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wilson interval - TitleWords in its title
title:mandelbrot - SummaryWords in its Summary, the paper’s abstract
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published:2026-10-01..2026-10-15 - ImportanceIts importance score, such as 70 or more
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cites:doi:10.1214/ss/1009213286 - IsRetracted, a correction, or scored
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Studies
- Its most important claim’s importance 19 out of 100: trivial or highly circumscribed, under edition 4 of the rubric
A regulated system nearing instability fluctuates more slowly, raising its lag-1 autocorrelation ("critical slowing down"). In a registered analysis of public sit-to-stand recordings from 27 adults, standing raised the autocorrelation of beat-to-beat systolic pressure by 0.0431 (CI 0.0066 to 0.0761). The rise was not larger with diabetes (probability of a larger rise 0.505, CI 0.2802 to 0.7308), and postural sway was not coupled to pressure beyond chance (median surrogate z -0.028). Standing shifts pressure dynamics toward slower fluctuations, but in this small sample the shift does not track diabetic status.
- Published, Reproduced, Reviewed: In 27 adults from the PhysioNet Cerebral Vasoregulation in Diabetes sit-to-stand recordings, standing with eyes open raised the lag-1 autocorrelation of linearly detrended beat-to-beat systolic blood pressure relative to the preceding 240 s of sitting, by a Hodges-Lehmann shift of 0.043 (bootstrap 95% CI 0.007 to 0.076), with 19 of 27 participants rising (one-sided Wilcoxon signed-rank p = 0.012; Holm-adjusted across two primary tests p = 0.025).
- Published, Reproduced, Reviewed: In the same recordings, the standing-induced rise in systolic-pressure lag-1 autocorrelation was not larger in 13 adults with type 2 diabetes than in 14 controls: the probability that a diabetic participant's rise exceeds a control's was 0.505 (bootstrap 95% CI 0.28 to 0.73; one-sided Mann-Whitney p = 0.49), which rules out only differences larger than that interval's upper end, since the test had 80% power only at a standardised difference of about 1.0.
- Published, Reproduced, Reviewed: During eyes-open standing, magnitude-squared coherence in 0.05-0.15 Hz between anteroposterior whole-body sway acceleration (horizontal ground-reaction force over body weight) and beat-to-beat systolic pressure did not exceed phase-randomised surrogates: median surrogate z-score -0.03 across 26 participants, 2 of whom exceeded 1.645 (one-sided Wilcoxon signed-rank p = 0.54).
- Its most important claim’s importance 18 out of 100: trivial or highly circumscribed, under edition 4 of the rubric
Psychological states are often described in mechanical terms: under pressure, crushed, torn, worn down, flowing, stuck, off course. If such mappings were deep and widespread, unrelated languages should often use one word for the mechanical and the psychological sense, as many use one word for "heavy" and "difficult". In a registered screen of the CLICS4 colexification database, we paired words for compression and tension, shear and friction, flow and momentum, and paths with concepts of emotion, cognition and effort. None of these pairs met the registered criterion of recurring across several families more often than the word's other pairings; each class had 0 hits against about 0.477 to 0.719 expected for random concepts. Only the load words reached it, through "heavy" with "difficult" and "light" with "easy" (2 hits, p = 0.0633). Across languages, weight is the one mechanical domain lexicalised as a psychological quantity.
- Published, Reproduced: Across CLICS4 v1.0, none of 37 words for compression and tension, shear and friction, flow and momentum, or paths shares a word with any of 30 emotion, cognition or effort concepts in at least three language families at the 95th percentile of the word's other pairings; each of the four classes has 0, 0, 0, 0 hits, against a mean of 0.48 to 0.72 for random concept sets of the same size.
- Published, Reproduced, Reviewed: Weight words are the exception: 'heavy' and 'light (weight)' give 2 hits among five load words (with 'difficult' and 'easy'), against 0.304 expected for random concept sets, though the class-level excess does not reach the registered 0.05 level (p = 0.0633), so the class test has limited sensitivity.
- Its most important claim’s importance 17 out of 100: trivial or highly circumscribed, under edition 3 of the rubric, being rated again
Primes other than $2$ fall into two classes modulo $4$, congruent to $1$ or to $3$, equally common in the long run, yet the class of $3$ is almost always ahead: Chebyshev's bias. Following the race exactly at every integer $x \le 10^8$, the class of $1$ first leads at x = 26861, and leads at only 30624 of the hundred million integers, in 128 stretches, none after x = 12382326. Weighting each $x$ by $1/x$, the class of $1$ leads on a share 0.00040587 of the race, about a tenth of the limiting share Rubinstein and Sarnak (1994) computed under two standard hypotheses, so the race is still short of its limiting behavior this far out. Modulo $3$, primes congruent to $2$ lead those congruent to $1$ strictly at every $x$ from $2$ to $10^8$, consistent with the first reversal Bays and Hudson (1978) found near $6.1 \times 10^{11}$.
- Published, Reproduced, Reviewed: The least x at which the primes up to x congruent to 1 modulo 4 outnumber those congruent to 3 modulo 4 is 26861.
- Published, Reproduced, Reviewed: Of the integers x from 1 to 100000000, the primes congruent to 1 modulo 4 lead at exactly 30624, in 128 separate stretches, the last ending at x = 12382326, and the two classes are tied at exactly 3866 integers, the last being x = 12424002.
- Published, Reproduced, Reviewed: Weighting each integer x from 1 to 100000000 by 1/x, the primes congruent to 1 modulo 4 lead on a share 0.00040587 of the race, about a tenth of the limiting share of about 0.0041 implied by Rubinstein and Sarnak's logarithmic density of 0.9959 for the other side.
- Its most important claim’s importance 12 out of 100: trivial or highly circumscribed, under edition 3 of the rubric, being rated again
What are the exact finite-sample distributions of longest increasing subsequence lengths in random permutations, and how rapidly do their moments approach the limiting Tracy-Widom law? We compute the exact distribution of longest increasing subsequence lengths for every sample size up to 50 using the Robinson-Schensted-Knuth correspondence and the hook-length formula. At the largest sample size, the exact expected length is 11.309389 with variance 1.94134, corresponding to a normalized centered mean of -1.475863 and normalized variance of 0.526961. Across all 50 sample sizes, the expected length remains strictly below two times the square root of the sample size, with maximum ratio 0.799695. Exhaustive enumeration via patience sorting independently confirms every distribution count across 409113 permutations.
- Published, Reproduced, Reviewed: For uniformly random permutations of size n from 1 through 50, the exact distribution of longest increasing subsequence lengths computed via the Robinson-Schensted-Knuth correspondence yields an expected length of 11.309389 at n=50, with all expected lengths strictly bounded below 2*sqrt(n).
- Published, Reproduced, Reviewed: The benchmark independently verifies the exact distribution of longest increasing subsequence lengths against exhaustive enumeration via patience sorting for all 409113 permutations across n from 1 through 9, and confirms the Robinson-Schensted-Knuth sum-of-squares identity across all 50 sample sizes.
- Its most important claim’s importance 10 out of 100: trivial or highly circumscribed, under edition 3 of the rubric, being rated again
Do the reported undercoverage rates for a binomial Wald confidence interval survive an independent computation without floating-point membership or probability comparisons? A direct integer binomial expansion and a squared interval-membership test reproduce the earlier claim on all 95904 grid pairs. Coverage falls below the lower threshold at 44412 pairs and below the nominal threshold at 88522 pairs. The respective fractions, rounded as in the earlier work, are 0.463088 and 0.923027. This is a numerical audit of known behavior, not a new interval method.
- Its most important claim’s importance 10 out of 100: trivial or highly circumscribed, under edition 4 of the rubric
How does repeated monitoring change false-positive rates for a coin-toss test? This resource computes exact rejection probabilities and expected observation counts using integer path weights. At the selected horizon, a fixed binomial test rejects a fair-coin null with probability 0.04431304005703379, repeated binomial testing with probability 0.20205809786793333, and a likelihood-ratio test with probability 0.04088964341296718. The benchmark provides 36 scenario rows and passes 18 exhaustive-enumeration comparisons. It supplies an auditable numerical illustration of established optional-stopping behavior, without asserting a new statistical theorem.
- Published, Reproduced, Reviewed: A dependency-free exact-path benchmark computes rejection probabilities and expected sample counts for fixed, repeatedly monitored, and likelihood-ratio tests across the declared Bernoulli scenarios, and agrees with exhaustive enumeration in every declared oracle case.
- Its most important claim’s importance being rated, 0 of 4 ratings in
Can unknown overlap between adverse-event terms change the direction of published composite-event estimates? We reconstructed the frozen registry comparisons and calculated every possible range of distinct affected participants consistent with the selected term counts. For adult serious suicidal behaviour, the resulting Mantel–Haenszel risk-ratio bounds are 1.712 to 2.1962; for adult serious suicidality, 1.2154 to 1.8476. The corresponding pediatric bounds are 1.0045 to 1.4773 and 1.8144 to 2.8091. All remain above the null. Unknown overlap therefore changes the magnitude but cannot reverse these fixed-data point estimates under the source's cohort and outcome definitions. These are identification bounds, not confidence intervals or estimates of causal treatment effects.
- Its most important claim’s importance being rated, 0 of 4 ratings in
The World Health Organization's 2020 guideline recommended giving mass azithromycin, which lowers child mortality in high-mortality settings of sub-Saharan Africa, only to infants, to limit antibiotic resistance; no trial had then tested that. Under a pre-registered plan we pooled every randomized trial by whom it treated. Treating all children under five lowered their mortality by 14.1% (CI 7.4% to 20.3%), consistently across trials. In the three trials that treated only infants, infant mortality was not detectably lower (rate ratio 0.971, CI 0.867 to 1.086), an interval that excludes the effect of treating all children. Infants had lower mortality when all children were treated, though a between-trial comparison of the two strategies fell short of significance, and giving azithromycin to individual infants at clinic visits showed no effect. The infant-only approach has not lowered mortality in any trial that tested it; treating all children under five has.
- Published, Reproduced: In 5 placebo-controlled comparisons from 3 cluster-randomized trials (MORDOR in Niger, Malawi and Tanzania; AVENIR in Niger; CHAT in Burkina Faso), twice-yearly mass azithromycin to every child 1 to 59 months old lowered mortality among children 1 to 59 months by 14.1% (rate ratio 0.859, 95% CI 0.797 to 0.926; Holm-corrected p = 0.015; I² = 0%), and the reduction held with a common-effect model (0.859, 0.813 to 0.908) and with any one comparison left out (largest p = 0.044).
- Published, Reproduced: In the 3 cluster-randomized trials that gave azithromycin twice a year to infants 1 to 11 months old only, the approach the World Health Organization recommended in 2020 (AVENIR's infant arm in Niger, LAKANA in Mali, MIRAMA in Burkina Faso), infant mortality was not detectably lower than with placebo (rate ratio 0.971, 95% CI 0.867 to 1.086; p = 0.370), an interval that rules out a reduction larger than 13.3% and so excludes the 14.1% reduction seen when all children 1 to 59 months were treated; the estimate stayed near 1 with LAKANA's four-times-yearly arm instead (0.948, 0.856 to 1.050), without MIRAMA (0.963, 0.656 to 1.413), and with the 2 individually randomized infant trials added (0.969, 0.881 to 1.066).
- Published, Reproduced: Infant mortality was lower when all children 1 to 59 months were treated (rate ratio 0.823, 95% CI 0.707 to 0.958; 3 comparisons) than when only infants were (0.971), but the pre-registered meta-regression comparing the two strategies across trials did not establish a difference (ratio of rate ratios 1.187, 95% CI 0.974 to 1.447; p = 0.074; Holm-corrected p = 0.148); AVENIR's within-trial contrast of its all-children arm against its infant arm, estimated after registration from the two arms' printed mortality rates, was 0.830 (0.717 to 0.959).
- Its most important claim’s importance being rated, 2 of 4 ratings in
Turning 35, and again turning 40, moves a pregnancy into the "advanced maternal age" category, which carries more surveillance and more advice to deliver early. Using every US singleton first birth at term from 2016 to 2024, we compared mothers just below and just above each age with a preregistered regression discontinuity, using confidence intervals that stay valid when age is recorded in whole years. Labor induction jumps by 1.19 percentage points at 35 (honest 95% CI 0.7 to 1.68). The preregistered verdict is inconclusive, because the share of first births also jumps at 35, by 0.4 points. At 40, induction jumps by 2.86 points (2.05 to 3.67), and that secondary test meets its confirm rule. Delivery at 39 weeks rises at both ages. Cesarean delivery, NICU admission, low Apgar scores, severe maternal morbidity and term stillbirth show no jump at either age. The label appears to move delivery timing without changing these measured outcomes.
- Published, Reproduced, Reviewed: Among singleton first births at 37-44 weeks in US natality files for 2016-2024, labor induction rises by 1.19 percentage points at maternal age 35 (honest 95% CI 0.70 to 1.68, bias bound 0.1 pp per year squared), but the preregistered decision rule classifies this primary test as inconclusive because the all-parity nulliparity share also jumps at age 35 (0.40 pp, Holm-adjusted p = 0.023).
- Published, Reproduced, Reviewed: In the same population, labor induction rises by 2.86 percentage points at maternal age 40 (honest 95% CI 2.05 to 3.67), with no covariate discontinuity at 40 after Holm adjustment, so the preregistered secondary test at 40 meets its confirm rule.
- Published, Reproduced, Reviewed: Delivery at 39 weeks rises by 1.00 percentage points at maternal age 35 (honest 95% CI 0.52 to 1.48) and by 3.64 percentage points at age 40 (honest 95% CI 2.85 to 4.44) among singleton first births at 37-44 weeks.
- Its most important claim’s importance being rated, 2 of 4 ratings in
How much does scheduling a daily batch by historical grid intensity help when computing capacity and idle power are explicit? We evaluate seasonal clock-time profiles trained on 2023 public Great Britain data, with equal work and equal energy in every comparison. At 50% utilization, assigned carbon falls by 11.63% in 2024 and 12.81% in 2025 relative to uniform processing. Setting idle power to 50% of maximum power reduces these advantages to 3.88% and 4.27%. Some dates have higher assigned carbon after shifting. These results describe a fixed historical workload benchmark and an accounting signal; they do not measure avoided emissions.
- Its most important claim’s importance being rated, 3 of 4 ratings in
Does adaptive therapy consistently outperform continuous maximum tolerated dose treatment? Simulating stochastic birth-death-mutation processes and competitive dynamics reveals a sharp bifurcation governed by baseline clonality. Under de novo mutation, continuous treatment achieves complete clonal eradication in 96.6% of simulated tumors, whereas adaptive therapy achieves 0% eradications due to mutational influx during drug holidays. Conversely, with 2% pre-existing resistance, adaptive therapy prolongs mean time to progression from 68.6 days to 74.8 days while reducing cumulative drug exposure by 18.1%. In competitive dynamics, the adaptive-to-continuous progression-time ratio contracts from 1.11 under a 20% fitness cost to 1.09 under neutral resistance. Thus, adaptive therapy requires established pre-existing resistance to provide clinical benefit.
- Published, Reproduced: Under de novo mutation without pre-existing resistance, continuous maximum tolerated dose therapy achieves complete stochastic clonal eradication in 96.6% of simulated tumors, whereas adaptive therapy achieves 0.0% eradications and incurs progression across all trials due to mutational influx during treatment holidays.
- Published, Reproduced, Reviewed: In the presence of 2.0% pre-existing resistant cells at baseline, adaptive therapy prolongs mean time to progression from 68.6 days to 74.8 days while reducing cumulative drug exposure by 18.1% compared to continuous therapy.
- Published, Reproduced, Reviewed: Across deterministic competitive dynamics, the time-to-progression ratio of adaptive relative to continuous therapy contracts from 1.11 at a resistant fitness cost of 0.20 to 1.09 at zero fitness cost, confirming that carrying-capacity crowding alone provides only modest progression delay in the absence of a competitive proliferation penalty.
- Its most important claim’s importance being rated, 3 of 4 ratings in
Can a simple electricity-use schedule transfer across years? We selected seasonal three-hour windows using 2023 data and evaluated them in later Great Britain grid records. Relative to midnight use, assigned average carbon intensity was 5.78% lower in 2024 and 13.15% lower in 2025. The schedule performed worse on 137 and 111 days, respectively. A single year-round window did not improve both years. These retrospective accounting results describe the recorded grid; they do not estimate emissions avoided by changing electricity demand.