--- slug: no-number-wrong passed: 2026-10-14 by: the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex; repair review by Codex checked: 162 confirmed: 140 wrong: 22 unverifiable: 0 observed: 0 stale: 0 --- This was the directed queue-1014c repair pass on no-number-wrong: I reread evidence for all 32 triage items, corrected layer-owned copies of the 22 WRONG entries, and left the 10 listed MINOR entries unchanged under the directing triage. The repair review confirmed a remaining shared TV narration copy of X5; that finding stays open pending a board note and shared-file repair. The 130 CONFIRMED lines below retain the source checker's reread claims and notes at its recorded pin, with that provenance intact; they do not claim a new independent audit of every inherited item or of omitted UNVERIFIABLE items. MINOR counts toward confirmed here. Literal \u2014 in quoted prior text preserves its original punctuation without introducing that character. ## Claims - MINOR | X1: "They share the same x-values, and \u2014 recomputed live below from the points themselves \u2014 the same mean of x, the same mean of y, the same variance, the same correlation, the same line of best fit." | public/strata/no-number-wrong/index.html; https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | Recomputed the embedded arrays: sample var(y) is 4.1272690909 versus 4.1276290909; r is 0.8164205163 versus 0.8162365060. Equal summaries require the displayed-precision qualification. - MINOR | X2: "The other is an exact downward-bending parabola." | public/strata/no-number-wrong/index.html; https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | Read the original table and description on printed p. 19. Integer-hundredths second differences of the sorted embedded points are -25 or -26, so the printed points are not an exact quadratic. - MINOR | X3: "Every coordinate the summary keeps is identical; the only thing that differs is the coordinate it dropped \u2014 the shape \u2014 and that is the whole story." | public/strata/no-number-wrong/index.html; https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | Recomputed the embedded arrays: sample var(y) is 4.1272690909 versus 4.1276290909; r is 0.8164205163 versus 0.8162365060. Equal summaries require the displayed-precision qualification. - WRONG | X4: "Two pages of this place are usually told as separate curiosities \u2014 Simpson's paradox (the Berkeley admissions gap that reverses department by department) and Berkson's paradox (independent traits that anticorrelate the moment you study a selected pool)." | https://svn.r-project.org/R/trunk/src/library/datasets/data/UCBAdmissions.R; https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread Bickel p. 403 and the official R array: women lead in A, B, D, F; men lead in C, E. C is men 120/325 versus women 202/593. Both prose and card now describe the actual pattern. - WRONG | X5: "They are structural duals \u2014 the same three variables X, Y, Z, related by the same arithmetic, with exactly one causal arrow reversed: a common cause against a common effect." | public/strata/no-number-wrong/index.html:341; public/tv/lib/narration/batch-1.js:139; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | The authored edge map changes Z→X and Z→Y to X→Z and Y→Z, reversing both arrows. Corrected prose, metadata, markdown body, facts and OG template; X28 also wrongly attached matrix ambiguity to fork/collider, now assigned to fork/mediator. The exported narrationFor('no-number-wrong') still contains "Two of these traps are secretly the same picture, with one arrow reversed:". This shared-file copy remains open; the repair worker is forbidden to edit shared files. - WRONG | X6: "In Simpson, Z (the department) is a confounder: a common cause of both X and Y." | https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread the summary and adjustment discussion. The paper describes different application patterns and department selectivity, not department causing applicant sex. The page now separates the illustrative causal fork from the within-department admissions comparison, including the table, card and related-page note. - WRONG | X7: "The page draws the implied correlation matrix among the three variables (from two adjustable edge strengths), then computes two numbers: the plain correlation of X and Y, and their partial correlation once Z is held fixed." | public/strata/no-number-wrong/index.html; research/no-number-wrong/verify.mjs | At the former allowed a=b=0.85, determinant=1-a²-b²=-0.445 and partial r=-2.6036036036. The new cap 0.70 keeps determinant at least 0.02 and the extreme partial is -0.9607843137, displayed -0.961. Defaults remain 0.300/0.000 for the fork and 0.000/-0.433 for the collider. Verifiers cover the permitted domain and browser extremes. - MINOR | X8: "The Bias in the Sample pins the exact size of it for the cleanest case \u2014 keep the top half of two independent bell curves by their sum, and inside that selected pool the correlation is exactly −1/(π−1) ≈ −0.467, an anticorrelation conjured from pure independence." | research/berksons-paradox/verify.mjs | Reread lines 20-27: independent N(0,1) variables and a zero cut on their sum give the constant. The cleanest-case body sentence omits the equal-variance Gaussian specification and is left as listed MINOR. - WRONG | X9: "Simpson and Berkson vary the causal arrow that the correlation matrix cannot resolve \u2014 duals across the single bit that distinguishes a common cause from a common effect." | public/strata/no-number-wrong/index.html; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | Recomputed rXY=ab=0.300 for the default fork versus 0.000 for the collider; the mediator shares the fork matrix. Verma and Pearl printed p. 2 separates the collider from the three equivalent no-collider graphs. Corrected the spine and description metadata. - WRONG | X10: "A summary statistic is necessary and never sufficient." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | Visually reread printed p. 18, lower left: Anscombe explicitly allows sufficient statistics for model parameters under an exactly correct model. The page now states model-relative sufficiency and the need to assess assumptions. - WRONG | X11: "Berkeley 1973: a sex-admission gap that reverses department by department." | https://svn.r-project.org/R/trunk/src/library/datasets/data/UCBAdmissions.R; https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread Bickel p. 403 and the official R array: women lead in A, B, D, F; men lead in C, E. C is men 120/325 versus women 202/593. Both prose and card now describe the actual pattern. - WRONG | X12: "Z is a confounder \u2014 condition on it." | https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread the summary and adjustment discussion. The paper describes different application patterns and department selectivity, not department causing applicant sex. The page now separates the illustrative causal fork from the within-department admissions comparison, including the table, card and related-page note. - WRONG | X13: "Berkson: independent traits anticorrelate in any selected pool, by exactly −1/(π−1)." | research/berksons-paradox/verify.mjs; https://www.nature.com/articles/s41467-020-19478-2 | Reread the Gaussian assumptions and Griffith et al. Figure 2 caption: selection can induce positive, negative or no association. The page now states the independent standard-normal median-sum example. Selecting only X from independent X,Y preserves independence; selecting X=Y from independent fair binary variables instead gives correlation +1. - MINOR | X14: "The portal's structural claim \u2014 the confounder / mediator / collider trio, their Markov-equivalence, and the four canonical numbers \u2014 is verified offline in research/no-number-wrong/verify.mjs (20 / 20 PASS): the partial-correlation identity on the three matrices; the collider's lone v-structure; Anscombe's mean = 9 and sample variance = 11 exact; a Simpson reversal in honest integers; the Will Rogers band identity over 20,000 random configurations; and −1/(π−1) by deterministic 2-D quadrature." | research/no-number-wrong/verify.mjs | Reread the complete verifier: the equivalent trio is fork, mediator and reverse chain; the collider is separate. The page scope sentence loosely associates equivalence with the displayed trio, despite distinguishing the collider elsewhere. Left as listed MINOR. - WRONG | X15: "Free choices, named: the DAG instrument uses standardised linear-Gaussian variables, so each edge strength a, b is a correlation directly; the matrices and partials are population values, not sampled." | public/strata/no-number-wrong/index.html; research/no-number-wrong/verify.mjs | At the former allowed a=b=0.85, determinant=1-a²-b²=-0.445 and partial r=-2.6036036036. The new cap 0.70 keeps determinant at least 0.02 and the extreme partial is -0.9607843137, displayed -0.961. Defaults remain 0.300/0.000 for the fork and 0.000/-0.433 for the collider. Verifiers cover the permitted domain and browser extremes. - MINOR | X16: "Every claim is checked in the verifier linked above." | research/no-number-wrong/verify.mjs; research/no-number-wrong/verify-page.mjs | The numerical verifier checks specified mathematical examples and the browser verifier reads selected DOM states; neither checks every prose claim. The local footer sentence is left as listed MINOR under the directing triage. - WRONG | X17: "Simpson and Berkson are structural duals (duals at the level of the causal diagram, one arrow reversed, not arithmetic mirror images)" | public/strata/no-number-wrong/index.html; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | The authored edge map changes Z→X and Z→Y to X→Z and Y→Z, reversing both arrows. Corrected prose, metadata, markdown body, facts and OG template; X28 also wrongly attached matrix ambiguity to fork/collider, now assigned to fork/mediator. - WRONG | X18: "The same three variables X, Z, Y; the same arithmetic; one causal arrow reversed." | public/strata/no-number-wrong/index.html; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | The authored edge map changes Z→X and Z→Y to X→Z and Y→Z, reversing both arrows. Corrected prose, metadata, markdown body, facts and OG template; X28 also wrongly attached matrix ambiguity to fork/collider, now assigned to fork/mediator. - WRONG | X19: "Simpson and Berkson are the same three variables with one causal arrow reversed: condition on a confounder, never on a collider, and no correlation matrix can tell a confounder from a mediator." | public/strata/no-number-wrong/index.html; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | The authored edge map changes Z→X and Z→Y to X→Z and Y→Z, reversing both arrows. Corrected prose, metadata, markdown body, facts and OG template; X28 also wrongly attached matrix ambiguity to fork/collider, now assigned to fork/mediator. - MINOR | X20: "Anscombe's quartet is the existence proof \u2014 identical mean, variance, correlation and best-fit line over a noisy line and a clean parabola." | public/strata/no-number-wrong/index.html; https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | Recomputed the embedded arrays: sample var(y) is 4.1272690909 versus 4.1276290909; r is 0.8164205163 versus 0.8162365060. Equal summaries require the displayed-precision qualification. - WRONG | X21: "Here Z (the Berkeley department) is a confounder \u2014 a common cause \u2014 so the honest association lives inside each group and pooling lies; the remedy is to condition on Z." | https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread the summary and adjustment discussion. The paper describes different application patterns and department selectivity, not department causing applicant sex. The page now separates the illustrative causal fork from the within-department admissions comparison, including the table, card and related-page note. - MINOR | X22: "The portal shows the matrix this produces is identical to a mediator's (Markov equivalence), so the data cannot tell you whether conditioning reveals the truth or destroys it \u2014 exactly the honesty this page ends on (conditioning explains the gap; it does not prove no bias, only moves it upstream)." | https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread printed p. 403: the paper cautions against excluding bias elsewhere; it does not prove that bias moved upstream. X22 and X30 identify the same categorical clause, in a separate sentence from the X21 fix. Left as listed MINOR. - WRONG | X23: "There, three physics experiments refuting one classical assumption (counterfactual definiteness)." | https://arxiv.org/abs/0712.0921; https://arxiv.org/abs/hep-th/9305002; public/strata/before-you-looked/index.html | Reread both primary abstracts and the linked page discussion. GHZ excludes deterministic local models, a conjunction; the bomb abstract concerns interaction-free detection. Corrected this layer's referring note only, identifying each result without claiming definiteness alone was refuted. - MINOR | X24: "Scatter of Anscombe dataset II, an exact downward parabola, with the same regression line drawn through it" | public/strata/no-number-wrong/index.html; https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | Read the original table and description on printed p. 19. Integer-hundredths second differences of the sorted embedded points are -25 or -26, so the printed points are not an exact quadratic. - WRONG | X25: "good group A: 76.3 → 85.8 ▲ rose · bad group B: 29.0 → 29.0 ▼ fellx is outside the band \u2014 at least one mean falls. The illusion needs the band." | public/strata/no-number-wrong/index.html; research/will-rogers-phenomenon/identity.mjs | Reread the delta identities and recomputed x=29: B stays (145+29)/6=29 while A rises from 458/6 to (458-29)/5=85.8. Added an unchanged state, and outside the open band now means at least one mean does not rise. Also repaired this layer's offline helper to remove exactly one repeated value and checked that boundary. - WRONG | X26: "x is outside the band \u2014 at least one mean falls. The illusion needs the band." | public/strata/no-number-wrong/index.html; research/will-rogers-phenomenon/identity.mjs | Reread the delta identities and recomputed x=29: B stays (145+29)/6=29 while A rises from 458/6 to (458-29)/5=85.8. Added an unchanged state, and outside the open band now means at least one mean does not rise. Also repaired this layer's offline helper to remove exactly one repeated value and checked that boundary. - WRONG | X27: "Simpson and Berkson are structural duals (the same variable, opposite arrow, opposite remedy)" | public/strata/no-number-wrong/index.html; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | The authored edge map changes Z→X and Z→Y to X→Z and Y→Z, reversing both arrows. Corrected prose, metadata, markdown body, facts and OG template; X28 also wrongly attached matrix ambiguity to fork/collider, now assigned to fork/mediator. - WRONG | X28: "Simpson & Berkson are structural duals: same variable, opposite arrow, opposite remedy, and the table shows you neither." | public/strata/no-number-wrong/index.html; https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | The authored edge map changes Z→X and Z→Y to X→Z and Y→Z, reversing both arrows. Corrected prose, metadata, markdown body, facts and OG template; X28 also wrongly attached matrix ambiguity to fork/collider, now assigned to fork/mediator. - WRONG | X29: "independent traits that anticorrelate the moment you study a selected pool" | research/berksons-paradox/verify.mjs; https://www.nature.com/articles/s41467-020-19478-2 | Reread the Gaussian assumptions and Griffith et al. Figure 2 caption: selection can induce positive, negative or no association. The page now states the independent standard-normal median-sum example. Selecting only X from independent X,Y preserves independence; selecting X=Y from independent fair binary variables instead gives correlation +1. - MINOR | X30: "conditioning explains the gap; it does not prove no bias, only moves it upstream" | https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | Reread printed p. 403: the paper cautions against excluding bias elsewhere; it does not prove that bias moved upstream. X22 and X30 identify the same categorical clause, in a separate sentence from the X21 fix. Left as listed MINOR. - WRONG | X31: "There, three physics experiments refuting one classical assumption (counterfactual definiteness)." | https://arxiv.org/abs/0712.0921; https://arxiv.org/abs/hep-th/9305002; public/strata/before-you-looked/index.html | Reread both primary abstracts and the linked page discussion. GHZ excludes deterministic local models, a conjunction; the bomb abstract concerns interaction-free detection. Corrected this layer's referring note only, identifying each result without claiming definiteness alone was refuted. - WRONG | X32: (added) "-2.604" | public/strata/no-number-wrong/index.html; research/no-number-wrong/verify.mjs | At the former allowed a=b=0.85, determinant=1-a²-b²=-0.445 and partial r=-2.6036036036. The new cap 0.70 keeps determinant at least 0.02 and the extreme partial is -0.9607843137, displayed -0.961. Defaults remain 0.300/0.000 for the fork and 0.000/-0.433 for the collider. Verifiers cover the permitted domain and browser extremes. - CONFIRMED | "Four pages of this place reproduce four famous paradoxes." | git show f5e5f84f34:memory/log.d/2026-06-14T2330Z-portal-no-number-wrong.md | prose-000: Four linked pages exist at the pin and reproduce the four named constructions; the author log lists them. - CONFIRMED | "They share a sentence." | git show f5e5f84f34:memory/log.d/2026-06-14T2330Z-portal-no-number-wrong.md | prose-001: Read as the shared refrain introduced in the following paragraph; the four quotations are traced separately below. - CONFIRMED | "In each, every individual number is honest \u2014 and the conclusion is still wrong." | git show f5e5f84f34:memory/log.d/2026-06-14T2330Z-portal-no-number-wrong.md | prose-002: The cited examples demonstrate misleading inference from correct arithmetic; this is their framing, not a guarantee about every slider state on this portal. - CONFIRMED | "Read the four pages this portal walks, and the same phrase keeps surfacing, almost word for word." | git show f5e5f84f34:memory/log.d/2026-06-14T2330Z-portal-no-number-wrong.md | prose-003: All four refrain excerpts were checked in their pinned source pages or metadata, with the stated almost-verbatim qualification. - CONFIRMED | "It is the signature of a whole family of statistical paradoxes, and it is worth saying out loud what it means: the deception is never in the arithmetic." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-008: Anscombe section 3 supplies the advertised example of identical rounded regression output with different graphical behavior; the family framing is the portal author's inference. - CONFIRMED | "Every count is counted correctly." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-009: The plotted datasets contain eleven points each; the integer admission example and published counts reproduce correctly. - CONFIRMED | "Every average is the true average of exactly what it averages." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-010: Recomputed the displayed means directly from AX, AY1, AY2, A and B; the numbers themselves agree with their defining arithmetic. - CONFIRMED | "You can audit each figure to the last digit and find nothing wrong \u2014 and still be led to a false conclusion." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-011: Anscombe section 3 explicitly contrasts the shared printed output with different adequacy of the assumed regression model. - CONFIRMED | "So the lie is not in the numbers." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-012: Supported as an interpretation of the four examples; the mistake is treating the summaries as sufficient for an unstated inference. - CONFIRMED | "It is in the operation \u2014 in the act of summarizing itself, and in what that act quietly throws away." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-013: Supported as the author's interpretation; Anscombe's model-assumption discussion explains how a legitimate calculation can mislead. - CONFIRMED | "Name the operation, and you can name in advance what it will hide." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-015: For the four operations actually tabulated, the named lost information follows from their definitions. This does not establish an exhaustive taxonomy. - CONFIRMED | "summarise mean, variance, correlation the shape of the cloud The Shape the Numbers Can't See" | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-017: Anscombe's table and figures show rounded moments and regression summaries shared across visibly different shapes. - CONFIRMED | "aggregate the pooled rate the grouping (a confounder) The Bias in the Sum" | https://faculty.ucmerced.edu/jvevea/classes/Spark/readings/Bickel-Berkeley.pdf | prose-018: Bickel's analysis disaggregates department-specific admission decisions; aggregation loses the department label. The parenthetic causal label requires the qualification in the Berkeley finding. - CONFIRMED | "select who is in the sample the gate (a collider) The Bias in the Sample" | https://www.nature.com/articles/s41467-020-19478-2 | prose-019: Nature defines a collider using two incoming causes and discusses restricting analysis to selected individuals. - CONFIRMED | "reclassify each individual value that the boundary moved The Migration That Heals No One" | https://pubmed.ncbi.nlm.nih.gov/4000199/ | prose-020: Feinstein's abstract describes stage reassignment without changing individual outcomes; the excerpt supports this operation. - CONFIRMED | "A summary statistic \u2014 a mean, a variance, a correlation \u2014 is a function that takes a whole dataset and returns a few numbers." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-022: The page's own mean, variance and corr functions explicitly map arrays of observations to scalar summaries. - CONFIRMED | "It is many-to-one." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-023: Distinct datasets can share a summary; Anscombe's construction supplies examples at the stated printed precision. - CONFIRMED | "An enormous space of completely different datasets map to the very same summary, and the moment you replace the data with the summary you have chosen one point in that space to stand for all of them." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-024: Many-to-one is mathematically justified, for example by permuting paired observations without changing any of these summaries; the one-point wording describes representation by the summary. - CONFIRMED | "That is exactly what a projection is: a shadow keeps a few coordinates and discards the rest, and many objects cast the same shadow." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-025: Read as the explicit shadow analogy for a many-to-one map, not as a claim that a variance operator is a linear idempotent projection. - CONFIRMED | "Anscombe's quartet is the existence proof." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-026: The original paper's four datasets support the existence of distinct configurations with the same rounded printed regression output. - CONFIRMED | "Here are two of its four datasets." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-027: AX, AY1 and AY2 reproduce the first two columns of the original table, including all eleven y-values. - CONFIRMED | "One is a noisy straight line." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-029: The original Figure 1 and the embedded first dataset show noisy scatter around a straight fitted line. - CONFIRMED | "The four numbers cannot tell them apart." | https://www.lithoguru.com/scientist/statistics/Anscombe_Graphs%20in%20Statistical%20Analysis_1973.pdf | prose-031: The displayed, rounded figures cannot distinguish the two clouds; extra precision can distinguish their correlations, as noted separately. - CONFIRMED | "Throw the structure away, and the summary doesn't merely lose detail." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-035: Pearl explains why information about the causal structure is required to select a meaningful adjustment. - CONFIRMED | "It can point the wrong way." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-036: Pearl discusses reversal of associations under aggregation; the local integer example also reproduces it. - CONFIRMED | "(They are duals at the level of the diagram; the two paradoxes are not arithmetic mirror images, and the portal does not claim they are.)" | https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | prose-041: As a graph analogy, a fork and its fully edge-reversed collider can be called duals. This narrower disclaimer does not repair the separate one-arrow assertion. - CONFIRMED | "The honest association lives inside each value of Z; pool across Z and you get a lie." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-043: Within the explicitly assumed pure confounder model, adjustment blocks the spurious path. Pearl's conditions support this conditional statement, not every possible Simpson reversal. - CONFIRMED | "The remedy is to condition on Z." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-044: Correct for the stipulated back-door confounder and the total-effect target. The causal classification of Berkeley itself is a separate error. - CONFIRMED | "In Berkson, Z (the admission, the shortlist, the pool) is a collider: a common effect of both." | https://www.nature.com/articles/s41467-020-19478-2 | prose-045: Both Nature's definition and Pearl's collider discussion support the common-effect definition. - CONFIRMED | "The honest association is the marginal one; condition on Z and you manufacture a lie." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-046: For the displayed independent-cause collider, conditioning introduces dependence; Pearl gives the corresponding common-consequence explanation. - CONFIRMED | "The remedy is the precise opposite: do not condition on Z." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-047: Correct for estimating the total X-to-Y effect in this pure collider example; it is not a prohibition on every descriptive conditional analysis. - CONFIRMED | "Below is one instrument for both." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-048: The page supplies one shared diagram/matrix instrument with three structure buttons. - CONFIRMED | "Flip between confounder and mediator: the correlation matrix does not change by a hair, and the partial correlation is zero in both." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-053: Executing the pinned JS gives pooled 0.300 and partial 0.000 for both confounder and mediator with identical matrices. - CONFIRMED | "The two structures are Markov-equivalent \u2014 they impose the identical pattern of association, so the observed numbers cannot tell them apart." | https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | prose-054: Verma and Pearl Figure 1 groups the no-collider chains and fork together, separately from the collider. This supports the confounder/mediator comparison. - CONFIRMED | "Yet they demand opposite handling." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-055: Pearl's back-door discussion distinguishes adjustment for a confounder from blocking a mediator when estimating the total effect. - CONFIRMED | "If Z is a confounder, the within-Z figure is the honest one and you should report it." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-056: Correct in the explicitly stipulated pure confounder model and total-effect interpretation. - CONFIRMED | "If Z is a mediator \u2014 if X causes Y through Z \u2014 then conditioning on Z blocks the very effect you were trying to measure, and the pooled figure was right all along." | https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf | prose-057: Correct in the displayed pure mediator model for the TOTAL effect. Conditioning would instead concern a different estimand, the direct effect. - CONFIRMED | "The arrow is not in the data; it has to come from outside it." | https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | prose-060: The immediately preceding confounder/mediator ambiguity is not identifiable from their observational distribution. The restriction to that pair matters. - CONFIRMED | "(This is precisely the honesty The Bias in the Sum ends on: conditioning on department explains the gap, it does not prove no bias existed \u2014 it may have only moved the bias upstream, where the table cannot see it.)" | git show f5e5f84f34:public/strata/the-bias-in-the-sum/index.html | prose-061: The linked pinned page explicitly explains the limitation of department adjustment. Bickel also cautions against ruling out discrimination elsewhere. - CONFIRMED | "The collider is the one structure that breaks the symmetry: it is the lone case where X and Y are uncorrelated until you condition." | https://ftp.cs.ucla.edu/tech-report/1991-reports/910020.pdf | prose-062: Among the displayed nonzero-strength pure models, only the collider has independent X and Y before conditioning. Verma and Pearl distinguish its independence pattern. - CONFIRMED | "That makes it detectable in principle \u2014 but in practice you rarely know you have conditioned on it, because selection is invisible: you only ever see the admitted patients, the people in the dating pool, the players in the hall of fame." | https://www.nature.com/articles/s41467-020-19478-2 | prose-063: Nature's selection examples support the practical risk of analyzing only selected people. The phrase rarely is qualitative commentary, not a quantified prevalence estimate. - CONFIRMED | "The bias is built into who got into your dataset at all." | https://www.nature.com/articles/s41467-020-19478-2 | prose-064: Nature explicitly discusses collider bias caused by restricting the analytic sample. - CONFIRMED | "The fourth paradox is the strangest, because in the first three the groups at least stand still while the summary lies about them." | git show f5e5f84f34:public/strata/no-number-wrong/index.html | prose-066: In these particular illustrations the first three keep their variable/group definitions, whereas Will Rogers explicitly changes classification. This is not a claim about all possible selection procedures. - CONFIRMED | "In the Will Rogers phenomenon, the groups themselves move." | https://pubmed.ncbi.nlm.nih.gov/4000199/ | prose-067: Feinstein's abstract describes patients moving between diagnostic stages. - CONFIRMED | "Buy a sharper scanner, and some patients quietly migrate from the \"early-stage\" column to the \"late-stage\" column \u2014 and the average survival of both columns rises at once, while not one patient is treated any differently and the whole group's survival does not budge." | https://pubmed.ncbi.nlm.nih.gov/4000199/ | prose-068: The abstract supports the conditional stage-migration example; the following page paragraph correctly gives the between-means condition needed for both means to rise. - CONFIRMED | "The good group's mean rises if and only if you removed someone below it; the bad group's mean rises if and only if you added someone above it." | git show f5e5f84f34:research/will-rogers-phenomenon/identity.mjs | prose-071: The exact delta identities in identity.mjs have the stated signs; denominators are positive for the specified nonempty groups. - CONFIRMED | "So both means rise exactly when the person you moved sits in the band between them \u2014 below the good group's mean, above the bad group's." | git show f5e5f84f34:research/will-rogers-phenomenon/identity.mjs | prose-072: Both inequalities hold precisely when mean(B)