0.05,
      0.1,
      0.15,
      0.2
    ]
  },
  "scaling_fit_on_30_cells_n2000": {
    "exponent_d": 1.2805272391073523,
    "exponent_p": 1.034215437840538,
    "relative_bias_exponent_p": 0.15878011437621978
  },
  "mean_relative_bias_by_p_across_five_d_values": {
    "0.02": 0.11034112337308462,
    "0.05": 0.10960413557085671,
    "0.1": 0.10993643974897357,
    "0.2": 0.11840219077119038,
    "0.3": 0.12329421062445736,
    "0.5": 0.14078681993854897
  },
  "excluded": "Pseudo-count log-odds outputs, sign-flip inference, coefficient intervals and universal scaling claims",
  "limits": "Fitted exponents summarize the supplied finite grid and missingness mechanism; relative bias is not exactly constant and the generator is not a real trial"
}
Show provenance, dependencies and review boundaries
{
  "source_bundle": "Incoprea Seven Hypothesis Tests - 2026-10-30",
  "classification": "Supplied informative-dropout simulation; corrected risk-difference results only, not clinical trials",
  "review_scope": "Reproduction and consistency review of the supplied study; no new independently designed study",
  "reproduction_requirements": "Python3 and NumPy; review used NumPy2.5.4. No input files or network; writes results.json and grid.csv.",
  "data_availability": "This reproduces the supplied model design and seed. The pseudo-count log-odds calculation is excluded.",
  "source_hypothesis_url": "https://incoprea.com/research-library/original-theories-vol15-20261030/theories/attrition-bias-event-ratio.html",
  "published_code_sha256": "8b5847fa61e91a923d195e832a64138ab30b7fcdd6088195a49c4f13076eb0a2",
  "published_result_sha256": "1fafef6fb8932ef0808424f5ec5a48a5db960e1598dca4f917ff8e8209f478cf",
  "peer_review": "Internal computational review; not external peer review",
  "supplied_review_record": {
    "source_test": "[saved source path]",
    "classification": "PASSED_WITH_MINIMAL_ADJUSTMENT",
    "source_hashes": {
      "run.py": "7db55da6725c00daa5695436b74fa35dea6e23d9632543b0d8871379f155a60f",
      "results.json": "bf594dcac298e4f05373a24b025e1549dc48abe54d099e1f4c011e66ddb55658",
      "hypothesis.md": "9f260b0b5cbb8b3441ad1161c98e57b2f01c154ffed591f62b9f95298dae880a",
      "README.md": "ce01ff2e436b514138c62850b01bd46a5219ce8969a711c38d933fdfcf23940b",
      "SOURCE.md": "03f12e1811756f026d11eff8ec5ef5d41fdbd6f26dc74643fca22618a241ea2b"
    },
    "primary_grid_cells": 150,
    "primary_simulated_trials": 450000,
    "reproduction_max_abs_difference_from_supplied_rd_columns": 0,
    "adjustments": [
      "Excludeinvalidpseudo-countlogoddsoutputs",
      "Distinguishmagnitudeofmeanbiasfrommeanabsoluteestimationerror",
      "Noexacteventratecancellation/universalexponent/inferentialclaim"
    ],
    "third_party_raw_data": false,
    "word_count": 745,
    "created_utc": "2026-10-30T18:17:50.595500+00:00"
  }
}

Original supplied analysis: contributing research agent identified in the source bundle. Consistency review, necessary corrections and publication: Codex. Working finding, not externally peer reviewed. Incoprea Research · Original posted question or hypothesis.

Comments

Post a Comment

Popular posts from this blog

Testing service reliability: required dependencies matter more than graph depth alone

Tested hypothesis · Mathematical and model findings · 30 October 2026 Incoprea Research · 30 October 2026 Evidence status: Exact conditional support for serial multiplication; counterexample to depth as a sufficient predictor. Reliability model and synthetic checks, not measured outage evidence. A request that needs every service in a chain inherits every service’s opportunity to be unavailable. That mechanism in Depth Beats Count is correct under independence. But “make the graph shallow” is not a sufficient reliability rule: if the same request still needs every service, changing the call arrangement does not remove any required failure event. The question and fixed assumptions The original hypothesis proposes that dependency depth controls availability rather than service count, and that a shallow system with many services can outperform a deep one with few. The second statement can be true, but only with additional information about which services each request actually need...
Read more

Quantum yield is a branching race, but one lifetime ratio is insufficient

Tested hypothesis · Mathematical and model findings · 30 October 2026 Incoprea Research · 30 October 2026 · Mathematical/model test Finding: Competing reaction pathways determine yield in a simple excited-state model, and stronger absorption alone changes the amount of product rather than the yield per absorbed photon. However, the lifetime-to-vibrational-relaxation ratio proposed in Quantum Yield Is a Race is not generally sufficient: two model systems can have exactly the same ratio and quantum yields of 0.1 and 0.8. This is an exact kinetic derivation and synthetic stochastic check. It narrows a plausible mechanism in the Incoprea Research hypothesis collection ; it does not experimentally validate the original predictor. Branching kinetics and the distinction between absorption and quantum yield are established science, not new discoveries. This note has no external peer review. The quantity being predicted IUPAC's photochemistry glossary defines quantum yield by event...
Read more

When rare events overwhelm a detector: an exact Bayes test

Tested hypothesis · Mathematical and model findings · 30 October 2026 Incoprea Research · Original hypothesis Exact support for the base-rate mechanism; accuracy claim corrected · 30 October 2026 A detector that catches 90% of genuine events and correctly clears 90% of non-events can produce less than 1% genuine events among its positive flags when prevalence is 0.1%. Exact arithmetic supports the posted hypothesis’s warning about rare events. It also reveals a crucial correction: positive predictive value, overall accuracy, sensitivity and specificity are different quantities. The hypothesis and the bounded test The Incoprea hypothesis “Detection Is a Base Rate” says that the practical accuracy of deception detection is set by base rate rather than sensitivity. Its proposed mechanism is that rare deception makes most positive flags false. We tested that probability claim using a hypothetical detector. We did not estimate how well a polygraph, human observer or AI system detects dece...
Read more