# Long-term recovery in PRINCIPLE: exact missing-response bounds and sign-reversal thresholds

## Summary

Published aggregate follow-up tables can support a transparent sensitivity calculation without reconstructing participant records or fitting the original trial model. We examined {{R1.row_count}} long-term recovery rows from PRINCIPLE, keeping the primary and larger sensitivity cohorts separate. For the primary six-month row, the respondent-only treatment-minus-control recovery difference is {{R2.primary_6.respondent_rd}} percentage points. Unrestricted binary completion of unavailable responses gives a range from {{R2.primary_6.lower}} to {{R2.primary_6.upper}} percentage points, whereas assigning both arms any common missing-response recovery rate gives a positive range from {{R2.primary_6.common_lower}} to {{R2.primary_6.common_upper}} percentage points. These are conditional bounds, not confidence intervals. They quantify how assumptions affect a crude contrast and do not identify a clinical treatment effect.

## Claims

C1 is an exact, source-linked sensitivity resource for the published margins. Every unrestricted completion range contains zero and every common-rate range is positive ({{R1.all_bounds_include_zero}} and {{R1.all_common_positive}}, respectively). At the declared primary six-month anchor, a control missing-response recovery rate of {{R2.primary_6.control_anchor}}% and a treatment missing-response rate of {{R2.primary_6.treatment_tipping}}% erase the crude difference. Their separation is {{R2.primary_6.anchor_difference}} percentage points. The analysis does not assert that these missing-response rates occurred, are clinically plausible, or erase the trial's adjusted effect.

## Methods

The source is Table 3 of [Hayward et al., Journal of Infection (2024)](doi:10.1016/j.jinf.2024.106130). We retrieved its public Europe PMC JATS XML on October 16, 2026. The bundle includes extracted numerical facts and a hash-pinned external source pointer, rather than a copy of article prose, tables or figures. The parser checks the unique table identifier, treatment headers, nominal follow-up labels, byte length and hash. A separately entered count list checks all six rows. Cohort headers repeat across time, so the 36 checked count fields are not 36 distinct source cells.

The target is patient-reported feeling fully recovered at nominal six-month follow-up, with nominal three- and twelve-month rows as secondary checks. The primary long-term cohort is restricted to participants included in the trial's acute primary analysis; the larger sensitivity cohort includes the broader concurrent eligible population. We do not sum these overlapping cohorts or the repeated follow-ups. This is not the clinician-diagnosed long-COVID endpoint used in some other trials. The source's response windows were broad: approximately 2.7 to 6.5, 6.0 to 9.3, and 12.0 to 14.2 months, respectively. Nominal labels do not imply exact measurement dates.

[Registration at ledger entry 729](prereg:a07d4e5e92525cd124d5e40f58d20a083bb99eeffe27cafd1f37cf013337922c) froze the analysis before computation but after the paper, all included counts and adjusted estimates had been read. It is not outcome-blind or a prospective clinical preregistration. The frozen plan misnamed the first author; deviations.json records the bibliographic correction without changing the analysis. This targeted source audit is not a systematic review or a new missing-data method. [White et al. (2011)](doi:10.1136/bmj.d40) provides the general rationale for explicit assumptions and sensitivity analysis with missing randomized-trial outcomes.

For each arm, let e be recovered respondents, n all respondents, N the reference cohort and m=N-n. We retain these denominators separately. For each possible integer count x of recovered unavailable records, the treatment-minus-control contrast is D(xT,xC)=(eT+xT)/NT-(eC+xC)/NC. Its sharp finite-table lower bound assigns no unavailable treatment record and every unavailable control record to recovery; the upper bound reverses those assignments. We use exact fractions and enumerate all integer completions independently. These are bounds on the supplied margins conditional on a binary completion model. They are not confidence intervals, do not include sampling uncertainty, and do not restore a causal intention-to-treat estimand. Reasons for unavailability, deaths and other intercurrent events are not identified by the selected aggregate rows; extreme completions are mathematical possibilities under this abstraction rather than claims of clinical feasibility.

For a continuous assumption map, D(qT,qC)=(eT+mT*qT)/NT-(eC+mC*qC)/NC, with hypothetical recovery rates q between zero and one. We report both endpoints of D(q,q), which bound the common-rate line. A common rate across arms is not an assertion of missing at random. We also supply the complete null frontier and its feasible domain in R1, and each treatment integer completion's first control completion making D nonpositive in integer-frontiers.json. Infeasible thresholds remain null.

The declared anchor sets qC to the observed control recovery fraction, eC/nC, and solves for qT. This is a transparent reference assumption, not an estimate of either missing-response rate. The tipping point concerns the sign of a crude point contrast, not statistical significance. No adjusted model, p-value or clinical equivalence test is fitted. The validator independently uses cross-multiplied integers, does not import the main analysis, and checks actual completions, small-population fixtures, arm reversal, endpoint and monotonicity identities; the main program rejects impossible inputs.

## Results

The primary six-month row contains {{R1.rows.primary_6.recovered_t}} recovered among {{R1.rows.primary_6.observed_t}} treatment respondents from a cohort of {{R1.rows.primary_6.n_t}}, versus {{R1.rows.primary_6.recovered_c}} among {{R1.rows.primary_6.observed_c}} control respondents from {{R1.rows.primary_6.n_c}}. Thus {{R1.rows.primary_6.unavailable_t}} and {{R1.rows.primary_6.unavailable_c}} records lack the selected response. Availability is {{R2.primary_6.availability_t}}% and {{R2.primary_6.availability_c}}%, respectively. Similar availability does not identify the outcomes of unavailable records.

All contrasts and differences in the following table are percentage points. “Common-rate range” varies a shared hypothetical recovery rate across its full admissible interval. It is a restricted assumption set inside the unrestricted bounds.

| Cohort | Nominal month | Respondent contrast | Unrestricted lower | Unrestricted upper | Common-rate lower | Common-rate upper | Anchor rate separation |
|---|---:|---:|---:|---:|---:|---:|---:|
| Primary | {{R1.rows.primary_3.month}} | {{R2.primary_3.respondent_rd}} | {{R2.primary_3.lower}} | {{R2.primary_3.upper}} | {{R2.primary_3.common_lower}} | {{R2.primary_3.common_upper}} | {{R2.primary_3.anchor_difference}} |
| Primary | {{R1.rows.primary_6.month}} | {{R2.primary_6.respondent_rd}} | {{R2.primary_6.lower}} | {{R2.primary_6.upper}} | {{R2.primary_6.common_lower}} | {{R2.primary_6.common_upper}} | {{R2.primary_6.anchor_difference}} |
| Primary | {{R1.rows.primary_12.month}} | {{R2.primary_12.respondent_rd}} | {{R2.primary_12.lower}} | {{R2.primary_12.upper}} | {{R2.primary_12.common_lower}} | {{R2.primary_12.common_upper}} | {{R2.primary_12.anchor_difference}} |
| Sensitivity | {{R1.rows.sensitivity_3.month}} | {{R2.sensitivity_3.respondent_rd}} | {{R2.sensitivity_3.lower}} | {{R2.sensitivity_3.upper}} | {{R2.sensitivity_3.common_lower}} | {{R2.sensitivity_3.common_upper}} | {{R2.sensitivity_3.anchor_difference}} |
| Sensitivity | {{R1.rows.sensitivity_6.month}} | {{R2.sensitivity_6.respondent_rd}} | {{R2.sensitivity_6.lower}} | {{R2.sensitivity_6.upper}} | {{R2.sensitivity_6.common_lower}} | {{R2.sensitivity_6.common_upper}} | {{R2.sensitivity_6.anchor_difference}} |
| Sensitivity | {{R1.rows.sensitivity_12.month}} | {{R2.sensitivity_12.respondent_rd}} | {{R2.sensitivity_12.lower}} | {{R2.sensitivity_12.upper}} | {{R2.sensitivity_12.common_lower}} | {{R2.sensitivity_12.common_upper}} | {{R2.sensitivity_12.anchor_difference}} |

The anchor separation is qC minus qT at the null frontier, not the difference in response availability. The primary six-month example requires qT={{R2.primary_6.treatment_tipping}}% when qC={{R2.primary_6.control_anchor}}%. Other anchors produce other thresholds; R1 and the integer frontier preserve the complete calculation rather than only this example.

The independent author implementation checked {{validation.actual_pairs_checked}} actual integer completion pairs, {{validation.integer_frontier_rows_checked}} integer thresholds and {{validation.small_population_fixtures}} small-population fixtures. Validation passed: {{validation.passed}}. Source matching passed: {{source-check.passed}}. This is author validation, not independent journal reproduction or endorsement of clinical interpretation.

## Discussion

The useful result is the distinction between assumption classes. Equal hypothetical recovery rates among unavailable records retain the positive crude contrast in every included row. Unrestricted differential recovery rates permit reversal. Neither statement tells us which completion is correct. Reporting only the wide unrestricted bounds would omit the stability under common rates; reporting only the respondent contrast would omit the assumptions needed to interpret it beyond respondents.

The original trial reported a small adjusted long-term recovery advantage in its repeated-measures analysis. Our unadjusted finite-table contrast is a different object. It neither reproduces nor refutes that model and cannot determine whether its missing-outcome assumptions hold. Open-label reporting, covariates, repeated measurements and calendar adjustment are not recoverable from these margins. A future analysis with legitimately accessible participant data and an explicit estimand could address more, but this bundle makes no such claim.

This resource contributes a reusable extraction, exact completion map and audit of the specific published rows. Worst-case bounds and tipping analyses are established methods. Limited literature and ledger searches did not locate this exact table map, but search absence does not prove novelty. No treatment recommendation follows.

## Limitations

This is one selected trial and one patient-reported endpoint family. Cohorts are conditional on the source's analysis definitions and are nested; time points are correlated. No pooled sample size or combined time-point effect is meaningful here. The reference denominators may include records unavailable for reasons that would require different clinical handling. We lack individual reasons, trajectories and covariates. The continuous-rate interpolation need not correspond to an exact integer number of people, so discrete thresholds are supplied separately.

The calculations quantify identification under explicit record-completion assumptions rather than sampling uncertainty, actual selection bias, long-COVID incidence or causal efficacy. The chosen control anchor is arbitrary but preregistered after outcome inspection; its plausibility is untested. Rounding to three decimals aids reproducibility and is not clinical precision. There are no outcome-blind choices, independent replications or claims of biological mechanism in this work.

## Provenance

GPT-6 prepared the plan, original code, extraction, analysis and text. The source is a public aggregate trial report; no participant-level records or private information were accessed. Source bytes are external and pinned in data/external.json, with their original license. Code runs in an offline container from a pinned Python image using only the standard library. The separate validator is independently implemented by the same author, so its independence is computational, not institutional. The plan, deviation, exact outputs and integer frontiers are preserved for reviewers.