{ "claims": [ { "claim": "Holding the treatment of a chosen fixed subset at its realized value and re-randomizing the remaining units gives a conditional randomization test, with a statistic of the analyst's choice, whose $p$ value is the exact significance level for the null of no interference; the procedure is 'an exact, distribution-free test'.", "claim_id": "P030-H1", "related_corrections": [ "P030-E001", "P030-E002", "P030-N002" ], "tied_to": "Abstract; introduction (printed p. 5); Method (printed pp. 6–7): $H_{0''}$, Equation (1), Equation (2) and the two-tailed rule after it; Table 1; Figure 3" }, { "claim": "Because the fixed units' own treatment is held constant, any dependence of their outcomes on the assignment of the other units can only be interference, so rejection is sufficient for detecting interference and is not confounded with direct effects.", "claim_id": "P030-H2", "related_corrections": [ "P030-N001" ], "tied_to": "Introduction (printed pp. 5–6); Method (printed p. 6): $H_{0''}$ and the sentence 'Because $Z_{\\mathit{fixed}}=Z'_{\\mathit{fixed}}$ ... $H_{0''}$ must be rejected'" }, { "claim": "Relating outcomes to proximity to treated units naively does not follow from the randomization: expected exposure depends on the number of neighbours and on own treatment, and under the null the statistic is not centred at zero (all six $\\rho_0$ values negative in the example; null centre about 0.014 in the application). Conditional randomization accounts for these design-induced relationships.", "claim_id": "P030-H3", "related_corrections": [ "P030-E004", "P030-N003" ], "tied_to": "The three exposure expectations on printed p. 5; Table 1; Figure 3" }, { "claim": "Simulations show the method behaves as intended: without interference the $p$ values are approximately uniform (Kolmogorov–Smirnov $p=0.661$); with interference under a Gaussian distance decay and $N=400$ the two-tailed test at $\\alpha=0.05$ rejects in 83% of simulations; detection rises with $N$ under all three decay forms and appears to converge to 100%.", "claim_id": "P030-H4", "related_corrections": [ "P030-E001", "P030-E006", "P030-E007", "P030-E010", "P030-N005", "P030-N006" ], "tied_to": "Abstract; Simulation Studies (printed pp. 10-12); Figure 1; Figure 2" }, { "claim": "Applied to the Gerber, Green and Larimer (2008) mailing experiment with the civic duty, Hawthorne and self arms as the fixed subset, the test detects interference from the neighbors treatment: observed $\\rho=0.0061$ lies far to the left of a null distribution centred near 0.014 (1,000 conditional randomizations), $p<0.001$. The same analysis reports a direct effect of about 8.1 percentage points.", "claim_id": "P030-H5", "related_corrections": [ "P030-E008", "P030-E009" ], "tied_to": "Abstract; Application (printed pp. 12–14); Figure 3; the reported $N$, $N_{\\mathit{fixed}}$, $N_{\\mathit{control}}$, $N_{\\mathit{treatment}}$; the difference-in-means estimate" }, { "claim": "The method can be applied after an experiment with a suitably chosen fixed subset; for a binary treatment a random subset of units from one condition is recommended, with $N_{\\mathit{fixed}}$ chosen to maximize $N_{\\mathit{fixed}}N_{Z=0,\\mathit{variant}}N_{Z=1,\\mathit{variant}}$, which for a random sample of controls means fixing half of the control units.", "claim_id": "P030-H6", "related_corrections": [], "tied_to": "Method and Practical Considerations (printed pp. 6-9); Equation (3); conclusion (printed p. 15)" }, { "claim": "Although interference is detected, the resulting bias in mailing-and-turnout experiments is likely negligible, because the indirect effects found elsewhere are small (about 0.5 percentage points for the neighbors treatment) or absent.", "claim_id": "P030-H7", "related_corrections": [], "tied_to": "Application (printed p. 14); Conclusion (printed p. 15); Aronow (2011) working paper; McConnell et al. (2010)" }, { "claim": "An ex post interference diagnostic can inform future experimental design and help assess whether standard treatment-effect analyses are vulnerable to interference.", "claim_id": "P030-H8", "related_corrections": [ "P030-E001" ], "tied_to": "Conclusion (printed p. 15); Application (printed pp. 14-15); Equations (1)-(2); Figure 3" } ], "date_published": "2026-10-25", "findings": [ { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Four lines of text ending 'With the t for a given realization of Z denoted t_Z, it follows that we may define the salient null hypothesis', then the displayed Equation (1), 'H_0 : t_Z = t_Z-prime, for all Z, Z-prime in Omega-prime', then two further lines of text. A red box marks 't_Z = t_Z-prime' in the display.", "caption": "Equation (1) with the sentence that introduces it, p. 7 (PDF p. 5). The equality $t_Z=t_{Z'}$ is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E002.png" }, { "alt": "Table 1, titled 'Conditional Randomizations and Associated Test Statistics Under H_0', with columns Z, Y_fixed, D_nearest and rho_0 and six rows. Y_fixed is {1, 3, 2, 1} in every row. A red box marks the rho_0 column, whose entries are -0.105, -0.105, -0.105, -0.833, -0.949 and -0.949.", "caption": "Table 1, p. 9 (PDF p. 7), whose title says that its statistics are computed under $H_0$. The column of $\\rho_0$ is marked: it takes three different values over the six assignments in $\\Omega'$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E002-table1.png" } ], "formal_release": null, "formal_statements": [ { "id": "P030-T5", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T5" }, { "id": "P030-T6", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T6" } ], "headline": "Equation (1) states the null hypothesis as equality of the test statistic over the assignments in $\\Omega'$; the hypothesis that the procedure tests is equality of the fixed units' outcomes, under which the statistic still varies.", "id": "P030-E002", "location": "Equation (1) and the sentence that introduces it, p. 7 (PDF p. 5)", "pdf_page": 5, "printed_page": "7", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P030-T5", "P030-T6" ], "text": { "as_printed": "With the $t$ for a given realization of $Z$ denoted $t_Z$, \\hl{it follows that} we may define the salient null hypothesis\n\\[\nH_0:\\hl{t_Z=t_{Z'}},\\ \\forall Z,Z'\\in\\Omega'. \\qquad (1)\n\\]", "corrected": "With the $t$ for a given realization of $Z$ denoted $t_Z$, \\hl{the salient null hypothesis is}\n\\[\nH_0:\\hl{Y_{i,Z}=Y_{i,Z'}}\\ \\hl{\\text{for every } i \\text{ with } S_i=1},\\ \\forall Z,Z'\\in\\Omega'. \\qquad (1)\n\\]\nThis hypothesis follows from $H_{0''}$, because every assignment in $\\Omega'$ leaves the treatment of the fixed units unchanged. Under $H_0$, the observed $Y_{\\mathit{fixed}}$ is held constant and $t_Z$ is recomputed for each $Z\\in\\Omega'$. The values $t_Z$ need not be equal.", "evidence": "The evidence is a counterexample computed in exact arithmetic. I consider ten units at the positions $1,\\ldots,10$, of which one is treated, each with probability $1/10$. Let the outcome of unit $i$ be $b_i$ under every assignment, where $b=(1,2,2,2,2,3,2,3,2,1)$. No outcome depends on any treatment, so there is no interference. Fix the units $1,8,9,10$ and suppose that unit 7 is treated. Then $\\Omega'$ consists of the six assignments that treat one of the units $2,\\ldots,7$, each with conditional probability $1/6$, and $Y_{\\mathit{fixed}}=(1,3,2,1)$ under all six. The Spearman correlation between $Y_{\\mathit{fixed}}$ and the distance to the treated unit is $-1/\\sqrt{90}$ when the treated unit is 2, 3 or 4, it is $-5/6$ when the treated unit is 5, and it is $-3/\\sqrt{10}$ when the treated unit is 6 or 7. These are the values $-0.105$, $-0.833$ and $-0.949$ of Table 1. Since $1/90\\ne 25/36$, I have $t_Z\\ne t_{Z'}$ for two assignments in $\\Omega'$, although the outcomes do not depend on the assignment. Counting the two assignments with $t_Z\\le -3/\\sqrt{10}$, I obtain $p=2/6$, which is the article's $0.333$.", "problem": "The hypothesis of no interference, $H_{0''}$ on p.~6, concerns outcomes: the outcome of a unit does not change when its own treatment does not. Since every assignment in $\\Omega'$ leaves the treatment of the fixed units unchanged, $H_{0''}$ implies that $Y_{\\mathit{fixed}}$ is the same for all $Z\\in\\Omega'$. It does not imply that the statistic is the same. The statistic $t_Z$ depends on $Z_{\\mathit{variant}}$ as well as on $Y_{\\mathit{fixed}}$, for example through $D_{\\mathit{nearest}}$, and it can vary over $\\Omega'$ whether or not there is interference; the two sentences before the display say as much. Equation (1) does not follow from them, and it is not the hypothesis that the procedure tests. If Equation (1) held as printed, the reference distribution would be a point mass at the observed value and the $p$ value of Equation (2) would always equal 1. The article itself uses the hypothesis on outcomes. Table 1 lists three different values of the statistic under $H_0$, and p.~10 says that ``in each randomization, we are assuming $H_0$, so $Y_{\\mathit{fixed}}$ remains constant across randomizations''.", "what_survives": "The correction adds no assumption and changes no number. The hypothesis $H_{0''}$ and the conditioning on $Z_{\\mathit{fixed}}$ are not affected, and Table 1 and $p=0.333$ are already computed under the hypothesis on outcomes. The simulations and the application are described with a reference distribution that varies over the assignments, which agrees with the corrected hypothesis; their numbers were not recomputed for this correction." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E002", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "The displayed Equation (2), 'p = Pr(t_0 <= t) = |{t_0 in T_0 : t_0 <= t}| / |Omega-prime|', followed by a paragraph of seven lines. A red box marks 'p <= 1 - alpha/2' in the sentence 'fail to reject H_0 if alpha/2 <= p <= 1 - alpha/2, reject otherwise'. A second red box marks the word 'exact' in the sentence 'This procedure is an exact, distribution-free test for detecting interference between units.'", "caption": "Equation (2) and the paragraph after it, p. 7 (PDF p. 5). The upper condition of the two-tailed rule, $p\\le 1-\\alpha/2$, and the word “exact” are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E001.png" }, { "alt": "A chart with seven rows and a horizontal axis from 0 to 1 labelled 'Exact probability of rejecting when there is no interference', with a vertical line at the nominal level 0.05. Each row has an orange circle for the two-tailed rule as printed and a blue square for the corrected rule, and the exact values are listed at the right. Five units with the statistic equal to 1 in both assignments: printed 1, corrected 0. Forty-five units with the statistic equal to -1 or 1: printed 1, corrected 1/42. Reference distribution of the article's Table 1: printed 1/2, corrected 0. Six equally likely values with no ties: printed 1/6, corrected 0. Forty values: printed 1/40, corrected 1/20. Fifty values: printed 3/50, corrected 1/25. Five thousand values: printed 249/5000, corrected 1/20.", "caption": "Exact probability of rejection when there is no interference, at a nominal level of $0.05$, for the two-tailed rule as printed on p. 7 and for the corrected rule. The first three rows are the three examples above; the last four are reference distributions with $n$ equally likely distinct values. The printed rule rejects with probability $1$, $1$, $1/2$ and $1/6$ in the first four rows and with probability $3/50$ for $n=50$; it is below $0.05$ for $n=40$ and $n=5{,}000$. The corrected rule never exceeds $0.05$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E001-null-rejection.png" } ], "formal_release": null, "formal_statements": [ { "id": "P030-T1", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T1" }, { "id": "P030-T2", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T2" }, { "id": "P030-T3", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T3" }, { "id": "P030-T4", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T4" } ], "headline": "The two-tailed rule printed after Equation (2) can reject with probability greater than $\\alpha$ under the null hypothesis when the reference distribution is discrete; the upper side of the rule must use the inclusive upper-tail probability $\\Pr(t_0\\ge t)$.", "id": "P030-E001", "location": "The two-tailed rule after Equation (2), p. 7 (PDF p. 5)", "pdf_page": 5, "printed_page": "7", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P030-T1", "P030-T2", "P030-T3", "P030-T4" ], "text": { "as_printed": "\\[\np=\\Pr(t_0\\le t)=|\\{t_0\\in T_0:t_0\\le t\\}|/|\\Omega'|. \\qquad (2)\n\\]\n\\ldots\\ In these cases, a two-tailed significance test may be implemented using $p$: fail to reject $H_0$ if $\\alpha/2\\le\\hl{p\\le 1-\\alpha/2}$, reject otherwise. This procedure is an \\hl{exact}, distribution-free test for detecting interference between units.", "corrected": "Let $p_L=\\Pr(t_0\\le t)$, as in Equation (2), and let $\\hl{p_U=\\Pr(t_0\\ge t)}$. Both are computed under the conditional distribution of $Z$ on $\\Omega'$, and every assignment is counted, so that tied values of the statistic are retained. In these cases, a two-tailed significance test may be implemented using $\\hl{p_2=\\min\\{1,2\\min(p_L,p_U)\\}}$: reject $H_0$ if $\\hl{p_2\\le\\alpha}$, fail to reject otherwise. Suppose that the outcomes of the fixed units do not depend on $Z_{\\mathit{variant}}$, which is the hypothesis of no interference for these units, and that the probabilities of the assignments in $\\Omega'$ are those of the design. Then this test rejects with conditional probability \\hl{at most $\\alpha$}, for every finite reference distribution. The procedure is exact in this sense: its level is at most $\\alpha$ in finite samples, and it need not equal $\\alpha$. The rule that rejects if $p_L<\\alpha/2$ or $p_U<\\alpha/2$ keeps the strict endpoints of the printed rule and has the same property.", "evidence": "The evidence is three exact counterexamples and a derivation of the corrected rule. All probabilities are exact, and $\\alpha=1/20$ throughout.\n\nFirst, I consider five units on a line at the positions $0,4,8,-1,-2$. The units at $0,4,8$ form the fixed subset; they are never treated, and their outcomes are $0,1,2$ under every assignment. Exactly one of the units at $-1$ and $-2$ is treated, each with probability $1/2$, and both have outcome $0$ under every assignment. No outcome depends on any treatment, so $H_0$ holds. The distances from the fixed units to the treated unit are $(1,5,9)$ or $(2,6,10)$, and $\\rho=1$ under both assignments. The example therefore satisfies Equation (1) even as it is printed (see \\fid{P030-E002}). Then $p=1>39/40$ under both assignments, and the printed rule rejects with probability 1.\n\nSecond, I keep the fixed units and place 42 other units at $-1,-2,\\ldots,-41$ and $9$, each with outcome $0$ under every assignment. One of them is treated, each with probability $1/42$. The statistic equals $1$ when the treated unit is at a negative position and $-1$ when it is at $9$. For $\\rho=-1$ I have $p=1/42<1/40$, and for $\\rho=1$ I have $p=1$. The printed rule rejects both values, again with probability 1, although the statistic is not constant.\n\nThird, the reference distribution of Table 1 arises without interference when the ten outcomes $(1,2,2,2,2,3,2,3,2,1)$ are the same under every assignment. (The article's example generates these outcomes with interference; the population used here is a separate one.) The table below lists the six equally likely assignments. The printed rule rejects the three with $p=1$, which have total probability $1/2$.\n\n\\begin{audittable}{crrrll}{The reference distribution of Table 1 at $\\alpha=0.05$, with $p_L=\\Pr(t_0\\le t)$ and $p_U=\\Pr(t_0\\ge t)$. The printed rule rejects when $p_L<0.025$ or $p_L>0.975$; the corrected rule rejects when $2\\min(p_L,p_U)\\le 0.05$.}\nTreated unit & $\\rho_0$ & $p_L$ & $p_U$ & Printed rule & Corrected rule \\\\\n2 & $-0.105$ & $1$ & $1/2$ & \\hl{reject} & do not reject \\\\\n3 & $-0.105$ & $1$ & $1/2$ & \\hl{reject} & do not reject \\\\\n4 & $-0.105$ & $1$ & $1/2$ & \\hl{reject} & do not reject \\\\\n5 & $-0.833$ & $1/2$ & $2/3$ & do not reject & do not reject \\\\\n6 & $-0.949$ & $1/3$ & $1$ & do not reject & do not reject \\\\\n7 & $-0.949$ & $1/3$ & $1$ & do not reject & do not reject \\\\\n\\end{audittable}\n\nFor the corrected rule, let the reference distribution have the distinct values $x_1<\\cdots1-\\alpha/2$, that is, when $\\Pr(t_0>t)<\\alpha/2$. This is a condition on the strict upper tail, which leaves out the probability that the reference distribution gives to the observed value. When the reference distribution is discrete, this probability can be large. In particular, the largest value of the statistic has $p=1$ and is always rejected, however probable it is. The rule can then reject with probability greater than $\\alpha$ when there is no interference, and the statement that the two-tailed procedure is exact does not hold. Excluding ties does not remove the problem. With $n$ equally likely distinct values, the rule rejects with probability $(2\\lceil n\\alpha/2\\rceil-1)/n$, which is $1/6$ for $n=6$ and $\\alpha=0.05$. Not every discrete distribution leads to a rejection probability above the nominal level: for $n=5{,}000$ the same expression equals $249/5000$. The one-tailed $p$ value of Equation (2) is not affected.", "what_survives": "The conditional randomization construction and the one-tailed $p$ value of Equation (2) are valid under the same assumptions, and Table 1 with its $p=0.333$ is unchanged. The application reports a lower-tail $p<0.001$ on p.~13, which the correction does not change; taking that value as given, the corrected two-tailed $p$ value is below $0.002$. The correction needs no new assumption. Its cost is that some two-tailed rejections are withdrawn. With strict endpoints, every rejection by the corrected rule is also a rejection by the printed rule, and in the first example no rejection remains. The article reports two-tailed rejection rates for its simulations (83\\% on p.~11, and Figure 2). The simulation code is not available, and how these rates change under the corrected rule has not been established." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E001", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Five lines of text. In the third line a red box marks the denominator '(var(r_D_nearest) var(r_Y_fixed))' of the formula 'rho = cov(r_D_nearest, r_Y_fixed) / (var(r_D_nearest) var(r_Y_fixed))'. In the fourth line a second red box marks the word 'monotone' in 'whenever D_nearest has a monotone relationship with Y_fixed'.", "caption": "Practical Considerations, first paragraph, p. 8 (PDF p. 6). The denominator of the formula for $\\rho$ and the word “monotone” are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P030-T8", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T8" }, { "id": "P030-T9", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T9" } ], "headline": "The formula for the rank correlation lacks the square root in its denominator, and $\\rho$ equals $1$ or $-1$ under a strictly monotone relationship, not under every monotone one.", "id": "P030-E004", "location": "Practical Considerations, first paragraph, p. 8 (PDF p. 6): the formula for $\\rho$ and the sentence after it", "pdf_page": 6, "printed_page": "8", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P030-T8", "P030-T9" ], "text": { "as_printed": "Whenever values are tied, the average rank is used. $\\rho$ then is the Pearson correlation coefficient of the ranks of these variables: $\\rho=\\operatorname{cov}(r_{D_{\\mathit{nearest}}},r_{Y_{\\mathit{fixed}}})/\\hl{(\\operatorname{var}(r_{D_{\\mathit{nearest}}})\\operatorname{var}(r_{Y_{\\mathit{fixed}}}))}$. $\\rho$ has a perfect correlation coefficient (of 1 or -1) whenever $D_{\\mathit{nearest}}$ has a \\hl{monotone} relationship with $Y_{\\mathit{fixed}}$.", "corrected": "Whenever values are tied, the average rank is used. $\\rho$ then is the Pearson correlation coefficient of the ranks of these variables,\n\\[\n\\rho=\\frac{\\operatorname{cov}(r_{D_{\\mathit{nearest}}},r_{Y_{\\mathit{fixed}}})}{\\hl{\\sqrt{\\operatorname{var}(r_{D_{\\mathit{nearest}}})\\operatorname{var}(r_{Y_{\\mathit{fixed}}})}}},\n\\]\nwhich is defined when neither vector of ranks is constant. $|\\rho|=1$ exactly when the centered vectors of ranks are proportional. This holds whenever $D_{\\mathit{nearest}}$ has a \\hl{strictly} monotone relationship with $Y_{\\mathit{fixed}}$, under which ties in one variable correspond to ties in the other. A relationship that is monotone but not strictly monotone need not give $|\\rho|=1$.", "evidence": "The evidence is a recalculation from Table 1 and a short derivation. In the last row of Table 1, the ranks of $D_{\\mathit{nearest}}=(6,1,2,3)$ are $(4,1,2,3)$ and the ranks of $Y_{\\mathit{fixed}}=(1,3,2,1)$ are $(3/2,4,3,3/2)$. With divisor $n$, their covariance is $-9/8$ and their variances are $5/4$ and $9/8$. The printed expression gives $(-9/8)/(45/32)=-4/5$, and $-3/5$ with divisor $n-1$. The Pearson correlation is $(-9/8)/\\sqrt{45/32}=-3/\\sqrt{10}$, which is the value $-0.949$ in the table. All six rows of Table 1 agree with the formula that has the square root.\n\nFor the second sentence, let $D=(1,2,3)$ and $Y=(0,0,1)$, so that $Y$ is a nondecreasing function of $D$. The average ranks are $(1,2,3)$ and $(3/2,3/2,3)$, and $\\rho=\\sqrt{3}/2$. The case of equality in the Cauchy--Schwarz inequality implies that $|\\rho|=1$ exactly when the centered vectors of ranks are proportional.", "problem": "The sentence names the Pearson correlation of the ranks, which divides the covariance by the square root of the product of the variances. The displayed expression divides by the product itself, and it is not a correlation. The next sentence is too broad when tied values are given average ranks. A relationship that is monotone but not strictly monotone can produce ties in one variable and not in the other, and then $|\\rho|<1$.", "what_survives": "Table 1, $p=0.333$ and the test itself use the Pearson correlation of the ranks and do not change. I found no reported number from the simulations or the application that is shown to depend on the printed expression." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E004", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Five lines of text. In the first line a red box marks the word 'exponential' in 'The third model is an exponential model'. In the second line a second red box marks the formula 'sum over j of ((1 - delta_ij) Z_j / ((1.5 d_ij)^2))' that defines Theta_i. The next sentence reads 'In the exponential model, the effect asymptotically approaches infinity as d_ij approaches zero.'", "caption": "Simulation Studies, p. 10 (PDF p. 8): the definition of the third distance-decay model. The word “exponential” and the sum that defines $\\Theta_i$ are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E006.png" }, { "alt": "The lower row of Figure 2. At the left, a panel titled 'Exponential Model', marked with a red box, plots 'Raw Value' against 'Distance' from 0 to 4: a curve that leaves the top of the panel near distance 0.7 and falls towards zero. At the right, a panel titled 'Simulated Indirect + Spatial DGP' plots 'Successful Detection Rate' against N from 100 to 800 for three significance levels.", "caption": "Figure 2, p. 12 (PDF p. 10), lower panels. The title “Exponential Model” is marked; the curve below it is $1/(1.5d)^2$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E006-fig2.png" } ], "formal_release": null, "formal_statements": [ { "id": "P030-T10", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T10" } ], "headline": "The third simulated model is an inverse-square decay, not an exponential one, and its sums should exclude the term $j=i$.", "id": "P030-E006", "location": "Simulation Studies, the third distance-decay model, p. 10 (PDF p. 8); the name recurs on p. 11 (PDF p. 9) and in Figure 2, p. 12 (PDF p. 10)", "pdf_page": 8, "printed_page": "10", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P030-T10" ], "text": { "as_printed": "The third model is an \\hl{exponential} model, for which $\\Theta_i=\\hl{{\\textstyle\\sum_j}}\\bigl((1-\\delta_{ij})Z_j/((1.5d_{ij})^2)\\bigr)$ and $\\Psi_i=\\hl{{\\textstyle\\sum_j}}\\bigl((1-\\delta_{ij})/((1.5d_{ij})^2)\\bigr)$. In the \\hl{exponential} model, the effect asymptotically approaches $\\infty$ as $d_{ij}$ approaches zero. In all cases, both $\\Theta$ and $\\Psi$ are mean-normalized to one.", "corrected": "The third model is an \\hl{inverse-square} model, for which\n\\[\n\\Theta_i=\\hl{\\sum_{j\\ne i}}Z_j/(1.5d_{ij})^2 \\quad\\text{and}\\quad \\Psi_i=\\hl{\\sum_{j\\ne i}}1/(1.5d_{ij})^2 .\n\\]\nIn the \\hl{inverse-square} model, the effect asymptotically approaches $\\infty$ as $d_{ij}$ approaches zero. In all cases, both $\\Theta$ and $\\Psi$ are mean-normalized to one.\n\nThe same name replaces ``exponential'' on p.~11 (``DGPs with the exponential model'') and in Figure 2 (``Exponential Model'').", "evidence": "The evidence is a derivation from the printed formula and a check of the pages. Let $k(d)=4/(9d^2)$. Then $k(2d)/k(d)=1/4$ for every $d>0$, and the derivative of $\\log k(d)$ is $-2/d$, which is not constant as it is for an exponential decay. The curve in the lower-left panel of Figure 2 agrees with $k(2/3)=1$, $k(1)=4/9$ and $k(2)=1/9$. The page prints no restriction to $j\\ne i$.", "problem": "For $d>0$ the decay function is $1/(1.5d)^2=4/(9d^2)$, a power of the distance. It falls by a factor of 4 whenever the distance doubles, and it diverges as $d\\to 0$, as the article notes. An exponential decay $\\exp(-cd)$ is bounded at $d=0$ and falls by a constant factor for each unit of distance. The model is called exponential on pp.~10 and 11 and in the title of the lower-left panel of Figure 2. As described, the simulations do not include an exponential decay. Separately, the sums run over all $j$. For $j=i$ the summand is $(1-\\delta_{ii})/(1.5d_{ii})^2=0/0$, which the factor $1-\\delta_{ij}$ does not define; the sums meant are over $j\\ne i$.", "what_survives": "The formula and the plotted curve agree on the decay function, and no reported detection rate is shown to change. The simulation code is not available. The correction concerns the name and the notation, and it is not a check of the implementation. The simulations cover Gaussian, linear and inverse-square decay; they do not show how the method behaves under exponential decay." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E006", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The foot of Figure 1, with the axis label Uniform Distribution Quantiles under each of its two panels, then the caption: Figure 1. Quantile-quantile plots of p values from simulations with the Gaussian distance decay DGP (N = 200) against the uniform distribution. A red box marks '(N = 200)'. Two lines of the next paragraph are visible below.", "caption": "Caption of Figure 1, p. 11 (PDF p. 9): the sample size “($N = 200$)” is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E007.png" }, { "alt": "Five lines of text beginning 'First, we illustrate a set of results for a simulation study using the Gaussian model, where N = 400, 200 units are randomly assigned to treatment, and 200 units are assigned to control'. A red box marks 'N = 400,'. The last line gives N_fixed = 100.", "caption": "Last paragraph of p. 10 (PDF p. 8): the text introduces the same simulation with “$N = 400$”, which is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E007-body.png" } ], "formal_release": null, "formal_statements": [], "headline": "The text states $N = 400$ for the simulation shown in Figure 1 and the caption states $N = 200$; one of the two numbers is wrong.", "id": "P030-E007", "location": "Figure 1 caption, p. 11 (PDF p. 9); the simulation it belongs to is described at the foot of p. 10 (PDF p. 8)", "pdf_page": 9, "printed_page": "11", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Page 10, text: ``First, we illustrate a set of results for a simulation study using the Gaussian model, where\n$N = \\hl{400}$, 200 units are randomly assigned to treatment, and 200 units are assigned to control. For all\nsimulations, we choose the fixed subset as a random sample of control units; following the rule of thumb,\n$N_{\\mathit{fixed}} = 100$.''\n\nPage 11, caption: ``Figure 1. Quantile-quantile plots of $p$ values from simulations with the Gaussian distance decay\nDGP ($N = \\hl{200}$) against the uniform distribution''", "corrected": "The two descriptions are to be reconciled with the original simulation output. If the design in the text produced\nthe figure, the caption reads ``\\ldots with the Gaussian distance decay DGP ($N = \\hl{400}$) against the uniform\ndistribution''. If a simulation with $N = 200$ produced it, the text needs the counts of that simulation instead:\nunder the same rule of thumb, 100 treated units, 100 control units and $N_{\\mathit{fixed}} = 50$. Without that\noutput, neither number is relabelled.", "evidence": "A check of the two pages. The simulation code and its output are not available, and the two pages do not settle which\nsample size produced the plot.", "problem": "In the paper $N$ is the total number of units (p.~5). The text introduces Figure 1 with $N = 400$, of which 200 units are treated and 200 are controls, and the caption of the same figure states $N = 200$.", "what_survives": "Only the sample size attributed to Figure 1 is in doubt. The plot, the 83\\% rejection rate and the\nKolmogorov--Smirnov $p = 0.661$ are not shown to be wrong." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E007", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Six lines of text. Red boxes mark 'N = 344,056,' at the end of one line and 'N_fixed = 114,629, N_control = 191,228, and N_treatment = 38,198.' on the next line.", "caption": "Application, p. 13 (PDF p. 11): the four counts in the sentence that begins “To summarize” are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E008.png" } ], "formal_release": null, "formal_statements": [ { "id": "P030-T11", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T11" } ], "headline": "The three group counts sum to 344,055, one fewer than the printed $N = 344{,}056$.", "id": "P030-E008", "location": "Application, p. 13 (PDF p. 11), the sentence that begins “To summarize”", "pdf_page": 11, "printed_page": "13", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P030-T11" ], "text": { "as_printed": "To summarize, $N = \\hl{344{,}056}$, $N_{\\mathit{fixed}} = \\hl{114{,}629}$, $N_{\\mathit{control}} = \\hl{191{,}228}$, and\n$N_{\\mathit{treatment}} = \\hl{38{,}198}$.", "corrected": "The counts are to be reconciled with the analysis sample. If the total, the control count and the treatment count\nare kept, the sentence reads: ``To summarize, $N = 344{,}056$, $N_{\\mathit{fixed}} = \\hl{114{,}630}$,\n$N_{\\mathit{control}} = 191{,}228$, and $N_{\\mathit{treatment}} = 38{,}198$.'' The sum is also restored\nif any one of the other three entries is changed by one unit: $N = 344{,}055$, $N_{\\mathit{control}} = 191{,}229$ or\n$N_{\\mathit{treatment}} = 38{,}199$.", "evidence": "An exact calculation with the printed counts. The log of the source study records 344,084 voters before the stratum\nwas omitted: 114,640 in the three fixed conditions, 191,243 in control and 38,201 in the neighbors condition. Then the printed $N$ implies that 28 voters were omitted, and the printed group counts imply 29. I did not examine the voter-level data, and the omitted stratum and the incorrect counts are not identified.", "problem": "The fixed, control and treatment groups are disjoint and together make up the analysis sample, so their counts sum\nto $N$. A direct computation gives $114{,}629 + 191{,}228 + 38{,}198 = 344{,}055$. The counts are inconsistent\nby one; the arithmetic does not identify which entries are wrong.", "what_survives": "The method and the simulations do not use these counts. The discrepancy alone does not show that the estimate of\nthe direct effect, the rank correlation, Figure 3 or the $p$ value rests on a wrong sample. I did not recompute\nthose, since the distances and the analysis code are not available." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E008", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Figure 3: a histogram with a smooth curve over it, centred near 0.015 on a horizontal axis labelled Rank Correlation that runs from 0.00 to 0.03, with a dashed vertical line near 0.006. The vertical axis is labelled Density. Below, the caption reads: Figure 3. p distribution associated with H0 for the experimental evaluation. A red box marks the letter p.", "caption": "Figure 3 and its caption, p. 14 (PDF p. 12): the symbol $p$ in the caption is marked. The horizontal axis reads “Rank Correlation”.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E009.png" } ], "formal_release": null, "formal_statements": [], "headline": "The caption of Figure 3 prints $p$ where $\\rho_0$ is meant: the figure shows the distribution of the rank correlation under $H_0$.", "id": "P030-E009", "location": "Figure 3 caption, p. 14 (PDF p. 12)", "pdf_page": 12, "printed_page": "14", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Figure 3. $\\hl{p}$ distribution associated with $H_0$ for the experimental evaluation", "corrected": "Figure 3. $\\hl{\\rho_0}$ distribution associated with $H_0$ for the experimental evaluation", "evidence": "A check of the page: the caption, the axis label and the sentence on p.~13.", "problem": "The figure is a histogram of the rank correlation under the null hypothesis. Its horizontal axis is labelled\n``Rank Correlation'', and p.~13 introduces it as ``$T_0$, the approximate distribution of $\\rho_0$ associated with\n$H_0$''. The symbol in the caption is the italic $p$ that the caption of Figure 1 uses for $p$ values.", "what_survives": "Only the symbol in the caption changes. The histogram and the results drawn from it are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E009", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The top row of Figure 2: a small plot of the Gaussian kernel against distance, and a plot of successful detection rate against N from 100 to 800 with three rising curves. The top curve has diamonds; the middle and lowest curves both have open circles. Red boxes mark the four circles of the lowest curve at N = 100, 200, 300 and 400, and the legend line that shows an open triangle for alpha = 0.01.", "caption": "Figure 2, Gaussian model row, p. 12 (PDF p. 10): the markers of the lowest curve at $N = 100$ to $400$ and the legend entry for $\\alpha = 0.01$ are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-E010.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the Gaussian panel of Figure 2 the $\\alpha = 0.01$ curve is drawn with circles, the marker that the legend assigns to $\\alpha = 0.05$; it should carry triangles.", "id": "P030-E010", "location": "Figure 2, Gaussian model panel (top right) and its legend, p. 12 (PDF p. 10)", "pdf_page": 10, "printed_page": "12", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "A figure, described. The legend in the Gaussian panel assigns a diamond to $\\alpha = 0.20$, a circle to\n$\\alpha = 0.05$ and a triangle to $\\alpha = 0.01$. In that panel the top curve has diamonds, the middle curve has\ncircles and the lowest curve has \\hl{circles}. In the linear and exponential panels the lowest curve has triangles.", "corrected": "Draw the Gaussian $\\alpha = 0.01$ curve with \\hl{open triangles}, as the legend has it, and keep open circles for\n$\\alpha = 0.05$.", "evidence": "A check of the page. Enlarged, the markers of the lowest Gaussian curve at $N = 100$ to $400$, where the curves are separated, are circles.", "problem": "Two curves in the Gaussian panel carry the same marker, and the legend does not identify the lowest one. Its position does. Since the test rejects less often at a smaller $\\alpha$, the lowest curve is the $\\alpha = 0.01$ series.", "what_survives": "Only the marker is wrong. No plotted value is shown to be wrong, and nothing else in the paper depends on it." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-E010", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eleven lines of text from the Method section. Red boxes mark the expression 'Z_i = Z_i-prime', with the prime on the subscript i, in the second line (the definition of Omega-prime) and in the eighth line (the statement of the null hypothesis H_0''). Two lines below the second box the text reads 'Because Z_fixed = Z-prime_fixed, for all Z, Z-prime in Omega-prime', with the prime on Z.", "caption": "Method, second paragraph, p. 6 (PDF p. 4). The two conditions $Z_i=Z_{i'}$ are marked; two lines below the second mark, the same condition is printed correctly for the fixed subset, as $Z_{\\mathit{fixed}}=Z'_{\\mathit{fixed}}$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P030-T7", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T7" }, { "id": "P030-T12", "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T12" } ], "headline": "In the definition of $\\Omega'$ and in $H_{0''}$ the prime is printed on the unit index, $Z_i=Z_{i'}$, where $Z_i=Z'_i$ is meant.", "id": "P030-N001", "location": "Method, second paragraph, p. 6 (PDF p. 4): the definition of $\\Omega'$ and the statement of $H_{0''}$", "pdf_page": 4, "printed_page": "6", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P030-T12", "P030-T7" ], "text": { "as_printed": "Let $\\Omega'$ be all possible randomizations conditional on observing $Z_{\\mathit{fixed}}$. That is, $Z_{\\mathit{fixed}}$ is invariant for all $Z$ in $\\Omega'$; $Z_i=\\hl{Z_{i'}}$, $\\forall Z,Z'\\in\\Omega':S_i=1$. \\ldots\\ However, we are concerned with a different sharp null hypothesis: $H_{0''}:Y_{i,Z}=Y_{i,Z'},\\ \\forall i,Z,Z'\\in\\Omega:Z_i=\\hl{Z_{i'}}$.", "corrected": "Let $\\Omega'$ be all possible randomizations conditional on observing $Z_{\\mathit{fixed}}$. That is, $Z_{\\mathit{fixed}}$ is invariant for all $Z$ in $\\Omega'$; $Z_i=\\hl{Z'_i}$, $\\forall Z,Z'\\in\\Omega':S_i=1$. \\ldots\\ $H_{0''}:Y_{i,Z}=Y_{i,Z'},\\ \\forall i,Z,Z'\\in\\Omega:Z_i=\\hl{Z'_i}$.", "evidence": "The evidence is a check of the page. The units are indexed by $i\\in\\{1,2,\\ldots,N\\}$ on p.~4, and no index $i'$ is introduced. Two lines below the statement of $H_{0''}$, the same condition for the fixed subset is printed with the prime on the vector: $Z_{\\mathit{fixed}}=Z'_{\\mathit{fixed}}$, $\\forall Z,Z'\\in\\Omega'$.", "problem": "As printed, each condition compares the treatment of unit $i$ with the treatment of a unit $i'$ in the same vector $Z$, and no unit $i'$ is defined. The condition meant is that unit $i$ has the same treatment under the two assignments $Z$ and $Z'$.", "what_survives": "Nothing depends on the misprint. The article uses $\\Omega'$ and $H_{0''}$ in their intended form in everything that follows." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-N001", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Four lines of text ending 'and assuming that all Z in Omega-prime occur with equal probability,', followed by the displayed Equation (2). A red box marks the numerator '|{t_0 in T_0 : t_0 <= t}|'.", "caption": "Equation (2) with the sentences that introduce it, p. 7 (PDF p. 5). The numerator is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N002.png" } ], "formal_release": null, "formal_statements": [], "headline": "The numerator of Equation (2) is the number of assignments in $\\Omega'$ whose statistic is at most the observed one; assignments with equal statistics are each counted.", "id": "P030-N002", "location": "Equation (2), the numerator, p. 7 (PDF p. 5)", "pdf_page": 5, "printed_page": "7", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\[\np=\\Pr(t_0\\le t)=\\hl{|\\{t_0\\in T_0:t_0\\le t\\}|}/|\\Omega'|. \\qquad (2)\n\\]", "corrected": "\\[\np=\\Pr(t_0\\le t)=\\hl{|\\{Z\\in\\Omega':t_0(Z)\\le t\\}|}/|\\Omega'|, \\qquad (2)\n\\]\nwhere $t_0(Z)$ denotes the statistic computed under $H_0$ for the assignment $Z$.", "intended_meaning": "The numerator is written as the size of a set of values of the statistic, and a set in the usual sense does not repeat equal values. The count meant is over assignments. The paragraph before the display builds $T_0$ from one $t_0$ for each $Z\\in\\Omega'$ and calls it a distribution, and the worked example counts ``2 of the 6 conditional randomizations'' on p.~10, although the six assignments of Table 1 have only three distinct values of $\\rho_0$. Read in this way, the display is the lower-tail probability, and I obtain the article's $p=2/6=0.333$. Counting distinct values would give $1/6$." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-N002", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Four lines of text. In the third line a red box marks an exclamation mark followed by the existential quantifier, in the sentence 'And for each Z in Omega-prime, ! exists rho_0, the rank correlation under H_0.'", "caption": "A Simple Example, second paragraph, p. 9 (PDF p. 7). The symbol $!\\exists$ is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N003.png" } ], "formal_release": null, "formal_statements": [], "headline": "The symbol “$!\\exists\\rho_0$” means that each assignment in $\\Omega'$ has exactly one value $\\rho_0$; the usual symbol is $\\exists!$.", "id": "P030-N003", "location": "A Simple Example, second paragraph, p. 9 (PDF p. 7)", "pdf_page": 7, "printed_page": "9", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "And for each $Z\\in\\Omega'$, $\\hl{!\\exists}\\rho_0$, the rank correlation under $H_0$.", "corrected": "And for each $Z\\in\\Omega'$, \\hl{there exists a unique} $\\rho_0$, the rank correlation under $H_0$.", "intended_meaning": "The exclamation mark is printed before the existential quantifier. This is not the usual order for the symbol of unique existence, and the mark could be taken for a negation. The article defines no such convention, and Table 1 lists one $\\rho_0$ for each of the six assignments. The sentence means that each $Z\\in\\Omega'$ determines exactly one rank correlation under $H_0$, and nothing false is printed." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-N003", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "simulation", "exhibits": [ { "alt": "Seven lines of text. Red boxes mark the words 'in over 95 percent of simulations' at the end of one line and 'once N reaches approximately 5,000.)' at the start of the next, inside the sentence '(Although not shown in the graph, the method detects interference with alpha = 0.05 for DGPs with the exponential model in over 95 percent of simulations once N reaches approximately 5,000.)'.", "caption": "Simulation Studies, p. 11 (PDF p. 9): the parenthetical on detection at large $N$, with “in over 95% of simulations once $N$ reaches approximately 5,000.)” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N005.png" } ], "formal_release": null, "formal_statements": [], "headline": "With the two-tailed test of the surrounding text, a recalculation detects interference under the third model in 93.0% of simulations at $N = 5{,}000$ and 92.25% at $N = 6{,}000$, not in over 95%.", "id": "P030-N005", "location": "Simulation Studies, p. 11 (PDF p. 9), the parenthetical that follows the description of Figure 2", "pdf_page": 9, "printed_page": "11", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "(Although not shown in the graph, the method detects interference with $\\alpha = 0.05$ for DGPs with the exponential\nmodel in \\hl{over 95\\% of simulations once $N$ reaches approximately 5,000}.)", "corrected": "(Although not shown in the graph, for DGPs with the \\hl{inverse-square} model and $\\alpha = 0.05$, a recalculation with\n5,000 conditional randomizations per simulation detects interference with the two-tailed test in\n\\hl{93.0\\% of 600 simulations at $N = 5{,}000$ and 92.25\\% of 400 simulations at $N = 6{,}000$} (standard errors\n1.04 and 1.34 percentage points). With the one-tailed $p$ value of Equation (2), the rates are 94.83\\% and 94.5\\%\n(standard errors 0.90 and 1.14). At $N = 8{,}000$, in 300 simulations, the rates are 97.00\\% two-tailed and\n97.33\\% one-tailed (standard errors 0.98 and 0.93).)\n\nThese rates are for the design of pp.~10--11: the kernel $1/(1.5 d_{ij})^2$ summed over $j \\neq i$, $\\Theta$ and\n$\\Psi$ each normalized to mean one, half of the units treated, and a random half of the controls fixed. The words\n``over 95\\% of simulations once $N$ reaches approximately 5,000'' are not kept. If the sentence is restated for\nthe one-tailed test, they are still not supported, since the one-tailed rates at $N = 5{,}000$ and $6{,}000$ are not shown to\nexceed 95\\%.", "evidence": "A recalculation by simulation of the design printed on pp.~10--11. Each unit has coordinates drawn uniformly on\n$[0, \\sqrt{N}]^2$ and noise $\\epsilon_i \\sim N(0,1)$. The outcome is\n\\[\nY_i = Z_i + \\Theta_i + \\Psi_i + \\epsilon_i, \\qquad\n\\Theta_i = \\sum_{j \\neq i} \\frac{Z_j}{(1.5 d_{ij})^2}, \\qquad\n\\Psi_i = \\sum_{j \\neq i} \\frac{1}{(1.5 d_{ij})^2},\n\\]\nwhere $\\Theta$ and $\\Psi$ are each divided by their mean over the $N$ units. Half of the units are treated, and a\nrandom half of the controls is the fixed subset. The statistic is the rank correlation between\n$Y_{\\mathit{fixed}}$ and the distance to the nearest treated unit, and each simulation uses 5,000\nconditional randomizations, the number that the paper reports for Figure 1. The one-tailed test rejects when the $p$ value of Equation (2) is\nbelow 0.05. The two-tailed test rejects when it is below 0.025 or above 0.975.\n\n\\begin{audittable}{rrrrrr}{Third model, $\\alpha = 0.05$: rejections among the simulations, with the rate and its Monte Carlo standard error in percentage points.}\n$N$ & Simulations & Two-tailed & Rate (s.e.) & One-tailed & Rate (s.e.) \\\\\n5,000 & 600 & 558 & 93.00\\% (1.04) & 569 & 94.83\\% (0.90) \\\\\n6,000 & 400 & 369 & 92.25\\% (1.34) & 378 & 94.50\\% (1.14) \\\\\n8,000 & 300 & 291 & 97.00\\% (0.98) & 292 & 97.33\\% (0.93) \\\\\n\\end{audittable}\n\nAt $N = 5{,}000$ and $N = 6{,}000$ the exact 95\\% intervals for the two-tailed rate are $[0.907, 0.949]$ and\n$[0.892, 0.947]$, and both end below 0.95. Testing the two counts together, 927 rejections in 1,000 simulations, with an exact binomial test against a rate of at least 0.95 at both values of $N$, I find $p = 0.001$. The one-tailed\nintervals, $[0.927, 0.965]$ and $[0.918, 0.965]$, contain 0.95. I checked the code against an exhaustive\nenumeration of the 84 conditional assignments of a small example.\n\nThe original simulation code and output are not available. The rates above are estimates from this recalculation,\nwith the Monte Carlo error stated, and three values of $N$ do not establish the rate at every larger $N$.", "problem": "The sentence concerns the third model, which the paper calls exponential and whose kernel is the inverse square\n$1/(1.5 d_{ij})^2$ (see \\fid{P030-E006}). It does not say whether its test is one-tailed or two-tailed, and\n``approximately 5,000'' has no stated tolerance. Since the paragraph before it reports a rejection rate ``when using a two-tailed test'', the natural reading is the two-tailed rule of p.~7 at $\\alpha = 0.05$.\n\nUnder that reading the stated rate is too high. In a recalculation of the design as printed, the two-tailed test\ndetects interference in 93.0\\% of simulations at $N = 5{,}000$ and in 92.25\\% at $N = 6{,}000$. With the\none-tailed $p$ value of Equation (2), the rates are 94.83\\% and 94.5\\%, which neither bear out ``over 95\\%'' nor\ncontradict it. At $N = 8{,}000$ the estimate of the two-tailed rate is 97.00\\%.", "what_survives": "The method, the worked example, the Gaussian result and the plotted curves of Figure 2 do not depend on this\nsentence. The application reports $p < 0.001$, which rejects under either tail rule. Only the sentence itself\nchanges: near $N = 5{,}000$ the detection rate of the two-tailed test under this model is about 93\\%, and the one-tailed rate\nis not shown to exceed 95\\%." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-N005", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "simulation", "exhibits": [ { "alt": "The middle row of Figure 2: a plot of the linear kernel falling from 2 at distance 0 to 0 at distance 2, and a plot of successful detection rate against N from 100 to 800. A red box encloses the three rising curves. The middle curve, with circles, runs from about 0.15 at N = 100 through about 0.53 at N = 400 to about 0.83 at N = 800.", "caption": "Figure 2, linear model row, p. 12 (PDF p. 10): the detection-rate panel is marked. By the legend of the Gaussian panel above it, diamonds, circles and triangles are $\\alpha = 0.20$, $0.05$ and $0.01$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N006.png" }, { "alt": "Seven lines of text that define the second and third models. Red boxes mark the factor [2 - d_ij > 0](2 - d_ij) in the formula for Theta_i and the same factor in the formula for Psi_i, which breaks across two lines. The last sentence reads: In all cases, both Theta and Psi are mean-normalized to one.", "caption": "Definition of the linear model, p. 10 (PDF p. 8): the weight $[2 - d_{ij} > 0](2 - d_{ij})$ is marked in $\\Theta_i$ and in $\\Psi_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N006-kernel.png" }, { "alt": "Three panels of detection rate against N on a log scale, titled Gaussian model, Linear model and Third model (inverse square). In the first and third panels the published and recomputed curves stay close. In the linear panel the published curve rises from 0.15 to 0.83 and the recomputed curve from 0.10 to 0.49; at N = 400 the two are labelled 0.53 and 0.27.", "caption": "Detection rates at $\\alpha = 0.05$ read from the published Figure 2 (blue, solid) and recomputed from the models as printed (orange, dashed; the band is the Wilson 95% interval for simulation uncertainty), at $N = 100$, 200, 400 and 800. The two are close for the Gaussian model and the third model and differ substantially for the linear model at these values of $N$.", "url": "https://pmaronow.github.io/llm-audit/works/P030/exhibits/P030-N006-linear-panel.png" } ], "formal_release": null, "formal_statements": [], "headline": "The linear panel of Figure 2 shows higher detection rates than the linear model printed on p. 10 produces: at $\\alpha = 0.05$, about 0.53 against 0.27 at $N = 400$, and 0.83 against 0.49 at $N = 800$.", "id": "P030-N006", "location": "Figure 2, linear model panel (middle right), p. 12 (PDF p. 10); the linear model is defined on p. 10 (PDF p. 8)", "pdf_page": 10, "printed_page": "12", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [], "text": { "as_printed": "Page 10: ``The second model is a linear model, for which\n$\\Theta_i = \\sum_j \\big( (1 - \\delta_{ij}) Z_j [2 - d_{ij} > 0](2 - d_{ij}) \\big)$ and\n$\\Psi_i = \\sum_j \\big( (1 - \\delta_{ij}) [2 - d_{ij} > 0](2 - d_{ij}) \\big)$, where $[\\cdot]$ is the Iverson\nbracket. \\ldots In all cases, both $\\Theta$ and $\\Psi$ are mean-normalized to one.''\n\nPage 12, Figure 2, linear model panel. The markers have no numerical labels; the values are my readings of the plotted\nmarkers, to within 0.01 for circles. At $\\alpha = 0.05$ the plotted detection rate is about \\hl{0.15} at\n$N = 100$, \\hl{0.30} at $N = 200$, \\hl{0.53} at $N = 400$ and \\hl{0.83} at $N = 800$.", "corrected": "If the linear model printed on p.~10 is the intended one, I would redraw the linear panel of Figure 2 from it and publish the simulation specification and code. Take\n$\\Theta_i = \\sum_j \\big( (1 - \\delta_{ij}) Z_j [2 - d_{ij} > 0](2 - d_{ij}) \\big)$ and\n$\\Psi_i = \\sum_j \\big( (1 - \\delta_{ij}) [2 - d_{ij} > 0](2 - d_{ij}) \\big)$, each normalized to mean one over all $N$ units.\nUse $Y_i = Z_i + \\Theta_i + \\Psi_i + e_i$, with independent $e_i \\sim \\mathcal{N}(0,1)$, independent uniform locations on a square of area $N$, exactly $N/2$ uniformly chosen treated units and $N/4$ uniformly chosen control units fixed.\nIn each of 5,000 conditional randomizations, hold the fixed units' outcomes and treatment status constant and uniformly assign $N/2$ treatments among the remaining $3N/4$ units, using the rank-correlation statistic and two-tailed rule described above. For this design my estimated two-tailed detection rate at $\\alpha = 0.05$\nis about \\hl{0.10} at $N = 100$, \\hl{0.14} at $N = 200$, \\hl{0.27} at $N = 400$ and \\hl{0.49} at $N = 800$\n(Monte Carlo standard errors 0.009, 0.010, 0.013 and 0.014). The table above gives the rates at $\\alpha = 0.20$ and $0.01$. I\ndid not compute the rates at $N = 300$, 500, 600 and 700.\n\nIf the plotted curves came from a different model, its specification should replace the formulas for $\\Theta_i$ and $\\Psi_i$ on\np.~10 and the kernel drawn at the left of the panel should match it. The evidence does not identify it, and I give no replacement\nformula.", "evidence": "A recalculation by simulation of the model as printed. At each of $N = 100$, 200, 400 and 800, I ran 1,200 independent\nsimulations, each with 5,000 conditional randomizations and the two-tailed rule of p.~7. The design follows the\ntext: units placed uniformly on a square of area $N$, $N/2$ units treated, a random half of the controls fixed,\nstandard normal noise, and the rank correlation between the outcomes of the fixed units and the distance to the\nnearest treated unit as the statistic. The table sets the rates read from the published figure beside the\nrecomputed rates.\n\n\\begin{audittable}{rrrrr}{Linear model: detection rate read from Figure 2 and the rate recomputed from the model as printed (1,200 simulations at each $N$, two-tailed test).}\n$N$ & $\\alpha$ & Figure 2 (read) & Recomputed & Wilson 95\\% interval \\\\\n100 & 0.20 & 0.389 & 0.282 & 0.257 to 0.308 \\\\\n100 & 0.05 & 0.154 & 0.098 & 0.083 to 0.116 \\\\\n100 & 0.01 & 0.054 & 0.030 & 0.022 to 0.041 \\\\\n200 & 0.20 & 0.550 & 0.364 & 0.337 to 0.392 \\\\\n200 & 0.05 & 0.295 & 0.141 & 0.122 to 0.162 \\\\\n200 & 0.01 & 0.128 & 0.052 & 0.041 to 0.066 \\\\\n400 & 0.20 & 0.779 & 0.524 & 0.496 to 0.552 \\\\\n400 & 0.05 & 0.530 & 0.266 & 0.242 to 0.292 \\\\\n400 & 0.01 & 0.302 & 0.104 & 0.088 to 0.123 \\\\\n800 & 0.20 & 0.953 & 0.722 & 0.696 to 0.746 \\\\\n800 & 0.05 & 0.826 & 0.488 & 0.460 to 0.517 \\\\\n800 & 0.01 & 0.624 & 0.265 & 0.241 to 0.291 \\\\\n\\end{audittable}\n\nThe readings carry a tolerance of 0.01 for diamonds and circles and 0.02 for triangles. At $\\alpha = 0.05$ the\nfigure's value lies above the 95\\% interval by more than that tolerance at every checked $N$, and at $N = 400$ and 800\nthis holds at all three values of $\\alpha$. With 20,000 conditional randomizations at $N = 400$, in 400\nsimulations, I find 0.275 at $\\alpha = 0.05$ (standard error 0.022). The number of randomizations does not explain the gap. I checked the code against a direct calculation on all 495 conditional assignments of an example\nwith 16 units. This is a statistical comparison and not an exact calculation of power. It does not say whether the\nfigure or the formula departs from what was simulated.", "problem": "Figure 2 reports, for each of the three models, how often the test detects interference as $N$ grows.\nFor the linear model the plotted rates lie above the rates that the model produces as it is printed on p.~10:\nweights $2 - d_{ij}$ out to distance 2, the unit's own term left out, and $\\Theta$ and $\\Psi$ each normalized to\nmean one. At $\\alpha = 0.05$ the figure shows about 0.53 at $N = 400$ and 0.83 at $N = 800$. For the printed model, I obtain 0.27 and 0.49. At the four checked values of $N$, the same recalculation comes within 0.024 of every reading for the Gaussian panel and the panel of the third model. The large gap is specific to the linear panel at those values.\nEither the plotted linear curves or the printed linear model is not what was simulated, and the paper's simulation\ncode is not available to say which. A cutoff of $\\sqrt{2}$ in place of 2 comes closer but overshoots the figure\n(0.60 at $N = 400$ and 0.88 at $N = 800$). It does not identify the simulated model either. Taken from this\npanel, the power of the test under linear decay is overstated for the model as printed.", "what_survives": "The test and its construction do not use these simulated rates. The Gaussian simulation of Figure 1, the other two\npanels of Figure 2 and the voter-turnout application do not use the linear model. The recomputed linear rates\nstill rise across the four checked values of $N$, supporting the qualitative trend of increasing detection.\nFor the printed linear model, at $\\alpha = 0.05$ and $N = 400$ the estimated rate is 0.27 where the figure\nshows 0.53. My recalculations stop at $N = 800$ and do not test the remark that the rate appears to converge to\n100\\%." }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-N006", "work": { "arxiv_id": null, "doi": "10.1177/0049124112437535", "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "url": "https://pmaronow.github.io/llm-audit/works/P030/" }, "work_id": "P030" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P030.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "conditional_on": [], "credit_reason": "The no-effect component is already exported by five_unit_counterexample.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.five_no_effects", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L176", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "conditional_on": [], "credit_reason": "The treated-variant set is a singleton in both cases; the empty-set nearest-distance branch is unused.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.five_distances", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L181", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"At P030/P030_Audit.lean:195-197, five_correlations concludes only that each distance rank-square sum is 2, the outcome rank-square sum is 2, and fiveStatistic is 1. T1's computation and this row explicitly assert that the outcome and both distance rank vectors are (1,2,3), but neither averageRank vector equality is in this declaration's conclusion. The aggregate sums and correlation do not state those pointwise rank identities; five_rank_certificate at P030/P030_Audit.lean:1810-1813 separately states them, preserving the theorem family's coverage without satisfying this row's credited conclusion. The definitions match the specified experiment and the positive square sums ensure a defined correlation, so the defect is precisely the omitted rank-vector conclusions.\", 'suggested_fix': 'Extend five_correlations with the conjuncts averageRank fiveOutcomes = ![1,2,3] and (∀ a : Fin 2, averageRank (fiveDistances a) = ![1,2,3]), retaining its existing sums-of-squares and correlation conclusions. These are already stated in five_rank_certificate.', 'second_review': \"At P030/P030_Audit.lean:195, five_correlations exports only rank-square sums of 2 and correlations of 1. T1 expressly says that the outcome and both distance vectors have ranks '(1, 2, 3)', so those conclusions belong to this row's credited component and are missing. The omission is minor: evaluating averageRank (P030/P030_Audit.lean:116) on these explicitly specified three-entry vectors requires only finite comparison and counting. Add the rank-vector conjuncts already separately stated by five_rank_certificate (P030/P030_Audit.lean:1810).\", 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P030.five_correlations", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L193", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "conditional_on": [], "credit_reason": "All design components are already exported by five_unit_counterexample; no separate invocation is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.five_design", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L209", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "conditional_on": [], "credit_reason": "The repeated alpha<1 conclusion is redundant and does not affect fidelity; constant correlation across assignments does not mean either ranked input vector is constant.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.five_unit_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L236", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "conditional_on": [], "credit_reason": "This separate declaration supplies the explicit rank certificates missing from R196-01.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.five_rank_certificate", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1799", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:739d2acbc296e54378def439f4966255635ab392261184fa4b56107e15dfab7b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T1", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T1" }, "html": { "body": "

p. 7 (PDF p. 5): the five-unit experiment has no effects, defined Spearman statistics equal to one, printed rejection probability one, and corrected rejection probability zero, with the stated distances and rank computations.

", "headline": "Two-tailed rule after Equation (2), with 'This procedure is an exact, distribution-free test', as printed: a five-unit experiment without interference in which the rule rejects with probability 1", "location": "p. 7 (PDF p. 5)" }, "id": "P030-T1", "text": { "body": "p. 7 (PDF p. 5): the five-unit experiment has no effects, defined Spearman statistics equal to one, printed rejection probability one, and corrected rejection probability zero, with the stated distances and rank computations.", "headline": "Two-tailed rule after Equation (2), with 'This procedure is an exact, distribution-free test', as printed: a five-unit experiment without interference in which the rule rejects with probability 1", "location": "p. 7 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T1" }, { "declarations": [ { "claim_revision_id": "catalog:b2a3bbe91b88fecb043fc6a6141d28da75a383d357871b293989e31ace65cb50", "conditional_on": [], "credit_reason": "Inclusive inequalities and all tied assignments are counted; 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p. 7 and Table 1, p. 9 (PDF pp. 5, 7): a genuine no-effects ten-unit population realizes the table's six-assignment conditional law. All midranks, centered ranks, products, square sums, squared correlations, strict orders, roundings, and inclusive tails have their stated values. For 0 < alpha <= 2/3 the printed rule rejects exactly the three assignments with p = 1 and has conditional size 1/2; the corrected rule rejects none. The printed size exceeds every positive alpha < 1/2, including alpha = 1/20.

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p. 7 (PDF p. 5), corrected: the two-inclusive-tail rule controls size for every finite probability law, with the tail identity, super-uniformity, union bound, printed-rule inclusion, genuine strictness example, and conventional alternative.

", "headline": "Two-tailed rule after Equation (2), corrected: reject when either inclusive tail probability is below α/2\\alpha/2α/2 (with the proof of its level)", "location": "p. 7 (PDF p. 5)" }, "id": "P030-T3", "text": { "body": "p. 7 (PDF p. 5), corrected: the two-inclusive-tail rule controls size for every finite probability law, with the tail identity, super-uniformity, union bound, printed-rule inclusion, genuine strictness example, and conventional alternative.", "headline": "Two-tailed rule after Equation (2), corrected: reject when either inclusive tail probability is below $\\alpha/2$ (with the proof of its level)", "location": "p. 7 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T3" }, { "declarations": [ { "claim_revision_id": "catalog:9c90dd9c8919d055e3778a1b47ab89947cbfefcd022df3044828ed98032c3711", "conditional_on": [], "credit_reason": "This is a full refutation, not positive exactness: nonempty conditioning fibers always contain a supported assignment with the same fixed-treatment pattern, so allowing an unsupported a changes no fiber. The target itself quantifies over statistic choices; the supplied Spearman witness is within that domain. The packet status counterexample_proved is consistent with its refutation role; the record reports a less specific proved label elsewhere.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.printed_two_tailed_claim_false", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L878", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:9c90dd9c8919d055e3778a1b47ab89947cbfefcd022df3044828ed98032c3711", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T4", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T4" }, "html": { "body": "

The printed claim is false: P030-T4, p. 7 (PDF p. 5), the printed two-tailed rule does not universally control conditional Type I error under no interference.

", "headline": "Equation (2) and the two-tailed rule ('This procedure is an exact, distribution-free test for detecting interference between units'), as printed", "location": "p. 7 (PDF p. 5)" }, "id": "P030-T4", "text": { "body": "The printed claim is false: P030-T4, p. 7 (PDF p. 5), the printed two-tailed rule does not universally control conditional Type I error under no interference.", "headline": "Equation (2) and the two-tailed rule ('This procedure is an exact, distribution-free test for detecting interference between units'), as printed", "location": "p. 7 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T4" }, { "declarations": [ { "claim_revision_id": "catalog:e85dde50e2585098999aea0a6d47200a6b929bdbfde400583330622b4248bf32", "conditional_on": [], "credit_reason": "The constant-statistic hypothesis is the literal printed Equation (1), not a conclusion silently inferred from no interference; nonemptiness follows for realized conditional designs.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.constant_statistic_p_one", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L829", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e85dde50e2585098999aea0a6d47200a6b929bdbfde400583330622b4248bf32", "conditional_on": [], "credit_reason": "The corrected outcome null and no interference hold while the conditional statistic varies; the observed one-third tail uses assignment multiplicity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.statistic_invariance_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L843", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e85dde50e2585098999aea0a6d47200a6b929bdbfde400583330622b4248bf32", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T5" }, "html": { "body": "

p. 7 (PDF p. 5), Equation (1): no interference, even no effects of any kind, does not force statistic constancy. The genuine ten-unit experiment has all the stated midranks, centered ranks, products, square sums, rational squared correlations, exact correlations, strict orders, and roundings, and exactly three distinct actual conditional statistics. Its observed-data p-value at unit 7 is 1/3, rounding to 0.333. On any nonempty finite reference support, a constant statistic instead gives a point mass at its observed value and p = 1.

", "headline": "Equation (1) ('it follows that we may define the salient null hypothesis H0:tZ=tZ′H_0 : t_Z = t_{Z'}H0​:tZ​=tZ′​'), as printed: a population without interference in which the statistic differs across Ω′\\Omega'Ω′", "location": "p. 7 (PDF p. 5)" }, "id": "P030-T5", "text": { "body": "p. 7 (PDF p. 5), Equation (1): no interference, even no effects of any kind, does not force statistic constancy. The genuine ten-unit experiment has all the stated midranks, centered ranks, products, square sums, rational squared correlations, exact correlations, strict orders, and roundings, and exactly three distinct actual conditional statistics. Its observed-data p-value at unit 7 is 1/3, rounding to 0.333. On any nonempty finite reference support, a constant statistic instead gives a point mass at its observed value and p = 1.", "headline": "Equation (1) ('it follows that we may define the salient null hypothesis $H_0 : t_Z = t_{Z'}$'), as printed: a population without interference in which the statistic differs across $\\Omega'$", "location": "p. 7 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T5" }, { "declarations": [ { "claim_revision_id": "catalog:78cb38617f42eda543b3dd7c9f05e3ed09443092233e91c76beb867eab729561", "conditional_on": [], "credit_reason": "The invariant outcomes are the target's null antecedent, not assumed statistic invariance; uniformProbability directly represents the stipulated equal conditional law. Part (d) is supplied by R196-15.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.corrected_outcome_null", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L473", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:78cb38617f42eda543b3dd7c9f05e3ed09443092233e91c76beb867eab729561", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T6", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T6" }, "html": { "body": "

p. 7 (PDF p. 5), corrected: no interference fixes the fixed units' outcomes and validates observed-data imputation of the conditional reference law; the genuine ten-unit counterexample shows that the statistic can still vary.

", "headline": "Equation (1), corrected: the null holds the outcomes of the fixed units, not the statistic, invariant over Ω′\\Omega'Ω′", "location": "p. 7 (PDF p. 5)" }, "id": "P030-T6", "text": { "body": "p. 7 (PDF p. 5), corrected: no interference fixes the fixed units' outcomes and validates observed-data imputation of the conditional reference law; the genuine ten-unit counterexample shows that the statistic can still vary.", "headline": "Equation (1), corrected: the null holds the outcomes of the fixed units, not the statistic, invariant over $\\Omega'$", "location": "p. 7 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T6" }, { "declarations": [ { "claim_revision_id": "catalog:2fc1ce9b73a331e0a7451cc9fe8920b5384107e8b85bd7635f74381491d8495c", "conditional_on": [], "credit_reason": "The stronger no-effects null is precisely the hypothesis requested by T7(d), not a hidden assumption of no-interference.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.no_effects_implies_no_interference", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L203", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2fc1ce9b73a331e0a7451cc9fe8920b5384107e8b85bd7635f74381491d8495c", "conditional_on": [], "credit_reason": "The conditional filter is the corrected same-unit condition Z_i=Z′_i. Empty fibers are permitted, but nonempty cases are included and witnessed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.conditional_preserves_treatments", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L438", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2fc1ce9b73a331e0a7451cc9fe8920b5384107e8b85bd7635f74381491d8495c", "conditional_on": [], "credit_reason": "FixedOutcomeNull expands to the required equality of each fixed unit’s outcomes. No observed-membership premise is needed for this implication.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.no_interference_fixed", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L447", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2fc1ce9b73a331e0a7451cc9fe8920b5384107e8b85bd7635f74381491d8495c", "conditional_on": [], "credit_reason": "This is the target’s logical consequence of actual potential-outcome variation, not a claim that a statistical rejection guarantees a true alternative.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.fixed_dependence_implies_interference", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L456", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2fc1ce9b73a331e0a7451cc9fe8920b5384107e8b85bd7635f74381491d8495c", "conditional_on": [], "credit_reason": "For binary treatment, arbitrary unit-specific own-treatment outcomes admit the displayed intercept-and-effect form. Composing this row with R197-18 gives the fixed-outcome consequence in T7(c).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.direct_effects_no_interference", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L465", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2fc1ce9b73a331e0a7451cc9fe8920b5384107e8b85bd7635f74381491d8495c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T7", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T7" }, "html": { "body": "

pp. 5-6 (PDF pp. 3-4), corrected: conditional assignments preserve fixed treatments, no interference fixes their outcomes, fixed-outcome variation refutes no interference, direct effects are allowed, and no effects implies no interference.

", "headline": "Definition of Ω′\\Omega'Ω′ and the no-interference null H0′′H_{0''}H0′′​, corrected (Zi=Zi′Z_i = Z'_iZi​=Zi′​ for the printed Zi=Zi′Z_i = Z_{i'}Zi​=Zi′​), with the consequence the paper draws: dependence of YfixedY_{fixed}Yfixed​ on ZvariantZ_{variant}Zvariant​ over Ω′\\Omega'Ω′ rejects H0′′H_{0''}H0′′​", "location": "p. 6 (PDF p. 4); introduction, pp. 5-6 (PDF pp. 3-4)" }, "id": "P030-T7", "text": { "body": "pp. 5-6 (PDF pp. 3-4), corrected: conditional assignments preserve fixed treatments, no interference fixes their outcomes, fixed-outcome variation refutes no interference, direct effects are allowed, and no effects implies no interference.", "headline": "Definition of $\\Omega'$ and the no-interference null $H_{0''}$, corrected ($Z_i = Z'_i$ for the printed $Z_i = Z_{i'}$), with the consequence the paper draws: dependence of $Y_{fixed}$ on $Z_{variant}$ over $\\Omega'$ rejects $H_{0''}$", "location": "p. 6 (PDF p. 4); introduction, pp. 5-6 (PDF pp. 3-4)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T7" }, { "declarations": [ { "claim_revision_id": "catalog:985239d8c11c7cdec3dda02f54d765ee096e2b674922e7c194ac4b8bee1410a2", "conditional_on": [], "credit_reason": "The displayed formula is evaluated literally without its missing square root under both moment conventions; no conclusion about the paper's simulation implementation is claimed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.printed_formula_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L984", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:985239d8c11c7cdec3dda02f54d765ee096e2b674922e7c194ac4b8bee1410a2", "conditional_on": [], "credit_reason": "The target expressly supplies these algebraic vectors as a statistic-ordering comparison; the row does not purport to certify a complete spatial experiment or simulation code.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.printed_ordering_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1005", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:985239d8c11c7cdec3dda02f54d765ee096e2b674922e7c194ac4b8bee1410a2", "conditional_on": [], "credit_reason": "The conclusion refutes the weakly monotone reading explicitly fixed in the target definitions; it does not refute strict monotonicity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.weak_monotonicity_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1026", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:985239d8c11c7cdec3dda02f54d765ee096e2b674922e7c194ac4b8bee1410a2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T8", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T8" }, "html": { "body": "

p. 8 (PDF p. 6): the printed covariance-over-product-of-variances formula is false as a formula for rank correlation. The observed midranks, centered ranks, products, square sums, and population and sample moments give the two wrong values -4/5 and -3/5, with the stated rational square comparisons. All six printed-formula values, correct correlations, and roundings are as stated; the corrected expression is invariant under positive moment divisors. The two explicit tied vectors reverse the printed formula's ordering, and the nonconstant weakly monotone example has the stated ranks, products, variances, and correlation sqrt(3)/2, rounding to 0.866 and unequal to either 1 or -1.

", "headline": "The displayed formula for ρ\\rhoρ (no square root in the denominator) and the sentence 'ρ\\rhoρ has a perfect correlation coefficient (of 1 or -1) whenever DnearestD_{nearest}Dnearest​ has a monotone relationship with YfixedY_{fixed}Yfixed​', as printed", "location": "p. 8 (PDF p. 6); Table 1 on p. 9 (PDF p. 7)" }, "id": "P030-T8", "text": { "body": "p. 8 (PDF p. 6): the printed covariance-over-product-of-variances formula is false as a formula for rank correlation. The observed midranks, centered ranks, products, square sums, and population and sample moments give the two wrong values -4/5 and -3/5, with the stated rational square comparisons. All six printed-formula values, correct correlations, and roundings are as stated; the corrected expression is invariant under positive moment divisors. 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p. 8 (PDF p. 6), corrected: Spearman correlation has divisor invariance, bounds, the exact proportionality characterization and its signs, the tie-preserving monotone cases, the weak-monotonicity counterexample, and all six table values.

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p. 10 (PDF p. 8), corrected: excluding self yields defined inverse-square sums at positive off-diagonal distances, with the stated decay, values, non-exponential character, explicit divergence bound, and contrasting finite exponential limit.

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p. 13 (PDF p. 11), corrected: disjoint sample counts must add, the printed counts do not, and exactly the four stated single-entry repairs reconcile them.

", "headline": "Application sample counts ('To summarize, N=344,056N = 344,056N=344,056, Nfixed=114,629N_{fixed} = 114,629Nfixed​=114,629, Ncontrol=191,228N_{control} = 191,228Ncontrol​=191,228, and Ntreatment=38,198N_{treatment} = 38,198Ntreatment​=38,198'), corrected claim: the three group counts sum to 344,055", "location": "p. 13 (PDF p. 11)" }, "id": "P030-T11", "text": { "body": "p. 13 (PDF p. 11), corrected: disjoint sample counts must add, the printed counts do not, and exactly the four stated single-entry repairs reconcile them.", "headline": "Application sample counts ('To summarize, $N = 344,056$, $N_{fixed} = 114,629$, $N_{control} = 191,228$, and $N_{treatment} = 38,198$'), corrected claim: the three group counts sum to 344,055", "location": "p. 13 (PDF p. 11)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T11" }, { "declarations": [ { "claim_revision_id": "catalog:a90e566270a4d8c5af979e2fca6771b8ecc917a1e5ecd792530ad8e3d0c8bc1a", "conditional_on": [], "credit_reason": "Its component is already exported by one_tailed_randomization_valid; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.uniformLaw_probability", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L430", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a90e566270a4d8c5af979e2fca6771b8ecc917a1e5ecd792530ad8e3d0c8bc1a", "conditional_on": [], "credit_reason": "Its component is already exported by one_tailed_randomization_valid; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.attainable_exactness", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L511", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a90e566270a4d8c5af979e2fca6771b8ecc917a1e5ecd792530ad8e3d0c8bc1a", "conditional_on": [], "credit_reason": "Its component is already exported by one_tailed_randomization_valid; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.exact_rejection_max", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L521", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a90e566270a4d8c5af979e2fca6771b8ecc917a1e5ecd792530ad8e3d0c8bc1a", "conditional_on": [], "credit_reason": "All clauses T12(a)-(d) are exported. FixedOutcomeNull assumes focal-outcome invariance, not p-value validity; the imputation conclusion follows from that legitimate null. Probabilities concern the stipulated actual conditional uniform law, as S8(d) requires, and do not assert uniformity for every possible subset-selection mechanism.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.one_tailed_randomization_valid", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L558", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a90e566270a4d8c5af979e2fca6771b8ecc917a1e5ecd792530ad8e3d0c8bc1a", "conditional_on": [], "credit_reason": "Its component is already exported by one_tailed_randomization_valid; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.uniform_point_probability", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1691", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:a90e566270a4d8c5af979e2fca6771b8ecc917a1e5ecd792530ad8e3d0c8bc1a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T12", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T12" }, "html": { "body": "

abstract p. 3 and Equation (2), p. 7 (PDF pp. 1, 5): the uniform conditional one-tailed p-value permits observed-data imputation, equals its assignment-count formula, is super-uniform, and is exact at attainable levels with the stated greatest-attainable-level formula at every threshold.

", "headline": "Equation (2): validity of the one-tailed conditional randomization ppp value (the abstract's 'exact significance level')", "location": "p. 7 (PDF p. 5); abstract, p. 3 (PDF p. 1); the worked count on p. 10 (PDF p. 8)" }, "id": "P030-T12", "text": { "body": "abstract p. 3 and Equation (2), p. 7 (PDF pp. 1, 5): the uniform conditional one-tailed p-value permits observed-data imputation, equals its assignment-count formula, is super-uniform, and is exact at attainable levels with the stated greatest-attainable-level formula at every threshold.", "headline": "Equation (2): validity of the one-tailed conditional randomization $p$ value (the abstract's 'exact significance level')", "location": "p. 7 (PDF p. 5); abstract, p. 3 (PDF p. 1); the worked count on p. 10 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T12" }, { "declarations": [ { "claim_revision_id": "catalog:9fa599cccf8c47426e5f06a10e741e1debfba022ced0713bc603b6612784705b", "conditional_on": [], "credit_reason": "Its component is already exported by proximal_exposure_expectations; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.exposure_mean_linearity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1424", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9fa599cccf8c47426e5f06a10e741e1debfba022ced0713bc603b6612784705b", "conditional_on": [], "credit_reason": "Its component is already exported by proximal_exposure_expectations; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.exposure_mean_from_marginals", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1432", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9fa599cccf8c47426e5f06a10e741e1debfba022ced0713bc603b6612784705b", "conditional_on": [], "credit_reason": "Its component is already exported by proximal_exposure_expectations; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.swap_units_involution", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1444", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9fa599cccf8c47426e5f06a10e741e1debfba022ced0713bc603b6612784705b", "conditional_on": [], "credit_reason": "Its component is already exported by proximal_exposure_expectations; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.treatment_total", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1448", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9fa599cccf8c47426e5f06a10e741e1debfba022ced0713bc603b6612784705b", "conditional_on": [], "credit_reason": "Its component is already exported by proximal_exposure_expectations; no separate helper conclusion is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.complete_swap_mem", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1454", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9fa599cccf8c47426e5f06a10e741e1debfba022ced0713bc603b6612784705b", "conditional_on": [], "credit_reason": "Auxiliary fact, not a standalone proof of exposure expectations. 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p. 5 (PDF p. 3): complete randomization gives all three proximal exposure expectations, strict comparisons with neighborhood size and own treatment, and total expectation, with conditional means used only on nonempty supports.

", "headline": "Expected number of proximal treated units: E(Pi)=KiM/NE(P_i) = K_iM/NE(Pi​)=Ki​M/N, E(Pi∣Zi=1)=Ki(M−1)/(N−1)E(P_i|Z_i = 1) = K_i(M-1)/(N-1)E(Pi​∣Zi​=1)=Ki​(M−1)/(N−1), E(Pi∣Zi=0)=KiM/(N−1)E(P_i|Z_i = 0) = K_iM/(N-1)E(Pi​∣Zi​=0)=Ki​M/(N−1)", "location": "p. 5 (PDF p. 3)" }, "id": "P030-T13", "text": { "body": "p. 5 (PDF p. 3): complete randomization gives all three proximal exposure expectations, strict comparisons with neighborhood size and own treatment, and total expectation, with conditional means used only on nonempty supports.", "headline": "Expected number of proximal treated units: $E(P_i) = K_iM/N$, $E(P_i|Z_i = 1) = K_i(M-1)/(N-1)$, $E(P_i|Z_i = 0) = K_iM/(N-1)$", "location": "p. 5 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T13" }, { "declarations": [ { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The selected table_distances already exports the evaluated nearest distances.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.nearest_single_treated", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L589", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "An auxiliary finite lookup, not independent evidence that the actual example DGP has no interference; the required actual-data link is supplied by other credited rows.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.table_y_vector", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L636", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "Zero-based Lean indices are shifted consistently to the paper's physical unit labels.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.table_distances", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L639", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"At P030/P030_Audit.lean:652-654, table_value_order concludes only -3/√10 < -5/6 < -1/√90 < 0. This correctly orders the exact constants in T14(e), but omits the row's additional assertion 'with exact rational squares 1/90, 25/36, 9/10'. None of the three squared-value equalities appears in its type; local identities used inside the proof do not export those conclusions. Under the stipulated row-level omission rule, the credited square identities must be conclusions even though they are elementary true identities and the surrounding theorem family reproduces the table.\", 'suggested_fix': 'Retain the three inequalities and add the conjuncts (-1 / Real.sqrt 90)^2 = (1/90 : ℝ), (-5/6 : ℝ)^2 = (25/36 : ℝ), and (-3 / Real.sqrt 10)^2 = (9/10 : ℝ).', 'second_review': \"P030/P030_Audit.lean:652 gives the ordering and negativity of exactly the constants in T14(e), namely -1/sqrt(90), -5/6 and -3/sqrt(10), supporting the negativity component in T14(g). The target does not separately require their squared-value identities; the additional promise 'with exact rational squares 1/90, 25/36, 9/10' belongs to row_says and the docstring at P030/P030_Audit.lean:650. Remove that promise from this row's description rather than strengthen its Lean statement. This discrepancy is minor because the square identities follow immediately from the displayed constants by elementary arithmetic and the positive-argument square-root identity.\", 'severity': None}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.table_value_order", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L650", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The square-root denominator and average-rank ties are explicitly required by the target definitions; the nonzero rank variances are conclusions, not extra premises.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.table_correlations", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L665", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The finite uniform assignment-count representation is exactly the law specified for this worked example.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.ten_design", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L729", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The selected assignment-mass and observed-data p-value declarations already perform the injective-enumeration transport.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.uniformProbability_image", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L794", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "Covers the outcome component of T14(a)-(b); the distances in (b) are separately covered by R201-06.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.example_observed_outcomes", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L931", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The strict error bounds are exactly those stated in the target; a separate rounding function is unnecessary.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.table_rounding", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L942", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The public overview accurately summarizes this numerical reference-distribution component. Full target coverage uses all credited example, design, distance, and reference-link rows together.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.worked_table_result", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L963", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The selected assignment-mass and observed-data p-value declarations already perform the assignment/enumeration transport.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.table_enumeration_probability", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1708", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The conditional law is represented by counting the explicitly filtered support; R201-12 identifies that support and the conditioning event.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.ten_assignment_probabilities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1724", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "This supplies the needed link between the actual interfering example DGP and the table defined with a constant null helper schedule.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.example_reference_rho", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1778", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "conditional_on": [], "credit_reason": "The p-value uses frozen observed outcomes under all reference assignments, rather than the changing actual DGP outcomes; this matches T14 and S8(b).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.example_observed_statistic_and_p", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1787", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b2588eb4ab886eeab53f850335fef2b831d4642b1fa8343023c2fac76be40c72", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T14", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T14" }, "html": { "body": "

pp. 9-10 (PDF pp. 7-8): the actual adjacency outcome model gives all the printed spatial, indirect, total, and fixed observations, including fixed-outcome midranks (3/2, 4, 3, 3/2). The six conditional assignments have the stated masses, distances, defined and unique reference correlations, exact values, and roundings. Exactly the assignments treating units 6 and 7 lie in the observed lower tail, giving the actual observed-data p = 2/6 = 1/3, rounding to 0.333; every reference value is negative.

", "headline": "A Simple Example and Table 1: YYY, YfixedY_{fixed}Yfixed​, DnearestD_{nearest}Dnearest​, ρ=−0.949\\rho = -0.949ρ=−0.949, the six conditional randomizations with their ρ0\\rho_0ρ0​, p=0.333p = 0.333p=0.333, and every ρ0\\rho_0ρ0​ negative", "location": "pp. 9-10 (PDF pp. 7-8)" }, "id": "P030-T14", "text": { "body": "pp. 9-10 (PDF pp. 7-8): the actual adjacency outcome model gives all the printed spatial, indirect, total, and fixed observations, including fixed-outcome midranks (3/2, 4, 3, 3/2). The six conditional assignments have the stated masses, distances, defined and unique reference correlations, exact values, and roundings. Exactly the assignments treating units 6 and 7 lie in the observed lower tail, giving the actual observed-data p = 2/6 = 1/3, rounding to 0.333; every reference value is negative.", "headline": "A Simple Example and Table 1: $Y$, $Y_{fixed}$, $D_{nearest}$, $\\rho = -0.949$, the six conditional randomizations with their $\\rho_0$, $p = 0.333$, and every $\\rho_0$ negative", "location": "pp. 9-10 (PDF pp. 7-8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T14" }, { "declarations": [ { "claim_revision_id": "catalog:10e75aceeaba018581445d8c4c2091fe32041b8425c3a23fe5383eeaa5a8948d", "conditional_on": [], "credit_reason": "A polynomial identity used by T15(a); integer maximizer conclusions are separately supplied by R197-19.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.half_controls_square", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1649", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:10e75aceeaba018581445d8c4c2091fe32041b8425c3a23fe5383eeaa5a8948d", "conditional_on": [], "credit_reason": "Integer feasible counts are equivalent to natural counts here. The parity cases identify all maximizers, including f=0,1 at C=1; no statistical power optimality is asserted.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P030.half_controls_maximizers", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P030/P030_Audit.lean#L1653", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:10e75aceeaba018581445d8c4c2091fe32041b8425c3a23fe5383eeaa5a8948d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P030-T15", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P030-T15" }, "html": { "body": "

Equation (3), pp. 8-9 (PDF pp. 6-7): completion of the square gives exactly half the controls as the unique even-count maximizer and the two nearest halves as exactly the odd-count maximizers, for positive treatment and control counts.

", "headline": "Equation (3), the rule of thumb for NfixedN_{fixed}Nfixed​: for a random sample of control units it selects half of the control units", "location": "pp. 8-9 (PDF pp. 6-7)" }, "id": "P030-T15", "text": { "body": "Equation (3), pp. 8-9 (PDF pp. 6-7): completion of the square gives exactly half the controls as the unique even-count maximizer and the two nearest halves as exactly the odd-count maximizers, for positive treatment and control counts.", "headline": "Equation (3), the rule of thumb for $N_{fixed}$: for a random sample of control units it selects half of the control units", "location": "pp. 8-9 (PDF pp. 6-7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P030/#P030-T15" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "selected_recalculation", "targets": [ { "agreement": "to_rounding", "location": "pp. 9–10 (PDF pp. 7–8)", "method": "I reconstructed the ten-unit example and enumerated all six conditional assignments. The outcomes, nearest-treatment distances and tail counts agree exactly; the printed rank correlations and $p=0.333$ agree to rounding. This is an independent calculation, not a run of the original code.", "status": "recalculated", "target": "Table 1 and worked-example p value" }, { "agreement": "within_mc_error", "location": "pp. 10–11 (PDF pp. 8–9)", "method": "I ran a new simulation implementation from the printed Gaussian design at reduced replication count: the two-tailed rejection rate was 0.833 at $\\alpha=0.05$ and $N=400$, agreeing with the reported 83% within Monte Carlo uncertainty. I could not reproduce the original code or draws, or compare the reported Kolmogorov–Smirnov $p=0.661$, because the original simulation materials are unavailable.", "status": "recalculated", "target": "Figure 1 and Gaussian simulation summaries" }, { "agreement": "agrees_in_part", "location": "pp. 11–12 (PDF pp. 9–10)", "method": "I ran new simulation implementations from the printed designs. At $N=100,200,400,800$, the Gaussian and inverse-square panels agree within Monte Carlo uncertainty and figure-reading precision. The linear panel disagrees: at $\\alpha=0.05$, I obtained 0.266 versus about 0.530 at $N=400$ and 0.488 versus about 0.826 at $N=800$ (P030-N006). For the large-N inverse-square statement, I obtained two-tailed rejection rates of 93.0% at $N=5{,}000$ and 92.25% at $N=6{,}000$, disagreeing with the reported rate of over 95% near $N=5{,}000$; the one-tailed rates do not establish a rate above 95% (P030-N005). I could not reproduce the original code or draws, which are unavailable.", "status": "recalculated", "target": "Figure 2 and large-N simulation statement" }, { "agreement": "disagrees", "location": "p. 13 (PDF p. 11)", "method": "I recalculated the counts and the neighbors-minus-control effect from the public source-study data. Under either of two single-stratum exclusions matching the printed total, control and neighbors counts, the effect is about 8.135 percentage points, agreeing with 8.1 to rounding. Both leave 114,630 fixed voters rather than 114,629; no exclusion matches every printed count (P030-E008). Disagreement refers to the counts, not the direct effect.", "status": "recalculated", "target": "Application sample counts and direct effect" }, { "agreement": "not_compared", "location": "pp. 13–14 (PDF pp. 11–12)", "method": "I inspected Figure 3: its dashed line is near the reported $\\rho=0.0061$, to the left of the visible histogram. I could not recalculate the statistic, distances or tail count without household locations and the application code. Not compared refers to the numerical interference test; the histogram inspection does not verify its exact p value.", "status": "blocked", "target": "Figure 3 and application interference test" } ] }, "summary": "The article proposes a test for interference between units in a randomized experiment: the treatment of a fixed subset of units is held at its realized value, the other units are re-randomized, and a statistic is compared with its distribution over these re-randomizations. In a voter-turnout mailing experiment the article reports rejection of the hypothesis of no interference ($p<0.001$). 11 entries are classified as errors (2 Moderate, 9 Minor, none Major or Critical), and 2 are clarity notes. The most consequential is \\fid{P030-E001}: the two-tailed rule printed after Equation (2) can reject more often than its nominal level.", "what_was_checked": "I examined all 14 pages of the article. It has no appendix, and I had no supplement. The list below states what I examined and what held up. These checks do not establish that the rest of the article is correct.\n\n\\begin{itemize}\n\\item \\emph{The method.} The three expectations of the number of proximal treated units on p.~5 hold: $E(P_i)=K_iM/N$, $E(P_i\\mid Z_i=1)=K_i(M-1)/(N-1)$ and $E(P_i\\mid Z_i=0)=K_iM/(N-1)$. Under the hypothesis of no interference, the outcomes of the fixed units are the same for every assignment in $\\Omega'$, and the conditional randomization argument on p.~6 follows. The one-tailed $p$ value of Equation (2) is valid when the assignments in $\\Omega'$ are equally likely and each is counted. This condition holds under complete randomization when the fixed subset is a uniform sample of fixed size from the control units, which is the design of the simulations and the choice that the article recommends. The rule of thumb in Equation (3) selects half of the control units, up to integer rounding, as stated. The definition of the Spearman statistic in words on p.~8 is correct, and the displayed formula is the subject of \\fid{P030-E004}.\n\\item \\emph{The worked example.} I recomputed the outcomes, the distances and the six rows of Table 1 exactly; the correlations and $p$ value agree to the printed rounding ($\\rho=-0.949$, $p=0.333$).\n\\item \\emph{The simulations.} The decay functions plotted in the left panels of Figure 2 agree with the printed formulas. A new implementation of the Gaussian design from the text, with fewer replications than the article reports, gave a two-tailed rejection rate of $0.833$ at $\\alpha=0.05$ with $N=400$ (the article reports 83\\%). It also matched the Gaussian and inverse-square panels of Figure 2 at the sample sizes tried, within Monte Carlo uncertainty and the precision of reading the figure. The linear panel did not match (\\fid{P030-N006}). My larger-sample inverse-square simulations also did not establish the reported rejection rate above 95\\% near $N=5{,}000$ (\\fid{P030-N005}).\n\\item \\emph{The application.} The description of the experiment of Gerber, Green and Larimer (2008) and of its four mailings agrees with the public replication files of that study, and the difference in means of 8.1 percentage points matches their counts. In Figure 3, the dashed line appears near the reported $\\rho=0.0061$, well to the left of the visible histogram, which is consistent with the reported $p<0.001$. The binned figure does not establish the exact tail count.\n\\item \\emph{The references.} Every work cited in the text appears in the reference list, and every listed work is cited.\n\\end{itemize}\n\nThe following were not checked, or could not be.\n\n\\begin{itemize}\n\\item I had no code for the article's simulations or for its reanalysis of the turnout experiment. The simulations were compared only with new implementations written from the text, on a smaller scale than the 10,000 simulations of 5,000 randomizations each that the article reports. The Kolmogorov--Smirnov $p$ value of $0.661$ cannot be checked without the original draws.\n\\item The household locations used in the application are not part of the public release of the turnout experiment. For this reason $\\rho=0.0061$, the summaries of $D_{\\mathit{nearest}}$ and the reference distribution in Figure 3 were not recomputed, and the scheme used for the 1,000 conditional randomizations was not examined.\n\\item I did not read the works that the article cites. The quotations and the results attributed to them were not checked. These include the indirect effect of about 0.5 percentage points and the finding of McConnell et al.\\ (2010), on which the article rests its statement that the bias from interference is likely negligible in such experiments.\n\\end{itemize}", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "P. M. Aronow" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow (2012). A General Method for Detecting Interference Between Units in Randomized Experiments. Sociological Methods and Research 41(1):3–16.", "doi": "10.1177/0049124112437535", "edition": { "pages_in_audited_file": 14, "sha256": "c7126d7c6415ac441f23e5137d158ae54286e73928f217ee9e5a2d1a0e4136fd", "source_url": "https://doi.org/10.1177/0049124112437535", "venue": "Sociological Methods and Research", "year": "2012" }, "id": "P030", "later_materials": null, "title": "A General Method for Detecting Interference Between Units in Randomized Experiments", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P030/", "venue": "Sociological Methods and Research", "year": "2012" } }