{"id":"5d07ca76-7ea2-4849-83c7-14439a4f0e76","arxiv_id":"2508.10223","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"For 90%, 95%, and 99% confidence intervals on a proportion, adding 3, 4, and 6 pseudo-observations respectively to the adjusted Wilson interval maximizes average coverage, beating Wald and plain Wilson intervals.","lead":"This paper runs an exhaustive numerical comparison of three common ways to build confidence intervals for a percentage, testing thousands of sample sizes and true percentages. It claims the best intervals come from slightly inflating the data, with the ideal amount of inflation depending on the chosen confidence level.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 3/4/6 pseudo-count optima are selected and scored on the same discrete p-grid under a coverage-only criterion; a sensitivity check is needed before treating them as general.","rationale":"The strongest claim cannot be checked from the supplied mojibake text, and the abstract alone provides only the finite-grid enumeration design. The reader already identified uniform grid weighting and exclusion of extreme p as the weakest assumption; my analysis agrees and makes it more pointed by noting that the grid is both the selection and evaluation set, so the optimum may be overfit. A sensitivity analysis is the natural adjudicator. Because this concern was already part of the reader's rationale for UNVERDICTED, it does not move the verdict; it reinforces it.","tokens_in":9092,"tokens_out":9258,"duration_ms":97208,"concrete_test":"Write an exact-coverage script and recompute the mean coverage for epsilon=0,...,10 at each confidence level on (i) the original grid, (ii) an extended grid p=0.001,0.002,...,0.999, and (iii) a random sample of p from Uniform(0,1), keeping n=1..1000. If the argmax stays 3/4/6 and the margin over neighbors is at least 0.001, the concern is resolved; if the argmax shifts or the margin is tiny, the claim must be qualified as grid-specific.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is an exhaustive enumeration, so it inherits every choice in the enumeration. The abstract reports the optimal adjusted-Wilson pseudo-counts as 3, 4 and 6 for the 90%, 95% and 99% levels 'using the mean coverage probability' over n=1..1000 and p=0.01..0.99, but it does not justify uniform weighting on this grid, nor does it test whether the optimum survives a different grid. Binomial coverage is a non-smooth function of p and n; the mean over 99 grid points and 1000 sample sizes is dominated by small-n behavior. Since the same grid is used to choose epsilon and to announce the winner, the headline values are in-sample fitted parameters: the coverage gain from epsilon=3 vs epsilon=2 or 4 could be a numerical accident of the grid. Moreover, 'performs better' is only about coverage; expected length is not reported, so an interval that is slightly more covering but wider is called better. This makes the central claim contingent on an arbitrary evaluation design.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript compares three families of confidence intervals for a binomial proportion—Wald, Wilson, and adjusted Wilson obtained by adding epsilon pseudo-observations—using mean coverage probability over n = 1,...,1000 and p = 0.01,...,0.99 at the 90%, 95%, and 99% confidence levels. It proposes pixel-color plots and a rainbow color code for visualizing coverage over the (p, n) grid. The headline finding is that epsilon = 3, 4, 6 is optimal for adjusted Wilson at 90%, 95%, and 99%, respectively, and that these adjusted intervals also have higher mean coverage than the corresponding Wald and Wilson intervals.","tokens_in":9305,"tokens_out":5297,"duration_ms":60263,"significance":"The paper's exhaustive finite-sample census is a useful empirical complement to the asymptotic literature; if the computations are exact and reproducible, the 90% and 99% pseudo-count recommendations (3 and 6) are new numerical results beyond the well-known Agresti-Coull epsilon = 4 at 95%. The proposed visualization may be instructive. However, the optimality claim is currently in-sample: epsilon is chosen by maximizing the same mean-coverage criterion on the same grid used for the final comparison. No sensitivity analysis, expected-length comparison, or out-of-sample check is visible in the supplied text. Therefore the significance of the headline constants is limited until the criterion dependence is quantified.","major_comments":[{"comment":"The headline epsilon values are selected by optimizing the mean coverage criterion on the same finite grid that is then used to declare the adjusted Wilson interval the winner. This is circular/in-sample. Report the mean-coverage curve as a function of epsilon for each level and test robustness to omitting small n, changing p-grid resolution, weighting, and exact vs simulated evaluation. In particular, the 99% optimum epsilon = 6 needs a sensitivity check.","section":"Abstract"},{"comment":"The coverage-only criterion is insufficient: a longer interval tends to have higher coverage, so 'performs better' is ambiguous. Report expected length or a coverage-length trade-off metric for the Wald, Wilson, and adjusted Wilson intervals at each level.","section":"Abstract"},{"comment":"The full text supplied for review is corrupted (mojibake), so definitions, equations, tables, and figures cannot be checked. The abstract alone does not state whether coverage is exact or Monte Carlo. Provide a clean version and state the computation method; if simulated, give Monte Carlo standard errors.","section":"Full text (as supplied)"},{"comment":"The grid p = 0.01,...,0.99 and n = 1,...,1000 is treated as exhaustive, but it excludes extreme tails and weights all n equally, so the mean is dominated by small-n behavior. Qualify all 'best' claims as conditional on this grid and justify the grid choice or add a sensitivity analysis.","section":"Abstract / grid definition"}],"minor_comments":[{"comment":"'Type 4' and 'type epsilon' are used without a formal definition; define the adjusted Wilson estimator explicitly.","section":"Abstract"},{"comment":"The rainbow color code should include a colorblind-safe option; pixel plots need clear axis labels and a legend.","section":"Visualization"},{"comment":"Citations to Agresti and Coull (1998) and Brown, Cai, and DasGupta (2001) should be included in the reference list; the supplied text does not show them.","section":"References"},{"comment":"The phrase 'comprehensively compare ... across all sample sizes' is misleading given the discrete grid; consider 'over the evaluated grid'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The supplied version is unreadable due to encoding corruption, making verification impossible. The main risk is that the 3/4/6 optima are artifacts of the chosen coverage criterion and grid. A sensitivity analysis and expected-length comparison would determine whether the paper is publishable as an empirical note. The topic is appropriate for a statistics education journal, but the contribution is incremental."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The usable claim is that the adjusted-Wilson interval with 3, 4, and 6 pseudo-observations is best under mean coverage for the 90%, 95%, and 99% levels. The 95% part is Agresti-Coull; the 90% and 99% values are new. But I can't verify the computations: the full text I was sent is mojibake from the first page on, with a stray line from another arXiv paper mixed in. So my verdict rests on the abstract and your summary.\n\nWhat the paper does well: it asks a clean, teachable question and answers it with a large explicit enumeration (n=1..1000, p=0.01..0.99). The pixel-color plot is a reasonable way to display coverage patterns across the grid. The abstract correctly credits Agresti-Coull rather than claiming the 95% add-4 rule as novel. That is honest and useful framing.\n\nThe soft spots are the ones you and the stress-test note flagged. The optimization criterion is uniform mean coverage on a discrete grid, and the same grid is used to select the pseudo-count and to declare the winner. That makes the 3/4/6 values in-sample fitted constants, not structural predictions. The choice of uniform weighting is not defended, and the grid excludes the extreme tails where coverage is worst. No interval-length criterion is reported, so \"better\" only means coverage on average. The claim that 6 beats 7 at 99% is counterintuitive and would need the actual coverage tables to be credible. These are not fatal flaws; they are contingencies that need to be stated more carefully and tested.\n\nWho benefits: instructors and textbook authors deciding what to teach; anyone who wants a convenient per-level pseudo-count recipe. For that audience, the 90% add-3 and 99% add-6 rules are potentially useful if the numbers hold up.\n\nMy recommendation: send it to peer review, but only after the authors provide a readable manuscript, full coverage tables, and a sensitivity analysis over grids, losses, and at least one tail region. A referee can check the enumeration in an afternoon; the question is whether the optimization is robust. As it stands, I would not cite it in my own work until I can verify the tables, but I would read the revised version.","headline":"A readable abstract, an unreadable body, and a headline 3/4/6 result that is plausible but contingent on the chosen coverage criterion and grid.","tokens_in":9864,"tokens_out":3178,"would_cite":false,"duration_ms":33300,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Measured by mean coverage over a dense grid of n and p, adjusted Wilson intervals with 3, 4, and 6 pseudo-observations are optimal at 90%, 95%, and 99% confidence, beating Wald and Wilson.","keywords":["binomial proportion","coverage probability","Wald interval","Wilson interval","adjusted Wilson interval","pseudo-observations","mean coverage","finite-sample comparison"],"falsifier":"Recompute the same mean-coverage comparison on an extended grid that includes $p=0.001,\\dots,0.999$; if the maximum moves away from $\\varepsilon=3/4/6$, the paper's optimality claim depends on the omitted tails. A cheaper check: use the same grid but minimize maximum coverage shortfall instead of mean coverage, and see whether the winning $\\varepsilon$ changes.","tokens_in":8917,"feed_emoji":"📊","tokens_out":11149,"duration_ms":119186,"temperature":0.7,"pith_summary":"The paper takes on a well-known problem in statistics teaching: the most commonly taught confidence interval for a proportion, the Wald interval (sample proportion plus or minus z times the standard error), has poor coverage for small samples and for proportions near 0 or 1. Setting out to find a simple fix, the paper evaluates a family of \"adjusted Wilson\" intervals in which a small number of pseudo-observations is added to the data before constructing the interval. Using mean coverage probability over every sample size $n=1,\\dots,1000$ and every proportion $p=0.01,\\dots,0.99$, it claims that the optimal number of pseudo-observations is 3 at 90% confidence, 4 at 95%, and 6 at 99%, and that at those settings the adjusted Wilson interval has higher mean coverage than either the plain Wald interval or the plain Wilson interval at the same level. If this holds, the practical prescription is nearly free: add a level-specific small number of fake successes and failures and use the Wilson formula. The paper also introduces rainbow pixel plots that show at a glance where each interval's coverage falls short.","feed_headline":"Adding 3, 4, or 6 pseudo-observations wins at 90, 95, 99%","feed_subtitle":"A 1000-by-99 grid comparison shows adjusted Wilson beats Wald and Wilson under mean coverage.","key_machinery":"The central object is the adjusted Wilson interval of type $\\varepsilon$: take the ordinary Wilson (score) interval for a proportion and first add $\\varepsilon/2$ successes and $\\varepsilon/2$ failures to the observed counts, so that $\\varepsilon$ pseudo-observations are added in total. $\\varepsilon=0$ is the plain Wilson interval, and $\\varepsilon=4$ is the Agresti-Coull-style correction. The argument is carried by exhaustive enumeration: for each $\\varepsilon$, the paper computes coverage for all $n$ and $p$ in the stated grid at each confidence level, averages over the grid, and compares the resulting mean coverage across the Wald, Wilson, and adjusted-Wilson families. The pixel plots are","core_discovery":"On a finite grid of all sample sizes $n=1,\\dots,1000$ and all population proportions $p=0.01,0.02,\\dots,0.99$, the paper computes the coverage probability of the Wald, Wilson, and adjusted-Wilson-of-type-$\\varepsilon$ intervals at the 90%, 95%, and 99% confidence levels. It reports that the mean coverage probability is maximized at $\\varepsilon=3$ for 90%, $\\varepsilon=4$ for 95%, and $\\varepsilon=6$ for 99%, and that each of these three adjusted intervals also dominates the corresponding Wald and Wilson intervals on the same grid. The result is a finite census rather than an asymptotic theorem: every point in the grid is evaluated directly, and the comparison is displayed as color-coded pix","pith_inferences":["The 3/4/6 optima are optima for the uniform mean-coverage loss on this particular grid; under a worst-case-coverage loss, or with p weighted toward realistic values, a different epsilon is likely to win.","The grid stops at p = 0.01 and p = 0.99; near the boundary, all three intervals behave poorly, so the paper's ranking should not be read as holding in the extreme tails.","The same enumeration could be run for interval length or for other families (e.g., Jeffreys or Clopper-Pearson); those extensions are not in the paper but follow naturally from its design."],"forward_implications":["At 95% confidence, the exhaustive census reproduces the Agresti-Coull finding: epsilon = 4 adjusted Wilson has higher mean coverage than plain Wilson, and both beat Wald.","At 90% and 99%, the same ranking holds with epsilon = 3 and epsilon = 6, so the pseudo-observation trick is not a 95%-only phenomenon.","Because the comparison is a finite census over $n=1,\\dots,1000$ and $p=0.01,\\dots,0.99$, the ranking is not an asymptotic approximation and covers small-sample cases that asymptotics miss.","The pixel-color displays give instructors and practitioners a direct visual map of where Wald coverage collapses, making the case for abandoning it easy to show in teaching."],"supporting_citations":[{"why":"Supplies the earlier 95% result that the type-4 adjusted Wilson interval can beat Wald and Wilson, which this paper extends to all three confidence levels and a full n, p grid.","marker":"Agresti and Coull"}],"fun_headline_variants":["Adjusted Wilson wins with 3, 4, or 6 extra observations","Optimal pseudo-observations: 3, 4, 6 for 90, 95, 99% CI","Adjusted Wilson beats Wald and Wilson on full grid","3, 4, 6 pseudo-obs yield best coverage for proportions","Color-coded grid crowns adjusted Wilson at 90, 95, 99%"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The entire ranking rests on one definition of \"best\": coverage averaged uniformly over $p=0.01,0.02,\\dots,0.99$ and $n=1,\\dots,1000$; if a different loss or grid is used, the winning number of pseudo-observations can change.","fun_headline_variants_meta":{"raw":{"variants":["Adjusted Wilson wins with 3, 4, or 6 extra observations","Optimal pseudo-observations: 3, 4, 6 for 90, 95, 99% CI","Adjusted Wilson beats Wald and Wilson on full grid","3, 4, 6 pseudo-obs yield best coverage for proportions","Color-coded grid crowns adjusted Wilson at 90, 95, 99%"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000126,"raw_usage":{"total_tokens":989,"prompt_tokens":824,"completion_tokens":165,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":65}},"tokens_in":568,"tokens_out":165,"duration_ms":2603,"temperature":1.0,"reasoning_tokens":65,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:34:51.127966+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the same mean-coverage comparison on an extended grid that includes $p=0.001,\\dots,0.999$; if the maximum moves away from $\\varepsilon=3/4/6$, the paper's optimality claim depends on the omitted tails. A cheaper check: use the same grid but minimize maximum coverage shortfall instead of mean coverage, and see whether the winning $\\varepsilon$ changes.","supporting_citations":[],"review_version":1}