{
 "claims": [
  {
   "claim": "Under the local randomization assumption, and allowing interference of arbitrary and unknown structure, the difference-in-means estimator applied to subjects inside the window around the cutoff is unbiased for a regime-specific causal effect $\\tau$.",
   "claim_id": "P014-H1",
   "related_corrections": [
    "P014-E001",
    "P014-E002"
   ],
   "tied_to": "Section 2, unnumbered displays on printed p. 131 (definitions of $\\overline{Y}_i(z)$, $\\tau$, $\\hat\\tau$ and '$\\mathrm{E}[\\hat\\tau] = \\ldots$'); local randomization paragraph on printed p. 130. The note has no numbered results or tables."
  },
  {
   "claim": "The effect the estimator targets is equivalent to the average direct effect of Hudgens and Halloran (2008) for the subpopulation of subjects within the window near the threshold.",
   "claim_id": "P014-H2",
   "related_corrections": [
    "P014-E001",
    "P014-N002"
   ],
   "tied_to": "Introduction, printed p. 129; Section 2, printed p. 131, 'our primary target is the average direct effect (Hudgens and Halloran, 2008)' and the definitions of $\\overline{Y}_i(z)$ and $\\tau$."
  },
  {
   "claim": "The estimand is regime specific: it depends on the joint distribution of $\\mathbf Z$ over the whole finite population, including the allocation of subjects outside the window, which the model leaves completely unspecified.",
   "claim_id": "P014-H3",
   "related_corrections": [
    "P014-E001",
    "P014-E002"
   ],
   "tied_to": "Section 2, printed p. 131, the paragraphs after the definitions of $\\overline{Y}_i(z)$ and of $\\tau$; local randomization clauses on printed p. 130."
  },
  {
   "claim": "Inference requires additional structure. Consistency at rate $n^{-1/2}$, asymptotic normality and conservative variance estimators are available with a growing number of bounded strata under partial interference and regularity conditions, or under stratified interference, with details in Liu and Hudgens (2014) and Aronow and Samii (2016). A Fisher-type exact test with more structure on effects is another option when $N_0$ is small.",
   "claim_id": "P014-H4",
   "related_corrections": [],
   "tied_to": "Section 2, last paragraph, printed pp. 131-132; no result in the note, cited works only."
  },
  {
   "claim": "The RD design can yield credible causal inferences in studies with interference, provided the estimated effect is given its regime-specific interpretation; the result may give a formal basis for the RD design where interference is widespread, for example in evaluating vaccines.",
   "claim_id": "P014-H5",
   "related_corrections": [
    "P014-E001",
    "P014-E002"
   ],
   "tied_to": "Introduction, printed pp. 129-130; Discussion, printed p. 132; rests on the Section 2 unbiasedness display, printed p. 131."
  }
 ],
 "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": "Nine lines of text from the final paragraph of Section 2. A red box marks the clause 'Pr(X = x) = 1/N0!, for all x in Supp(X)', with X and x in bold. The line continues 'where N0 is the sum of the indicators that Xi is in W0: implying that all permutations of the running variable values (and therefore treatment values) within the window W0 are equiprobable'.",
     "caption": "Section 2, final paragraph, p. 130 (PDF p. 3). The marked clause assigns probability $1/N_0!$ to each support point of the full vector $\\mathbf{X}$. The words after the colon draw the conclusion that the permutations within the window are equiprobable.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E002.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P014-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T2"
    },
    {
     "id": "P014-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T3"
    },
    {
     "id": "P014-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T5"
    },
    {
     "id": "P014-T6",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T6"
    },
    {
     "id": "P014-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T7"
    }
   ],
   "headline": "The probability condition in the local randomization assumption should be placed on the running variables of the subjects in the window, $\\Pr(\\mathbf{X}_S=a)=1/N_0!$ for every arrangement $a$ of their values, and not on the full vector $\\mathbf{X}$.",
   "id": "P014-E002",
   "location": "Section 2, final paragraph, p. 130 (PDF p. 3), sentence beginning “Our local randomization assumption requires that”; related sentence on p. 131 (PDF p. 4), beginning “While the treatment allocation scheme”",
   "pdf_page": 3,
   "printed_page": "130",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P014-T2",
    "P014-T3",
    "P014-T5",
    "P014-T6",
    "P014-T7"
   ],
   "text": {
    "as_printed": "Our local randomization assumption requires that $F_{\\mathbf{x},W_0}(t)=F_0(t)$, $\\forall\\mathbf{x}\\in\\operatorname{Supp}(\\mathbf{X})$ and $\\forall t$; $\\Pr(X_i\\in W_0)\\in\\{0,1\\}$, $\\forall i\\in\\{1,\\ldots,N\\}$, $\\Pr(\\hl{\\mathbf{X}=\\mathbf{x}})=\\frac{1}{N_0!}$, $\\forall\\hl{\\mathbf{x}\\in\\operatorname{Supp}(\\mathbf{X})}$, where $N_0=\\sum_{i=1}^{N}\\mathrm{I}(X_i\\in W_0)$: implying that all permutations of the running variable values (and therefore treatment values) within the window $W_0$ are equiprobable. Further assume that $0<\\sum_{i:X_i\\in W_0}Z_i<N_0$, so that at least one subject within the window is treated, and at least one subject within the window is in the control group.",
    "corrected": "Let $S=\\{i:\\Pr(X_i\\in W_0)=1\\}$ be the set of subjects in the window, and let $n=N_0$ be its size. Our local randomization assumption requires that $\\Pr(X_i\\in W_0)\\in\\{0,1\\}$, $\\forall i\\in\\{1,\\ldots,N\\}$; that $F_{\\mathbf{x},W_0}(t)=F_0(t)$, $\\forall\\mathbf{x}\\in\\operatorname{Supp}(\\mathbf{X})$ and $\\forall t$, where $F_0$ is the empirical distribution function of $n$ distinct values $v_1,\\ldots,v_n$; and that\n\\[\n\\Pr(\\hl{\\mathbf{X}_S=a})=\\frac{1}{n!}\\quad\\text{for every arrangement } \\hl{a} \\text{ of } v_1,\\ldots,v_n \\text{ among the subjects in } S,\n\\]\nwhere $\\mathbf{X}_S=(X_i)_{i\\in S}$. That is, all permutations of the running variable values (and therefore treatment values) within the window $W_0$ are equiprobable. The distribution of the running variables outside the window, given $\\mathbf{X}_S$, is restricted only by $\\Pr(X_i\\in W_0)=0$ for $i\\notin S$, and it need not be independent of $\\mathbf{X}_S$. Further assume that the number $m$ of nonnegative values among $v_1,\\ldots,v_n$ satisfies $0<m<n$, so that at least one subject within the window is treated, and at least one subject within the window is in the control group. Then $\\sum_{i\\in S}Z_i=m$ with probability one, and $\\Pr(Z_i=1)=m/n$ for every $i\\in S$.",
    "evidence": "The evidence is an exact counterexample to the implication, together with a counting argument for the corrected condition.\n\n\\emph{The counterexample.} Let $N=4$, $b=2$ and $W_0=[-2,2]$, and assign probability $1/6$ to each of the six vectors\n\\[\n(1,-1,-2,3),\\quad (-1,1,-2,3),\\quad (-1,-2,1,3),\\quad (-1,-2,1,4),\\quad (-1,-2,1,5),\\quad (-1,-2,1,6).\n\\]\nSubjects 1, 2 and 3 are in the window with probability one and subject 4 is never in it, so that $N_0=3$. In each vector the values in the window are $-2$, $-1$ and $1$. They are distinct, and the empirical distribution function $F_{\\mathbf{x},W_0}$ is the same for all six. Each support point has probability $1/6=1/N_0!$, and exactly one subject in the window is treated. Every printed clause holds. However, the arrangement $(-1,-2,1)$ has probability $2/3$, the arrangements $(1,-1,-2)$ and $(-1,1,-2)$ have probability $1/6$ each, and the other three have probability zero. Subjects 1, 2 and 3 are treated with probabilities $1/6$, $1/6$ and $2/3$. Let $y_1(\\mathbf{z})=z_1$, and let all other potential outcomes be zero, so that there is no interference. Then $\\tau=1/3$, while $\\hat\\tau=Z_1$ and $\\mathrm{E}[\\hat\\tau]=1/6$.\n\n\\emph{The counting argument.} Let $S$ be the set of subjects in the window, write $n=N_0$, and let $\\mathbf{X}_S=(X_i)_{i\\in S}$. For an arrangement $a$, let $k(a)$ be the number of support points $\\mathbf{x}$ of $\\mathbf{X}$ such that $\\mathbf{x}_S=a$. Under the printed clauses, I have $\\Pr(\\mathbf{X}_S=a)=k(a)/n!$, and nothing determines $k(a)$. If in addition $\\Pr(\\mathbf{X}_S=a)=1/n!$ for every $a$, then $k(a)=1$ for every $a$, so that the values outside the window are a function of $\\mathbf{X}_S$. For instance, let the arrangement be uniform over the six arrangements of $-2$, $-1$ and $1$, and let $X_4$ be independent of it, equal to $-3$ with probability $1/3$ and to $3$ with probability $2/3$. Then $\\mathbf{X}$ has twelve support points, with probabilities $1/18$ and $1/9$, and none has probability $1/6$. Under the corrected condition, let $m$ be the number of nonnegative values in the window. A subject $i\\in S$ receives a nonnegative value in $m(n-1)!$ of the $n!$ arrangements, which implies that $\\Pr(Z_i=1)=m/n$.",
    "problem": "Under the marked clause, every support point of the full vector $\\mathbf{X}=(X_1,\\ldots,X_N)$ has probability $1/N_0!$. The sentence then says that its clauses imply that all permutations of the running variable values within the window are equiprobable. They do not. I define an arrangement to be an assignment of the $N_0$ values in the window to the $N_0$ subjects in the window. Two support points of $\\mathbf{X}$ can have the same arrangement and differ outside the window. The clauses fix the number of support points at $N_0!$, but they do not fix how many support points have a given arrangement. Some arrangements can have probability zero and others probability above $1/N_0!$, and the subjects in the window can then have unequal probabilities of treatment.\n\nThe clause is also too strong once the conclusion drawn from it is granted. If the $N_0!$ arrangements are equiprobable and every support point of $\\mathbf{X}$ has probability $1/N_0!$, then the running variables outside the window are a function of the arrangement. An allocation outside the window that remains random given the arrangement is excluded, although p.~131 says that this allocation is ``left completely unspecified''. The words after the colon state the intended assumption, which is a uniformly random arrangement, and the correction states that assumption directly.",
    "what_survives": "The potential-outcome model, the threshold rule that sets $Z_i=1$ when $X_i\\ge 0$, the fixed set of subjects in the window and the difference-in-means estimator are unchanged. The corrected condition is the assumption that the prose of the sentence already states, and it changes neither the estimator nor the intended estimand. Under it every subject in the window is treated with probability $m/n$, and without interference the unbiasedness argument on p.~131 holds as printed. This correction does not repair the definition of the target under interference, which is the subject of \\fid{P014-E001}."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-E002",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
   },
   "work_id": "P014"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Two lines of text introduce the individual average potential outcome under treatment z, followed by a display. A red box marks the fraction that defines Y-bar i of z. Its numerator is the sum over z in Supp(Z) of y i of z times the indicator that z i equals z, and its denominator is the sum over z in Supp(Z) of that indicator. Two lines below the display begin 'Note that this estimand is regime specific'.",
     "caption": "Section 2, first display, p. 131 (PDF p. 4). The marked fraction defines $\\overline{Y}_i(z)$ as an average over the assignment vectors in $\\operatorname{Supp}(\\mathbf{Z})$. No probability appears in either sum.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E001.png"
    },
    {
     "alt": "Three lines of text say that under local randomization the difference-in-means estimator is unbiased for the average direct effect for subjects near the threshold. A red box marks the display 'E of tau-hat equals one over N0 times, in square brackets, the sum over i with Xi in W0 of Y-bar i of 1 minus the sum over i with Xi in W0 of Y-bar i of 0'. A second red box marks the sentence 'where the result trivially follows from linearity of expectations.' Three lines of the next paragraph, on inference, follow.",
     "caption": "Section 2, p. 131 (PDF p. 4). The marked display equates $\\mathrm{E}[\\hat\\tau]$ with the target built from the printed $\\overline{Y}_i(z)$, and the marked sentence attributes the identity to linearity of expectations.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E001-b.png"
    },
    {
     "alt": "Bar chart with three groups, one for each assignment vector with z1 equal to 0. The groups are (0, 0, 1, 0), where the outcome of subject 1 is 0; (0, 0, 1, 1), where it is 1; and (0, 1, 0, 0), where it is 0. In each group a hatched blue bar shows the weight in the printed average, which is 1/3 in all three groups, and an orange bar shows the probability given Z1 equal to 0, which is 1/4, 1/4 and 1/2. The legend says that the printed weights give Y-bar 1 of 0 equal to 1/3 and that the probabilities give the conditional expectation 1/4. The title reads 'The printed average weights assignment vectors equally; their probabilities under the design differ'.",
     "caption": "The first example, subject 1 under control. The printed $\\overline{Y}_1(0)$ assigns weight $1/3$ to each of the three assignment vectors with $z_1=0$ and equals $1/3$. Given $Z_1=0$, these vectors have probabilities $1/4$, $1/4$ and $1/2$ under the design, and the expected outcome is $1/4$. All values are exact.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E001-weights.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P014-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T1"
    },
    {
     "id": "P014-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T2"
    },
    {
     "id": "P014-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T3"
    },
    {
     "id": "P014-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T4"
    },
    {
     "id": "P014-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T7"
    },
    {
     "id": "P014-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T9"
    },
    {
     "id": "P014-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T10"
    }
   ],
   "headline": "The difference-in-means estimator is unbiased for the probability-weighted average direct effect, in which $\\overline{Y}_i(z)=\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=z]$, and not in general for the target printed on p. 131, which weights the assignment vectors in $\\operatorname{Supp}(\\mathbf{Z})$ equally.",
   "id": "P014-E001",
   "location": "Section 2, p. 131 (PDF p. 4): the first display, which defines $\\overline{Y}_i(z)$; the display that defines $\\tau$; and the display for $\\mathrm{E}[\\hat\\tau]$ with the sentence “where the result trivially follows from linearity of expectations”. The corresponding claim is in the Introduction, p. 129 (PDF p. 2)",
   "pdf_page": 4,
   "printed_page": "131",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [
    "P014-T1",
    "P014-T10",
    "P014-T2",
    "P014-T3",
    "P014-T4",
    "P014-T7",
    "P014-T9"
   ],
   "text": {
    "as_printed": "Denote the individual average potential outcome under treatment $z$ for all subjects $i:X_i\\in W_0$,\n\\[\n\\overline{Y}_i(z)=\\frac{\\sum_{\\mathbf{z}\\in\\operatorname{Supp}(\\mathbf{Z})}\\hl{y_i(\\mathbf{z})\\mathrm{I}(\\mathbf{z}_i=z)}}{\\sum_{\\mathbf{z}\\in\\operatorname{Supp}(\\mathbf{Z})}\\hl{\\mathrm{I}(\\mathbf{z}_i=z)}}.\n\\]\n\n\\ldots\n\nThen our primary target is the average direct effect (Hudgens and Halloran, 2008) of treatment for subjects who are close to the threshold:\n\\[\n\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}\\overline{Y}_i(1)-\\overline{Y}_i(0).\n\\]\n\n\\ldots\n\nFormally, the difference-in-means estimator for subjects local to the threshold,\n\\[\n\\hat\\tau=\\frac{\\sum_{i:X_i\\in W_0}Y_iZ_i}{\\sum_{i:X_i\\in W_0}Z_i}-\\frac{\\sum_{i:X_i\\in W_0}Y_i(1-Z_i)}{\\sum_{i:X_i\\in W_0}(1-Z_i)}.\n\\]\nThen, under the assumption of local randomization, the difference-in-means estimator as applied to subjects local to the threshold is unbiased for the average direct effect for subjects near the threshold:\n\\[\n\\mathrm{E}[\\hat\\tau]=\\frac{1}{N_0}\\Biggl[\\sum_{i:X_i\\in W_0}\\hl{\\overline{Y}_i(1)}-\\sum_{i:X_i\\in W_0}\\hl{\\overline{Y}_i(0)}\\Biggr],\n\\]\nwhere the result \\hl{trivially} follows from linearity of expectations.",
    "corrected": "Let $S=\\{i:X_i\\in W_0\\}$ be the fixed set of subjects in the window and $n=N_0$. Assume local randomization: the $n$ distinct, fixed values of the running variable in the window, of which $m$ are nonnegative with $0<m<n$, are arranged among the subjects in $S$ uniformly at random, so that $\\sum_{i\\in S}Z_i=m$ with probability one and $\\Pr(Z_i=1)=m/n$ for every $i\\in S$. The allocation outside the window is not restricted. The potential outcomes $y_i(\\mathbf{z})$ are fixed and may depend on the whole assignment vector. Denote the individual average potential outcome under treatment $z$ for all subjects $i:X_i\\in W_0$,\n\\[\n\\overline{Y}_i(z)=\\frac{\\sum_{\\mathbf{z}\\in\\operatorname{Supp}(\\mathbf{Z})}y_i(\\mathbf{z})\\mathrm{I}(\\mathbf{z}_i=z)\\,\\hl{\\Pr(\\mathbf{Z}=\\mathbf{z})}}{\\sum_{\\mathbf{z}\\in\\operatorname{Supp}(\\mathbf{Z})}\\mathrm{I}(\\mathbf{z}_i=z)\\,\\hl{\\Pr(\\mathbf{Z}=\\mathbf{z})}}\\hl{=\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=z]}.\n\\]\nThen our primary target is the average direct effect (Hudgens and Halloran, 2008) of treatment for subjects who are close to the threshold:\n\\[\n\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}\\bigl[\\overline{Y}_i(1)-\\overline{Y}_i(0)\\bigr].\n\\]\nThe difference-in-means estimator $\\hat\\tau$ is as printed. Then, under the assumption of local randomization, the difference-in-means estimator as applied to subjects local to the threshold is unbiased for the average direct effect for subjects near the threshold:\n\\[\n\\mathrm{E}[\\hat\\tau]=\\frac{1}{N_0}\\Biggl[\\sum_{i:X_i\\in W_0}\\overline{Y}_i(1)-\\sum_{i:X_i\\in W_0}\\overline{Y}_i(0)\\Biggr],\n\\]\nwhere the result follows from \\hl{the fixed numbers of treated and control subjects in the window, their common probability of treatment, and} linearity of expectations. The brackets in the display for $\\tau$ are those of \\fid{P014-N004}. The treated-minus-control contrast uses the reverse of Hudgens and Halloran's sign convention (\\fid{P014-N002}).\n\nAlternatively, the definition of $\\overline{Y}_i(z)$ printed on p.~131 can be kept under the additional assumption that $\\mathbf{Z}$ is uniformly distributed on $\\operatorname{Supp}(\\mathbf{Z})$, which holds, for instance, when the assignments of the subjects outside the window are fixed. The two definitions then agree.",
    "evidence": "The evidence is two exact counterexamples to the printed identity, and a derivation of the corrected one.\n\n\\emph{The first example.} Let $N=4$, $b=2$ and $W_0=[-2,2]$. Let $(X_1,X_2,X_3)$ be a uniformly random arrangement of the values $-2$, $-1$ and $1$, and set $X_4=3$ if $(X_1,X_2,X_3)=(-2,-1,1)$ and $X_4=-3$ otherwise. The vector $\\mathbf{X}$ has six support points, each of probability $1/6=1/N_0!$. Subjects 1, 2 and 3 are always in the window and subject 4 never is, the values in the window are distinct and the same in every case, and exactly one subject in the window is treated. Every condition printed on p.~130 holds, with the no-ties condition read for distinct subjects (\\fid{P014-E003}), and in addition all six arrangements are equiprobable. Each subject in the window is treated with probability $1/3$. For all sixteen assignment vectors $\\mathbf{z}$, set $y_1(\\mathbf{z})=y_2(\\mathbf{z})=y_3(\\mathbf{z})=z_4$ and $y_4(\\mathbf{z})=0$.\n\\begin{audittable}{crrrr}{The first example: the four distinct assignment vectors $\\mathbf{z}=(z_1,z_2,z_3,z_4)$. The second column counts the arrangements of the running variable that produce $\\mathbf{z}$; each arrangement has probability $1/6$. The fourth column is the outcome of subjects 1, 2 and 3.}\n$\\mathbf{z}$ & Arrangements & $\\Pr(\\mathbf{Z}=\\mathbf{z})$ & $y_i(\\mathbf{z})=z_4$ & $\\hat\\tau$ \\\\\n$(0,0,1,0)$ & 1 & $1/6$ & 0 & 0 \\\\\n$(0,0,1,1)$ & 1 & $1/6$ & 1 & 0 \\\\\n$(0,1,0,0)$ & 2 & $1/3$ & 0 & 0 \\\\\n$(1,0,0,0)$ & 2 & $1/3$ & 0 & 0 \\\\\n\\end{audittable}\nThe three subjects in the window always have the same outcome, so that $\\hat\\tau=0$ for every assignment and $\\mathrm{E}[\\hat\\tau]=0$. The printed $\\overline{Y}_i(z)$ counts each row of the table once. For subject 1 under control, the rows with $z_1=0$ are the first three, with outcomes $0$, $1$ and $0$, and the printed average is $1/3$. Given $Z_1=0$, these rows have probabilities $1/4$, $1/4$ and $1/2$, and the conditional expectation is $1/4$.\n\\begin{audittable}{crrrr}{The first example: the individual average potential outcomes under the printed definition and under the probability-weighted definition.}\nSubject $i$ & Printed $\\overline{Y}_i(1)$ & Printed $\\overline{Y}_i(0)$ & $\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=1]$ & $\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=0]$ \\\\\n1 & 0 & $1/3$ & 0 & $1/4$ \\\\\n2 & 0 & $1/3$ & 0 & $1/4$ \\\\\n3 & $1/2$ & 0 & $1/2$ & 0 \\\\\n\\end{audittable}\nFrom the second and third columns, I obtain the printed target, and from the last two the probability-weighted target:\n\\[\n\\tau=\\frac13\\Bigl(-\\frac13-\\frac13+\\frac12\\Bigr)=-\\frac{1}{18},\n\\qquad\n\\frac13\\Bigl(-\\frac14-\\frac14+\\frac12\\Bigr)=0=\\mathrm{E}[\\hat\\tau].\n\\]\nThe printed identity fails by $1/18$.\n\n\\emph{The second example.} Let $N=4$, $b=3$ and $W_0=[-3,3]$. Let $(X_1,X_2,X_3)$ be a uniformly random arrangement of $-1$, $1$ and $2$, so that two subjects in the window are treated, and set $X_4=4$ if $(X_1,X_2,X_3)=(-1,1,2)$ and $X_4=-4$ otherwise. Set $y_2(\\mathbf{z})=12z_2z_4$ and all other potential outcomes equal to zero. The same conditions hold. The assignment vectors $(0,1,1,0)$ and $(0,1,1,1)$ have probability $1/6$ each, and $(1,0,1,0)$ and $(1,1,0,0)$ have probability $1/3$ each. The estimator equals $6$ at $(0,1,1,1)$ and $0$ elsewhere, so that $\\mathrm{E}[\\hat\\tau]=1$. The printed $\\overline{Y}_2(1)$ is $(0+12+0)/3=4$, and I find $\\tau=4/3$. The conditional expectation $\\mathrm{E}[y_2(\\mathbf{Z})\\mid Z_2=1]$ is $3$, and the probability-weighted target is $1$.\n\\begin{audittable}{lrrr}{The two exact examples. In each, the six arrangements of the running variable in the window are equiprobable and the conditions printed on p.~130 hold, with the no-ties condition read for distinct subjects.}\nExample & Printed $\\tau$ & $\\mathrm{E}[\\hat\\tau]$ & Probability-weighted target \\\\\nFirst & $-1/18$ & 0 & 0 \\\\\nSecond & $4/3$ & 1 & 1 \\\\\n\\end{audittable}\n\n\\emph{The corrected identity.} Let $S$ be the set of subjects in the window and $n=N_0$. Suppose that $\\sum_{i\\in S}Z_i=m$ with probability one, where $0<m<n$, and that $\\Pr(Z_i=1)=m/n$ for every $i\\in S$. Both hold when the $n$ distinct values in the window, of which $m$ are nonnegative, are arranged uniformly at random: a subject receives each value in $(n-1)!$ of the $n!$ arrangements. I write $p(\\mathbf{z})=\\Pr(\\mathbf{Z}=\\mathbf{z})$ and\n\\[\n\\overline{Y}_i(a)=\\frac{\\sum_{\\mathbf{z}}p(\\mathbf{z})\\,y_i(\\mathbf{z})\\,\\mathrm{I}(z_i=a)}{\\sum_{\\mathbf{z}}p(\\mathbf{z})\\,\\mathrm{I}(z_i=a)}=\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=a],\\qquad a\\in\\{0,1\\}.\n\\]\nSince the numbers of treated and control subjects in the window are fixed, I may write\n\\[\n\\hat\\tau=\\frac1m\\sum_{i\\in S}y_i(\\mathbf{Z})Z_i-\\frac{1}{n-m}\\sum_{i\\in S}y_i(\\mathbf{Z})(1-Z_i).\n\\]\nFor each $i\\in S$, by the definition of $\\overline{Y}_i(a)$ and the identity $\\Pr(Z_i=1)=m/n$, I have\n\\[\n\\mathrm{E}[y_i(\\mathbf{Z})Z_i]=\\sum_{\\mathbf{z}}p(\\mathbf{z})\\,y_i(\\mathbf{z})\\,\\mathrm{I}(z_i=1)=\\frac{m}{n}\\,\\overline{Y}_i(1),\n\\qquad\n\\mathrm{E}[y_i(\\mathbf{Z})(1-Z_i)]=\\frac{n-m}{n}\\,\\overline{Y}_i(0).\n\\]\nSubstituting these into the expectation of the previous display, I obtain\n\\[\n\\mathrm{E}[\\hat\\tau]=\\frac1n\\sum_{i\\in S}\\bigl\\{\\overline{Y}_i(1)-\\overline{Y}_i(0)\\bigr\\}.\n\\]\nNote that this derivation uses no assumption on the allocation outside the window. In both examples the difference between $\\mathrm{E}[\\hat\\tau]$ and the probability-weighted target vanishes identically as a linear form in the 48 potential outcomes of the three subjects in the window, and the difference between $\\mathrm{E}[\\hat\\tau]$ and the printed target does not.\n\n\\emph{When the printed target is correct.} If $\\mathbf{Z}$ is uniform on $\\operatorname{Supp}(\\mathbf{Z})$, then the two definitions of $\\overline{Y}_i(z)$ agree. This is the case when the assignments outside the window are fixed, since each assignment vector is then produced by exactly $m!(n-m)!$ arrangements. With $X_4$ fixed in either example, the printed identity holds for all potential outcomes. The two definitions also agree when there is no interference, since $y_i(\\mathbf{z})$ then depends on $z_i$ alone.",
    "problem": "The first display defines $\\overline{Y}_i(z)$ as an average of $y_i(\\mathbf{z})$ over the distinct assignment vectors in $\\operatorname{Supp}(\\mathbf{Z})$ whose $i$th component is $z$. Each such vector counts once, whatever its probability. The expectation of the difference-in-means estimator weights each assignment vector by its probability. Under local randomization, by linearity of expectation, I have\n\\[\n\\mathrm{E}[\\hat\\tau]=\\frac{1}{N_0}\\sum_{i:X_i\\in W_0}\\bigl\\{\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=1]-\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=0]\\bigr\\}.\n\\]\nThis equals the printed target for all potential outcomes only if, for each subject $i$ in the window and each $z$, the conditional distribution of $\\mathbf{Z}$ given $Z_i=z$ is uniform on its support. Local randomization does not imply this. Under it the arrangements of the running variable in the window are equiprobable, so that the subjects in the window have equal probabilities of treatment. However, the allocation outside the window is ``left completely unspecified'', and the number of arrangements that produce a particular assignment vector of the whole population can vary from one vector to another. The distinct assignment vectors can then have unequal probabilities, so the identity $\\mathrm{E}[\\hat\\tau]=\\tau$ need not hold for the printed $\\tau$ and cannot be justified by linearity of expectations alone.\n\nThe prose around the displays describes the probability-weighted quantity: the target is ``the average outcome we would expect to see in treated subjects (across all subjects within the window and across all randomizations)'' minus the same for control subjects. The display can disagree with the prose when assignment vectors have unequal probabilities, and the text does not settle whether the probabilities were omitted from the display or an assumption of uniform assignment was intended. To keep the allocation outside the window unrestricted, the correction inserts $\\Pr(\\mathbf{Z}=\\mathbf{z})$ in both sums of the first display. The estimator is unchanged and the estimand changes.",
    "what_survives": "The difference-in-means estimator, the potential-outcome model with arbitrary interference and the local randomization design are unchanged. The estimator is exactly unbiased for the probability-weighted target with no restriction on the allocation outside the window. This implies that the statement of the Introduction, that it is unbiased for a regime-specific causal effect, holds for that target. The printed target and identity hold as printed when $\\mathbf{Z}$ is uniform on its support, and when there is no interference. The definition of the estimand changes; in the first example its value changes from $-1/18$ to $0$. The paper reports no simulations, application or numerical results that depend on the definition. The paragraph on inference on pp.~131--132 refers to the same target, and its sufficient conditions were not examined for this correction."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-E001",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
   },
   "work_id": "P014"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Three lines of text. A red box marks the formula 'Pr(Xi = Xj) = 0, for all i, j : Xi, Xj in W0'. It is followed by the words 'so that no two observed age values within the window are exactly identical'.",
     "caption": "Section 2, final paragraph, p. 130 (PDF p. 3). The marked condition is quantified over all $i,j$ such that $X_i,X_j\\in W_0$, with no restriction to $i\\neq j$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E003.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P014-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T1"
    },
    {
     "id": "P014-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T4"
    },
    {
     "id": "P014-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T5"
    },
    {
     "id": "P014-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T7"
    },
    {
     "id": "P014-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T9"
    }
   ],
   "headline": "The no-ties condition should be restricted to distinct subjects: $\\Pr(X_i=X_j)=0$ for all $i\\neq j$ such that $X_i,X_j\\in W_0$.",
   "id": "P014-E003",
   "location": "Section 2, final paragraph, p. 130 (PDF p. 3), sentence beginning “To simplify our definition of local randomization”",
   "pdf_page": 3,
   "printed_page": "130",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P014-T1",
    "P014-T4",
    "P014-T5",
    "P014-T7",
    "P014-T9"
   ],
   "text": {
    "as_printed": "To simplify our definition of local randomization, without loss of generality, assume that $\\Pr(X_i=X_j)=0,\\ \\hl{\\forall i,j}:X_i,X_j\\in W_0$, so that no two observed age values within the window are exactly identical.",
    "corrected": "To simplify our definition of local randomization, without loss of generality, assume that $\\Pr(X_i=X_j)=0,\\ \\forall i,j:\\hl{i\\neq j},\\ X_i,X_j\\in W_0$, so that no two observed age values within the window are exactly identical.",
    "evidence": "The evidence is a check of the page and a one-line derivation. No restriction $i\\neq j$ is printed. The window contains at least one subject, since the same paragraph assumes $0<\\sum_{i:X_i\\in W_0}Z_i<N_0$. For such a subject $i$, the printed condition with $j=i$ requires $\\Pr(X_i=X_i)=0$, while the event $\\{X_i=X_i\\}$ has probability one.",
    "problem": "The quantifier ranges over all pairs $i,j$ of subjects in the window, including $j=i$. Since $\\Pr(X_i=X_i)=1$, the condition cannot hold as printed. The words that follow it, ``no two observed age values'', state the intended condition, which concerns distinct subjects.",
    "what_survives": "Adding $i\\neq j$ expresses the intended no-ties condition without changing the intended local randomization design. I did not examine whether ``without loss of generality'' is justified for every model of the running variable."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-E003",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
   },
   "work_id": "P014"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Two entries of the reference list. In the first, Cattaneo, Frandsen and Titiunik (2015), a red box marks the closing quotation mark after the word Senate at the end of the title. In the second, Cattaneo, Titiunik and Vasquez-Bare (2016), a red box marks the surname Vasquez-Bare, spelled with an s.",
     "caption": "References, p. 132 (PDF p. 5). The marks show the closing quotation mark after the title of the Cattaneo, Frandsen and Titiunik entry, which has no opening mark, and the spelling “Vasquez-Bare”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E004.png"
    },
    {
     "alt": "Four entries of the reference list: Lee (2008), Liu and Hudgens (2014), Neyman (1923) and Rosenbaum (2007). A red box marks the closing quotation mark after 'elections.' in the Lee entry. Two red boxes mark the third and fourth lines of the Neyman entry, which read 'erzy Splawa, D. M. Dabrowska, and T. P. Speed [1923.] 1990. On the application of probability theory to agricultural experiments: Essay on principles, section 9.' A red box marks the words 'American American' in the journal name of the Rosenbaum entry.",
     "caption": "References, p. 133 (PDF p. 6). The marks show the unmatched closing quotation mark in the Lee entry; the two lines of the Neyman entry that hold the fragment “erzy Splawa”, the date “[1923.] 1990” and the repeated title; and the repeated word “American” in the Rosenbaum entry.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E004-b.png"
    },
    {
     "alt": "The three lines of the entry Thistlethwaite and Campbell (1960). A red box marks the closing quotation mark after the word experiment at the end of the title.",
     "caption": "References, p. 133 (PDF p. 6). The mark shows the unmatched closing quotation mark in the Thistlethwaite and Campbell entry.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-E004-c.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Six entries of the reference list contain misprints: a misspelled surname, a repeated word, three closing quotation marks without an opening mark, and a Neyman entry that prints its title twice and sets a fragment of the author’s name among the translators.",
   "id": "P014-E004",
   "location": "References, pp. 132–133 (PDF pp. 5–6): the entries Cattaneo, Frandsen and Titiunik (2015) and Cattaneo, Titiunik and Vasquez-Bare (2016) on p. 132; Lee (2008), Neyman (1923), Rosenbaum (2007), and Thistlethwaite and Campbell (1960) on p. 133",
   "pdf_page": 5,
   "printed_page": "132–133",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\begin{itemize}\n\\item Cattaneo, M.D., B.R. Frandsen, and R. Titiunik (2015). Randomization inference in the regression discontinuity design: an application to party advantages in the U.S. Senate.\\hl{''} \\emph{Journal of Causal Inference}. 3(1): 1 --24.\n\\item Cattaneo, M.D., R. Titiunik, and G. \\hl{Vasquez-Bare} (2016). Inference in regression discontinuity designs under local randomization. \\emph{The Stata Journal}. 16(2): 331 -- 367.\n\\item Lee, D. (2008) Randomized experiments from non-random selection in U.S. House elections.\\hl{''} \\emph{Journal of Econometrics}. 142: 675 -- 697.\n\\item Neyman, J.S. (1923). On the application of probability theory to agricultural experiments: Essay on principles, section 9. Translated and reprinted by \\hl{erzy Splawa,} D. M. Dabrowska, and T. P. Speed [1923.] 1990. \\hl{On the application of probability theory to agricultural experiments: Essay on principles, section 9.} \\emph{Statistical Science} \\textbf{5}: 465--480.\n\\item Rosenbaum, P.R (2007). Interference between units in randomized experiments. \\emph{Journal of the American \\hl{American} Statistical Association}. 102(477): 191 -- 200.\n\\item Thistlethwaite, D. and D. Campbell. (1960). Regression-discontinuity analysis: an alternative to the ex post facto experiment.\\hl{''} \\emph{Journal of Educational Psychology}. 51: 309 -- 317.\n\\end{itemize}",
    "corrected": "\\begin{itemize}\n\\item Cattaneo, M.D., B.R. Frandsen, and R. Titiunik (2015). Randomization inference in the regression discontinuity design: an application to party advantages in the U.S. Senate. \\emph{Journal of Causal Inference}. 3(1): 1 --24.\n\\item Cattaneo, M.D., R. Titiunik, and G. \\hl{Vazquez-Bare} (2016). Inference in regression discontinuity designs under local randomization. \\emph{The Stata Journal}. 16(2): 331 -- 367.\n\\item Lee, D. (2008) Randomized experiments from non-random selection in U.S. House elections. \\emph{Journal of Econometrics}. 142: 675 -- 697.\n\\item \\hl{Splawa-Neyman, J. (1923 [1990]).} On the application of probability theory to agricultural experiments. Essay on principles. Section 9. \\hl{Translated and edited by D. M. Dabrowska and T. P. Speed.} \\emph{Statistical Science} 5(4): 465--472 (with discussion, 465--480).\n\\item Rosenbaum, P.R (2007). Interference between units in randomized experiments. \\emph{Journal of the American Statistical Association}. 102(477): 191 -- 200.\n\\item Thistlethwaite, D. and D. Campbell. (1960). Regression-discontinuity analysis: an alternative to the ex post facto experiment. \\emph{Journal of Educational Psychology}. 51: 309 -- 317.\n\\end{itemize}",
    "evidence": "The evidence is a check of the pages against public sources. The article page of \\emph{The Stata Journal} names the third author as Gonzalo Vazquez-Bare. The title page of the 1990 translation in \\emph{Statistical Science} names the author as Jerzy Splawa-Neyman, carries the credit ``Translated and edited by D. M. Dabrowska and T. P. Speed'', and has the masthead ``1990, Vol. 5, No. 4, 465-480''. The contents of that issue list the comment by D. B. Rubin separately at pp.~472--480.",
    "problem": "The surname in the 2016 entry is spelled ``Vasquez-Bare'', while the text on pp.~129 and 132 has ``Vazquez-Bare'', which is the spelling of the journal's byline. The Rosenbaum entry prints ``American'' twice. Three titles end with a closing quotation mark that has no opening mark. The Neyman entry runs two citations of one item together. Its title is printed twice, the date appears as ``[1923.] 1990'', and ``erzy Splawa'', a fragment of the author's name Jerzy Splawa-Neyman, stands among the translators. The page range 465--480 of that entry is not an error: it is the range on the masthead of the translation, which covers the translation (465--472) and the discussion that follows it.",
    "what_survives": "Every cited work remains identifiable, and no mathematical content or result of the paper is affected."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-E004",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
   },
   "work_id": "P014"
  },
  {
   "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 followed by a display. A red box marks the words 'average direct effect (Hudgens and Halloran, 2008)' in the sentence 'Then our primary target is the average direct effect (Hudgens and Halloran, 2008) of treatment for subjects who are close to the threshold'. The display reads 'tau equals N0 to the minus 1 times the sum over i with Xi in W0 of Y-bar i of 1 minus Y-bar i of 0'.",
     "caption": "Section 2, p. 131 (PDF p. 4). The marked words identify the target with the average direct effect of Hudgens and Halloran (2008). The display below them is treated minus control.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-N002.png"
    },
    {
     "alt": "Four lines of text from the end of the Introduction. Red boxes mark the words 'This causal effect is equivalent to Hudgens and Halloran (2008)'s average direct effect', which run over two lines. The sentence continues 'for the subpopulation of subjects within the window near the threshold'.",
     "caption": "Introduction, p. 129 (PDF p. 2). The marked sentence says that the causal effect is equivalent to the average direct effect of Hudgens and Halloran (2008).",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-N002-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P014-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T7"
    },
    {
     "id": "P014-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T9"
    },
    {
     "id": "P014-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T10"
    }
   ],
   "headline": "The target is the average direct effect of Hudgens and Halloran (2008) for the subjects in the window once $\\overline{Y}_i(z)$ is the conditional expectation $\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=z]$, and then with the opposite sign: their direct effect is control minus treatment.",
   "id": "P014-N002",
   "location": "Introduction, p. 129 (PDF p. 2), sentence beginning “This causal effect is equivalent to”; Section 2, p. 131 (PDF p. 4), sentence beginning “Then our primary target is the average direct effect”",
   "pdf_page": 2,
   "printed_page": "129",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P014-T10",
    "P014-T7",
    "P014-T9"
   ],
   "text": {
    "as_printed": "\\emph{Introduction (p.~129).} This causal effect is \\hl{equivalent to} Hudgens and Halloran (2008)'s average direct effect for the subpopulation of subjects within the window near the threshold.\n\n\\emph{Section 2 (p.~131).} Then our primary target is \\hl{the average direct effect (Hudgens and Halloran, 2008)} of treatment for subjects who are close to the threshold:\n\\[\n\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}\\overline{Y}_i(1)-\\overline{Y}_i(0).\n\\]",
    "corrected": "\\emph{Introduction (p.~129).} With the \\hl{probability-weighted averages} of \\fid{P014-E001}, this causal effect is, \\hl{up to the sign convention,} Hudgens and Halloran (2008)'s average direct effect for the subpopulation of subjects within the window near the threshold.\n\n\\emph{Section 2 (p.~131).} With $\\overline{Y}_i(z)=\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=z]$ as in \\fid{P014-E001}, our primary target is the average direct effect (Hudgens and Halloran, 2008) of treatment for subjects who are close to the threshold, \\hl{written as treated minus control, which is the reverse of their sign convention}:\n\\[\n\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}\\bigl[\\overline{Y}_i(1)-\\overline{Y}_i(0)\\bigr].\n\\]",
    "evidence": "The evidence is a reading of the cited paper and an exact example. In the public author manuscript of Hudgens and Halloran (2008), the definition in Section 3.1 multiplies each potential outcome by $\\Pr_\\psi(\\mathbf{Z}_{i(j)}=\\omega\\mid Z_{ij}=z)$ and is followed by the statement that the ``individual average potential outcome is the conditional expectation''. In Section 3.2, equation (3) defines the individual direct causal effect as $\\overline{Y}_{ij}(0;\\psi)-\\overline{Y}_{ij}(1;\\psi)$. I did not check the page and equation numbers of the version of record.\n\nFor the example, let $N=4$, $b=1$ and $W_0=[-1,1]$. Let $(X_1,X_2,X_3)$ be a uniformly random arrangement of $-1$, $-1/2$ and $1/2$, and set $X_4=2$ if $(X_1,X_2,X_3)=(-1,-1/2,1/2)$ and $X_4=-2$ otherwise. Set $y_1(\\mathbf{z})=z_4$ and all other potential outcomes equal to zero. The conditions printed on p.~130 hold and the six arrangements are equiprobable. The assignment vectors $(0,0,1,0)$ and $(0,0,1,1)$ have probability $1/6$ each, and $(0,1,0,0)$ and $(1,0,0,0)$ have probability $1/3$ each. The printed $\\overline{Y}_1(0)$ is $1/3$, while $\\mathrm{E}[y_1(\\mathbf{Z})\\mid Z_1=0]=1/4$, and all other averages are zero. I obtain\n\\[\n\\tau=-\\frac19,\\qquad \\frac13\\bigl\\{\\mathrm{E}[y_1(\\mathbf{Z})\\mid Z_1=1]-\\mathrm{E}[y_1(\\mathbf{Z})\\mid Z_1=0]\\bigr\\}=-\\frac{1}{12}=\\mathrm{E}[\\hat\\tau],\n\\]\nand the average direct effect in the convention of Hudgens and Halloran is $1/12$. The printed $\\tau$ equals neither $1/12$ nor $-1/12$. To isolate the sign, fix the assignment of subject 4 at control and set $y_i(\\mathbf{z})=z_i$ for the subjects in the window. Then the two definitions of $\\overline{Y}_i(z)$ agree, $\\tau=\\mathrm{E}[\\hat\\tau]=1$, and the average direct effect in their convention is $-1$.",
    "problem": "Hudgens and Halloran (2008) define the individual average potential outcome as a conditional expectation. The outcomes of an individual are averaged over the assignments of the other individuals, each weighted by its conditional probability given the individual's own treatment. They define the individual direct causal effect as the average under control minus the average under treatment. The target $\\tau$ differs from the average of that effect over the subjects in the window in two ways. First, the printed $\\overline{Y}_i(z)$ weights the distinct assignment vectors equally and is not their conditional expectation in general; this is the error corrected in \\fid{P014-E001}. Second, $\\tau$ is treated minus control. Once the probabilities are inserted, $\\tau$ is therefore the negative of their average direct effect for the subjects in the window. The sign is a convention. The paper uses treated minus control in both $\\tau$ and $\\hat\\tau$, and no result on estimation depends on it.",
    "what_survives": "The estimator, the design and, with the definition corrected in \\fid{P014-E001}, the unbiasedness of the estimator are unaffected. The reference to Hudgens and Halloran (2008) remains the right one for the probability-weighted target. The identification with their average direct effect needs a remark on the sign."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-N002",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
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   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Three lines of text followed by a display and two more lines. The display reads 'tau equals N0 to the minus 1 times the sum over i with Xi in W0 of Y-bar i of 1 minus Y-bar i of 0'. A red box marks the two terms after the summation sign, 'Y-bar i of 1 minus Y-bar i of 0', which are not enclosed in brackets.",
     "caption": "Section 2, p. 131 (PDF p. 4), the display that defines $\\tau$. The marked summand $\\overline{Y}_i(1)-\\overline{Y}_i(0)$ has no brackets.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-N004.png"
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   "formal_statements": [
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     "id": "P014-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T4"
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    {
     "id": "P014-T8",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-T8"
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   "headline": "The summand in the definition of $\\tau$ should be bracketed: $\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}[\\overline{Y}_i(1)-\\overline{Y}_i(0)]$.",
   "id": "P014-N004",
   "location": "Section 2, p. 131 (PDF p. 4), the display that defines $\\tau$",
   "pdf_page": 4,
   "printed_page": "131",
   "reaches": [],
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   "severity": "Minor",
   "subject": "paper",
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   "text": {
    "as_printed": "\\[\n\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}\\hl{\\overline{Y}_i(1)-\\overline{Y}_i(0)}.\n\\]",
    "corrected": "\\[\n\\tau=N_0^{-1}\\sum_{i:X_i\\in W_0}\\hl{\\bigl[}\\overline{Y}_i(1)-\\overline{Y}_i(0)\\hl{\\bigr]}.\n\\]",
    "evidence": "The evidence is a check of the page. The display that defines $\\tau$ has neither brackets nor a second summation sign. The later display reads $\\mathrm{E}[\\hat\\tau]=N_0^{-1}[\\sum_{i:X_i\\in W_0}\\overline{Y}_i(1)-\\sum_{i:X_i\\in W_0}\\overline{Y}_i(0)]$, and by linearity of the finite sum its right side equals the bracketed form below.",
    "problem": "The two terms after the summation sign are not bracketed. Read literally, the sum and the factor $N_0^{-1}$ apply to $\\overline{Y}_i(1)$ alone, and the index $i$ in $\\overline{Y}_i(0)$ is free. The prose that follows, and the display for $\\mathrm{E}[\\hat\\tau]$ on the same page, show that both terms belong to the summand.",
    "what_survives": "The intended target and its agreement with the display for $\\mathrm{E}[\\hat\\tau]$ are unaffected, and nothing depends on the missing brackets. The definition of $\\overline{Y}_i(z)$ inside the sum is corrected separately in \\fid{P014-E001}."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-N004",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
   },
   "work_id": "P014"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Two entries of the reference list, Rosenbaum (2007) and Sobel (2006). A red box marks the page range 1398-1408 at the end of the Sobel entry.",
     "caption": "References, p. 133 (PDF p. 6). The marked page range of the Sobel (2006) entry ends at 1408.",
     "url": "https://pmaronow.github.io/llm-audit/works/P014/exhibits/P014-N006.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The page range of Sobel (2006) is 1398–1407, not 1398–1408.",
   "id": "P014-N006",
   "location": "References, p. 133 (PDF p. 6), the entry Sobel (2006)",
   "pdf_page": 6,
   "printed_page": "133",
   "reaches": [],
   "related_entries": [],
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   "severity": "Minor",
   "subject": "paper",
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   "text": {
    "as_printed": "Sobel, M.E. (2006). What do randomized studies of housing mobility demonstrate?: Causal inference in the face of interference. \\emph{Journal of the American Statistical Association} 101 (476): 1398--\\hl{1408}.",
    "corrected": "Sobel, M.E. (2006). What do randomized studies of housing mobility demonstrate?: Causal inference in the face of interference. \\emph{Journal of the American Statistical Association} 101 (476): 1398--\\hl{1407}.",
    "evidence": "The evidence is a check against the publisher. Its table of contents for the \\emph{Journal of the American Statistical Association} 101(476) lists Sobel's article at pages 1398--1407 and the next article, ``Exceedance Control of the False Discovery Proportion'', at pages 1408--1417. The publisher's page for the article has the same range.",
    "problem": "The range is one page too long. Page 1408 of that issue is the first page of the next article.",
    "what_survives": "The cited work remains identifiable, and nothing in the paper depends on the last page."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P014/#P014-N006",
   "work": {
    "arxiv_id": null,
    "doi": "10.1353/obs.2017.0001",
    "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
    "url": "https://pmaronow.github.io/llm-audit/works/P014/"
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     "body": "<p>p. 131 (PDF p. 4), assumptions p. 130: with a fixed interior treated count and common propensities, probability-weighted own-arm averages are conditional expectations with positive denominators, and the unchanged difference in means is unbiased for their bracketed window contrast. All four displayed derivation steps hold. Corrected uniform local score arrangements imply these hypotheses and this identity for arbitrary potential outcomes, with the outside allocation and its dependence unrestricted.</p>",
     "headline": "Section 2, the individual average potential outcome, the target <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>τ</mi></mrow><annotation encoding=\"application/x-tex\">\\tau</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ</span></span></span></span> and the unbiasedness display, corrected: probability-weighted averages, same estimator",
     "location": "p. 131 (PDF p. 4)"
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    "text": {
     "body": "p. 131 (PDF p. 4), assumptions p. 130: with a fixed interior treated count and common propensities, probability-weighted own-arm averages are conditional expectations with positive denominators, and the unchanged difference in means is unbiased for their bracketed window contrast. All four displayed derivation steps hold. Corrected uniform local score arrangements imply these hypotheses and this identity for arbitrary potential outcomes, with the outside allocation and its dependence unrestricted.",
     "headline": "Section 2, the individual average potential outcome, the target $\\tau$ and the unbiasedness display, corrected: probability-weighted averages, same estimator",
     "location": "p. 131 (PDF p. 4)"
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     "body": "<p>p. 131 (PDF p. 4): the bracketed window mean of differences is the difference of the two sums divided by the nonzero window size.</p>",
     "headline": "Section 2, the display defining the target <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>τ</mi></mrow><annotation encoding=\"application/x-tex\">\\tau</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.4306em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ</span></span></span></span>, corrected: the summand bracketed",
     "location": "p. 131 (PDF p. 4)"
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     "body": "p. 131 (PDF p. 4): the bracketed window mean of differences is the difference of the two sums divided by the nonzero window size.",
     "headline": "Section 2, the display defining the target $\\tau$, corrected: the summand bracketed",
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     "body": "<p>p. 130 (PDF p. 3): uniform local score arrangements give fixed interior treatment counts, common propensities, fixed window membership, the stated empirical CDF and no distinct-subject ties. Adding discreteness and the printed full-vector mass clause forces exactly one full supported vector per local arrangement and makes the outside scores an almost surely deterministic function of that arrangement. Conversely such an outside function gives n! full support points of mass 1/n!. The specified independent outside example has twelve points, six of mass 1/18 and six of mass 1/9, and satisfies the correction while failing the printed full-vector mass clause.</p>",
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     "body": "p. 130 (PDF p. 3): uniform local score arrangements give fixed interior treatment counts, common propensities, fixed window membership, the stated empirical CDF and no distinct-subject ties. Adding discreteness and the printed full-vector mass clause forces exactly one full supported vector per local arrangement and makes the outside scores an almost surely deterministic function of that arrangement. Conversely such an outside function gives n! full support points of mass 1/n!. The specified independent outside example has twelve points, six of mass 1/18 and six of mass 1/9, and satisfies the correction while failing the printed full-vector mass clause.",
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 "summary": "The note asks what the regression discontinuity design estimates when subjects interfere with one another. Under a local randomization assumption, in which the values of the running variable are randomly permuted among the subjects in a window around the cutoff, it states that the difference in means within the window is unbiased for a regime-specific average direct effect. 7 entries are classified as errors (1 Moderate, 6 Minor). The Moderate error is that unbiasedness holds for the probability-weighted average direct effect, and not in general for the target as printed (\\fid{P014-E001}).",
 "what_was_checked": "I checked the following parts of the model and argument.\n\\begin{itemize}\n\\item The model of Section 2, p.~130: potential outcomes indexed by all $2^N$ assignment vectors allow arbitrary interference, and the count $2^N$ and the example $y_2((0,1,0,\\ldots,0))$ are correct.\n\\item The threshold rule determines $\\mathbf{Z}$ from $\\mathbf{X}$. Reading $Z_i$ as eligibility under noncompliance gives intention-to-treat effects, as the paper says.\n\\item With the set of subjects in the window and the values in the window fixed, the numbers of treated and control subjects in the window are fixed, so the denominators of $\\hat\\tau$ are constants and linearity of expectation applies.\n\\item With the arrangement in the window uniformly random, each subject in the window is treated with the same probability, and $\\mathrm{E}[\\hat\\tau]$ is the average over the window of $\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=1]-\\mathrm{E}[y_i(\\mathbf{Z})\\mid Z_i=0]$, for arbitrary potential outcomes and arbitrary allocation outside the window. This was derived and agreed with exact enumeration of small designs.\n\\item Under uniformly random arrangements in the window, the printed support averages give unbiasedness when treatment assignments outside the window are fixed or are a function of treatment assignments in the window; $\\mathbf{Z}$ is then uniform on its support. It uses no restriction on interference.\n\\item The estimator on p.~131 is the ordinary difference in means over the window, and the condition $0<\\sum_{i:X_i\\in W_0}Z_i<N_0$ is what it needs to be defined.\n\\item The verbal description of $\\tau$ on p.~131, and the statement that the estimand is regime specific, agree with the probability-weighted target of \\fid{P014-E001}.\n\\item The Discussion presents estimators under imperfect compliance and the estimation of indirect, total and overall effects as future work, and it makes no claim to identify them.\n\\item Each of the 15 works cited in the text has an entry in the reference list, and each entry is cited.\n\\item The definitions of Hudgens and Halloran (2008) that the paper relies on were read in the public author manuscript (\\fid{P014-N002}). The bibliographic details in \\fid{P014-E004} and \\fid{P014-N006} were checked against public sources.\n\\end{itemize}\nMy checks had the following limits. The paragraph on inference on pp.~131--132 has no stated theorem or proof in the paper, and the results it cites from Liu and Hudgens (2014) and Aronow and Samii (2016) were not checked against the design of the paper; a possible mismatch between its conditions and that design was examined and not confirmed. The other cited works were not examined beyond their bibliographic details. Whether the no-ties condition is ``without loss of generality'' was not examined. The volume and year of the article, on which the publisher's cover sheet and the running head disagree, could not be settled. The paper has no tables, figures, estimates, simulations or appendix, and no supplement or replication files were available, so there was nothing numerical to reproduce.",
 "work": {
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  "audit_complete": true,
  "authors": [
   "P. M. Aronow",
   "Nicole E. Basta",
   "M. Elizabeth Halloran"
  ],
  "category": "Articles - Theory and Methodology",
  "citation": "P. M. Aronow, Nicole E. Basta, and M. Elizabeth Halloran (2017). The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach. Observational Studies.",
  "doi": "10.1353/obs.2017.0001",
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   "sha256": "61e58d20510ba3fd7ee73ab1a459085137910b3d2435e5c20a2f5ce925b6482a",
   "source_url": "https://doi.org/10.1353/obs.2017.0001",
   "venue": "Observational Studies",
   "year": "2017"
  },
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  "later_materials": null,
  "title": "The Regression Discontinuity Design under Interference: A Local Randomization-Based Approach",
  "type": "Journal article",
  "url": "https://pmaronow.github.io/llm-audit/works/P014/",
  "venue": "Observational Studies",
  "year": "2017"
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