{
 "claims": [
  {
   "claim": "For $n$ independent rows and $k$ variables, if each conditional observation probability is at most $q_*<1$ whenever the preceding variables are all observed with positive probability, the probability that listwise deletion removes every row is at least $(1-q_*^k)^n$.",
   "claim_id": "P007-H1",
   "related_corrections": [
    "P007-E003"
   ],
   "tied_to": "Lemma 3; Assumptions 1 and 2 (p. 150); proof in the Appendix (p. 154)"
  },
  {
   "claim": "Under independent rows and a fixed bound $q_*<1$ on conditional observation probabilities, the probability that listwise deletion removes all rows tends to 1 when the number of variables grows faster than $\\log n$.",
   "claim_id": "P007-H2",
   "related_corrections": [
    "P007-E003",
    "P007-N001"
   ],
   "tied_to": "Proposition 5; Assumptions 1, 2 and 4; Lemma 3; abstract and Introduction (p. 149); proof p. 154"
  },
  {
   "claim": "The finite-sample lower bound is substantial when idiosyncratic missingness is 10% or 25%, and can be extremely small when it is 1%, even with $k=150$ and $n=100$.",
   "claim_id": "P007-H3",
   "related_corrections": [
    "P007-E005",
    "P007-E006"
   ],
   "tied_to": "Section 2.1, last paragraph (p. 151); Supporting Information A, Figures 3-5; Numerical_Illustration.R"
  },
  {
   "claim": "For the January 2021 Quality of Government Standard dataset, the paper reports 194 countries, 351 variables and 35.9% missing cells. With 2 randomly chosen variables, expected row loss exceeds 50%; with 17 it exceeds 99%. The probability of losing all rows exceeds $0.5$ at 14 variables and $0.99$ at 52.",
   "claim_id": "P007-H4",
   "related_corrections": [
    "P007-E009"
   ],
   "tied_to": "Table 1 (Quality of Government column); Figure 1; Section 3.2 (p. 153); Simcode_Qog.R"
  },
  {
   "claim": "For the State Failure data, the paper reports 8,580 country-years, 1,205 variables and 66.8% missing cells. Expected row loss exceeds 50% with one randomly chosen variable and 99% with more than three; the probability of losing all rows exceeds $0.5$ at 6 variables and $0.99$ with more than 17.",
   "claim_id": "P007-H5",
   "related_corrections": [
    "P007-E010",
    "P007-E014"
   ],
   "tied_to": "Table 1 (State Failure column); Figure 2; Section 3.2 (p. 153); Simcode_Statefailure.R"
  },
  {
   "claim": "The results extend to groups of variables that share one missingness pattern, with the number of groups $g$ in place of the number of variables $k$.",
   "claim_id": "P007-H6",
   "related_corrections": [
    "P007-E003"
   ],
   "tied_to": "Section 2 (p. 150); Supporting Information B: Assumptions 6, 7, 9; Lemma 8; Proposition 10"
  },
  {
   "claim": "Listwise deletion is not viable in many data-analytic settings; researchers using it may have to forgo variables they need even when $n$ is large, and the method for handling missing data should be chosen for the problem at hand.",
   "claim_id": "P007-H7",
   "related_corrections": [
    "P007-E010",
    "P007-E009"
   ],
   "tied_to": "Abstract; end of Introduction (p. 150); Section 4 (p. 154); Proposition 5; Figures 1 and 2; Table 1"
  }
 ],
 "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": "Crop of the proof of Lemma 3 under the heading 'Appendix: Proofs'. Seven red outlines mark: the words 'which is equivalent to say that all variables are fully missing' (two outlines, over a line break); the expression 'q i j less than or equal to q star, for all j in 1, 2, up to k'; and four occurrences of the comma-separated list 'q i 1, q i 2, up to q i k'.",
     "caption": "Proof of Lemma 3, p. 154 (PDF p. 6). Marked: the sentence on the case $q_* = 0$, the bound $q_{ij} \\leq q_*$ asserted for all $j$, and the four comma lists printed for the product $q_{i1} q_{i2} \\cdots q_{ik}$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E003.png"
    },
    {
     "alt": "Crop of three lines of the proof of Proposition 5, beginning 'Since q star in (0,1) and k = f(n)'. Three red outlines mark a greater-than-or-equal sign with a subscript star after 'k = f(n)', a second such sign after the n-th power of one minus q star to the f(n), and the expression one minus q star to the power 'f(n' with no closing parenthesis, raised to the power n.",
     "caption": "Proof of Proposition 5, p. 154 (PDF p. 6). Marked: the two signs $\\geq$ printed with a subscript star, and the expression $(1 - q_*^{f(n})^n$, in which the exponent $f(n)$ lacks its closing parenthesis.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E003-b.png"
    },
    {
     "alt": "Crop of Lemma 8 and its proof in the supplement. Three red outlines mark the words 'Since this entails that all groups of variables are completely missing,' and, over a line break, the expression 'q i j less than or equal to q star, for all j in 1, 2, up to g'.",
     "caption": "Supplementary Material B, p. 4: Lemma 8 and its proof. Marked: the sentence on the case $q_* = 0$ and the bound $q_{ij} \\leq q_*$ asserted for all $j$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E003-c.png"
    },
    {
     "alt": "Crop of two lines of the proof of Proposition 10, beginning 'Since q star in (0,1) and k = l(n)'. Four red outlines mark the letter k, two greater-than-or-equal signs that carry a subscript star, and the expression one minus q star to the power 'l(n' with no closing parenthesis, raised to the power n.",
     "caption": "Supplementary Material B, foot of p. 4: proof of Proposition 10. Marked: $k$ printed for $g$, the two signs $\\geq$ with a subscript star, and the expression $(1 - q_*^{l(n})^n$ with the unclosed exponent.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E003-d.png"
    },
    {
     "alt": "Crop of two paragraphs of Supplementary Material B that define g, the number of variable groups, and the indicator for the j-th group in the i-th row. In the sentence 'we use M i j to represent the random vector collecting the missingness indicators up to variable j', the word 'variable' is outlined in red.",
     "caption": "Supplementary Material B, p. 3: definitions for groups of variables. The marked word “variable” should be “group”: in this section $j$ indexes groups.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E003-e.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P007-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T3"
    },
    {
     "id": "P007-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T4"
    },
    {
     "id": "P007-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T5"
    },
    {
     "id": "P007-T6",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T6"
    }
   ],
   "headline": "The proofs of Lemmas 3 and 8 wrongly read $q_* = 0$ as every variable being missing and omit the case of a conditioning event of probability zero, and several symbols are misprinted; the two lemmas and Propositions 5 and 10 hold as stated.",
   "id": "P007-E003",
   "location": "Appendix, proofs of Lemma 3 and Proposition 5, p. 154 (PDF p. 6); Supplementary Material B, p. 3 (definition of $\\mathbf{M}_{ij}$) and p. 4 (proofs of Lemma 8 and Proposition 10)",
   "pdf_page": 6,
   "printed_page": "154",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P007-T3",
    "P007-T4",
    "P007-T5",
    "P007-T6"
   ],
   "text": {
    "as_printed": "\\begin{itemize}\n\\item Proof of Lemma 3, p.~154: ``First suppose $q_* = 0$, \\hl{which is equivalent to say that all variables are fully missing}. Then $p_{all} = 1 = (1 - q_*^k)^n$. Now suppose $q_* \\in (0,1)$. By the independence assumption, $p_{all} = \\Pi_{i=1}^{n}(1 - \\Pr(\\mathbf{M}_{ik} = 0))$. Denote $q_{ij} = \\Pr(M_{ij} = 0 | \\mathbf{M}_{i(j-1)} = 0)$ if $j > 1$, else $q_{ij} = \\Pr(M_{ij} = 0)$. By Assumption 2, $\\hl{q_{ij} \\leq q_*}$, for all $j \\in \\{1, 2, \\ldots, k\\}$. By the chain rule of conditional probability, $\\Pr(\\mathbf{M}_{ik} = 0) = \\hl{q_{i1}, q_{i2}, \\ldots, q_{ik}}$. This means that the probability of a single observation containing at least one missing entry is $(1 - \\hl{q_{i1}, q_{i2}, \\ldots, q_{ik}})$. Since $q_* \\geq q_{ij}$ for all $j \\in \\{1, 2, \\ldots, k\\}$, $q_*^k \\geq \\hl{q_{i1}, q_{i2}, \\ldots, q_{ik}}$. Thus $(1 - q_*^k) \\leq (1 - \\hl{q_{i1}, q_{i2}, \\ldots, q_{ik}})$.''\n\\item Proof of Proposition 5, p.~154: ``Since $q_* \\in (0,1)$ and $k = f(n) \\hl{\\geq_*} 1$, $-q_*^{f(n)} > -1$ and $1 - q_*^{f(n)} \\leq 1$. By Bernoulli's Inequality, since $n \\in \\mathbb{N}$, $(1 - q_*^{f(n)})^n \\hl{\\geq_*} 1 + n(-q_*^{f(n)}) = 1 - nq_*^{f(n)}$. Thus $1 - nq_*^{f(n)} \\leq \\hl{(1 - q_*^{f(n})^n} \\leq 1$ in the common domain $n \\in \\mathbb{N}$.''\n\\item Supplement, p.~3: ``we use $\\mathbf{M}_{ij}$ to represent the random vector collecting the missingness indicators up to \\hl{variable} $j$, $(M_{i1}, M_{i2}, \\ldots, M_{ij})$.''\n\\item Supplement, proof of Lemma 8, p.~4: ``Suppose $q_* = 0$. \\hl{Since this entails that all groups of variables are completely missing}, $p_{all} = 1 = (1 - q_*^g)^n$. For the second case suppose $q_* \\in (0,1)$. By the group independence assumption, $p_{all} = \\Pi_{i=1}^{n}(1 - \\Pr(\\mathbf{M}_{ig} = 0))$. Denote $q_{ij} = \\Pr(M_{ij} = 0 | \\mathbf{M}_{i(j-1)} = 0)$ if $j > 1$, else $q_{ij} = \\Pr(M_{ij} = 0)$. By Assumption 7, $\\hl{q_{ij} \\leq q_*}$, for all $j \\in \\{1, 2, \\ldots, g\\}$.''\n\\item Supplement, proof of Proposition 10, p.~4: ``Since $q_* \\in (0,1)$ and $\\hl{k} = l(n) \\hl{\\geq_*} 1$, $-q_*^{l(n)} > -1$ and $1 - q_*^{l(n)} \\leq 1$. By Bernoulli's Inequality, since $n \\in \\mathbb{N}$, $(1 - q_*^{l(n)})^n \\hl{\\geq_*} 1 + n(-q_*^{l(n)}) = 1 - nq_*^{l(n)}$. Thus $1 - nq_*^{l(n)} \\leq \\hl{(1 - q_*^{l(n})^n} \\leq 1$ in the common'' (the sentence runs on to p.~5).\n\\end{itemize}",
    "corrected": "The statements of Lemma 3, Lemma 8, Proposition 5 and Proposition 10 stand as printed. The proofs are corrected as follows. For the supplement, read groups for variables, $g$ for $k$, $l(n)$ for $f(n)$, and Assumptions 6, 7 and 9 for Assumptions 1, 2 and 4.\n\n\\emph{Proof of Lemma 3.} Assume that the rows are mutually independent (Assumption 1) and that there exists $q_* \\in [0,1)$ such that, for all $i$, $\\Pr(M_{i1} = 0) \\leq q_*$ and $\\Pr(M_{ij} = 0 | \\mathbf{M}_{i(j-1)} = 0) \\leq q_*$ for all $j \\in \\{2, \\ldots, k\\}$ such that $\\Pr(\\mathbf{M}_{i(j-1)} = 0) > 0$ (Assumption 2). Fix a row $i$. Let $s_{i0} = 1$ and, for $j \\in \\{1, \\ldots, k\\}$, let $s_{ij} = \\Pr(\\mathbf{M}_{ij} = 0)$ be the probability that the first $j$ entries of row $i$ are observed. Then\n\\[\ns_{ij} \\leq q_* \\, s_{i(j-1)} \\quad \\text{for all } j \\in \\{1, \\ldots, k\\}.\n\\]\nFor $j = 1$ this is the first part of Assumption 2. For $j \\geq 2$ with $\\hl{s_{i(j-1)} = 0}$, the event $\\{\\mathbf{M}_{ij} = 0\\}$ is contained in $\\{\\mathbf{M}_{i(j-1)} = 0\\}$, so $s_{ij} = 0$. For $j \\geq 2$ with $s_{i(j-1)} > 0$, the second part of Assumption 2 gives $s_{ij} / s_{i(j-1)} = \\Pr(M_{ij} = 0 | \\mathbf{M}_{i(j-1)} = 0) \\leq q_*$. By induction on $j$, this implies that $s_{ik} \\leq q_*^k$. Since the rows are independent,\n\\[\np_{all} = \\prod_{i=1}^{n} (1 - s_{ik}) \\geq (1 - q_*^k)^n.\n\\]\nIf $q_* = 0$, then $s_{i1} = 0$ for every $i$: \\hl{the first variable is missing with probability one in every row}, and $p_{all} = 1$. Later variables need not be missing.\n\n\\emph{Proof of Proposition 5.} If $\\hl{q_* = 0}$, Lemma 3 gives $p_{all} = 1$ for every $n$ with $f(n) \\geq 1$. Suppose that $q_* \\in (0,1)$. The limit $\\lim_{n \\to \\infty} n q_*^{f(n)} = 0$ is obtained as printed. The paragraph that follows should read: ``Since $q_* \\in (0,1)$ and $k = f(n) \\hl{\\geq} 1$, $-q_*^{f(n)} > -1$ and $1 - q_*^{f(n)} \\leq 1$. By Bernoulli's Inequality, since $n \\in \\mathbb{N}$, $(1 - q_*^{f(n)})^n \\hl{\\geq} 1 + n(-q_*^{f(n)}) = 1 - nq_*^{f(n)}$. Thus $1 - nq_*^{f(n)} \\leq \\hl{(1 - q_*^{f(n)})^n} \\leq 1$ in the common domain $n \\in \\mathbb{N}$.'' The rest of the proof is as printed.\n\n\\emph{Other changes.} Each of the four comma lists in the printed proof of Lemma 3 is the product $\\hl{q_{i1} q_{i2} \\cdots q_{ik}}$. The proof of Proposition 10 should read $\\hl{g} = l(n) \\geq 1$, with the same three repairs as in the proof of Proposition 5. On p.~3 of the supplement, read ``up to \\hl{group} $j$''. In the proof of Lemma 8, ``the group independence assumption'' is Assumption 6, which is independence across rows.",
    "evidence": "The evidence consists of exact counterexamples to the two statements in the printed proofs, and a derivation of the lemmas and propositions under the stated assumptions, given under ``Corrected''.\n\nLet $n = 1$ and $k = 2$, and let the row satisfy $(M_{11}, M_{12}) = (1, 0)$ with probability one. Assumption 1 holds, since there is only one row. With $q_* = 0$, Assumption 2 holds: I have $\\Pr(M_{11} = 0) = 0 \\leq q_*$, and no condition is imposed for $j = 2$ because $\\Pr(M_{11} = 0) = 0$. Since the second variable is always observed, the variables are not all fully missing, while $p_{all} = 1 = (1 - q_*^2)^1$. With two groups of one variable each, the same example applies to the proof of Lemma 8.\n\nFor the same row and $q_* = 1/2$, Assumption 2 holds again, but $q_{12} = \\Pr(M_{12} = 0 | M_{11} = 0)$ is a ratio $0/0$. The lemma's conclusion holds, since $p_{all} = 1 \\geq 3/4$. The event of probability zero need not involve the first variable alone. Let $k = 3$ and $q_* = 1/2$, and let a row equal $(1,0,0)$ or $(0,1,0)$ with probability $1/2$ each. Then $\\Pr(M_{i1} = 0) = 1/2$ and $\\Pr(M_{i1} = M_{i2} = 0) = 0$, and $q_{i3}$ is undefined. For two independent rows with this distribution, $p_{all} = 1 \\geq 49/64$.\n\nA row equal to $(1,1)$ with probability one satisfies Assumption 2 with $q_* = 1/2$ as well as with $q_* = 0$. Complete missingness does not force $q_* = 0$. For the index in the proof of Proposition 10, two groups of two variables each give $g = 2$ and $k = 4$. I read the misprints from the printed pages. The supplement prints the product in the proof of Lemma 8 correctly, as $q_{i1} q_{i2} \\cdots q_{ig}$.",
    "problem": "The printed proofs have two defects of argument and several misprints. None of them makes a stated result false.\n\n\\emph{The case $q_* = 0$.} Both proofs read $q_* = 0$ as saying that all variables (in the supplement, all groups) are fully missing. Assumption 2 with $q_* = 0$ says only that the first variable is missing with probability one in every row. The conditions on later variables are then vacuous, because each is imposed only when the earlier variables are all observed with positive probability, and later variables may be observed. The conclusion $p_{all} = 1$ is still correct, since every row has a missing first entry. The printed equivalence also fails in the other direction. Since $q_*$ is an upper bound and need not be the smallest one, data in which every variable is always missing satisfy Assumption 2 with $q_* = 1/2$ as well.\n\n\\emph{Conditioning on an event of probability zero.} In the second case the proofs define $q_{ij}$, for $j > 1$, as the probability that entry $j$ is observed given that the first $j - 1$ entries are observed, and assert that $q_{ij} \\leq q_*$ for all $j$. Assumption 2 (in the supplement, Assumption 7) bounds this conditional probability only when the conditioning event has positive probability. When that event has probability zero, $q_{ij}$ is a ratio $0/0$ and the proofs state no convention for it. Neither the bound nor the product formula is then available as written. The split into $q_* = 0$ and $q_* \\in (0,1)$ does not cover this, since it can happen in either case.\n\n\\emph{Misprints.} In the proof of Lemma 3 the product $q_{i1} q_{i2} \\cdots q_{ik}$ is printed four times as a list separated by commas. In the proofs of Propositions 5 and 10 the sign $\\geq$ twice carries a subscript star, and in the sandwich inequality the exponent $f(n)$ (in the supplement, $l(n)$) lacks its closing parenthesis. The proof of Proposition 10 writes $k = l(n)$ where the number of groups, $g = l(n)$, is meant; $k$ is the number of variables, and $k \\geq g$. On p.~3 of the supplement, ``up to variable $j$'' should be ``up to group $j$''. The proofs of the two propositions also begin the logarithm argument with $q_* \\in (0,1)$ without saying that the case $q_* = 0$ follows from the lemma.",
    "what_survives": "Lemma 3, Lemma 8, Proposition 5 and Proposition 10 hold as stated, under the stated assumptions; no assumption is added and no rate changes. No number changes: the numerical illustrations of Supplementary Material A use the bound of Lemma 3 directly, and the simulations of Section 3 count complete rows directly. What changes is the wording of the case $q_* = 0$, one step in the proofs of the two lemmas, and the misprinted symbols."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E003",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the paragraph beginning 'We also assume that there is some idiosyncratic missingness in each variable', followed by Assumption 2. Red boxes mark the words 'one variable is always observed whenever another variable is observed', which run over two lines. Assumption 2 below bounds Pr(M_i1 = 0) and Pr(M_ij = 0 given that the bold vector M_i(j-1) equals 0) by q_*.",
     "caption": "Section 2, p. 150 (PDF p. 2): the paragraph before Assumption 2 and the assumption. The marked words describe what the assumption rules out.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E002.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Assumption 2 rules out a variable that is always observed whenever all the variables before it in the ordering are observed; it does not rule out the reverse, as the sentence before the assumption can suggest.",
   "id": "P007-E002",
   "location": "Section 2, p. 150 (PDF p. 2), paragraph before Assumption 2, last sentence",
   "pdf_page": 2,
   "printed_page": "150",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "``Namely, we will assume that there is a factorization of the data such that all conditional probabilities that an observation is missing are bounded away from zero, given whenever prior variables are observed. Substantively, this assumption rules out the case where, for example, \\hl{one variable is always observed whenever another variable is observed}.''",
    "corrected": "``Substantively, this assumption rules out the case where, for example, \\hl{a variable is always observed whenever all preceding variables in the ordering are observed, provided that this event has positive probability}.''",
    "intended_meaning": "Assumption 2 bounds by $q_*<1$ the probability that variable $j$ is observed given that variables $1,\\dots,j-1$ are all observed, whenever that event has positive probability. The sentence describes this bound, and its example is to be read in the direction of the ordering: a variable cannot be always observed whenever all the variables before it are observed. Read in this way, the sentence is correct.\n\nThe assumption allows the reverse. For example, let $k=2$, and let $(M_{i1},M_{i2})$ take the values $(1,1)$, $(0,1)$ and $(0,0)$ with probabilities $1/2$, $1/4$ and $1/4$, where $M_{ij}=0$ means that the entry is observed. Then $\\Pr(M_{i1}=0)=1/2$ and $\\Pr(M_{i2}=0\\mid M_{i1}=0)=1/2$, and Assumption 2 holds with $q_*=1/2$. However, $\\Pr(M_{i1}=0\\mid M_{i2}=0)=1$: the first variable is always observed whenever the second variable is observed. No result uses the sentence."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E002",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the third paragraph of Section 3.1. Three terms are outlined in red: 'out' in 'how out theoretical assumptions', 'i.i.d.' in 'Assumption 1 asserts i.i.d. missingness across rows', and 'i.i.d.' in 'Our theoretical assumption of i.i.d. missingness across rows'.",
     "caption": "Section 3.1, third paragraph, p. 152 (PDF p. 4). The misprint “out” and the two places where Assumption 1 is called “i.i.d.” are marked.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E004.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P007-T11",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T11"
    }
   ],
   "headline": "Assumption 1 requires mutually independent rows, not i.i.d. rows, and “out theoretical assumptions” should read “our theoretical assumptions”.",
   "id": "P007-E004",
   "location": "Section 3.1, third paragraph, p. 152 (PDF p. 4), first, second and last sentences",
   "pdf_page": 4,
   "printed_page": "152",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P007-T11"
   ],
   "text": {
    "as_printed": "``We briefly discuss how \\hl{out} theoretical assumptions align with this setting. Assumption 1 asserts \\hl{i.i.d.} missingness across rows, which is unlikely to be met in this setting. [\\ldots] Our theoretical assumption of \\hl{i.i.d.} missingness across rows can therefore be seen as optimistic when faced with real-world data and, all else equal, the probability that all data will be lost may be higher than theory might dictate.''",
    "corrected": "``We briefly discuss how \\hl{our} theoretical assumptions align with this setting. Assumption 1 asserts \\hl{mutually independent} missingness across rows, which is unlikely to be met in this setting. [\\ldots] Our theoretical assumption of \\hl{mutually independent} missingness across rows can therefore be seen as optimistic when faced with real-world data and, all else equal, the probability that all data will be lost may be higher than theory might dictate.''",
    "evidence": "The evidence is a check of the page against Assumption 1 as printed, and an example. Two independent rows with one variable each, observed with probabilities $1/4$ and $1/2$, satisfy Assumptions 1 and 2 with $q_* = 1/2$ and are not identically distributed. Lemma 3 holds for them, since $p_{all} = (3/4)(1/2) = 3/8 \\geq 1/4$.",
    "problem": "Assumption 1 (p.~150) states that the rows are mutually independent. It does not ask that they be identically distributed, and the proof of Lemma 3 works with row-specific probabilities. The paragraph twice describes the assumption as ``i.i.d.'', which adds a condition that the paper neither assumes nor uses. The first sentence also prints ``out'' for ``our''.",
    "what_survives": "Nothing depends on the label. Assumption 1 as stated, Lemma 3, Proposition 5 and the paragraph's point about clustered rows are unaffected."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E004",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of two paragraphs. In the first, a red box marks the words 'smallest possible' in 'we will get the smallest possible k for each p_all that k =', followed by the floor of log(1 - p_all^(1/n)) over log(q_*). In the second, red boxes mark 'k = 33' in the example with n = 10,000 and q_* = 0.75 and 'k = 952' in the example with n = 10,000 and q_* = 0.99.",
     "caption": "Supporting Information A, supplement p. 2: the two paragraphs that describe Figure 4. The marked words call the plotted value the smallest possible $k$, and the marked values are given as yielding a 50% probability.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E005.png"
    },
    {
     "alt": "Crop of the subcaptions '(a) p_all at most 0.5' and '(b) p_all at most 0.99' and of the caption 'Figure 4. Largest k such that p_all is at most 0.5, 0.99 plotted against n and q_*'. Red boxes mark the three occurrences of p_all, none of which is underlined.",
     "caption": "Supplement p. 2: the subcaptions and the caption of Figure 4. The caption says “Largest $k$”; the three marked symbols are printed without the underline that denotes the lower bound.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E005-caption.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P007-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T7"
    },
    {
     "id": "P007-T8",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T8"
    }
   ],
   "headline": "The formula plotted in Figure 4 gives the largest $k$ at which the lower bound $(1-q_*^k)^n$ does not exceed the target, not the smallest $k$ that reaches it; at $k=33$ and $k=952$ the bound is 47.1% and 49.7%, and it first reaches 50% at $k=34$ and $k=953$.",
   "id": "P007-E005",
   "location": "Supporting Information A, supplement p. 2: the two paragraphs before Figure 4, and the caption and subcaptions of Figure 4",
   "pdf_page": 2,
   "printed_page": "2",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P007-T7",
    "P007-T8"
   ],
   "text": {
    "as_printed": "``Figure 4 illustrates, for a given $n,q_*$, how large $k$ can be while still ensuring that $\\hl{p_{all}}\\leq 0.5,0.99$. We compute this using the result from Lemma 3, $\\underline{p_{all}}=(1-q_*^k)^n$. Since $(1-q_*^k)^n$ is strictly increasing in $k$, solving for equality $\\hl{p_{all}}=(1-q_*^k)^n$ we will get the \\hl{smallest possible} $k$ for each $\\underline{p_{all}}$ that\n\\[\nk=\\left\\lfloor\\frac{\\log(1-\\underline{p_{all}}^{1/n})}{\\log(q_*)}\\right\\rfloor.\n\\]\n\\ldots\n\nWith more missingness, $q_*=0.75,0.90$, even relatively small $k$ can yield missingness of $\\underline{p_{all}}$. For example, even with $n=10,000$ and $q_*=0.75$ we need only $\\hl{k=33}$ to have a 50\\% probability that all rows will be missing. However, when missingness is very low, $k$ needs to be very large to cause all data to be missing. For example, with $n=10,000$ and $q_*=0.99$, we need $\\hl{k=952}$ to have a 50\\% probability that all rows will be missing.''\n\nSubcaptions and caption of Figure 4: ``(a) $\\hl{p_{all}}\\leq 0.5$'', ``(b) $\\hl{p_{all}}\\leq 0.99$'', ``Figure 4. Largest $k$ such that $\\hl{p_{all}}\\leq 0.5,0.99$ plotted against $n$ and $q_*$.''",
    "corrected": "``Figure 4 shows, for each $n$ and $q_*$, the \\hl{largest} nonnegative integer $k$ for which the sharp lower bound $\\underline{p_{all}}=(1-q_*^k)^n$ does not exceed $a$, with $a=0.5$ or $0.99$. Solving $(1-q_*^r)^n=a$ gives\n\\[\nr=\\frac{\\log(1-a^{1/n})}{\\log(q_*)},\n\\]\nand the plotted value is $k=\\lfloor r\\rfloor$. This calculation concerns the lowest possible probability that listwise deletion removes all rows under Assumptions 1 and 2, not an upper bound on the actual probability. \\ldots\n\n\\ldots\\ For $n=10,000$, the lower bound is \\hl{approximately 47.08\\%} at $q_*=0.75$, $k=33$, and \\hl{approximately 49.69\\%} at $q_*=0.99$, $k=952$. The smallest integers at which this lower bound is at least 50\\% are \\hl{34 and 953}, respectively.''\n\nSubcaptions and caption of Figure 4: ``(a) $\\hl{\\underline{p_{all}}}\\leq 0.5$'', ``(b) $\\hl{\\underline{p_{all}}}\\leq 0.99$'', ``Figure 4. Largest $k$ such that $\\hl{\\underline{p_{all}}}\\leq 0.5,0.99$ plotted against $n$ and $q_*$.'' In the titles of both panels, ``probability all data missing'' is replaced by ``lower bound on probability all data missing''.\n\nThe statement behind the figure is the following. Let $n\\ge1$, $0<q_*<1$ and $0<a<1$, and let $r$ be as in the display. The largest integer $k\\ge0$ such that $(1-q_*^k)^n\\le a$ is $\\lfloor r\\rfloor$, and the smallest integer $k\\ge0$ such that $(1-q_*^k)^n\\ge a$ is $\\lceil r\\rceil$.",
    "evidence": "The evidence is a derivation and a calculation in exact rational arithmetic. Let $n\\ge1$, $0<q_*<1$ and $0<a<1$, and set $r=\\log(1-a^{1/n})/\\log q_*$. For real $k\\ge0$, I have\n\\[\n\\begin{aligned}\nB(k)\\le a&\\iff 1-q_*^k\\le a^{1/n}\\iff q_*^k\\ge 1-a^{1/n}\\\\\n&\\iff k\\log q_*\\ge\\log(1-a^{1/n})\\iff k\\le r,\n\\end{aligned}\n\\]\nwhere the last step divides by $\\log q_*<0$. Reversing the inequalities, I obtain that $B(k)\\ge a$ if and only if $k\\ge r$. The largest integer $k$ such that $B(k)\\le a$ is $\\lfloor r\\rfloor$, and the smallest integer $k$ such that $B(k)\\ge a$ is $\\lceil r\\rceil$. For $n=10{,}000$ and $a=1/2$, the values of $B$ on both sides of $r$ are as follows.\n\n\\begin{audittable}{lrr}{The lower bound $B(k)=(1-q_*^k)^n$ at $n=10{,}000$, computed in exact rational arithmetic and rounded to five places.}\n$q_*$ & $k$ & $B(k)$ \\\\\n0.75 & 33 & 0.47075 \\\\\n0.75 & 34 & 0.56833 \\\\\n0.99 & 952 & 0.49689 \\\\\n0.99 & 953 & 0.50037 \\\\\n\\end{audittable}\n\nThe bound $B(k)$ is attained when all entries are independent and each is observed with probability $q_*$. In that case the probability that listwise deletion removes all rows is 47.1\\% at $k=33$ and 49.7\\% at $k=952$, below the 50\\% of the text. The statements printed without the underline fail as well. Let $n=10{,}000$, $q_*=0.75$ and $k=33$, and let all entries be independent, the first variable observed with probability $3/4$ and each of the other 32 variables with probability $1/4$. Assumptions 1 and 2 hold, $B(33)<0.5$, and $p_{all}=(1-3\\cdot4^{-33})^{10000}>0.99$. All 59,406 values plotted in Figure 4 satisfy $B(k)<a<B(k+1)$.",
    "problem": "Let $B(k)=(1-q_*^k)^n$ denote the lower bound of Lemma 3 on the probability $p_{all}$ that listwise deletion removes all rows; the supplement writes it $\\underline{p_{all}}$. For a target $a$, the floor in the display is the largest integer $k$ such that $B(k)\\le a$. The titles of the two panels (the first reads ``Largest k such that probability all data missing is $\\leq$ 0.50'') and the caption say so, and the code computes this value. The text calls it the smallest possible $k$, and the two examples present $k=33$ and $k=952$ as the numbers of variables needed for a 50\\% probability that all rows are missing. At these values the lower bound is below 50\\%: for $n=10{,}000$, I find $B(33)=0.4708$ at $q_*=0.75$ and $B(952)=0.4969$ at $q_*=0.99$. The smallest integers at which the lower bound is at least 50\\% are 34 and 953.\n\nThe first sentence, the subcaptions and the caption also print $p_{all}$ where the lower bound $\\underline{p_{all}}$ is meant. A lower bound on $p_{all}$ cannot ensure that $p_{all}\\le a$, and Figure 4 shows the largest $k$ such that $\\underline{p_{all}}\\le a$.",
    "what_survives": "The plotted values in Figure 4 and the code that produces them are correct when read as the largest $k$ at which the lower bound does not exceed the target, and no plotted value changes. Lemma 3, Proposition 5 and the two applications do not use this inversion. The corrected text is a statement about the lower bound of Lemma 3, and it contains no upper bound on $p_{all}$. The contrast between high and low rates of missingness stands."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E005",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of three lines beginning 'Our final numerical illustration considers an upper bound on the expected proportion of observations that are missing, 1 - q_*^k, which does not depend on n'. A red box marks the words 'upper bound'.",
     "caption": "Supporting Information A, supplement p. 2, last paragraph. The marked words should read “lower bound”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E006.png"
    },
    {
     "alt": "Crop of Figure 5. The title reads 'Lower bound on expected proportion of data missing'. Three curves, labelled q_* = 0.75, 0.90 and 0.99, show the lower bound on the expected proportion missing, from 0 to 1, against the number of variables, from 0 to 150; the curve for 0.99 ends near 0.78. In the caption below, 'Figure 5. Lower bound of expected proportion of all rows missing (p_all underlined) plotted against values of n,k and q_*', red boxes mark the underlined symbol p_all in parentheses and the letters 'n,k'.",
     "caption": "Supplement p. 3: Figure 5 with its title and caption. The title says “Lower bound on expected proportion of data missing”; the marked symbol denotes the bound of Figure 3, and the marked parameter list incorrectly includes $n$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E006-figure5.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P007-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T9"
    },
    {
     "id": "P007-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T10"
    }
   ],
   "headline": "The quantity $1-q_*^k$ of Figure 5 is a lower bound, not an upper bound, on the expected proportion of rows removed, and the caption of Figure 5 labels it with the symbol $\\underline{p_{all}}$ and the dependence on $n$ of a different quantity.",
   "id": "P007-E006",
   "location": "Supporting Information A, supplement p. 2, last paragraph, first sentence; supplement p. 3, caption of Figure 5",
   "pdf_page": 2,
   "printed_page": "2",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P007-T10",
    "P007-T9"
   ],
   "text": {
    "as_printed": "Supplement p.~2: ``Our final numerical illustration considers an \\hl{upper bound} on the expected proportion of observations that are missing, $1-q_*^k$, which does not depend on $n$. Figure 5 plots the expected proportion of data missing versus the number of variables $k$.''\n\nSupplement p.~3: ``Figure 5. Lower bound of expected proportion of all rows missing ($\\hl{\\underline{p_{all}}}$) plotted against values of $\\hl{n},k$ and $q_*$.''",
    "corrected": "Supplement p.~2: ``Our final numerical illustration considers a \\hl{lower bound} on the expected proportion of rows removed by listwise deletion, $b_k=1-q_*^k$, which does not depend on $n$. Figure 5 plots this lower bound versus the number of variables $k$.''\n\nSupplement p.~3: ``Figure 5. Lower bound $\\hl{b_k=1-q_*^k}$ on the expected proportion of rows removed by listwise deletion, plotted against $\\hl{k}$ for $q_*\\in\\{0.75,0.90,0.99\\}$.''\n\nThe statement behind the figure is the following. Under Assumption 2, for all $n\\ge1$ and $k\\ge1$, the expected proportion of rows removed by listwise deletion is at least $1-q_*^k$. Equality holds when all entries are independent and each is observed with probability $q_*$. The lower bound $(1-q_*^k)^n$ of Lemma 3 on the probability that all rows are removed is strictly smaller than $1-q_*^k$ when $n>1$ and $\\hl{0<q_*<1}$.",
    "evidence": "The evidence is a derivation and an exact counterexample. Let $F_n$ denote the proportion of the $n$ rows that are removed. By Assumption 2 and the chain rule, each row is complete with probability at most $q_*^k$; a row for which some conditioning event of the assumption has probability zero is complete with probability zero. By linearity of expectation, I obtain $\\mathbb{E}[F_n]\\ge1-q_*^k$, and Assumption 1 is not used. For the counterexample to the upper bound, let $n=k=2$ and $q_*=3/4$, and let the four entries be independent, the first variable observed with probability $3/4$ and the second with probability $1/2$ in each row. Assumptions 1 and 2 hold, and each row is complete with probability $3/8$. I find $\\mathbb{E}[F_2]=5/8$, which exceeds $1-q_*^2=7/16$. In the same example the lower bound on the probability that both rows are removed is $(7/16)^2=49/256$, which differs from $7/16$. The code computes the plotted curve as $1-q_*^k$ for $k=1,\\dots,150$ and the three values of $q_*$, and all 1,350 values of its grid agree with this formula. At $q_*=0.99$ and $k=150$ the curve equals $0.7785$, while $(1-q_*^k)^n=1.3\\times10^{-11}$ for $n=100$.",
    "problem": "Under Assumption 2, each row is complete with probability at most $q_*^k$, and the expected proportion of rows removed by listwise deletion is at least $1-q_*^k$. The quantity is a lower bound, as the title of Figure 5 says, and the first sentence of the paragraph calls it an upper bound. The caption attaches to the curve the symbol $\\underline{p_{all}}$, which the supplement defines as $(1-q_*^k)^n$, the lower bound on the probability that all rows are removed, and it lists $n$ among the quantities that vary. The plotted quantity is $1-q_*^k$, which does not depend on $n$, and the figure varies $k$ and $q_*$ only. The two quantities agree when $n=1$, as the text after the paragraph notes.",
    "what_survives": "The curves of Figure 5 are correct and do not change. The correction replaces one word and the caption, and no assumption, proof or calculation in the applications depends on either."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E006",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "recalculation_from_source",
   "exhibits": [
    {
     "alt": "Crop of the heading '3 Application' and the six lines below it. A red box marks the words '(Teorell et al. 2021) standard' in the phrase 'the January 2021 QoG (Teorell et al. 2021) standard cross-sectional dataset'.",
     "caption": "Section 3, p. 151 (PDF p. 3), first paragraph. The marked words name and cite the Standard dataset.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E009.png"
    },
    {
     "alt": "Crop of three reference entries, the last partly shown. In the entry for Teorell, Sundström, Holmberg, Rothstein, Pachon and Dalli, 2021, a red box marks the word 'Standard' in the title 'The Quality of Government Standard Dataset, Version Jan21'.",
     "caption": "Reference list, p. 155 (PDF p. 7), entry for Teorell et al. The marked word is in the title of the dataset.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E009-reference.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The Quality of Government data analysed in Section 3 are the January 2021 Basic cross-sectional dataset, not the Standard dataset that the article names and cites; the reported numbers stand for the Basic file.",
   "id": "P007-E009",
   "location": "Section 3, p. 151 (PDF p. 3), first paragraph; reference list, p. 155 (PDF p. 7), entry for Teorell et al.; replication files, readme.txt and Simcode_Qog.R",
   "pdf_page": 3,
   "printed_page": "151",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Section 3, p.~151: ``\\ldots\\ we considered two prominent datasets in use in the fields of comparative politics and international relations: the January 2021 QoG (\\hl{Teorell \\textit{et al.} 2021}) \\hl{standard} cross-sectional dataset, and the State Failure dataset \\ldots''\n\nReference list, p.~155: Teorell, J., A. Sundström, S. Holmberg, B. Rothstein, N. A. Pachon, and C. M. Dalli, 2021. ``The Quality of Government \\hl{Standard} Dataset, Version Jan21.'' University of Gothenburg, The Quality of Government Institute.\n\nReplication files, \\texttt{readme.txt}: ``\\texttt{qog\\_bas\\_cs\\_jan21.dta} available at: \\href{https://www.qogdata.pol.gu.se/data/qog_std_cs_jan21.dta}{https://www.qogdata.pol.gu.se/data/qog\\_\\hl{std}\\_cs\\_jan21.dta} (last accessed Jun 25, 2021)'', followed by the citation of ``The Quality of Government \\hl{Standard} Dataset, version Jan21'' with \\texttt{doi:10.18157/qog\\hl{std}jan21}.",
    "corrected": "Section 3, p.~151: ``\\ldots\\ the January 2021 QoG \\hl{Basic} cross-sectional dataset, and the State Failure dataset \\ldots''\n\nThe citation ``Teorell \\textit{et al.} 2021'' in Section 3 and in the Introduction (p.~149), and the entry in the reference list, are replaced by the reference for the Quality of Government Basic Dataset, Version Jan21. The file is \\texttt{qog\\_bas\\_cs\\_jan21.dta}, with version field \\texttt{QoGBasCSJan21}. I have not confirmed the author list and the DOI of that reference against the documentation of the Quality of Government Institute, and I do not give them here.\n\nIn the replication files, the download address and the citation in \\texttt{readme.txt} and in the comment of \\texttt{Simcode\\_Qog.R} are replaced by those of the Basic file. The Quality of Government column of Table 1 and Figure 1 keep their values, and their source is the Basic dataset.",
    "evidence": "The evidence is a recalculation from the replication files. The code drops six identifier columns (\\texttt{ccode}, \\texttt{cname}, \\texttt{ccodealp}, \\texttt{ccodecow}, \\texttt{ccodewb}, \\texttt{version}), which leaves 194 rows and 351 columns with 24,424 of 68,094 cells missing. The column with the most missing values has 176 of them, and six columns have none. These counts give the five numerical entries of the Quality of Government column of Table 1. Let $r_i$ denote the number of observed columns in row $i$. When $k$ of the 351 columns are drawn uniformly without replacement, the expected proportion of complete rows is\n\\[\n\\frac{1}{194}\\sum_{i=1}^{194}\\binom{r_i}{k}\\Big/\\binom{351}{k},\n\\]\nwhich is approximately $0.4364$ at $k=2$ and $0.00913$ at $k=17$. I ran the simulation code on the Basic file with the seed of the replication files (213) and 25,000 draws at each of the 100 values of $k$. The run reproduces all 100 estimates of expected proportions and all 100 probability estimates of the saved log at its printed precision. I did not have the Standard file, and I did not compute its dimensions, its missingness or its simulation curves.\n\n\\begin{audittable}{lll}{The Quality of Government numbers of the article, as printed and as recomputed from the Basic file. The first five rows are the numerical entries of the column of Table 1 (p.~152), and the last four are the thresholds of Section 3.2 (p.~153). The expected proportions are calculated exactly and rounded here, and the two probabilities are simulation estimates from 25,000 draws.}\nQuantity & As printed & From the Basic file \\\\\nNumber of observations & 194 & 194 \\\\\nNumber of variables & 351 & 351 \\\\\nProportion of missing values, average & 35.9\\% & 35.87\\% (24,424 of 68,094) \\\\\nProportion of missing values, maximum & 90.7\\% & 90.72\\% (176 of 194) \\\\\nNumber of variables fully observed & 6 & 6 \\\\\nExpected proportion of rows removed, 2 variables & more than 50\\% & 56.36\\% \\\\\nExpected proportion of rows removed, 17 variables & more than 99\\% & 99.09\\% \\\\\nProbability that all rows are removed, 14 variables & greater than 0.5 & 0.5076 \\\\\nProbability that all rows are removed, 52 variables & more than 0.99 & 0.9902 \\\\\n\\end{audittable}",
    "problem": "Section 3 describes the first dataset as the January 2021 QoG standard cross-sectional dataset, and the entry in the reference list is for the Standard dataset. The file that the replication code reads is \\texttt{qog\\_bas\\_cs\\_jan21.dta}. The data label stored in this file is ``Quality of Government Basic dataset 2021 - Cross-Section'', and its field \\texttt{version} equals \\texttt{QoGBasCSJan21} in all 194 rows. The Quality of Government column of Table 1, Figure 1 and the thresholds 2, 17, 14 and 52 of Section 3.2 are computed from this Basic file. The article attributes these results to a dataset that the replication files do not analyse. The readme and a comment in the script pair the name of the Basic file with the download address and the citation of the Standard file. The citation and download address identify a different edition. The simulation draws variables at random from the 351 variables of the Basic file, and its expected proportions and probabilities refer to that set of variables. The replication files contain no result for the Standard dataset.\n\n\\begin{audittable}{ll}{How each source identifies the Quality of Government data.}\nSource & Identification \\\\\nSection 3, p.~151 & ``standard cross-sectional dataset'' \\\\\nReference list, p.~155 & ``The Quality of Government Standard Dataset, Version Jan21'' \\\\\nReadme, download address & \\texttt{qog\\_std\\_cs\\_jan21.dta} \\\\\nReadme, citation & Standard Dataset, \\texttt{doi:10.18157/qogstdjan21} \\\\\nFile read by the code & \\texttt{qog\\_bas\\_cs\\_jan21.dta} \\\\\nData label stored in the file & ``Quality of Government Basic dataset 2021 - Cross-Section'' \\\\\nField \\texttt{version}, all 194 rows & \\texttt{QoGBasCSJan21} \\\\\n\\end{audittable}",
    "what_survives": "The Quality of Government column of Table 1 and the simulation estimates in Figure 1 and Section 3.2 stand for the Basic file. Section 2, the Appendix, the supplement and the State Failure application do not use this file. The correction changes the dataset, and with it the set of variables, that the Quality of Government application describes."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E009",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "historical_lean_attribution",
   "exhibits": [
    {
     "alt": "Crop of Table 1, summary statistics for the QoG and State Failure datasets. In the State Failure column, the entry 66.8 per cent in the row 'Proportion of missing values (avg.)' is outlined in red. The other entries in that column are Country-years, 8,580, 1,205, 99.99 per cent and 79.",
     "caption": "Table 1, p. 152 (PDF p. 4). The marked entry, 66.8%, is an average over 1,224 variables; the rest of the column describes the 1,205 variables that remain after 19 were removed.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E010.png"
    },
    {
     "alt": "Crop of the third paragraph of Section 3.2 and the start of the fourth. In the sentence 'With more than three variables, the loss is more than 99 per cent.' the words 'more than three' are outlined in red.",
     "caption": "Section 3.2, p. 153 (PDF p. 5), third paragraph. The marked words place the 99% threshold at four variables; in the simulation it is first passed at six.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E010-b.png"
    },
    {
     "alt": "Line chart with a logarithmic vertical axis. Seven blue points give the estimated share of rows retained with one to seven variables: 33.6, 13.5, 6.06, 2.82, 1.37, 0.66 and 0.32 per cent. A dashed horizontal line marks 1 per cent of rows retained, that is 99 per cent lost. A shaded orange region below that line, beginning between three and four variables, marks the printed statement. The points for four and five variables lie above the line, and the points for six and seven lie below it. A black diamond at four variables marks the exact lower bound, 1.27 per cent.",
     "caption": "State Failure data: estimated proportion of rows retained after listwise deletion, by number of randomly selected variables, from the archived simulation log (logarithmic scale). The shaded region is the printed statement, a loss above 99% with more than three variables. The estimates for four and five variables lie outside it, and the exact lower bound at four variables (diamond) is 1.27%.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E010-loss-threshold.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P007-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T1"
    },
    {
     "id": "P007-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T2"
    },
    {
     "id": "P007-T12",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T12"
    },
    {
     "id": "P007-T13",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T13"
    }
   ],
   "headline": "For the State Failure data, the average proportion of missing values is 66.3%, not 66.8%, and in the simulation the estimated loss of rows exceeds 99% with six or more variables, not with more than three.",
   "historical_evidence_type": "lean_proof",
   "id": "P007-E010",
   "location": "Table 1, p. 152 (PDF p. 4), State Failure column, row “Proportion of missing values (avg.)”; Section 3.2, p. 153 (PDF p. 5), third paragraph, last sentence",
   "pdf_page": 4,
   "printed_page": "152",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P007-T1",
    "P007-T12",
    "P007-T13",
    "P007-T2"
   ],
   "text": {
    "as_printed": "Table 1, State Failure column: Number of observations 8,580; Number of variables 1,205; Proportion of missing values (avg.) \\hl{66.8\\%}; Proportion of missing values (max) 99.99\\%; Number of variables fully observed 79.\n\nSection 3.2: ``The researcher can expect to lose more than 50\\% of the data if the only variable included is randomly selected. With \\hl{more than three} variables, the loss is more than 99\\%.''",
    "corrected": "Table 1, State Failure column: Proportion of missing values (avg.) \\hl{66.3\\%}.\n\nSection 3.2: ``The researcher can expect to lose more than 50\\% of the data if the only variable included is randomly selected. \\hl{In the simulation, the estimated} loss exceeds 99\\% with \\hl{six or more} variables.''\n\nIn the replication script, recompute the average after the wholly missing columns are removed, with \\texttt{prop.missing <- mean(colMeans(isnamatrix))} after line 44.",
    "evidence": "I checked the replication script and its saved log, recalculated the average, and derived an exact bound.\n\n\\emph{The average.} The script computes the average at line 29, removes the wholly missing variables at lines 39--40, and builds the table column at line 55 from the reduced data and the earlier average. The log prints the average as 0.668390827582004, taken over $8{,}580 \\times 1{,}224 = 10{,}501{,}920$ cells. Allowing for rounding, the only whole number of missing cells consistent with that value is 7,019,387. The 19 removed variables hold $19 \\times 8{,}580 = 163{,}020$ missing cells and no observed ones. The average over the remaining variables is\n\\[\n\\frac{7{,}019{,}387 - 163{,}020}{8{,}580 \\times 1{,}205} = \\frac{6{,}856{,}367}{10{,}338{,}900} = 0.66316\\ldots,\n\\]\nwhich rounds to 66.3\\%. I infer these counts from the log. My recalculation from the data file gives the same average, $0.66316\\ldots$, under the code's definition of a missing cell, as described under ``What was checked''.\n\n\\emph{The threshold.} Let $r_i$ be the number of observed cells in row $i$ among the 1,205 variables, so that $\\sum_i r_i = 10{,}338{,}900 - 6{,}856{,}367 = 3{,}482{,}533$. When four variables are drawn uniformly without replacement, row $i$ survives for $\\binom{r_i}{4}$ of the $\\binom{1{,}205}{4}$ equally likely draws. Since the function $r \\mapsto \\binom{r}{4}$ is convex on the integers, the sum $\\sum_i \\binom{r_i}{4}$ is smallest when the $r_i$ are as equal as possible, which means 405 in 947 rows and 406 in 7,633 rows. I obtain\n\\[\n\\begin{aligned}\n\\mathbb{E}[\\text{proportion of rows retained}] &= \\frac{\\sum_i \\binom{r_i}{4}}{8{,}580 \\binom{1{,}205}{4}} \\geq \\frac{947 \\binom{405}{4} + 7{,}633 \\binom{406}{4}}{8{,}580 \\binom{1{,}205}{4}} \\\\\n&= \\frac{24{,}513{,}560{,}217}{1{,}923{,}069{,}889{,}510} = 0.01274\\ldots.\n\\end{aligned}\n\\]\nThis implies that the expected loss with four variables is at most 98.73\\%, however the missing cells are arranged. The bound uses only the dimensions of the data and the number of observed cells.\n\nThe estimates of the archived simulation, with 25,000 draws for each number of variables, are in the table below. A Chernoff bound for the mean of 25,000 independent draws in $[0,1]$ applies to the two estimates nearest the threshold. If the expected proportion retained with five variables were at most 1\\%, an estimate as large as the one in the log would have probability at most $2.8 \\times 10^{-7}$. If it were at least 1\\% with six variables, an estimate as small as the one in the log would have probability at most $9.6 \\times 10^{-8}$.\n\n\\begin{audittable}{rrr}{State Failure data: estimates in the archived simulation log, by number of randomly selected variables.}\nVariables & Proportion of rows retained & Loss of rows \\\\\n1 & 0.3361604 & 66.38\\% \\\\\n2 & 0.1345654 & 86.54\\% \\\\\n3 & 0.06058382 & 93.94\\% \\\\\n4 & 0.02820753 & 97.18\\% \\\\\n5 & 0.01365003 & 98.63\\% \\\\\n6 & 0.006640252 & 99.34\\% \\\\\n7 & 0.003187953 & 99.68\\% \\\\\n\\end{audittable}",
    "problem": "Table 1 describes the State Failure data after the 19 variables with 100\\% missingness were removed: 8,580 country-years and 1,205 variables. The replication script computes the average proportion of missing values before it removes those 19 variables and does not compute it again. The printed 66.8\\% is an average over 1,224 variables, while the number of variables and the maximum in the same column refer to the 1,205 that remain. For those 1,205 variables the average is 66.3\\%.\n\nThe sentence in Section 3.2 places the 99\\% threshold too early. In the simulation that the article reports, the estimated loss of rows is 97.2\\% with four variables and 98.6\\% with five, and it first exceeds 99\\% with six. The gap at four variables is far larger than the simulation error. An exact bound shows that, for data of these dimensions with this many observed cells, the expected loss with four variables cannot reach 99\\%.",
    "what_survives": "Figure 2 and the simulation estimates are unchanged: the simulation uses the 1,205 variables that remain and never reads the average. The other State Failure entries of Table 1 do not change as a consequence. The QoG results, Lemma 3, Proposition 5 and the numerical illustrations do not depend on either number. The simulation still supports the conclusion that a few variables are enough for rapid loss of rows. One table entry and one sentence change. The script's part in the table entry is the subject of \\fid{P007-E014}."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E010",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
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  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "recalculation_from_source",
   "exhibits": [
    {
     "alt": "Crop of Table 1 and the sentence that follows it. In the State Failure column, the entry 66.8 per cent in the row 'Proportion of missing values (avg.)' is outlined in red. The sentence below the table ends 'we removed 19 variables that exhibited 100 per cent missingness.'",
     "caption": "Table 1 and the sentence below it, p. 152 (PDF p. 4). The script computes the marked average before the removal of the 19 variables that the sentence describes.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E014.png"
    }
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   "formal_release": null,
   "formal_statements": [
    {
     "id": "P007-T12",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T12"
    },
    {
     "id": "P007-T13",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-T13"
    }
   ],
   "headline": "The script behind the State Failure column of Table 1 averages missingness before it drops the 19 wholly missing variables, and it counts an empty text entry imported from Stata as observed, a convention that the article does not state.",
   "id": "P007-E014",
   "location": "Table 1, p. 152 (PDF p. 4), State Failure column; replication script Simcode_Statefailure.R, lines 13, 28–29, 39–47 and 55",
   "pdf_page": 4,
   "printed_page": "152",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P007-T12",
    "P007-T13"
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   "text": {
    "as_printed": "Table 1, State Failure column: Number of observations 8,580; Number of variables 1,205; Proportion of missing values (avg.) \\hl{66.8\\%}; Proportion of missing values (max) 99.99\\%; Number of variables fully observed 79. Below the table: ``and in the case of the State Failure data, to apply the principle of charity, we removed 19 variables that exhibited 100\\% missingness.''\n\nReplication script \\texttt{Simcode\\_Statefailure.R}:\n\\begin{itemize}\n\\item line 13: \\texttt{statefailure <- read\\_dta(\"sort11v3.dta\")}\n\\item line 28: \\texttt{isnamatrix <- is.na(statefailurepure)}\n\\item line 29: \\texttt{prop.missing <- mean(colMeans(isnamatrix))}\n\\item line 44: \\texttt{isnamatrix <- is.na(statefailure\\_final)}\n\\end{itemize}",
    "corrected": "Table 1, State Failure column: Proportion of missing values (avg.) \\hl{66.3\\%}.\n\nIn the script, add \\texttt{prop.missing <- mean(colMeans(isnamatrix))} after line 44.\n\nAdd to the description of the data: ``A cell is counted as missing when \\texttt{is.na} is true after import with \\texttt{read\\_dta}; an empty text entry is counted as observed.'' An analysis that counts empty text entries as missing should be reported separately, with its own definition of a missing cell and its own set of variables, after the data and their documentation have been inspected.",
    "evidence": "The evidence is a derivation for the average and a test of the import on a small constructed file.\n\nLet $q_{\\text{before}}$ and $q_{\\text{after}}$ be the average proportions of missing cells over the 1,224 and the 1,205 variables. Since the 19 removed variables are wholly missing and all variables have the same 8,580 rows, I have\n\\[\n1{,}224 \\, q_{\\text{before}} = 19 + 1{,}205 \\, q_{\\text{after}}.\n\\]\nA value that rounds to 66.8\\% satisfies $q_{\\text{before}} < 0.6685$. Then the identity implies that\n\\[\nq_{\\text{after}} < \\frac{1{,}224 \\times 0.6685 - 19}{1{,}205} = \\frac{199{,}811}{301{,}250} < 0.6675,\n\\]\nand the average over the 1,205 variables cannot round to 66.8\\%. Substituting the value in the saved log, $q_{\\text{before}} = 0.668390827582004$, into the identity, I obtain $q_{\\text{after}} = 0.66316\\ldots$, which rounds to 66.3\\%. The log implies 6,856,367 missing cells out of 10,338,900 retained cells. The pre-removal average exceeds the retained average by 0.52 percentage points.\n\nFor the import, a Stata file with four rows was written and read back with \\texttt{read\\_dta} (haven 2.5.5; the authors report version 2.4.0). Its text variable held an empty string, ``0'', ``1'' and a missing value, and its numeric variable had no missing value. The text values came back as an empty string, ``0'', ``1'' and an empty string. Under \\texttt{is.na}, none of the 8 cells is missing and all 4 rows are complete. When empty strings are counted as missing, 2 cells are missing and 2 rows are complete. The test shows how the convention works. It does not show that the State Failure file has empty text entries.",
    "problem": "Two things are at issue. The first is the average in Table 1, corrected in \\fid{P007-E010}. The script computes it at line 29, over 1,224 variables, and line 55 reports it for the 1,205 variables that remain after the 19 wholly missing ones are dropped.\n\nThe second is how the script decides that a cell is missing. It applies \\texttt{is.na} to data read from a Stata file with \\texttt{read\\_dta}. An empty text entry in a Stata file arrives in R as an empty string, which \\texttt{is.na} does not count as missing. An empty text entry is then counted as an observed value. The article does not state this convention. I have not established whether the State Failure file contains empty text entries or how counting them as missing would change the results. For a fixed set of variables, counting them as missing adds missing cells and removes none. It could only show more rows lost.",
    "what_survives": "Recomputing the average changes no simulation result, since the simulation uses the 1,205 variables that remain and never reads the average. Figure 2 is unchanged. Under the script's own definition of a missing cell, the count of 79 fully observed variables is unaffected. The results of Section 2, the numerical illustrations and the QoG calculations do not depend on this script. If the State Failure file does hold empty text entries, the count of fully observed variables and the curves of Figure 2 could change under a definition that counts them as missing; no such change is established here."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E014",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
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  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the first paragraph of the Introduction and the first two lines of the next paragraph. Red boxes mark the words 'the glmnet function in the R package of the same name' at the end of one line and '(Friedman, Hastie, and Tibshirani 2010)' at the start of the next line.",
     "caption": "Introduction, p. 149 (PDF p. 1), first paragraph. The marked words are the third software example and its citation.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E001.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The glmnet function did not omit rows with missing data by default when the article appeared, and it should be removed from the list of software examples.",
   "id": "P007-E001",
   "location": "Introduction, p. 149 (PDF p. 1), first paragraph, second sentence",
   "pdf_page": 1,
   "printed_page": "149",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
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   "text": {
    "as_printed": "``It constitutes the default behavior for standard data analyses in popular software packages: for example, rows with any missing data are by default omitted by the \\textit{lm} function in R (R Core Team 2020), the \\textit{regress} command in Stata (Stata.com 2020), and \\hl{the \\textit{glmnet} function in the R package of the same name (Friedman, Hastie, and Tibshirani 2010)}.''",
    "corrected": "``It constitutes the default behavior for standard data analyses in popular software packages: for example, rows with any missing data are by default omitted by the \\textit{lm} function in R (R Core Team 2020) \\hl{and} the \\textit{regress} command in Stata (Stata.com 2020).''\n\nThe citation of Friedman, Hastie, and Tibshirani (2010) is removed with the example. It is the only citation of that reference in the article, and the entry in the reference list is removed as well.",
    "evidence": "The evidence is a check of the package source as released. I read the files \\texttt{R/glmnet.R} and \\texttt{R/elnet.R} of glmnet 4.1-3, released on CRAN on 2 November 2021 (\\url{https://raw.githubusercontent.com/cran/glmnet/4.1-3/R/glmnet.R}), and the source of releases 4.1, 4.1-1, 4.1-2 and 1.1-4, the version named in Friedman, Hastie, and Tibshirani (2010). None of them removes incomplete rows by default. I also ran the R code of 4.1-3 on eight rows, with the compiled fitting routine replaced by a function that records its arguments. With one missing predictor value, all eight rows reach the fitting routine. With one missing response value, the call stops with the error ``missing value where TRUE/FALSE needed''. Seven rows reach the fitting routine only when the incomplete row is removed beforehand. I did not run the compiled package of 2021, and I have not established what its fitting routine did with a missing predictor value. I did not check the \\textit{lm} and \\textit{regress} examples against the releases of that date.",
    "problem": "The sentence says that three functions omit incomplete rows by default, and the third does not. In glmnet 4.1-3, the release current when the article was published online on 2 March 2022, the function \\textit{glmnet} contains no step that removes rows. For a Gaussian response, it keeps the number of rows of its input and passes the matrix of predictors, with any missing values, to the fitting routine. A missing value in the response stops the function with an error. Rows are omitted only if the user removes them, from the predictors and the response together, before the call.",
    "what_survives": "No result, table or figure depends on the example, and the replication code does not use glmnet. The other two examples and the point that the sentence makes about default behavior are not affected."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E001",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
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   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of three entries in the reference list. In the entry for Berk (1983), the words 'A. B. A. Peter, H. Rossi, and J. D. Wright,' after 'edited by' are outlined in red.",
     "caption": "References, p. 155 (PDF p. 7), entry for Berk (1983). The marked editor names are garbled.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-E013.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The editors of the Handbook of Survey Research are P. H. Rossi, J. D. Wright and A. B. Anderson.",
   "id": "P007-E013",
   "location": "References, p. 155 (PDF p. 7), entry for Berk (1983)",
   "pdf_page": 7,
   "printed_page": "155",
   "reaches": [],
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   "text": {
    "as_printed": "Berk, R. 1983. ``Applications of the General Linear Model to Survey Data.'' In \\emph{Handbook of Survey Research}, edited by \\hl{A. B. A. Peter, H. Rossi, and J. D. Wright}, pp.~495--546. Quantitative Studies in Social Relations. New York: Academic Press.",
    "corrected": "Berk, R. 1983. ``Applications of the General Linear Model to Survey Data.'' In \\emph{Handbook of Survey Research}, edited by \\hl{P. H. Rossi, J. D. Wright, and A. B. Anderson}, pp.~495--546. Quantitative Studies in Social Relations. New York: Academic Press.",
    "evidence": "The evidence is a check of a public source. The \\href{https://www.sciencedirect.com/book/edited-volume/9780125982269/handbook-of-survey-research}{publisher's catalogue entry} for the 1983 Academic Press edition (ISBN 9780125982269) lists the editors as Peter H. Rossi, James D. Wright and Andy B. Anderson, in that order, and the title page reproduced in a public preview of the book has the same names in the same order. The catalogue entry lists Berk's chapter as chapter 13, pp.~495--546.",
    "problem": "The entry names the editors of the volume as ``A. B. A. Peter, H. Rossi, and J. D. Wright''. The editors of the 1983 edition are Peter H. Rossi, James D. Wright and Andy B. Anderson.",
    "what_survives": "The chapter title, pages, series and publisher in the entry are as printed, and no result depends on it. I did not read the chapter itself."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-E013",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
   },
   "work_id": "P007"
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   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the abstract. A red box marks the clause 'for large n, listwise deletion will drop all rows with probability 1' in the second sentence.",
     "caption": "Abstract, p. 149 (PDF p. 1). The marked clause states the main result.",
     "url": "https://pmaronow.github.io/llm-audit/works/P007/exhibits/P007-N001.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "In the abstract, “for large $n$, listwise deletion will drop all rows with probability 1” means that the probability of dropping all rows tends to 1 as $n$ grows, which is the statement of Proposition 5.",
   "id": "P007-N001",
   "location": "Abstract, p. 149 (PDF p. 1), second sentence",
   "pdf_page": 1,
   "printed_page": "149",
   "reaches": [],
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   "text": {
    "as_printed": "``We show that when (i) all data have some idiosyncratic missingness and (ii) the number of variables grows superlogarithmically in $n$, then, \\hl{for large $n$, listwise deletion will drop all rows with probability 1}.''",
    "corrected": "``We show that when (i) all data have some idiosyncratic missingness and (ii) the number of variables grows superlogarithmically in $n$, then \\hl{the probability that listwise deletion drops all rows tends to 1 as $n$ grows}.''",
    "intended_meaning": "The clause summarizes Proposition 5 (p.~151), which states that $\\lim_{n\\to\\infty}p_{all}=1$ under Assumptions 1, 2 and 4, where $p_{all}$ is the probability that listwise deletion removes all rows. This statement is correct. The clause is not a guarantee at a finite $n$: if all entries are independent and each is observed with probability $q\\in(0,1)$, then $p_{all}=(1-q^k)^n<1$ for every $n$ and $k$. For instance, $p_{all}=9/16$ when $q=1/2$ and $n=k=2$.\n\nA stronger reading also holds, although the article does not state it. Suppose that the datasets for all $n$ are defined on one probability space and that the bound $q_*$ of Assumption 2 does not depend on $n$. By Assumption 2, each row is complete with probability at most $q_*^{f(n)}$, and the probability that some row of the dataset with $n$ rows is complete is at most $nq_*^{f(n)}$. Since $f(n)/\\log n\\to\\infty$, I have $nq_*^{f(n)}\\le n^{-2}$ for all large $n$. These bounds are summable, and the first Borel--Cantelli lemma implies that, with probability 1, listwise deletion removes all rows for all sufficiently large $n$. The corresponding sentence of the Introduction says ``asymptotically'' and needs no change."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P007/#P007-N001",
   "work": {
    "arxiv_id": "2101.11470",
    "doi": "10.1017/pan.2022.5",
    "title": "Listwise Deletion in High Dimensions",
    "url": "https://pmaronow.github.io/llm-audit/works/P007/"
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     "body": "<p>repaired proof of Proposition 5, p. 154 (PDF p. 6), and Proposition 10, Supplement pp. 4-5. The corrected finite Bernoulli sandwich, zero endpoint, positive-domain exponential identity and two-variable group counts hold.</p>",
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     "location": "p. 154 (PDF p. 6), Appendix: Proofs, Proof of Proposition 5, paragraph beginning “Since <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>q</mi><mo>∗</mo></msub><mo>∈</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">q_* \\in (0,1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1757em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span></span></span></span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi><mo>=</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">k = f(n)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">n</span><span class=\"mclose\">)</span></span></span></span>”; Supplement p. 4 (foot) and p. 5, Proof of Proposition 10, paragraph beginning “Since <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>q</mi><mo>∗</mo></msub><mo>∈</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo separator=\"true\">,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">q_* \\in (0,1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1757em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">∗</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">1</span><span class=\"mclose\">)</span></span></span></span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi><mo>=</mo><mi>l</mi><mo stretchy=\"false\">(</mo><mi>n</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">k = l(n)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.0197em;\">l</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">n</span><span class=\"mclose\">)</span></span></span></span>”"
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     "body": "repaired proof of Proposition 5, p. 154 (PDF p. 6), and Proposition 10, Supplement pp. 4-5. The corrected finite Bernoulli sandwich, zero endpoint, positive-domain exponential identity and two-variable group counts hold.",
     "headline": "Proofs of Proposition 5 and Proposition 10, third paragraph (the Bernoulli step), as repaired. The statements of the two propositions are unchanged",
     "location": "p. 154 (PDF p. 6), Appendix: Proofs, Proof of Proposition 5, paragraph beginning “Since $q_* \\in (0,1)$ and $k = f(n)$”; Supplement p. 4 (foot) and p. 5, Proof of Proposition 10, paragraph beginning “Since $q_* \\in (0,1)$ and $k = l(n)$”"
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     "body": "<p>Figure 4 and its discussion, Supplement p. 2. All specified equality, rounding, worked-value and actual-probability upper-guarantee counterexamples hold for the displayed normalized independent-cell laws.</p>",
     "headline": "Supporting Information A, the two paragraphs on Figure 4 (“the smallest possible <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi></mrow><annotation encoding=\"application/x-tex\">k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k</span></span></span></span>”; “we need only <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi><mo>=</mo><mn>33</mn></mrow><annotation encoding=\"application/x-tex\">k = 33</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">33</span></span></span></span>”; “we need <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi><mo>=</mo><mn>952</mn></mrow><annotation encoding=\"application/x-tex\">k = 952</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">952</span></span></span></span>”; “ensuring that <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>p</mi><mrow><mi>a</mi><mi>l</mi><mi>l</mi></mrow></msub><mo>≤</mo><mn>0.5</mn><mo separator=\"true\">,</mo><mn>0.99</mn></mrow><annotation encoding=\"application/x-tex\">p_{all} \\leq 0.5, 0.99</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.8304em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">a</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0197em;\">l</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0197em;\">l</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≤</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\"></span><span class=\"mord\">0.5</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">0.99</span></span></span></span>”) and the caption of Figure 4. The witnesses are the validator&#x27;s and are not in the paper",
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    "text": {
     "body": "Figure 4 and its discussion, Supplement p. 2. All specified equality, rounding, worked-value and actual-probability upper-guarantee counterexamples hold for the displayed normalized independent-cell laws.",
     "headline": "Supporting Information A, the two paragraphs on Figure 4 (“the smallest possible $k$”; “we need only $k = 33$”; “we need $k = 952$”; “ensuring that $p_{all} \\leq 0.5, 0.99$”) and the caption of Figure 4. The witnesses are the validator's and are not in the paper",
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     "body": "<p>corrected Figure 4 and its discussion, Supplement p. 2. The real inverse, integer floor and ceiling, sharp probability guarantees and both worked values describe the lower bound, with explicit attaining laws and an upper-guarantee counterexample.</p>",
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    "text": {
     "body": "corrected Figure 4 and its discussion, Supplement p. 2. The real inverse, integer floor and ceiling, sharp probability guarantees and both worked values describe the lower bound, with explicit attaining laws and an upper-guarantee counterexample.",
     "headline": "Supporting Information A, the two paragraphs on Figure 4 and the caption, subcaptions and plot titles of Figure 4, as repaired by Sol",
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     "body": "<p>Figure 5 and its discussion, Supplement pp. 2-3. The specified two-row law has expected deleted fraction 5/8, exceeding 7/16, and distinguishes that fraction bound from both all-deletion probabilities.</p>",
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     "body": "<p>corrected Figure 5 and its discussion, Supplement pp. 2-3. The usual measure-theoretic conditions are assumed. For positive integers n,k and any measurable missingness array on a probability space, Assumption 2 gives E[D_i]=1-s_ik &gt;= b_k and E[F_n]=(sum_i (1-s_ik))/n &gt;= b_k, where b_k=1-q^k is independent of n. These conclusions require no row independence. Every law with mutually independent cells of observation probability q satisfies Assumptions 1 and 2 and attains E[F_n]=b_k; a normalized independent-cell law exists for every q in [0,1). The all-deletion lower bound is b_k^n, equals b_k at n=1, and is strictly smaller for n&gt;=2 and 0&lt;q&lt;1, when 0&lt;b_k&lt;1. On the entire domain q in [0,1), the bounds coincide if and only if n=1 or q=0; at q=0 both bounds are one.</p>",
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     "body": "corrected Figure 5 and its discussion, Supplement pp. 2-3. The usual measure-theoretic conditions are assumed. For positive integers n,k and any measurable missingness array on a probability space, Assumption 2 gives E[D_i]=1-s_ik >= b_k and E[F_n]=(sum_i (1-s_ik))/n >= b_k, where b_k=1-q^k is independent of n. These conclusions require no row independence. Every law with mutually independent cells of observation probability q satisfies Assumptions 1 and 2 and attains E[F_n]=b_k; a normalized independent-cell law exists for every q in [0,1). The all-deletion lower bound is b_k^n, equals b_k at n=1, and is strictly smaller for n>=2 and 0<q<1, when 0<b_k<1. On the entire domain q in [0,1), the bounds coincide if and only if n=1 or q=0; at q=0 both bounds are one.",
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     "body": "<p>Section 3.1, p. 152 (PDF p. 4), referring to Assumption 1, p. 150 (PDF p. 2). Independent rows need not have identical distributions, as the explicit witness and general product of row-specific deletion probabilities show.</p>",
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     "body": "<p>preprocessing, pp. 151-152 (PDF pp. 3-4), and Table 1, p. 152 (PDF p. 4). Removing wholly missing columns gives the exact average identities, rules out the printed retained average, and yields the corrected average and overstatement under the specified missing-cell count.</p>",
     "headline": "Table 1, State Failure column, “Proportion of missing values (avg.)”, corrected for the 1,205 retained variables (the part of <code class=\"fid\">P007-E014</code> that is established)",
     "location": "Table 1, p. 152 (PDF p. 4); preprocessing sentence, pp. 151-152 (PDF pp. 3-4); replication script Simcode_Statefailure.R, lines 28-55"
    },
    "id": "P007-T12",
    "text": {
     "body": "preprocessing, pp. 151-152 (PDF pp. 3-4), and Table 1, p. 152 (PDF p. 4). Removing wholly missing columns gives the exact average identities, rules out the printed retained average, and yields the corrected average and overstatement under the specified missing-cell count.",
     "headline": "Table 1, State Failure column, “Proportion of missing values (avg.)”, corrected for the 1,205 retained variables (the part of P007-E014 that is established)",
     "location": "Table 1, p. 152 (PDF p. 4); preprocessing sentence, pp. 151-152 (PDF pp. 3-4); replication script Simcode_Statefailure.R, lines 28-55"
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    "html": {
     "body": "<p>No current theorem is supplied for this target.</p>",
     "headline": "Table 1, State Failure column, and the State Failure sentences on expected loss in Section 3.2, as printed",
     "location": "Table 1, p. 152 (PDF p. 4); Section 3.2, third paragraph, p. 153 (PDF p. 5); Section 3.1, p. 152 (PDF p. 4)"
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    "text": {
     "body": "No current theorem is supplied for this target.",
     "headline": "Table 1, State Failure column, and the State Failure sentences on expected loss in Section 3.2, as printed",
     "location": "Table 1, p. 152 (PDF p. 4); Section 3.2, third paragraph, p. 153 (PDF p. 5); Section 3.1, p. 152 (PDF p. 4)"
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      "credit_reason": "Finite normalized weights are an exact representation of the distribution of finitely many Boolean indicators on any underlying probability space. All such joint laws are included, including q=0 and null prefixes; there is no added finite-support restriction on the underlying data.",
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    },
    "html": {
     "body": "<p>Lemma 3, p. 150 (PDF p. 2). Under mutual row independence and the printed conditional observation bound, the probability of deleting all rows is at least (1-q^k)^n.</p>",
     "headline": "Lemma 3",
     "location": "p. 150 (PDF p. 2), Section 2.1; proof in the Appendix, p. 154 (PDF p. 6)"
    },
    "id": "P007-T14",
    "text": {
     "body": "Lemma 3, p. 150 (PDF p. 2). Under mutual row independence and the printed conditional observation bound, the probability of deleting all rows is at least (1-q^k)^n.",
     "headline": "Lemma 3",
     "location": "p. 150 (PDF p. 2), Section 2.1; proof in the Appendix, p. 154 (PDF p. 6)"
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     "release_id": "sap-2026-11-08-assured-release",
     "target_id": "P007-T15"
    },
    "html": {
     "body": "<p>Lemma 8, Supporting Information B, Supplement p. 4. Under mutual row independence and the printed group observation bound, the probability of deleting all rows is at least (1-q^g)^n.</p>",
     "headline": "Lemma 8 (Supporting Information B)",
     "location": "Supplement p. 4 (publisher-supplement.pdf p. 4), Supporting Information B; proof on the same page"
    },
    "id": "P007-T15",
    "text": {
     "body": "Lemma 8, Supporting Information B, Supplement p. 4. Under mutual row independence and the printed group observation bound, the probability of deleting all rows is at least (1-q^g)^n.",
     "headline": "Lemma 8 (Supporting Information B)",
     "location": "Supplement p. 4 (publisher-supplement.pdf p. 4), Supporting Information B; proof on the same page"
    },
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      "conditional_on": [],
      "credit_reason": "This row proves the printed proposition itself. ConditionalObservationBound preserves the positive-prefix proviso exactly: the ratio is tested only when its denominator is positive, and the null-prefix cases remain allowed. The one q is fixed over every positive n, precisely as specified in the standing assumptions; no identical row law or cross-n coupling is required. All custom definitions were expanded, and none assumes the limit or an intermediate result. No metadata inconsistency was found.",
      "credit_status": "ok",
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      "lean_name": "AronowAudit.P007.proposition5",
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      "credit_reason": "Auxiliary analytic decay is already used internally by the headline proposition; the Challenge states its probability limit.",
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     "decision": "whole",
     "release_id": "sap-2026-11-08-assured-release",
     "target_id": "P007-T16"
    },
    "html": {
     "body": "<p>Proposition 5, p. 151 (PDF p. 3). Arbitrary changing joint laws with independent rows, one fixed conditional observation bound q&lt;1 and superlogarithmic variable counts have all-deletion probability tending to one, including q=0.</p>",
     "headline": "Proposition 5",
     "location": "p. 151 (PDF p. 3), Section 2.1; proof in the Appendix, p. 154 (PDF p. 6)"
    },
    "id": "P007-T16",
    "text": {
     "body": "Proposition 5, p. 151 (PDF p. 3). Arbitrary changing joint laws with independent rows, one fixed conditional observation bound q<1 and superlogarithmic variable counts have all-deletion probability tending to one, including q=0.",
     "headline": "Proposition 5",
     "location": "p. 151 (PDF p. 3), Section 2.1; proof in the Appendix, p. 154 (PDF p. 6)"
    },
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      "conditional_on": [],
      "credit_reason": "Component: the whole printed Proposition 10. The single q is outside all n-dependent assumptions, and convergence is along the full natural sequence, not a subsequence.",
      "credit_status": "ok",
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    },
    "html": {
     "body": "<p>Proposition 10, Supporting Information B, Supplement p. 4. Arbitrary changing group-indicator laws with independent rows, one fixed conditional observation bound q&lt;1 and superlogarithmic group counts have all-deletion probability tending to one, including q=0.</p>",
     "headline": "Proposition 10 (Supporting Information B)",
     "location": "Supplement p. 4 (publisher-supplement.pdf p. 4), Supporting Information B; proof on pp. 4-5"
    },
    "id": "P007-T17",
    "text": {
     "body": "Proposition 10, Supporting Information B, Supplement p. 4. Arbitrary changing group-indicator laws with independent rows, one fixed conditional observation bound q<1 and superlogarithmic group counts have all-deletion probability tending to one, including q=0.",
     "headline": "Proposition 10 (Supporting Information B)",
     "location": "Supplement p. 4 (publisher-supplement.pdf p. 4), Supporting Information B; proof on pp. 4-5"
    },
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 },
 "missing_sources": [],
 "reproduction": {
  "agreement": "agrees_in_part",
  "status": "full",
  "targets": [
   {
    "agreement": "to_rounding",
    "location": "p. 152 (main PDF p. 4)",
    "method": "I reran the supplied code with changes. It reproduces the printed Table 1 cells to rounding, including the erroneous 66.8% average computed before variables were removed. For the 1,205 retained State Failure variables, the corrected average is 66.3% (P007-E010; P007-E014).",
    "status": "reproduced",
    "target": "Table 1"
   },
   {
    "agreement": "to_rounding",
    "location": "p. 153 (main PDF p. 5), Figure 1 and results prose",
    "method": "I reran the supplied code with changes on the QoG Basic file. All 2.5 million surviving-row counts match the saved output exactly; the plotted curves agree at displayed precision and all four reported thresholds agree. The dataset name is corrected in P007-E009.",
    "status": "reproduced",
    "target": "Figure 1 and QoG thresholds"
   },
   {
    "agreement": "agrees_in_part",
    "location": "p. 153 (main PDF p. 5), Figure 2 and results prose",
    "method": "I reran the supplied code with changes. All 2.5 million surviving-row counts match the saved output exactly, and the Figure 2 curves agree at displayed precision. The prose threshold for expected loss above 99% disagrees: the first crossing is at six variables, rather than four (P007-E010). The reported probabilities of losing all rows agree.",
    "status": "reproduced",
    "target": "Figure 2 and State Failure thresholds"
   },
   {
    "agreement": "to_rounding",
    "location": "Supplement A pp. 1-2 (supplement PDF pp. 1-2); main p. 151 (main PDF p. 3)",
    "method": "I reran the supplied numerical code with changes and recalculated all 1,350 grid points from the formula. The plotted curves and approximate endpoints agree at displayed precision.",
    "status": "reproduced",
    "target": "Figure 3"
   },
   {
    "agreement": "exact",
    "location": "Supplement A p. 2 (supplement PDF p. 2)",
    "method": "I reran the supplied numerical code with changes and recalculated the threshold grids. The values 33 and 952 reproduce exactly as the largest $k$ with lower bound at most $0.5$; the smallest integers reaching $0.5$ are 34 and 953. The interpretation in the text and caption is corrected in P007-E005.",
    "status": "reproduced",
    "target": "Figure 4 and integer examples"
   },
   {
    "agreement": "to_rounding",
    "location": "Supplement A pp. 2-3 (supplement PDF pp. 2-3)",
    "method": "I reran the supplied numerical code with changes and recalculated the curves of $1-q_*^k$. The plotted curves and approximate endpoints agree at displayed precision; the bound is a lower bound on the expected proportion of rows removed (P007-E006).",
    "status": "reproduced",
    "target": "Figure 5"
   },
   {
    "agreement": "exact",
    "location": "p. 151 (main PDF p. 3), paragraph before Assumption 4",
    "method": "I recalculated $\\lfloor(\\log n)^{1.1}\\rfloor$ at the three stated values $n=10$, $n=1{,}000$ and $n=1{,}000{,}000$, obtaining the printed integers $2$, $8$ and $17$.",
    "status": "recalculated",
    "target": "Superlogarithmic growth example"
   }
  ]
 },
 "summary": "The paper studies listwise deletion when the number of variables $k$ grows with the number of observations $n$. Its main result, Proposition 5, states that under Assumptions 1 and 2, with $q_*<1$ fixed as $n$ grows, the probability that listwise deletion removes all rows tends to 1 when $k$ grows faster than $\\log n$. Simulations on two datasets illustrate the loss. 9 entries are classified as errors (9 Minor), and 2 are clarity notes. Two entries overlap: \\fid{P007-E010} and \\fid{P007-E014} both correct the State Failure average in Table 1; the latter also explains the script ordering and the treatment of empty text entries. The Quality of Government results come from the Basic dataset, not the Standard dataset named and cited (\\fid{P007-E009}). No finding changes the statement of Lemma 3 or of Proposition 5.",
 "what_was_checked": "I checked the following, subject to the corrections cited.\n\\begin{itemize}\n\\item Lemma 3 and Proposition 5. I followed both proofs step by step and re-derived the statements with the bound $q_*$ fixed along the sequence. The bound of Lemma 3 is attained when all entries are independent and each is observed with probability $q_*$. Lemma 8 and Proposition 10 of the supplement follow by the same argument with groups in place of variables. The proofs require the corrections in \\fid{P007-E003}.\n\\item Section 2.1, p.~151. The values $\\lfloor(\\log n)^{1.1}\\rfloor=2$, 8 and 17 at $n=10$, 1,000 and 1,000,000 are correct, and the lower bound of Lemma 3 is approximately $1.34\\times10^{-11}$ at 1\\% idiosyncratic missingness, $k=150$ and $n=100$.\n\\item Supporting Information A. All values plotted in Figures 3, 4 and 5 were recomputed from the formulas and from the replication code, and they agree. The text around Figures 4 and 5 is corrected in \\fid{P007-E005} and \\fid{P007-E006}.\n\\item Quality of Government. The Table 1 column and saved surviving-row counts reproduce from the Basic file (\\fid{P007-E009}) with the original simulation seed. Estimated row loss first exceeds 50\\% at 2 variables and 99\\% at 17; the estimated probability of losing every row first exceeds 0.5 at 14 variables and 0.99 at 52.\n\\item State Failure. The data file \\texttt{sort11v3.dta} is among the replication files. I ran the simulation code on this file with the seed of the replication files (213) and 25,000 draws at each of the 100 values of $k$. The run reproduces the saved numbers of surviving rows. The simulation estimates an expected proportion of rows removed above 50\\% with one variable and above 99\\% first with 6 variables. The estimated probability that all rows are removed first exceeds 0.5 with 6 variables and 0.99 with 17; the printed condition of more than 17 variables is sufficient. The other cells of the State Failure column of Table 1 match the output of the code. The average missingness and the 99\\% threshold for the expected proportion of rows removed are corrected in \\fid{P007-E010}. The evidence given in \\fid{P007-E010} and \\fid{P007-E014} does not use the data file, and the counts of missing cells in those findings are inferred from the saved log of the simulation code. With the code's definition of a missing cell, I find from the data file that the average missingness over the 1,205 variables that remain is $0.66316\\ldots$, which rounds to the corrected entry of 66.3\\%.\n\\item The simulation code matches the description in Section 3.1: 25,000 replications, a fresh draw of $k$ variables without replacement in each, and the share of complete rows recorded.\n\\item The numbering of assumptions, results and figures, and the cross-references to them, are consistent. The six figure files of the replication files match the published figures by eye.\n\\end{itemize}\n\nI did not check all the cited books and articles. The quotation from Lehmann (1999) and the page citations to Allison, Berk, Cameron and Trivedi, Little and Rubin, and Schafer are unchecked. The source checks cover glmnet and the package version named in its cited article (\\fid{P007-E001}) and the editors of the volume that contains Berk (1983) (\\fid{P007-E013}). I could not read Lai, Robbins, and Wei (1978) beyond its abstract, and whether that paper supports the remark on p.~151 that cites it is not settled. I did not establish whether the State Failure file contains empty text fields that the code counts as observed (\\fid{P007-E014}).\n\nThe Standard dataset, which the article names, is not among the replication files, and no analysis of it was made. These checks cover the items listed, and they are not a verification of the paper as a whole.",
 "work": {
  "arxiv_id": "2101.11470",
  "audit_complete": true,
  "authors": [
   "J. Sophia Wang",
   "P. M. Aronow"
  ],
  "category": "Articles - Theory and Methodology",
  "citation": "J. Sophia Wang and P. M. Aronow (2023). Listwise Deletion in High Dimensions. Political Analysis 31(1):149–155.",
  "doi": "10.1017/pan.2022.5",
  "edition": {
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   "sha256": "995b10fbd583f2f43964658e09c3d03a8059a504ed6df543b8038edded5ee3cb",
   "source_url": "https://doi.org/10.1017/pan.2022.5",
   "venue": "Political Analysis",
   "year": "2023"
  },
  "id": "P007",
  "later_materials": null,
  "title": "Listwise Deletion in High Dimensions",
  "type": "Journal article",
  "url": "https://pmaronow.github.io/llm-audit/works/P007/",
  "venue": "Political Analysis",
  "year": "2023"
 }
}