{
 "claims": [
  {
   "claim": "The Horvitz-Thompson difference-of-totals estimator (eq. [8]) is unbiased for the finite-population ATE in any randomised experiment whose assignment probabilities are known and strictly between 0 and 1, whatever the clustering, the heterogeneity of probabilities, the dependence in assignment or the form of the effects.",
   "claim_id": "P026-H1",
   "related_corrections": [],
   "tied_to": "eqs. [5]-[8]; eqs. [11]-[12]; Table 2 column HT"
  },
  {
   "claim": "Under complete randomisation of units the HT estimator is the difference in means; under complete randomisation of clusters it need not be, and difference-in-means, IPW and regression estimators can be biased in clustered or unequal-probability designs or with covariate adjustment.",
   "claim_id": "P026-H2",
   "related_corrections": [
    "P026-E003"
   ],
   "tied_to": "eqs. [11], [12]; Table 2 columns IPW, FE, FE (Cov.)"
  },
  {
   "claim": "The generalised difference estimator (eqs. [13]–[16]) gives unbiased covariate adjustment with any outcome model $f$ provided $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$ for every $i$. The paper states that this holds when $\\theta_i$ is fixed in advance or computed only from outcomes of units whose assignments are each independent of $T_i$.",
   "claim_id": "P026-H3",
   "related_corrections": [
    "P026-E001",
    "P026-N001",
    "P026-N004"
   ],
   "tied_to": "eqs. [13]-[19]; Appendix B (eqs. [39]-[40]); Section 5.1 and footnote 5; Table 2 columns G (Prior), G (Linear), G (Logit)"
  },
  {
   "claim": "The HT estimator is not location invariant, whereas the generalised difference estimator is when the fitted function transforms with the outcome.",
   "claim_id": "P026-H4",
   "related_corrections": [],
   "tied_to": "Appendix A (eqs. [36]-[38]); Appendix C (eqs. [41]-[42])"
  },
  {
   "claim": "The paper gives closed-form expressions for the sampling variance of the HT estimator and, with fixed $f$, of the generalised difference estimator, at unit level and at the level of cluster totals.",
   "claim_id": "P026-H5",
   "related_corrections": [
    "P026-E002",
    "P026-E005"
   ],
   "tied_to": "eqs. [20]-[30]"
  },
  {
   "claim": "The variance estimator of eq. [35] is conservative (nonnegatively biased) for any experimental design when $f$ is fixed, and is exactly unbiased under the sharp null when all pairs of clusters have positive probability of joint treatment and of joint control.",
   "claim_id": "P026-H6",
   "related_corrections": [
    "P026-E002",
    "P026-E004",
    "P026-E005",
    "P026-N002"
   ],
   "tied_to": "eqs. [31]-[35]; Appendices D and E; footnote 9"
  },
  {
   "claim": "In the 16-unit, 10-cluster, 2-block example under the sharp null, only the HT-based estimators are unbiased; the regression-based ATE and variance estimators are negatively biased; the covariate-adjusted HT-based estimators have lower RMSE than the regression-based ones; and the HT-based variance estimators are unbiased.",
   "claim_id": "P026-H7",
   "related_corrections": [
    "P026-E003"
   ],
   "tied_to": "Table 1; Table 2"
  },
  {
   "claim": "Outcome modelling within the proposed class can improve precision while retaining unbiasedness under the stated restrictions.",
   "claim_id": "P026-H8",
   "related_corrections": [
    "P026-E003",
    "P026-A001"
   ],
   "tied_to": "Section 5; Section 7 and Table 2"
  }
 ],
 "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 six lines of the paragraph beginning 'Following the basic logic of Williams'. Red boxes mark the words 'any', 'or all of the elements of the set of Y-j such that T-i is independent of T-j', and 'then T-i is independent of theta-i'.",
     "caption": "Section 5.1, second paragraph, p. 142 (PDF p. 8). The marked words define the admissible outcomes by independence of $T_i$ from each $T_j$ separately and conclude that $T_i$ is independent of $\\theta_i$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E001.png"
    },
    {
     "alt": "Crop of the four lines of footnote 5. Red boxes mark the words 'without loss of generality,', the expression 'f of X-i and g of Z and Y-j', which has a single outcome Y-j as its argument, and the step 'T-i independent of T-j implies T-i independent of f-prime of T-j'.",
     "caption": "Footnote 5, p. 142 (PDF p. 8). The argument is written for a single outcome $Y_j$ and is said to lose no generality.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E001-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T4"
    },
    {
     "id": "P026-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T5"
    },
    {
     "id": "P026-T18",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T18"
    }
   ],
   "headline": "A sufficient condition for fitting $\\theta_i$ from other units' outcomes while preserving unbiasedness is independence of $T_i$ from the whole vector of assignments used in the fit; independence from each assignment separately does not suffice.",
   "id": "P026-E001",
   "location": "Section 5.1, second paragraph, and footnote 5, p. 142 (PDF p. 8)",
   "pdf_page": 8,
   "printed_page": "142",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [
    "P026-T18",
    "P026-T4",
    "P026-T5"
   ],
   "text": {
    "as_printed": "``If $\\theta_i$ is a function of \\hl{any or all} of the elements of the set $\\hl{\\{Y_j:T_i\\perp\\!\\!\\!\\perp T_j\\}}$ and no other random variables, then $T_i\\perp\\!\\!\\!\\perp\\theta_i$. It follows that $T_i\\perp\\!\\!\\!\\perp f(\\mathbf{X}_i,\\theta_i)$ and therefore $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$.${}^{5}$''\n\nFootnote 5: ``More formally and \\hl{without loss of generality}, let $f(\\mathbf{X}_i,\\theta_i)=f(\\mathbf{X}_i,g(\\mathbf{Z},\\hl{Y_j}))=f(\\mathbf{X}_i,g(\\mathbf{Z},\\hl{h(Y_{0j},Y_{1j};T_j)})))$, where $\\mathbf{Z}$ is a matrix of pretreatment covariates (that may or may not coincide with $\\mathbf{X}_j$), $g(.)$ is an arbitrary function (e.g., the least squares fit), and $h(.)$ is the function implied by eq. [1]. Since only $T_j$ is a random variable, the random variable $f(\\mathbf{X}_i,\\theta_i)$ equals some function $f'(T_j)$. $T_i\\perp\\!\\!\\!\\perp T_j$ implies $T_i\\perp\\!\\!\\!\\perp f'(T_j)$ (equivalently $T_i\\perp\\!\\!\\!\\perp f(\\mathbf{X}_i,\\theta_i)$) which, in turn, implies $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$.''",
    "corrected": "``If, for a set $J_i$ of units fixed before the randomization, \\hl{the assignment} $T_i$ \\hl{is independent of the vector} $\\hl{(T_j)_{j\\in J_i}}$, and $\\theta_i$ is a function of the elements of the set $\\hl{\\{Y_j:j\\in J_i\\}}$, of fixed pretreatment quantities and of no other random variables, then $T_i\\perp\\!\\!\\!\\perp\\theta_i$. It follows that $T_i\\perp\\!\\!\\!\\perp f(\\mathbf{X}_i,\\theta_i)$ and therefore $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$.''\n\nFootnote 5: ``More formally, let\n\\[\nf(\\mathbf{X}_i,\\theta_i)=f\\left(\\mathbf{X}_i,g\\left(\\mathbf{Z},\\hl{(Y_j)_{j\\in J_i}}\\right)\\right)=f\\left(\\mathbf{X}_i,g\\left(\\mathbf{Z},\\hl{(h(Y_{0j},Y_{1j};T_j))_{j\\in J_i}}\\right)\\right),\n\\]\nwhere $\\mathbf{Z}$ is a matrix of pretreatment covariates, $g(.)$ is an arbitrary function (e.g., the least squares fit), and $h(.)$ is the function implied by eq. [1]. Since only the $T_j$ are random variables, the random variable $f(\\mathbf{X}_i,\\theta_i)$ equals some function $\\hl{f'((T_j)_{j\\in J_i})}$. \\hl{Independence of} $T_i$ \\hl{and the vector} $(T_j)_{j\\in J_i}$ implies $T_i\\perp\\!\\!\\!\\perp f'((T_j)_{j\\in J_i})$ (equivalently $T_i\\perp\\!\\!\\!\\perp f(\\mathbf{X}_i,\\theta_i)$) which, in turn, implies $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$.''\n\nThe premise can be replaced by a direct check that $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$ for all $i$. It also suffices that $T_i$ be independent of the vector of observed outcomes $(Y_j)_{j\\in J_i}$.",
    "evidence": "This correction rests on an exact counterexample, computed in rational arithmetic, and on a derivation of the corrected condition.\n\n\\emph{Counterexample.} Let $N=6$, and let the randomization $(T_1,\\ldots,T_6)$ take the four values $010101$, $011010$, $100110$, $101001$ with probability $1/4$ each. Equivalently, $T=(A,1-A,B,1-B,C,1-C)$, where $A$ and $B$ are independent fair bits and $C=A+B-2AB$. Every unit has $\\pi_{1i}=\\pi_{0i}=1/2$. For two units in different pairs among $\\{1,2\\}$, $\\{3,4\\}$, $\\{5,6\\}$, each of the four joint outcomes of their assignments has probability $1/4$, and the two assignments are independent. The printed set then contains $Y_3,Y_4,Y_5,Y_6$ for units 1 and 2, $Y_1,Y_2,Y_5,Y_6$ for units 3 and 4, and $Y_1,Y_2,Y_3,Y_4$ for units 5 and 6. The assignment $T_5$ is not independent of the pair $(T_1,T_3)$: $\\Pr(T_5=0,T_1=0,T_3=0)=1/4$, while $\\Pr(T_5=0)\\Pr(T_1=0,T_3=0)=1/8$.\n\nLet $Y_{0i}=0$, $Y_{1i}=1$ and $X_i=1$ for every unit, and let $f(X,\\theta)=X\\theta$. Then $Y_i=T_i$ and $\\Delta=1$. Let $Q(u,v)=u+v-2uv$, and fix the rules\n\\[\n\\begin{aligned}\n\\theta_1&=Q(Y_3,Y_5),&\\qquad\\theta_3&=Q(Y_1,Y_5),&\\qquad\\theta_5&=Q(Y_1,Y_3),\\\\\n\\theta_2&=1-Q(Y_3,Y_5),&\\qquad\\theta_4&=1-Q(Y_1,Y_5),&\\qquad\\theta_6&=1-Q(Y_1,Y_3).\n\\end{aligned}\n\\]\nEach rule uses two outcomes of the printed set for its unit and no other random variable. Since $Q(B,C)=A$, $Q(A,C)=B$ and $Q(A,B)=C$, I have $f(X_i,\\theta_i)=\\theta_i=T_i$ for every unit in every randomization. Hence $\\operatorname{Cov}(T_i,f(X_i,\\theta_i))=1/2-1/4=1/4$ for all $i$. Substituting $\\pi_{1i}=\\pi_{0i}=1/2$ and $f(X_i,\\theta_i)=T_i$ into [13] and [14], I obtain\n\\[\n\\widehat{Y_1^{T*}}=\\sum_{i=1}^{6}\\left(2T_i-2T_i^2+T_i\\right)=3,\\qquad\\widehat{Y_0^{T*}}=\\sum_{i=1}^{6}\\left(-2(1-T_i)T_i+T_i\\right)=3,\n\\]\nso that $\\widehat{\\Delta}_G=(3-3)/6=0$ in every randomization, and the bias is $-1$. By equation [19], the bias is again $-(1/6)\\cdot 6\\cdot(1/4)\\cdot(2+2)=-1$. The Horvitz--Thompson estimate equals $1$ in every randomization.\n\\begin{audittable}{llrrrr}{The four randomizations of the design with six units, each with probability $1/4$. The true totals are $Y_1^T=6$ and $Y_0^T=0$, and $\\Delta=1$.}\nRandomization & Predictions $f(X_i,\\theta_i)$, $i=1,\\ldots,6$ & $\\widehat{Y_1^{T*}}$ & $\\widehat{Y_0^{T*}}$ & $\\widehat{\\Delta}_G$ & $\\widehat{\\Delta}_{HT}$ \\\\\n$010101$ & $(0,1,0,1,0,1)$ & $3$ & $3$ & $0$ & $1$ \\\\\n$011010$ & $(0,1,1,0,1,0)$ & $3$ & $3$ & $0$ & $1$ \\\\\n$100110$ & $(1,0,0,1,1,0)$ & $3$ & $3$ & $0$ & $1$ \\\\\n$101001$ & $(1,0,1,0,0,1)$ & $3$ & $3$ & $0$ & $1$ \\\\\n\\end{audittable}\n\n\\emph{Derivation of the corrected condition.} Fix a set $J_i$ of units, and let $V_i=(T_j)_{j\\in J_i}$. Suppose that $T_i$ is independent of $V_i$ and that $\\theta_i$ depends only on $(Y_j)_{j\\in J_i}$ and on fixed quantities. Since the potential outcomes are fixed, equation [1] implies that $f(\\mathbf{X}_i,\\theta_i)=H_i(V_i)$ for a fixed function $H_i$. For $a\\in\\{0,1\\}$ and any set $D$ of values,\n\\[\n\\begin{aligned}\n\\Pr(T_i=a,\\,H_i(V_i)\\in D)&=\\sum_{v:H_i(v)\\in D}\\Pr(T_i=a,\\,V_i=v)\\\\\n&=\\Pr(T_i=a)\\sum_{v:H_i(v)\\in D}\\Pr(V_i=v)\\\\\n&=\\Pr(T_i=a)\\Pr(H_i(V_i)\\in D).\n\\end{aligned}\n\\]\nThe sums are finite. Hence $T_i$ and $f(\\mathbf{X}_i,\\theta_i)$ are independent, and their covariance is zero. Equation [19] then implies that $\\widehat{\\Delta}_G$ is unbiased.",
    "problem": "The set $\\{Y_j:T_i\\perp\\!\\!\\!\\perp T_j\\}$ is defined by the independence of $T_i$ from each $T_j$ separately. A parameter $\\theta_i$ fitted from several of these outcomes is a function of several assignments jointly, and $T_i$ can be independent of each of them and still depend on the vector they form. The conclusion $T_i\\perp\\!\\!\\!\\perp\\theta_i$ does not follow from the stated premise, and neither does $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$. Footnote 5 treats a fit that uses one outcome $Y_j$, in which case $\\theta_i$ is a function of $T_j$ alone, and for that case its argument is correct. The reduction to that case is not without loss of generality when the fit uses several outcomes.\n\nWhen the premise holds only pairwise, the generalized difference estimator can be biased. In the design with six units described below, every randomization has three treated units and three controls, every outcome used in a fit belongs to the printed set, the estimate is $0$ in every randomization, and the average treatment effect is $1$.",
    "what_survives": "The Horvitz--Thompson estimator, the definitions [13]--[16], the bias identity [19], and the sufficient condition $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$ with its proof in Appendix B are unaffected. The two fitting examples in Section 5.1 satisfy the corrected condition. These are designs in which units are assigned independently and the fit for unit $i$ leaves out unit $i$, and designs in which blocks are randomized independently and the fit leaves out the block of unit $i$, as in Section 7. Under the sharp null the observed outcomes are fixed, and so is any fit from them. A prediction that is additive in the outcomes with fixed coefficients, $f(\\mathbf{X}_i,\\theta_i)=a_i+\\sum_jb_{ij}Y_j$, has zero covariance with $T_i$ under the printed pairwise rule, because $\\operatorname{Cov}(T_i,Y_j)=(Y_{1j}-Y_{0j})\\operatorname{Cov}(T_i,T_j)=0$. What changes is the general rule for an arbitrary fit: the stronger premise gives a valid guarantee of unbiasedness and excludes designs such as the one above."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-E001",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop showing equation 29, the three lines of text that follow it, equation 30 and the line after it. A red box marks the clause 'then pi-prime-1k0l is greater than zero for all l not equal to k', with footnote mark 6, in the sentence that begins 'Since, for all pairs k, l such that k is not equal to l, the probability that T-prime-k differs from T-prime-l is positive'.",
     "caption": "Section 6.3, p. 145 (PDF p. 11), from equation [29] to equation [30]. The marked clause infers $\\pi'_{1k0l}>0$ for all $l\\ne k$ from the fact that distinct cluster indicators differ with positive probability.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E002.png"
    },
    {
     "alt": "Crop of the three lines that introduce the covariance estimator, equation 34, and the two lines after it, which refer to Appendix E for the bound on the expectation of the estimator and say that the estimator is unbiased under the sharp null. A red box marks the denominator pi-prime-1k0l of the first fraction in equation 34.",
     "caption": "Equation [34], p. 146 (PDF p. 12). The marked joint probability is a denominator, and the estimator is undefined for a pair $k\\ne l$ with $\\pi'_{1k0l}=0$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E002-d.png"
    },
    {
     "alt": "Crop of the three rows on the right-hand side of equation 35 and the four lines of text below it. Red boxes mark the term 'minus two times T-prime-k times one minus T-prime-l, over pi-prime-1k0l plus epsilon-1k0l, and so on' in the second row; the sentence in parentheses 'epsilon-akbl is included to avoid division by zero'; the words 'conservative has been established'; and the words 'the estimator is exactly unbiased'.",
     "caption": "Equation [35] and the sentences after it, p. 147 (PDF p. 13). The marked cross term is set to zero by $\\varepsilon_{1k0l}$ when $\\pi'_{1k0l}=0$, and no term bounds the product it leaves out; the marked sentences state that the estimator is conservative and, under the sharp null, exactly unbiased.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E002-b.png"
    },
    {
     "alt": "Crop of five lines of the paragraph on variance estimation in the review of the literature. A red box marks the words 'conservative for any experimental design,', with 'any' in italics.",
     "caption": "Fourth paragraph of p. 137 (PDF p. 3). The marked words state the scope claimed for the variance estimator.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E002-c.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T1"
    },
    {
     "id": "P026-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T2"
    },
    {
     "id": "P026-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T3"
    },
    {
     "id": "P026-T8",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T8"
    },
    {
     "id": "P026-T17",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T17"
    },
    {
     "id": "P026-T24",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T24"
    }
   ],
   "headline": "Two distinct clusters need not have $\\pi'_{1k0l}>0$; when $\\pi'_{1k0l}=0$ for some $k\\ne l$, the covariance estimator [34] is undefined and the variance estimator [35] can be biased downward, also under the sharp null. Adding one term to [35] makes it conservative for every design.",
   "id": "P026-E002",
   "location": "Section 6.3, sentence between equations [29] and [30], p. 145 (PDF p. 11); equation [34], p. 146 (PDF p. 12); equation [35], the sentences after it and footnote 9, p. 147 (PDF p. 13); Appendix E, p. 152 (PDF p. 18); p. 137 (PDF p. 3)",
   "pdf_page": 11,
   "printed_page": "145",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [
    "P026-T1",
    "P026-T17",
    "P026-T2",
    "P026-T24",
    "P026-T3",
    "P026-T8"
   ],
   "text": {
    "as_printed": "p.~145: ``Since, for all pairs $k,l$ such that $k\\ne l$, $\\Pr(T'_k\\ne T'_l)>0$, \\hl{then} $\\hl{\\pi'_{1k0l}>0}$ \\hl{for all} $\\hl{l\\ne k}$.${}^{6}$ Therefore, the covariance expression may be reduced further.''\n\np.~146, equation [34]:\n\\[\n\\begin{aligned}\n\\widehat{\\operatorname{Cov}}_C\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)={}&\\sum_{k=1}^{M}\\sum_{l\\ne k}\\frac{T'_k(1-T'_l)}{\\hl{\\pi'_{1k0l}}}(\\pi'_{1k0l}-\\pi'_{1k}\\pi'_{0l})\\frac{Y'_kY'_l}{\\pi'_{1k}\\pi'_{0l}}\\\\\n&-\\sum_{k=1}^{M}T'_k\\frac{(Y'_k)^2}{2\\pi'_{1k}}-\\sum_{k=1}^{M}(1-T'_k)\\frac{(Y'_k)^2}{2\\pi'_{0k}}.\n\\end{aligned}\n\\]\n\np.~147, equation [35]:\n\\[\n\\begin{aligned}\n&\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})=\\frac{1}{N^2}\\sum_{k=1}^{M}\\Bigg[T'_k\\left(\\frac{Y'_k}{\\pi'_{1k}}\\right)^2+(1-T'_k)\\left(\\frac{Y'_k}{\\pi'_{0k}}\\right)^2\\\\\n&\\quad+\\sum_{l\\ne k}\\Bigg(\\frac{T'_kT'_l}{\\pi'_{1k1l}+\\varepsilon_{1k1l}}(\\pi'_{1k1l}-\\pi'_{1k}\\pi'_{1l})\\frac{Y'_kY'_l}{\\pi'_{1k}\\pi'_{1l}}+\\frac{(1-T'_k)(1-T'_l)}{\\pi'_{0k0l}+\\varepsilon_{0k0l}}(\\pi'_{0k0l}-\\pi'_{0k}\\pi'_{0l})\\frac{Y'_kY'_l}{\\pi'_{0k}\\pi'_{0l}}\\\\\n&\\qquad\\qquad-2\\frac{T'_k(1-T'_l)}{\\pi'_{1k0l}+\\varepsilon_{1k0l}}(\\pi'_{1k0l}-\\pi'_{1k}\\pi'_{0l})\\frac{Y'_kY'_l}{\\pi'_{1k}\\pi'_{0l}}\\Bigg)\\\\\n&\\quad+\\sum_{\\forall l:\\pi'_{1k1l}=0}\\left(T'_k\\frac{(Y'_k)^2}{2\\pi'_{1k}}+T'_l\\frac{(Y'_l)^2}{2\\pi'_{1l}}\\right)+\\sum_{\\forall l:\\pi'_{0k0l}=0}\\left((1-T'_k)\\frac{(Y'_k)^2}{2\\pi'_{0k}}+(1-T'_l)\\frac{(Y'_l)^2}{2\\pi'_{0l}}\\right)\\Bigg],\n\\end{aligned}\n\\]\n``where $\\varepsilon_{akbl}=1[\\pi'_{akbl}=0]$. ($\\varepsilon_{akbl}$ is included to avoid division by zero.) The fact that $\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})$ \\hl{is conservative has been established}. But also note that when, for all $k,l$, $\\pi'_{1k1l}>0$ and $\\pi'_{0k0l}>0$, and there is no treatment effect whatsoever, \\hl{the estimator is exactly unbiased}.''\n\np.~137: ``Our article extends this prior work by proposing a new Horvitz--Thompson-based variance estimator that is \\hl{conservative for \\emph{any} experimental design}, though additional regularity conditions would be required for use in constructing confidence intervals and hypothesis tests.''",
    "corrected": "p.~145: ``For all pairs $k,l$ such that $k\\ne l$, $\\Pr(T'_k\\ne T'_l)=\\pi'_{1k0l}+\\pi'_{0k1l}>0$, \\hl{and at least one of} $\\pi'_{1k0l}$ \\hl{and} $\\pi'_{0k1l}$ \\hl{is positive}. \\hl{A pair} $k\\ne l$ \\hl{with} $\\pi'_{1k0l}=0$ \\hl{contributes} $-Y'_{1k}Y'_{0l}$ \\hl{in both forms of eq. [29]}. Therefore, the covariance expression may be reduced further.'' Equation [30] stands as printed.\n\nEquation [34], for every design with $\\pi'_{1k}>0$ and $\\pi'_{0k}>0$ for all $k$:\n\\[\n\\begin{aligned}\n\\widehat{\\operatorname{Cov}}_C\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)={}&\\sum_{k=1}^{M}\\sum_{\\hl{\\scriptstyle l\\ne k:\\pi'_{1k0l}>0}}\\frac{T'_k(1-T'_l)}{\\pi'_{1k0l}}(\\pi'_{1k0l}-\\pi'_{1k}\\pi'_{0l})\\frac{Y'_kY'_l}{\\pi'_{1k}\\pi'_{0l}}\\\\\n&-\\sum_{k=1}^{M}T'_k\\frac{(Y'_k)^2}{2\\pi'_{1k}}-\\sum_{k=1}^{M}(1-T'_k)\\frac{(Y'_k)^2}{2\\pi'_{0k}}\\\\\n&{}\\hl{-\\frac12\\sum_{k=1}^{M}\\sum_{l\\ne k:\\pi'_{1k0l}=0}\\left(T'_k\\frac{(Y'_k)^2}{\\pi'_{1k}}+(1-T'_l)\\frac{(Y'_l)^2}{\\pi'_{0l}}\\right)}.\n\\end{aligned}\n\\]\nIts expectation is at most $\\operatorname{Cov}(\\widehat{Y_1^T},\\widehat{Y_0^T})$, and the covariance minus its expectation is\n\\[\n\\frac12\\sum_{k=1}^{M}(Y'_{1k}-Y'_{0k})^2+\\frac12\\sum_{k=1}^{M}\\sum_{l\\ne k:\\pi'_{1k0l}=0}(Y'_{1k}-Y'_{0l})^2.\n\\]\nIn Appendix E the bound [43] is replaced accordingly: for each pair $k\\ne l$ with $\\pi'_{1k0l}=0$, Young's inequality replaces $-Y'_{1k}Y'_{0l}$ in the lower bound by $\\hl{-\\left((Y'_{1k})^2+(Y'_{0l})^2\\right)/2}$. The corrected [34] estimates this bound without bias.\n\nEquation [35], with $\\varepsilon_{akbl}=1[\\pi'_{akbl}=0]$:\n\\[\n\\begin{aligned}\n&\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})=\\frac{1}{N^2}\\sum_{k=1}^{M}\\Bigg[T'_k\\left(\\frac{Y'_k}{\\pi'_{1k}}\\right)^2+(1-T'_k)\\left(\\frac{Y'_k}{\\pi'_{0k}}\\right)^2\\\\\n&\\quad+\\sum_{l\\ne k}\\Bigg(\\frac{T'_kT'_l}{\\pi'_{1k1l}+\\varepsilon_{1k1l}}(\\pi'_{1k1l}-\\pi'_{1k}\\pi'_{1l})\\frac{Y'_k}{\\pi'_{1k}}\\frac{Y'_l}{\\pi'_{1l}}+\\frac{(1-T'_k)(1-T'_l)}{\\pi'_{0k0l}+\\varepsilon_{0k0l}}(\\pi'_{0k0l}-\\pi'_{0k}\\pi'_{0l})\\frac{Y'_k}{\\pi'_{0k}}\\frac{Y'_l}{\\pi'_{0l}}\\\\\n&\\qquad\\qquad-2\\frac{T'_k(1-T'_l)}{\\pi'_{1k0l}+\\varepsilon_{1k0l}}(\\pi'_{1k0l}-\\pi'_{1k}\\pi'_{0l})\\frac{Y'_kY'_l}{\\pi'_{1k}\\pi'_{0l}}\\Bigg)\\\\\n&\\quad+\\sum_{\\forall l:\\pi'_{1k1l}=0}\\left(T'_k\\frac{(Y'_k)^2}{2\\pi'_{1k}}+T'_l\\frac{(Y'_l)^2}{2\\pi'_{1l}}\\right)+\\sum_{\\forall l:\\pi'_{0k0l}=0}\\left((1-T'_k)\\frac{(Y'_k)^2}{2\\pi'_{0k}}+(1-T'_l)\\frac{(Y'_l)^2}{2\\pi'_{0l}}\\right)\\\\\n&\\quad\\hl{+\\sum_{l\\ne k:\\pi'_{1k0l}=0}\\left(T'_k\\frac{(Y'_k)^2}{\\pi'_{1k}}+(1-T'_l)\\frac{(Y'_l)^2}{\\pi'_{0l}}\\right)}\\Bigg].\n\\end{aligned}\n\\]\nHere $N$ is the number of units, $T'_k$ is the assignment of cluster $k$, $Y'_k$ is its observed outcome total, and $\\pi'_{1k}$, $\\pi'_{0k}$, $\\pi'_{1k1l}$, $\\pi'_{0k0l}$, $\\pi'_{1k0l}$ are the marginal and joint assignment probabilities of the clusters.\n\nStatement after [35]: for every design with fixed potential outcomes and $\\pi'_{1k}>0$, $\\pi'_{0k}>0$ for all $k$, the corrected estimator is conservative, $\\mathrm{E}[\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})]\\ge\\operatorname{Var}(\\widehat{\\Delta}_{HT})$. When, in addition, $\\pi'_{1k1l}>0$ and $\\pi'_{0k0l}>0$ for all $k,l$ and there is no treatment effect whatsoever,\n\\[\n\\mathrm{E}\\left[\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})\\right]-\\operatorname{Var}(\\widehat{\\Delta}_{HT})=\\hl{\\frac{1}{N^2}\\sum_{k=1}^{M}\\sum_{l\\ne k:\\pi'_{1k0l}=0}(Y'_k-Y'_l)^2},\n\\]\nand the estimator is exactly unbiased if and only if $\\hl{Y'_k=Y'_l}$ for every pair $k\\ne l$ with $\\hl{\\pi'_{1k0l}=0}$. This holds in particular when $\\pi'_{1k0l}>0$ for all $k\\ne l$. The same statements hold for $\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_G)$ with fixed $f$ (footnote 9), with the cluster totals of $U_i=Y_i-f(\\mathbf{X}_i,\\theta_i)$ in place of $Y'_k$.\n\nAlternatively, keep [34] and [35] as printed and add the assumption that $\\pi'_{1k0l}>0$ for all $k\\ne l$. The covariance and HT variance guarantees after [34] and [35], Appendix E's expectation argument for [34], and footnote 9's sharp-null claim for fixed $f$ are then restored, and ``conservative for any experimental design'' on p.~137 becomes ``conservative for any experimental design with $\\pi'_{1k0l}>0$ for all $k\\ne l$''.",
    "evidence": "This correction rests on an exact counterexample, computed in rational arithmetic, and on a derivation of the bias identity and of the corrected estimator.\n\n\\emph{Counterexample.} Let $N=M=6$, with each unit its own cluster, and let all twelve potential outcomes equal $1$, so that the sharp null holds. Let the randomization be drawn uniformly from the fourteen randomizations such that exactly three units are treated and $T'_1\\le T'_2$. The marginal probabilities of treatment are $(2/7,5/7,1/2,1/2,1/2,1/2)$. Every joint probability $\\pi'_{1k1l}$ and $\\pi'_{0k0l}$ is positive, the smallest being $1/14$, and every two indicators differ with probability at least $3/7$. The probability $\\Pr(T'_1=1,T'_2=0)$ is zero, and it is the only $\\pi'_{1k0l}$ with $k\\ne l$ that vanishes. All hypotheses of the printed statements hold. Enumerating the fourteen randomizations, I find\n\\[\n\\operatorname{Var}(\\widehat{\\Delta}_{HT})=\\frac{9}{700},\\qquad\\mathrm{E}\\left[\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})\\right]=-\\frac{269}{6300},\n\\]\na bias of $-1/18=-2\\cdot 1\\cdot 1/6^2$, in agreement with the identity above. The covariance of the two estimated totals is $27/175$, and [34] with its undefined term omitted has expectation $202/175$, which exceeds the covariance by $1$. With the added term the expectation of the variance estimator is $9/700$.\n\\begin{audittable}{lrrrrrr}{The design with six units under the sharp null with all outcomes equal to $1$. The statistics are constant within each value of $(T'_1,T'_2)$. The true variance is $9/700$.}\n$(T'_1,T'_2)$ & Randomizations & Probability & $\\widehat{\\Delta}_{HT}$ & [35] as printed & Added term & [35] corrected \\\\\n$(0,0)$ & $4$ & $2/7$ & $-3/20$ & $-619/3600$ & $7/72$ & $-269/3600$ \\\\\n$(0,1)$ & $6$ & $3/7$ & $0$ & $7/54$ & $0$ & $7/54$ \\\\\n$(1,1)$ & $4$ & $2/7$ & $3/20$ & $-619/3600$ & $7/72$ & $-269/3600$ \\\\\nExpectation & $14$ & $1$ & $0$ & $-269/6300$ & $1/18$ & $9/700$ \\\\\n\\end{audittable}\nThe corrected estimator is not unbiased under the sharp null in general. In the same design with both potential outcomes equal to $(1,2,1,1,1,1)$, the variance is $2759/12600$, the expectation of [35] as printed is $151/1400$, and the expectation of the corrected estimator is $3109/12600$. Its bias is $1/36=(1-2)^2/6^2$. Under complete randomization of three of six units every $\\pi'_{1k0l}$ with $k\\ne l$ equals $3/10$, and with sharp-null outcomes $(1,2,3,-1,0,4)$ the variance and the expectation of [35] both equal $7/3$.\n\n\\emph{Derivation.} I take expectations in $N^2$ times [35] term by term and compare with $N^2\\operatorname{Var}(\\widehat{\\Delta}_{HT})$, whose off-diagonal part is the sum over ordered pairs $k\\ne l$ of\n\\[\n\\left(\\frac{\\pi'_{1k1l}}{\\pi'_{1k}\\pi'_{1l}}-1\\right)x_kx_l+\\left(\\frac{\\pi'_{0k0l}}{\\pi'_{0k}\\pi'_{0l}}-1\\right)z_kz_l-2\\left(\\frac{\\pi'_{1k0l}}{\\pi'_{1k}\\pi'_{0l}}-1\\right)x_kz_l.\n\\]\nThe diagonal terms of [35] have expectation $x_k^2/\\pi'_{1k}+z_k^2/\\pi'_{0k}$, which exceeds the diagonal part of the variance by $(x_k-z_k)^2$. A product term with a positive joint probability is unbiased for the corresponding term of the variance. For $(k,l)\\in Z_{11}$ the variance has the term $-x_kx_l$, while [35] has a product term equal to zero and a bounding term with expectation $(x_k^2+x_l^2)/2$. The difference is $(x_k+x_l)^2/2$, and the same holds for $Z_{00}$ with $z$ in place of $x$. For $(k,l)\\in Z_{10}$ the variance has the term $+2x_kz_l$ and [35] has zero. Summing these contributions, I obtain the identity stated above. The added term has expectation $N^{-2}\\sum_{Z_{10}}(x_k^2+z_l^2)$, and $x_k^2+z_l^2-2x_kz_l=(x_k-z_l)^2$. For the corrected estimator the last term of the identity becomes $+\\sum_{Z_{10}}(x_k-z_l)^2$, and every term is nonnegative. Under the sharp null, with $x_k=z_k=Y'_k$ and $Z_{11}$, $Z_{00}$ empty, the bias reduces to $N^{-2}\\sum_{Z_{10}}(Y'_k-Y'_l)^2$.",
    "problem": "For binary indicators, $\\Pr(T'_k\\ne T'_l)=\\pi'_{1k0l}+\\pi'_{0k1l}$. Since clusters $k$ and $l$ are distinct, at least one of the two probabilities on the right is positive, and not necessarily both. If cluster $k$ can be treated only when cluster $l$ is treated, then $\\pi'_{1k0l}=0$ although $k\\ne l$. The paper assumes that the marginal probabilities are positive. It does not assume that they are equal for the two clusters, in which case both probabilities would be positive.\n\nThree statements rest on the deduction. First, the estimator [34] divides by $\\pi'_{1k0l}$ and is undefined for such a pair. Second, equation [35] sets the corresponding cross term to zero through $\\varepsilon_{1k0l}$ and adds nothing in its place. The variance of $\\widehat{\\Delta}_{HT}$ contains the term $2Y'_{1k}Y'_{0l}/N^2$ for that pair, and [35] neither estimates nor bounds it, so that [35] is not conservative in general. Third, under the sharp null with $\\pi'_{1k1l}>0$ and $\\pi'_{0k0l}>0$ for all $k,l$, [35] is not unbiased in general. Appendix E is affected in the same way, since the bound [43] contains the products $Y'_{1k}Y'_{0l}$ for such pairs and these cannot be estimated by weighting with inverse probabilities. The same holds for footnote 9, already for $f\\equiv 0$, and for the statements of the abstract, of p.~137 and of the discussion that the variance estimation is conservative.\n\nThe bias of [35] is as follows. Let $x_k=Y'_{1k}$ and $z_k=Y'_{0k}$, and let $Z_{11}$, $Z_{00}$ and $Z_{10}$ be the sets of ordered pairs $(k,l)$ with $k\\ne l$ such that $\\pi'_{1k1l}=0$, $\\pi'_{0k0l}=0$ and $\\pi'_{1k0l}=0$, respectively. Then\n\\[\nN^2\\left\\{\\mathrm{E}\\left[\\widehat{\\operatorname{Var}}_C(\\widehat{\\Delta}_{HT})\\right]-\\operatorname{Var}(\\widehat{\\Delta}_{HT})\\right\\}=\\sum_{k=1}^{M}(x_k-z_k)^2+\\frac12\\sum_{Z_{11}}(x_k+x_l)^2+\\frac12\\sum_{Z_{00}}(z_k+z_l)^2-2\\sum_{Z_{10}}x_kz_l.\n\\]\nThe first three terms are nonnegative. The last can be negative when the outcomes are positive, and it can outweigh the others. In the design with six units described below the sharp null holds, the variance is $9/700$, and the expectation of [35] is $-269/6300$.\n\nThere are two ways to correct this, and each has a cost. The first is to add a term to the estimator, given below. The corrected estimator is conservative for every design with positive marginal probabilities, but under the sharp null with positive same-arm joint probabilities it is unbiased only when the cluster totals agree across every pair with $\\pi'_{1k0l}=0$. The second is to keep [34] and [35] and to assume that $\\pi'_{1k0l}>0$ for all $k\\ne l$. The printed statements then hold, for a smaller class of designs that excludes the one below.",
    "what_survives": "The population variance and covariance identities [20]--[30] are unaffected by this error. In particular [30] is correct, because a pair $k\\ne l$ with $\\pi'_{1k0l}=0$ contributes $-Y'_{1k}Y'_{0l}$ to either form of the covariance. Only the sentence that justifies it changes. The Young bound [43] and the estimators [32] and [33] of the variances of the two estimated totals are unaffected, and so are the point estimators and their unbiasedness. In every design with $\\pi'_{1k0l}>0$ for all $k\\ne l$, the covariance and HT variance guarantees of [34] and [35] hold. The example of Section 7 is such a design: its within-block probabilities $\\pi'_{1k0l}$ are $1/3$ and $4/15$ and its cross-block probabilities are $1/3$ and $1/6$, and the added term is zero there. Three things change. The estimator [35] gains a term in designs where one cluster can be treated only if another is. The statement that the variance estimator is conservative for any design holds for the corrected estimator and not for [35] as printed. Exact unbiasedness under the sharp null requires a further condition."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-E002",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the sentence 'Cov-C of the two estimated totals may be estimated without bias', the two-row display that follows it, and three lines of the proof. One red box encloses the first row of the display, whose left side reads 'Var-hat-C of Y-hat-1-T' and whose two terms contain T-prime-k and the product T-prime-k T-prime-l with pi-prime-1k1l. A second red box marks the identity 'E of T-prime-k times one minus T-prime-l equals pi-prime-1k0l' at the start of the proof.",
     "caption": "Appendix E, p. 152 (PDF p. 18). The first row of the display, marked, has the label and the two leading terms of the variance estimator [32]; the identity marked below it, on which the proof relies, concerns the factor $T'_k(1-T'_l)$, which the display does not contain.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E004.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T8",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T8"
    },
    {
     "id": "P026-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T9"
    }
   ],
   "headline": "The display in Appendix E that should state the estimator of the covariance bound carries the label and the first two terms of the variance estimator for the treatment total; it should be equation [34].",
   "id": "P026-E004",
   "location": "Appendix E, unnumbered display after “$\\operatorname{Cov}_C(\\widehat{Y_1^T},\\widehat{Y_0^T})$ may be estimated without bias:”, p. 152 (PDF p. 18)",
   "pdf_page": 18,
   "printed_page": "152",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P026-T8",
    "P026-T9"
   ],
   "text": {
    "as_printed": "``$\\operatorname{Cov}_C\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)$ may be estimated without bias:''\n\\[\n\\begin{aligned}\n{}\\hl{\\widehat{\\operatorname{Var}}_C(\\widehat{Y_1^T})}={}&{}\\hl{\\sum_{k=1}^{M}T'_k(1-\\pi'_{1k})\\left(\\frac{Y'_k}{\\pi'_{1k}}\\right)^2}+\\hl{\\sum_{k=1}^{M}\\sum_{l\\ne k:\\pi'_{1k1l}>0}\\frac{T'_kT'_l}{\\pi'_{1k1l}}(\\pi'_{1k1l}-\\pi'_{1k}\\pi'_{1l})\\frac{Y'_k}{\\pi'_{1k}}\\frac{Y'_l}{\\pi'_{1l}}}\\\\\n&-\\sum_{k=1}^{M}T'_k\\frac{(Y'_k)^2}{2\\pi'_{1k}}-\\sum_{k=1}^{M}(1-T'_k)\\frac{(Y'_k)^2}{2\\pi'_{0k}},\n\\end{aligned}\n\\]\n``by $\\mathrm{E}[T'_k(1-T'_l)]=\\pi'_{1k0l}$, $\\mathrm{E}[T'_k]=\\pi'_{1k}$, $\\mathrm{E}[1-T'_k]=\\pi'_{0k}$ and eq. [1].''",
    "corrected": "Suppose that $\\pi'_{1k0l}>0$ for all $k\\ne l$. Then ``$\\operatorname{Cov}_C\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)$ may be estimated without bias:''\n\\[\n\\begin{aligned}\n{}\\hl{\\widehat{\\operatorname{Cov}}_C\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)}={}&{}\\hl{\\sum_{k=1}^{M}\\sum_{l\\ne k}\\frac{T'_k(1-T'_l)}{\\pi'_{1k0l}}(\\pi'_{1k0l}-\\pi'_{1k}\\pi'_{0l})\\frac{Y'_kY'_l}{\\pi'_{1k}\\pi'_{0l}}}\\\\\n&-\\sum_{k=1}^{M}T'_k\\frac{(Y'_k)^2}{2\\pi'_{1k}}-\\sum_{k=1}^{M}(1-T'_k)\\frac{(Y'_k)^2}{2\\pi'_{0k}},\n\\end{aligned}\n\\]\n``by $\\mathrm{E}[T'_k(1-T'_l)]=\\pi'_{1k0l}$, $\\mathrm{E}[T'_k]=\\pi'_{1k}$, $\\mathrm{E}[1-T'_k]=\\pi'_{0k}$ and eq. [1].'' This is equation [34]. The rest of the proof stands: $\\mathrm{E}[\\widehat{\\operatorname{Cov}}_C(\\widehat{Y_1^T},\\widehat{Y_0^T})]=\\operatorname{Cov}_C(\\widehat{Y_1^T},\\widehat{Y_0^T})\\le\\operatorname{Cov}(\\widehat{Y_1^T},\\widehat{Y_0^T})$, with equality when $Y'_{0k}=Y'_{1k}$ for all $k$.",
    "evidence": "This correction rests on an exact counterexample to the display as printed and on a derivation for the corrected display.\n\n\\emph{Counterexample.} Let $N=M=4$, with each unit its own cluster, and let two of the four units be treated, each of the six subsets having probability $1/6$. Let $Y'_{1k}=Y'_{0k}$ take the values $1,2,4,7$. Then $\\pi'_{1k}=\\pi'_{0k}=1/2$, and for $k\\ne l$ the joint probabilities are $\\pi'_{1k1l}=\\pi'_{0k0l}=1/6$ and $\\pi'_{1k0l}=1/3$, all positive. If $u$ and $v$ are the two treated outcomes, the printed right side equals $2(u-v)^2-70$, and equation [34] equals the product of the treated and the control outcome sums minus $70$. The printed right side has expectation $-42$. The covariance of the two estimated totals, the bound [43] and the expectation of [34] all equal $-28$.\n\\begin{audittable}{lrrrr}{The six randomizations of the design with four units, each with probability $1/6$, under the sharp null with outcomes $1,2,4,7$.}\nTreated units & $\\widehat{Y_1^T}$ & $\\widehat{Y_0^T}$ & Display as printed & Equation [34] \\\\\n$\\{1,2\\}$ & $6$ & $22$ & $-68$ & $-37$ \\\\\n$\\{1,3\\}$ & $10$ & $18$ & $-52$ & $-25$ \\\\\n$\\{1,4\\}$ & $16$ & $12$ & $2$ & $-22$ \\\\\n$\\{2,3\\}$ & $12$ & $16$ & $-62$ & $-22$ \\\\\n$\\{2,4\\}$ & $18$ & $10$ & $-20$ & $-25$ \\\\\n$\\{3,4\\}$ & $22$ & $6$ & $-52$ & $-37$ \\\\\nExpectation & $14$ & $14$ & $-42$ & $-28$ \\\\\n\\end{audittable}\n\n\\emph{Derivation.} Suppose that $\\pi'_{1k0l}>0$ for all $k\\ne l$. Since $T'_k(1-T'_l)Y'_kY'_l=T'_k(1-T'_l)Y'_{1k}Y'_{0l}$, the cross term of [34] has expectation $(\\pi'_{1k0l}-\\pi'_{1k}\\pi'_{0l})Y'_{1k}Y'_{0l}/(\\pi'_{1k}\\pi'_{0l})$. The two remaining sums have expectations $\\sum_k(Y'_{1k})^2/2$ and $\\sum_k(Y'_{0k})^2/2$. Summing, I obtain $\\mathrm{E}[\\widehat{\\operatorname{Cov}}_C]=\\operatorname{Cov}_C$, the right side of [43]. Equations [30] and [43] imply that\n\\[\n\\operatorname{Cov}\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)-\\operatorname{Cov}_C\\left(\\widehat{Y_1^T},\\widehat{Y_0^T}\\right)=\\frac12\\sum_{k=1}^{M}(Y'_{1k}-Y'_{0k})^2\\ge 0,\n\\]\nwith equality under the sharp null.",
    "problem": "The sentence before the display and the proof after it concern the covariance bound $\\operatorname{Cov}_C$ of equation [43]. The left side of the display is the label of the variance estimator for the treatment total, and its first row has the two leading terms of that estimator, equation [32]. Only the last two sums belong to the covariance estimator. The expectation of the expression as printed is not $\\operatorname{Cov}_C$, and the identity $\\mathrm{E}[T'_k(1-T'_l)]=\\pi'_{1k0l}$ invoked after the display concerns a factor that the display does not contain. The estimator meant is equation [34] of the main text.",
    "what_survives": "Equation [34] is printed correctly in the main text, and equation [35] is built from it and not from this display, so that the misprint does not reach the variance estimator. The identity [30] and the bound [43] are unaffected. In the example of Section 7 the variance of $\\widehat{\\Delta}_{HT}$ and the expectation of [35] both equal $353/1920$, which rounds to the $0.184$ of Table 2. The corrected display has the stated expectation when $\\pi'_{1k0l}>0$ for all $k\\ne l$. The case in which some $\\pi'_{1k0l}$ vanishes is the subject of \\fid{P026-E002}."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-E004",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of Table 2, which has eight columns of estimator properties: IPW, FE, FE (Cov.), RE (Cov.), HT, G (Prior), G (Linear) and G (Logit), and ten rows. Red boxes mark the whole columns FE (Cov.), G (Linear) and G (Logit), whose SE entries are 0.191, 0.162 and 0.170.",
     "caption": "Table 2, p. 149 (PDF p. 15). The three marked columns do not follow from the population of Table 1 and the estimators of Section 7.1; within the two G columns the entries equal to 0.000 stand.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E003.png"
    },
    {
     "alt": "Crop of Table 1, which lists for each of 16 units its block, cluster, two potential outcomes, the covariate X-i and the probability of treatment. A red box marks the column X-i, with the values 4, 0, 4, 1, 4, 2, 4, 1, 2, 5, 4, 1, 4, 2, 2, 3.",
     "caption": "Table 1, p. 147 (PDF p. 13). The marked column is the covariate from which the three columns of Table 2 were recomputed.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E003-b.png"
    },
    {
     "alt": "Dot plot with one row for each of seven estimators and a horizontal axis from 0 to .46 for the standard error of the ATE estimator. For IPW, FE, HT and G (Prior) the open circle for the printed value and the filled diamond for the recomputed value nearly coincide, at about .28, .28, .43 and .30. For FE (Cov.) the printed value is .191 and the recomputed value .279; for G (Linear), .162 and .246; for G (Logit), .170 and .247.",
     "caption": "The SE row of Table 2 as printed and as recomputed from Table 1 over the 90 randomizations, for the seven estimators recomputed. The four columns that do not use $X$ agree with the table; FE (Cov.), G (Linear) and G (Logit) do not. RE (Cov.) is not shown because its recalculation is not settled.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E003-table2-se.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T6",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T6"
    },
    {
     "id": "P026-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T7"
    },
    {
     "id": "P026-T14",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T14"
    },
    {
     "id": "P026-T19",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T19"
    }
   ],
   "headline": "Three columns of Table 2 do not follow from the population printed in Table 1: for that population the standard errors of FE (Cov.), G (Linear) and G (Logit) are .279, .246 and .247, not .191, .162 and .170, and the gain in precision from covariate adjustment is smaller than the table shows.",
   "id": "P026-E003",
   "location": "Table 2, columns FE (Cov.), G (Linear) and G (Logit), p. 149 (PDF p. 15); compare Table 1, column $X_i$, p. 147 (PDF p. 13), and Section 7.1, p. 148 (PDF p. 14)",
   "pdf_page": 15,
   "printed_page": "149",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [
    "P026-T14",
    "P026-T19",
    "P026-T6",
    "P026-T7"
   ],
   "text": {
    "as_printed": "Table 1 gives $X_i=(4,0,4,1,4,2,4,1,2,5,4,1,4,2,2,3)$ for units 1 to 16.\n\\begin{audittable}{lrrr}{Table 2, p.~149: three of the eight columns as printed.}\n & FE (Cov.) & G (Linear) & G (Logit) \\\\\n$\\Delta$ & 0.000 & 0.000 & 0.000 \\\\\n$\\mathrm{E}[\\widehat{\\Delta}]$ & $\\hl{-0.012}$ & 0.000 & 0.000 \\\\\nBias & $\\hl{-0.012}$ & 0.000 & 0.000 \\\\\nSE & \\hl{0.191} & \\hl{0.162} & \\hl{0.170} \\\\\nRMSE & \\hl{0.191} & \\hl{0.162} & \\hl{0.170} \\\\\nVar & \\hl{0.036} & \\hl{0.026} & \\hl{0.029} \\\\\n$\\mathrm{E}[\\widehat{\\operatorname{Var}}]$ & \\hl{0.031} & \\hl{0.026} & \\hl{0.029} \\\\\nBias & $\\hl{-0.005}$ & 0.000 & 0.000 \\\\\nSE & \\hl{0.011} & \\hl{0.007} & \\hl{0.007} \\\\\nRMSE & \\hl{0.012} & \\hl{0.007} & \\hl{0.007} \\\\\n\\end{audittable}",
    "corrected": "For the population printed in Table 1, the three columns of Table 2 read as follows.\n\\begin{audittable}{lrrr}{Table 2, three columns recomputed from Table 1.}\n & FE (Cov.) & G (Linear) & G (Logit) \\\\\n$\\Delta$ & 0.000 & 0.000 & 0.000 \\\\\n$\\mathrm{E}[\\widehat{\\Delta}]$ & $\\hl{-0.014}$ & 0.000 & 0.000 \\\\\nBias & $\\hl{-0.014}$ & 0.000 & 0.000 \\\\\nSE & \\hl{0.279} & \\hl{0.246} & \\hl{0.247} \\\\\nRMSE & \\hl{0.279} & \\hl{0.246} & \\hl{0.247} \\\\\nVar & \\hl{0.078} & \\hl{0.060} & \\hl{0.061} \\\\\n$\\mathrm{E}[\\widehat{\\operatorname{Var}}]$ & \\hl{0.072} & \\hl{0.060} & \\hl{0.061} \\\\\nBias & $\\hl{-0.006}$ & 0.000 & 0.000 \\\\\nSE & \\hl{0.019} & \\hl{0.012} & \\hl{0.013} \\\\\nRMSE & \\hl{0.020} & \\hl{0.012} & \\hl{0.013} \\\\\n\\end{audittable}\nThe \\hl{FE (Cov.)} variance-estimator rows here use the cluster-robust estimator with the factor $G(N-1)/((G-1)(N-K))$, for $G=10$ clusters, $N=16$ units and $K$ regressors, \\hl{including the intercept}, and this convention should be stated. In Section 7.2 the comparison becomes: for this population, G (Linear) and G (Logit) are unbiased and have lower RMSE than IPW, FE and FE (Cov.). The RE (Cov.) column, and the statement that the variance estimators of all regression-based estimators are negatively biased, should be reconciled with the original inputs and settings. If Table 1 and not Table 2 carries the misprint, the correction belongs in the column $X_i$ of Table 1, and it requires the original inputs.",
    "evidence": "This correction rests on a recalculation over all 90 randomizations. For G (Linear) the recalculation is exact. Least squares on the opposite block gives the predictions $1/4+X_i/8$ for the units of block 1 and $70/93+X_i/31$ for the units of block 2. The estimator has mean $0$ and variance $38307/634880\\approx .060337$, and the variance estimator [35] has mean $38307/634880$ and standard error $.012370$. For G (Logit) both logistic fits have unique finite maximum-likelihood estimates, and the moments are computed to high precision. The mean and standard error of FE (Cov.) do not depend on the variance-estimator convention. I computed its variance-estimator rows using the cluster-robust estimator with the finite-sample factor $(10/9)\\cdot 15/(16-K)$, where $K$ is the number of regressors, the intercept included. The paper does not state this factor. With it, the recalculation reproduces the IPW and FE columns. A second, independent calculation agrees with these values.\n\\begin{audittable}{lrrrrrr}{Table 2 as printed and as recomputed from the population of Table 1, for the three columns that differ. Each column of the table is followed by its recomputed values.}\n & FE (Cov.) & recomputed & G (Linear) & recomputed & G (Logit) & recomputed \\\\\n$\\Delta$ & .000 & .000 & .000 & .000 & .000 & .000 \\\\\n$\\mathrm{E}[\\widehat{\\Delta}]$ & $-.012$ & $-.014$ & .000 & .000 & .000 & .000 \\\\\nBias & $-.012$ & $-.014$ & .000 & .000 & .000 & .000 \\\\\nSE & .191 & .279 & .162 & .246 & .170 & .247 \\\\\nRMSE & .191 & .279 & .162 & .246 & .170 & .247 \\\\\nVar & .036 & .078 & .026 & .060 & .029 & .061 \\\\\n$\\mathrm{E}[\\widehat{\\operatorname{Var}}]$ & .031 & .072 & .026 & .060 & .029 & .061 \\\\\nBias & $-.005$ & $-.006$ & .000 & .000 & .000 & .000 \\\\\nSE & .011 & .019 & .007 & .012 & .007 & .013 \\\\\nRMSE & .012 & .020 & .007 & .012 & .007 & .013 \\\\\n\\end{audittable}",
    "problem": "Table 2 reports, for the population of Table 1 and the estimators of Section 7.1, the moments of each estimator over the 90 possible randomizations. When they are recomputed from the printed population, three of the four columns that use the covariate $X$ differ from the table. For G (Linear) the standard error is $.246$ and the variance $.060$, where the table has $.162$ and $.026$. For G (Logit) the standard error is $.247$, and for FE (Cov.) it is $.279$. The variances differ by a factor of about two, which rounding cannot explain. All 40 entries of the four columns that do not use $X$ (IPW, FE, HT and G (Prior)) agree with the table to the third decimal.\n\nThe example overstates the gain from covariate adjustment. Relative to IPW, the standard error of G (Linear) is lower by about 11\\%, and not by the 41\\% that the printed $.162$ and $.276$ imply.\n\nI have not established which of the two tables is misprinted. The commands and inputs behind Table 2 are not among the replication files. If $X$ is changed to $2,3,3,3$ for units 2, 4, 8 and 12, where Table 1 has $0,1,1,1$, the printed columns are nearly reproduced, but one entry is not (the RMSE of the FE (Cov.) variance estimator is $.01255$, which rounds to $.013$ against the printed $.012$), and I have found no evidence that these values were used. The RE (Cov.) column is not settled either. The version of \\texttt{lme4} that the paper cites (0.999375-37) could not be run, and version 1.1-37 does not reproduce the column. With REML the estimator has mean $-.008$ and standard error $.303$, and its variance estimator has bias $+.002$. With maximum likelihood the variance estimator has negative bias. Whether ``all of the regression-based estimators, including variance estimators, are negatively biased'' (Section 7.2) holds for the variance estimator of the random-effects fit remains open.",
    "what_survives": "The four columns that do not use $X$ stand as printed. For the printed population the estimators G (Linear) and G (Logit) remain unbiased, and so do their variance estimators. They still have lower RMSE than IPW, FE and FE (Cov.), by a smaller margin: $.246$ and $.247$ against $.277$, $.283$ and $.279$. They also have lower RMSE than the random-effects fits obtained with version 1.1-37 of \\texttt{lme4}. The comparison with the RE (Cov.) column as printed is not settled. No formula, proof or theoretical result uses the entries of Table 2."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-E003",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of the two-row display that rewrites the Horvitz-Thompson estimator of the treatment total, with the two lines of text above it. A red box marks the fraction 'one over pi-i' in the first sum; the second sum has 'one over pi-1i'.",
     "caption": "Section 6.3, p. 144 (PDF p. 10). In the first sum the marked denominator is $\\pi_i$, a symbol that the paper does not define; the next sum has $\\pi_{1i}$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E005.png"
    },
    {
     "alt": "Crop of footnote 8 with the last line of footnote 7 above it. A red box marks the reference number '[40]' after the name Imai.",
     "caption": "Footnote 8, p. 145 (PDF p. 11). The marked reference number should be 39.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E005-b.png"
    },
    {
     "alt": "Crop of the display for the conservative variance estimator in Appendix D, with the line above it and the four lines of text below it. Red boxes mark the symbols T-k and T-l in the last row, printed without primes; the first row has T-prime-k and T-prime-l.",
     "caption": "Appendix D, p. 152 (PDF p. 18). In the last row the two marked indicators lack the prime that the cluster indicators carry in the first row.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E005-c.png"
    },
    {
     "alt": "Crop of the three rows of equation 24. A red box marks a full stop at the end of the second row, after the fraction Y-0j over pi-0j; the third row continues with minus two times a double sum and ends with the closing parenthesis.",
     "caption": "Equation [24], p. 143 (PDF p. 9). The marked full stop ends the second row although the expression continues on the third.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E005-d.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T10"
    },
    {
     "id": "P026-T11",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T11"
    },
    {
     "id": "P026-T12",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T12"
    }
   ],
   "headline": "Four misprints: $\\pi_i$ should read $\\pi_{1i}$ in the cluster-total display, “Imai [40]” should read “Imai [39]” in footnote 8, $T_k$ and $T_l$ should read $T'_k$ and $T'_l$ in the last row of the Appendix D display, and the full stop inside equation [24] should be deleted.",
   "id": "P026-E005",
   "location": "Section 6.3, display that rewrites $\\widehat{Y^T_{1,HT}}$, p. 144 (PDF p. 10); footnote 8, p. 145 (PDF p. 11); Appendix D, last row of the estimator display, p. 152 (PDF p. 18); equation [24], second row, p. 143 (PDF p. 9)",
   "pdf_page": 10,
   "printed_page": "144",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P026-T10",
    "P026-T11",
    "P026-T12"
   ],
   "text": {
    "as_printed": "(a) p.~144:\n\\[\n\\widehat{Y^T_{1,HT}}=\\sum_{i=1}^{N}\\frac{1}{\\hl{\\pi_i}}T_iY_i=\\sum_{k=1}^{M}\\sum_{i\\in K_k}\\frac{1}{\\pi_{1i}}T_iY_i=\\sum_{k=1}^{M}\\sum_{i\\in K_k}\\frac{1}{\\pi'_{1k}}T'_kY_i=\\sum_{k=1}^{M}\\frac{1}{\\pi'_{1k}}T'_k\\sum_{i\\in K_k}Y_i=\\sum_{k=1}^{M}\\frac{1}{\\pi'_{1k}}T'_kY'_k,\n\\]\n\n(b) p.~145, footnote 8: ``The variance estimators derived in this paper do not reduce to those proposed by Neyman [49], Imai \\hl{[40]} or Middleton and Aronow [5], due to differences in how the covariance term is approximated.''\n\n(c) p.~152, Appendix D, last row of the display for $\\widehat{\\operatorname{Var}}_C(\\widehat{Y_1^T})$:\n\\[\n+\\sum_{k=1}^{M}\\sum_{\\forall l:\\pi'_{1k1l}=0}\\left(\\hl{T_k}\\frac{(Y'_k)^2}{2\\pi'_{1k}}+\\hl{T_l}\\frac{(Y'_l)^2}{2\\pi'_{1l}}\\right),\n\\]\n\n(d) p.~143, second row of equation [24]:\n\\[\n+\\sum_{i=1}^{N}\\pi_{0i}(1-\\pi_{0i})\\left(\\frac{Y_{0i}}{\\pi_{0i}}\\right)^2+\\sum_{i=1}^{N}\\sum_{j\\ne i}(\\pi_{0i0j}-\\pi_{0i}\\pi_{0j})\\frac{Y_{0i}Y_{0j}}{\\pi_{0i}\\pi_{0j}}\\hl{.}\n\\]\nThe third row continues the expression inside the same parenthesis.",
    "corrected": "(a)\n\\[\n\\widehat{Y^T_{1,HT}}=\\sum_{i=1}^{N}\\frac{1}{\\hl{\\pi_{1i}}}T_iY_i=\\sum_{k=1}^{M}\\sum_{i\\in K_k}\\frac{1}{\\pi_{1i}}T_iY_i=\\cdots=\\sum_{k=1}^{M}\\frac{1}{\\pi'_{1k}}T'_kY'_k.\n\\]\n\n(b) ``\\ldots those proposed by Neyman [49], Imai \\hl{[39]} or Middleton and Aronow [5] \\ldots''\n\n(c)\n\\[\n+\\sum_{k=1}^{M}\\sum_{\\forall l:\\pi'_{1k1l}=0}\\left(\\hl{T'_k}\\frac{(Y'_k)^2}{2\\pi'_{1k}}+\\hl{T'_l}\\frac{(Y'_l)^2}{2\\pi'_{1l}}\\right),\n\\]\n\n(d) The second row of [24] ends without a full stop:\n\\[\n+\\sum_{i=1}^{N}\\pi_{0i}(1-\\pi_{0i})\\left(\\frac{Y_{0i}}{\\pi_{0i}}\\right)^2+\\sum_{i=1}^{N}\\sum_{j\\ne i}(\\pi_{0i0j}-\\pi_{0i}\\pi_{0j})\\frac{Y_{0i}}{\\pi_{0i}}\\frac{Y_{0j}}{\\pi_{0j}}\n\\]",
    "evidence": "This correction rests on a check of the pages against the adjacent formulas, the definitions on pp.~138 and 144, equations [6], [26] and [32], and the reference list on pp.~153--154.",
    "problem": "(a) The paper defines $\\pi_{1i}$ and $\\pi_{0i}$ and no $\\pi_i$, and the next expression in the chain and equation [6] have $\\pi_{1i}$. (b) Reference 40 is Hansen and Bowers. Imai's paper on variance identification is reference 39, and p.~137 cites it correctly. (c) The cluster indicators are $T'_k$. The first row of the same display and the corresponding term of equation [32] carry the primes, and the argument that follows uses $\\mathrm{E}[T'_k]=\\pi'_{1k}$. (d) The parenthesis opened in the first row of [24] closes at the end of the third row.",
    "what_survives": "Everything else is unaffected. The cluster-total identity, the bound of Appendix D and equation [24] are correct after these repairs."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-E005",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of five lines of the paragraph on Lin and on Miratrix, Sekhon and Yu. A red box marks the word 'than' in 'a special case of the regression-based estimator than Lin proposes'.",
     "caption": "Paragraph on Lin, p. 137 (PDF p. 3). The marked word should read “that”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E006.png"
    },
    {
     "alt": "Crop of the four lines that end Section 5 before the heading of Section 5.1. A red box marks the word 'auxillary' in 'derived from an auxillary or prior source'.",
     "caption": "Last sentence before Section 5.1, p. 141 (PDF p. 7). The marked word is misspelled.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E006-b.png"
    },
    {
     "alt": "Crop of the heading '6 Sampling variance' and the two lines below it. A red box marks the words 'In the section,' at the start of the first line.",
     "caption": "First sentence of Section 6, p. 142 (PDF p. 8). The marked words should read “In this section”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E006-c.png"
    },
    {
     "alt": "Crop of the reference list from entry 16 to entry 21. Red boxes mark '17, 1995' in the Godambe entry, number 17, and '663-84' in the Horvitz and Thompson entry, number 20.",
     "caption": "References 16 to 21, p. 153 (PDF p. 19). In reference 17 the marked volume number is repeated and the marked year should be 1955; in reference 20 the marked page range should end at 685.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E006-d.png"
    },
    {
     "alt": "Crop of reference entries 36 and 37. A red box marks the word 'post-stratication' in the title of the entry for Miratrix, Sekhon and Yu.",
     "caption": "References 36 and 37, p. 153 (PDF p. 19). The marked word in the title is misspelled.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-E006-e.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Three words in the text and three entries of the reference list are misprinted: “than” should read “that”, “auxillary” should read “auxiliary”, “In the section” should read “In this section”, Godambe’s article is dated 1955 and its volume number is printed twice, Horvitz and Thompson’s article ends on p. 685, and “post-stratication” should read “post-stratification”.",
   "id": "P026-E006",
   "location": "p. 137 (PDF p. 3), paragraph on Lin [9]; p. 141 (PDF p. 7), last sentence before Section 5.1; p. 142 (PDF p. 8), first sentence of Section 6; references 17, 20 and 37, p. 153 (PDF p. 19)",
   "pdf_page": 3,
   "printed_page": "137",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\begin{itemize}\n\\item (a) p.~137: ``(and, in fact, the post-stratification-based estimator may be shown to be a special case of the regression-based estimator \\hl{than} Lin proposes).''\n\\item (b) p.~141: ``The simplest way for this assumption to hold is for $\\theta_i$ to be derived from an \\hl{auxillary} or prior source, but we examine this selection process further in Section 5.1.''\n\\item (c) p.~142: ``In \\hl{the} section, we will provide expressions for the sampling variance of the HT estimator and the generalized difference estimator.''\n\\item (d) p.~153: ``17. Godambe VP. A unified theory of sampling from finite populations. J R Stat Soc Ser B Methodol \\hl{17, 1995};17(2):269--278.''\n\\item (e) p.~153: ``20. Horvitz DG, Thompson DJ. A generalization of sampling without replacement from a finite universe. J Am Stat Assoc 1952;47:663--\\hl{84}.''\n\\item (f) p.~153: ``37. Miratrix L, Sekhon J, Yu B. Adjusting treatment effect estimates by \\hl{post-stratication} in randomized experiments. J R Stat Soc Ser B Methodol, 2013;75(2):369--396.''\n\\end{itemize}",
    "corrected": "\\begin{itemize}\n\\item (a) ``\\ldots a special case of the regression-based estimator \\hl{that} Lin proposes).''\n\\item (b) ``\\ldots derived from an \\hl{auxiliary} or prior source \\ldots''\n\\item (c) ``In \\hl{this} section, \\ldots''\n\\item (d) ``17. Godambe VP. A unified theory of sampling from finite populations. J R Stat Soc Ser B Methodol \\hl{1955};17(2):269--278.''\n\\item (e) ``\\ldots J Am Stat Assoc 1952;47:663--\\hl{85}.''\n\\item (f) ``\\ldots Adjusting treatment effect estimates by \\hl{post-stratification} in randomized experiments \\ldots''\n\\end{itemize}",
    "evidence": "This correction rests on a check of the pages and on the publication records of two cited articles. The table of contents of the journal issue lists Godambe's article as Volume 17, Issue 2, July 1955, pp.~269--278. The JSTOR record of Horvitz and Thompson's article has Vol.~47, No.~260 (Dec., 1952), pp.~663--685. The paper spells ``auxiliary'' and ``post-stratification'' correctly on pp.~142 and 137.",
    "problem": "These are six misprints. In (d) the year should be 1955, and the volume number 17 appears twice. In (e) the article ends on p.~685. The meaning is clear in each case, and each cited work can still be identified.",
    "what_survives": "Everything else is unaffected, since no result depends on these words or entries."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-E006",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the first four lines of the page. A red box marks the inequality 'Var of Y-hat-1-T-star is less than Var of Y-hat-1-T', which follows the words 'will be correlated across randomizations, thus yielding'.",
     "caption": "Top of p. 141 (PDF p. 7), the end of the sentence that begins on p. 140. The marked inequality is stated as a consequence of correlation alone.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-A001.png"
    },
    {
     "alt": "Crop of five lines of the paragraph beginning 'Consider two ways that one might derive theta-i'. A red box marks the words 'may increase the variance of the estimate,'.",
     "caption": "Section 5.1, paragraph after equation [19], p. 142 (PDF p. 8). The marked words are the paper’s own warning that a poor choice of $\\theta_i$ can raise the variance.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-A001-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The intuition on pp. 140–141 passes from correlation to a strict reduction in variance; with predictions fixed across randomizations, the variance falls exactly when $2\\operatorname{Cov}(H,F)>\\operatorname{Var}(F)$ for the outcome total $H$ and the prediction total $F$.",
   "id": "P026-A001",
   "location": "Section 5, sentence after equations [13] and [14], pp. 140–141 (PDF pp. 6–7); compare the warning after equation [19], p. 142 (PDF p. 8)",
   "pdf_page": 6,
   "printed_page": "140–141",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "pp.~140--141: ``In inspecting eq. [13], one intuition is that if $f(\\mathbf{X}_i,\\theta_i)$ predicts the value of $Y_{1i}$, then across units in the study population $f(\\mathbf{X}_i,\\theta_i)$ and $Y_{1i}$ will be correlated. By implication then, $\\sum_{i=1}^{N}T_i\\frac{Y_{1i}}{\\pi_{1i}}$ and $\\sum_{i=1}^{N}T_i\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}$ will be correlated across randomizations, thus yielding $\\widehat{Y_1^{T*}}$ such that $\\hl{\\operatorname{Var}\\left(\\widehat{Y_1^{T*}}\\right)<\\operatorname{Var}\\left(\\widehat{Y_1^{T}}\\right)}$. The same intuition holds for eq. [14], so that both estimators will typically have precision gains.''\n\np.~142: ``The choice of $\\theta_i$ may be suboptimal and, if chosen poorly, may increase the variance of the estimate, but, so long as the analyst does not use the data at hand in forming a judgment, there will be no consequence for bias.''",
    "corrected": "``\\ldots will be correlated across randomizations. \\hl{Let} $H$ \\hl{and} $F$ \\hl{denote these two totals. When the predictions are fixed across randomizations, the adjusted total has smaller variance than the unadjusted total exactly when} $\\hl{2\\operatorname{Cov}(H,F)>\\operatorname{Var}(F)}$. \\hl{Correlation alone does not ensure this, and a poor prediction can increase the variance (see p.~142).} The same holds for eq. [14], and both estimators will typically have precision gains when the predictions are good.''",
    "intended_meaning": "The passage states an intuition and not a theorem: a prediction that tracks the outcome usually lowers the variance. The words ``one intuition'' and ``typically'' qualify it, and p.~142 warns that a poorly chosen $\\theta_i$ may increase the variance. Read in this way the passage is correct. Read literally, the step from ``correlated'' to a strict inequality leaves out a condition. Let\n\\[\nH=\\sum_{i=1}^{N}T_i\\frac{Y_{1i}}{\\pi_{1i}},\\qquad F=\\sum_{i=1}^{N}T_i\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}},\n\\]\nand suppose that the predictions are fixed across randomizations, so that $\\sum_{i=1}^{N}f(\\mathbf{X}_i,\\theta_i)$ is a constant. By equation [13], I have $\\widehat{Y_1^{T*}}=H-F+\\sum_{i=1}^{N}f(\\mathbf{X}_i,\\theta_i)$, and therefore\n\\[\n\\operatorname{Var}\\left(\\widehat{Y_1^{T*}}\\right)-\\operatorname{Var}\\left(\\widehat{Y_1^{T}}\\right)=\\operatorname{Var}(F)-2\\operatorname{Cov}(H,F).\n\\]\nThe variance falls exactly when $2\\operatorname{Cov}(H,F)>\\operatorname{Var}(F)$, and a positive correlation does not ensure this. Suppose that the population has two units, with $Y_{11}=1$ and $Y_{12}=2$, that exactly one of them is treated, each with probability $1/2$, and that the predictions are fixed at $f=3Y_{1i}$. Then $H$ takes the values $2$ and $4$, $F$ takes the values $6$ and $12$, and the correlation of $H$ and $F$ is $1$. A direct computation gives $\\operatorname{Var}(H)=1$, $\\operatorname{Var}(F)=9$ and $\\operatorname{Cov}(H,F)=3$, and the adjusted total takes the values $5$ and $1$, with variance $4$. If the predictions are random, the same identity holds with $F-\\sum_{i=1}^{N}f(\\mathbf{X}_i,\\theta_i)$ in place of $F$. The definitions of the estimators, the results on unbiasedness and the warning on p.~142 are unaffected, and no reported number depends on this sentence."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-A001",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of equation 17 with the two lines of text above it and the words 'and, likewise,' below it. A red box encloses the expression that begins 'E, open square bracket' and runs through the third sum; no closing bracket appears inside the box or after it.",
     "caption": "Equation [17], p. 141 (PDF p. 7). The marked expectation opens a bracket that is not closed before the equals sign.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-N001.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T13",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T13"
    }
   ],
   "headline": "In equation [17] the bracket of the second expectation is not closed; it closes after the third sum.",
   "id": "P026-N001",
   "location": "Section 5.1, equation [17], p. 141 (PDF p. 7)",
   "pdf_page": 7,
   "printed_page": "141",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P026-T13"
   ],
   "text": {
    "as_printed": "\\[\n\\mathrm{E}\\left[\\widehat{Y_1^{T*}}\\right]=\\mathrm{E}\\hl{\\Bigg[}\\sum_{i=1}^{N}T_i\\frac{Y_{1i}}{\\pi_{1i}}-\\sum_{i=1}^{N}T_i\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}+\\sum_{i=1}^{N}f(\\mathbf{X}_i,\\theta_i)=Y_1^T-\\sum_{i=1}^{N}\\operatorname{Cov}\\left(T_i,\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}\\right),\\qquad\\text{[17]}\n\\]",
    "corrected": "\\[\n\\mathrm{E}\\left[\\widehat{Y_1^{T*}}\\right]=\\mathrm{E}\\Bigg[\\sum_{i=1}^{N}T_i\\frac{Y_{1i}}{\\pi_{1i}}-\\sum_{i=1}^{N}T_i\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}+\\sum_{i=1}^{N}f(\\mathbf{X}_i,\\theta_i)\\hl{\\Bigg]}=Y_1^T-\\sum_{i=1}^{N}\\operatorname{Cov}\\left(T_i,\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}\\right).\n\\]",
    "evidence": "This correction rests on a check of the page and on a short derivation. Since $\\mathrm{E}[T_i]=\\pi_{1i}$, I have\n\\[\n\\mathrm{E}\\left[T_i\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}\\right]=\\operatorname{Cov}\\left(T_i,\\frac{f(\\mathbf{X}_i,\\theta_i)}{\\pi_{1i}}\\right)+\\mathrm{E}\\left[f(\\mathbf{X}_i,\\theta_i)\\right],\n\\]\nwhether or not $f(\\mathbf{X}_i,\\theta_i)$ is random. The first sum has expectation $Y_1^T$, and the terms $\\mathrm{E}[f(\\mathbf{X}_i,\\theta_i)]$ cancel against the expectation of the third sum. Summing over the units, I obtain the right side of [17].",
    "problem": "The bracket that opens after the second $\\mathrm{E}$ has no closing bracket, and as printed the expectation appears to extend over the equals sign. The identity is correct when the bracket closes after the third sum.",
    "what_survives": "Everything else is unaffected. The sign and the denominator of the covariance term are correct as printed, and equations [18] and [19] do not change."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-N001",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the five lines that precede equation 31. A red box marks the words 'the same treatment condition.' at the end of the sentence 'This assumption is equivalent to assuming that all pairs of units have nonzero probability of being assigned to the same treatment condition'.",
     "caption": "Section 6.4, first paragraph on p. 146 (PDF p. 12). The marked words paraphrase the assumption stated in symbols in the sentence before.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-N002.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "“The same treatment condition” means each condition: every pair must have positive probability of being jointly treated and positive probability of being jointly in control.",
   "id": "P026-N002",
   "location": "Section 6.4, first paragraph on p. 146 (PDF p. 12), second sentence, before equation [31]",
   "pdf_page": 12,
   "printed_page": "146",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "``First, let us derive an unbiased estimator of $\\operatorname{Var}(\\widehat{Y_1^T})$ under the assumption that, for all pairs $k,l$, $\\pi'_{1k1l}$ and $\\pi'_{0k0l}>0$. This assumption is equivalent to assuming that all pairs of units have nonzero probability of being assigned to \\hl{the same treatment condition}.''",
    "corrected": "``This assumption is equivalent to assuming that every pair of units has positive probability of being \\hl{jointly assigned to treatment and positive probability of being jointly assigned to control}.''",
    "intended_meaning": "The assumption is the one printed in symbols: $\\pi'_{1k1l}>0$ and $\\pi'_{0k0l}>0$ for all pairs. The second sentence restates it loosely. The example that follows it, of pair-randomized studies in which the two units of a pair are never jointly treated, shows that joint assignment to each condition is meant, and equation [31] is unbiased under that assumption. Read literally, the second sentence says only that $\\Pr(T'_k=T'_l)=\\pi'_{1k1l}+\\pi'_{0k0l}>0$, which is weaker. With three clusters of one unit each, exactly one of which is treated, each with probability $1/3$, every pair of distinct clusters has $\\pi'_{0k0l}=1/3$ and $\\pi'_{1k1l}=0$. The pair is then assigned to the same condition with probability $1/3$, and the assumption fails. That the sentence speaks of pairs of units and not of clusters is not an error. Two units in different clusters have the joint probabilities of their clusters, and two units of one cluster are jointly treated and jointly in control with the positive marginal probabilities of that cluster. No formula or result changes."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-N002",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Listing in monospaced type of parts of the help source for the function estate: the description, the usage line, the item for the argument X and the item for the argument HT, with line numbers. Red boxes mark the words 'unbiased Horvitz-Thompson estimation' in the description and the words 'when HT=TRUE, invokes the Horvitz-Thompson (difference-in-totals) estimator' in the item for HT.",
     "caption": "Package ri, file man/estate.Rd, lines 7–12, 20–22 and 29–31, typeset from the source file with line numbers added; a row of dots stands for the lines left out. The marked words call Horvitz–Thompson estimation unbiased and describe HT=TRUE without separating the case in which X is supplied.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-N004.png"
    },
    {
     "alt": "Listing in monospaced type of lines 28 to 36 of the function estate, with line numbers: the branch commented 'HT' and the branch commented 'Raj-Within'. A red box marks line 33, 'Ypre2 is assigned the fitted values of lm of Yd on X'.",
     "caption": "Package ri, file R/estate.R, lines 28–36, typeset from the source file with line numbers added. With HT == TRUE and X supplied, the marked line regresses Y (or Y-Ypre when Ypre is supplied) on X, using all units.",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/exhibits/P026-N004-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P026-T15",
     "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-T15"
    }
   ],
   "headline": "The help page of estate should say that with HT=TRUE and covariates X supplied the function fits a regression using all units, and that the resulting estimator is not unbiased in general.",
   "id": "P026-N004",
   "location": "R package ri (version 0.9), help page of estate (man/estate.Rd, description and argument HT) and R/estate.R, lines 31–36; no passage of the article",
   "pdf_page": null,
   "printed_page": null,
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P026-T15"
   ],
   "text": {
    "as_printed": "Help page of \\texttt{estate}, description: ``Function for estimating the average treatment effect (ATE). Permits regression adjustment for covariates, difference estimation (with a pretreatment measure of the outcome variable), inverse probability weighting, and \\hl{unbiased Horvitz-Thompson estimation}.''\n\nHelp page, argument \\texttt{HT}: ``when \\texttt{HT=TRUE}, \\hl{invokes the Horvitz-Thompson (difference-in-totals) estimator}. When \\texttt{HT=FALSE}, invokes the inverse-probability-weighted regression estimator''\n\nFile \\texttt{R/estate.R}, lines 31 to 36: when \\texttt{HT == TRUE} and \\texttt{X} is supplied, \\texttt{Ypre2} receives the fitted values from regressing \\texttt{Yd} on \\texttt{X} with \\texttt{lm}. The code sets \\texttt{Yd2} to \\texttt{Yd - Ypre2} and computes \\texttt{ate} as the weighted difference of the treatment and control residual totals divided by the number of units. The source-code exhibit below shows these lines verbatim.",
    "corrected": "Description: ``\\ldots inverse probability weighting, and Horvitz-Thompson estimation, \\hl{which is unbiased when X is NULL, the assignment probabilities are correct, and any Ypre adjustment is fixed before randomization}.''\n\nArgument \\texttt{HT}: ``\\hl{When HT=TRUE and X=NULL, computes the Horvitz-Thompson difference of totals divided by N, applied to Y (or Y-Ypre). With the correct assignment probabilities and fixed pretreatment Ypre, this is unbiased for the ATE. When HT=TRUE and X is supplied, first fits a least squares regression of Y (or Y-Ypre when Ypre is supplied) on X using all units and applies the same weighted contrast to its residuals. This within-sample covariate adjustment is not generally unbiased for the ATE. To use externally computed adjustments satisfying the covariance condition of the article, pass their prediction vector as Ypre and leave X NULL.} When \\texttt{HT=FALSE}, invokes the inverse-probability-weighted regression estimator.''\n\nThe code does not change.",
    "evidence": "This correction rests on an exact counterexample, computed in rational arithmetic and reproduced by running the package's code. Suppose that the population has three units, with $X=(0,1,2)$, $Y_{0i}=0$ and $Y_{1i}=1$, so that the average treatment effect is $1$, and that exactly one unit is treated, each with probability $1/3$. With \\texttt{prob} equal to $(1/3,1/3,1/3)$, \\texttt{HT=TRUE} and \\texttt{X} supplied, the function returns $1/4$, $1$ and $1/4$ for the three randomizations. Its expectation is $1/2$ and its bias is $-1/2$. The covariances of $T_i$ with the fitted values are $(1/6,0,1/6)$, and by equation [19] the bias is again $-(1/3)(3+3/2)(1/6+0+1/6)=-1/2$. Without \\texttt{X} the function returns $1$ for every randomization. With \\texttt{Ypre} fixed at $(0,1,2)$ and \\texttt{X=NULL} it returns $5/2$, $1$ and $-1/2$, with expectation $1$.",
    "problem": "With \\texttt{HT=TRUE} and \\texttt{X} supplied, \\texttt{estate} regresses \\texttt{Y} (or \\texttt{Y-Ypre} when \\texttt{Ypre} is supplied) on \\texttt{X}, using all units, including unit $i$, and applies the Horvitz--Thompson contrast to the residuals. This is the generalized difference estimator with $\\theta$ fitted from the data at hand. Section 5.1 of the article says that unbiasedness is then not guaranteed, and equation [19] gives the bias. The help page calls Horvitz--Thompson estimation unbiased and describes \\texttt{HT=TRUE} as invoking the Horvitz--Thompson estimator, and it does not separate the branch that uses \\texttt{X}. A user who combines the two arguments is not told that unbiasedness is no longer guaranteed. The code defines a valid estimator, which adjusts by residuals fitted within the sample. What is missing is the qualification in the help page.",
    "what_survives": "No result of the article depends on this branch. Section 7.1 fits its predictions on the opposite block, and its example imposes the sharp null. The article's results on the Horvitz--Thompson estimator and on the generalized difference estimator under $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$ are not contradicted, since the bias found is the one that equation [19] predicts. With \\texttt{X=NULL}, correct probabilities and a fixed \\texttt{Ypre}, the function is unbiased. Predictions that satisfy the covariance condition can be computed outside the function and passed as \\texttt{Ypre}. Not every call with \\texttt{X} is biased: under the sharp null the fitted values are fixed. The functions \\texttt{gendist} and \\texttt{estlate} pass \\texttt{X} to the same branch."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P026/#P026-N004",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/jci-2012-0009",
    "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P026/"
   },
   "work_id": "P026"
  }
 ],
 "lean": {
  "applicable": true,
  "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P026.md",
  "repository": "https://github.com/pmaronow/llm-audit-lean",
  "targets": [
   {
    "declarations": [
     {
      "claim_revision_id": "catalog:672148ac412163ab64cdc1b4d60f052a5f9ccc8825b30740ff960b40cdd47fdb",
      "conditional_on": [],
      "credit_reason": "The empty realized arms on 00 and 11 are allowed by the printed positive-marginal assumptions; the six-cluster witness additionally handles fixed nonempty arms. The undefined covariance ratio is not silently treated as the literal printed statistic. No claim-stage verdict changed after reading the record.",
      "credit_status": "ok",
      "fidelity_status_baseline": "ok",
      "lean_name": "AronowAudit.P026.asymmetricTwo_counterexample",
      "manifest_use": "used",
      "named_axioms": [],
      "says": null,
      "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P026/P026_FiniteDesign.lean#L251",
      "status": "counterexample_proved",
      "support_scope": "historical declaration",
      "used_in_solution": true
     },
     {
      "claim_revision_id": "catalog:672148ac412163ab64cdc1b4d60f052a5f9ccc8825b30740ff960b40cdd47fdb",
      "conditional_on": [],
      "credit_reason": "This is a witness refuting the printed support and eq. [35] claims, not an assertion that literal eq. [34] has a defined value. The exact marginal/minimum and grouped per-allocation details in the target are supplementary to the component credited to this row. No claim-stage verdict changed after reading the record.",
      "credit_status": "ok",
      "fidelity_status_baseline": "ok",
      "lean_name": "AronowAudit.P026.asymmetricSix_counterexample",
      "manifest_use": "used",
      "named_axioms": [],
      "says": null,
      "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P026/P026_FiniteDesign.lean#L450",
      "status": "counterexample_proved",
      "support_scope": "historical declaration",
      "used_in_solution": true
     }
    ],
    "formalization": {
     "claim_revision_id": "catalog:672148ac412163ab64cdc1b4d60f052a5f9ccc8825b30740ff960b40cdd47fdb",
     "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P026-T1",
     "decision": "whole",
     "release_id": "sap-2026-11-08-assured-release",
     "target_id": "P026-T1"
    },
    "html": {
     "body": "<p>pp. 145-147 (PDF pp. 11-13). The printed claim is false: the uniform sharp-null designs on 00,01,11 and on the fourteen six-bit allocations with three treated and T1 ≤ T2 satisfy positive marginals, positive same-arm joints and distinct cluster indicators, but have Pr(T1=1,T2=0)=0. Literal equation [34] contains 0/0 and is outside its domain. For the two-cluster design the marginals are (1/3,2/3) and (2/3,1/3), both same-arm joints and disagreement are 1/3, HT=(-9/4,0,9/4), and [35]= (57/16,3/2,57/16). Its true variance is 27/8, the expected [35] is 23/8 and the bias is -1/2; [34] with the unavailable term omitted is always -9/4, exceeding the true covariance -13/4 by 1. For the six-cluster design the treatment marginals are (2/7,5/7,1/2,1/2,1/2,1/2), the same-arm minimum is 1/14 and the disagreement minimum is 3/7 at {1,2}. The 00,01,11 groups have sizes 4,6,4, HT values -3/20,0,3/20 and [35] values -619/3600,7/54,-619/3600. The true variance is 9/700, the expected [35] is -269/6300 and the bias is -1/18; the covariance is 27/175 and the zero-omitted [34] has expectation 202/175. Both covariance biases are +1. These facts refute the support deduction, conservativeness and the stated sharp-null equalities. Usual measure-theoretic conditions are assumed and hold automatically for these finite laws.</p>",
     "headline": "Section 6.3, sentence before eq. [30] (&#x27;Since, for all pairs k,l such that k != l, Pr(T&#x27;_k != T&#x27;_l) &gt; 0, then pi&#x27;_{1k0l} &gt; 0 for all l != k&#x27;), and the claims that rest on it: eq. [34] is nonpositively biased, eq. [35] is conservative, and eq. [35] is exactly unbiased under the sharp null; as printed",
     "location": "p. 145 (PDF p. 11); p. 146 (PDF p. 12); p. 147 (PDF p. 13)"
    },
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     "body": "<p>Table 2, p. 149 (PDF p. 15); Table 1, p. 147 (PDF p. 13); Section 7.1, p. 148 (PDF p. 14). For the printed population and ninety equiprobable assignments, the opposite-block lines are the unique OLS fits, and FE (Cov.) is the unique OLS regression on (1,T,block2,X). G (Linear) has mean zero, variance and expected equation [35] variance estimate 38307/634880, and variance-estimator variance 2400328658503/15687787688755200. Its complete corrected column is (0,0,0,.246,.246,.060,.060,0,.012,.012). FE (Cov.) has corrected ATE mean and bias -.014, SE and RMSE .279, and variance .078, with the stated strict mean and variance intervals; G (Linear) has the smaller RMSE. All columns without X are unchanged: HT (0,0,0,.429,.429,.184,.184,0,.037,.037), G (Prior) (0,0,0,.302,.302,.091,.091,0,.020,.020), IPW (0,-.014,-.014,.276,.277,.076,.071,-.005,.017,.018), and FE (0,-.016,-.016,.283,.283,.080,.074,-.006,.019,.020). HT and prior are unbiased, with variance equal to expected equation [35] variance estimate 353/1920 and 233/2560, respectively. Their variance-estimator variances are 10147/7372800 and 20083/52428800. IPW is WLS with inverse observed-arm inclusion weights, and FE is OLS on (1,T,block2); both fits are unique. Their clustered variance estimates use the explicit HC1 convention (10/9)*15/(16-k), k=2,3. All inverses and arm denominators exist on every assignment. The usual measure-theoretic conditions are assumed; here every quantity is a moment of a function on a fixed finite probability space.</p>",
     "headline": "Table 2, columns G (Linear) and FE (Cov.), corrected for the population printed in Table 1",
     "location": "Table 2, p. 149 (PDF p. 15); Table 1, p. 147 (PDF p. 13); Section 7.1, p. 148 (PDF p. 14)"
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     "body": "Table 2, p. 149 (PDF p. 15); Table 1, p. 147 (PDF p. 13); Section 7.1, p. 148 (PDF p. 14). For the printed population and ninety equiprobable assignments, the opposite-block lines are the unique OLS fits, and FE (Cov.) is the unique OLS regression on (1,T,block2,X). G (Linear) has mean zero, variance and expected equation [35] variance estimate 38307/634880, and variance-estimator variance 2400328658503/15687787688755200. Its complete corrected column is (0,0,0,.246,.246,.060,.060,0,.012,.012). FE (Cov.) has corrected ATE mean and bias -.014, SE and RMSE .279, and variance .078, with the stated strict mean and variance intervals; G (Linear) has the smaller RMSE. All columns without X are unchanged: HT (0,0,0,.429,.429,.184,.184,0,.037,.037), G (Prior) (0,0,0,.302,.302,.091,.091,0,.020,.020), IPW (0,-.014,-.014,.276,.277,.076,.071,-.005,.017,.018), and FE (0,-.016,-.016,.283,.283,.080,.074,-.006,.019,.020). HT and prior are unbiased, with variance equal to expected equation [35] variance estimate 353/1920 and 233/2560, respectively. Their variance-estimator variances are 10147/7372800 and 20083/52428800. IPW is WLS with inverse observed-arm inclusion weights, and FE is OLS on (1,T,block2); both fits are unique. Their clustered variance estimates use the explicit HC1 convention (10/9)*15/(16-k), k=2,3. All inverses and arm denominators exist on every assignment. The usual measure-theoretic conditions are assumed; here every quantity is a moment of a function on a fixed finite probability space.",
     "headline": "Table 2, columns G (Linear) and FE (Cov.), corrected for the population printed in Table 1",
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     "body": "Appendix E, p. 152 (PDF p. 18); eq. [34], p. 146 (PDF p. 12). The corrected Appendix E display estimates the covariance lower bound under directed support, with the exact gap and equality characterization.",
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     "body": "<p>Appendix E, p. 152 (PDF p. 18). The printed claim is false: for four singleton clusters with two treated uniformly, and sharp-null outcomes (1,2,4,7), the literal Appendix E display has the six values (-68,-52,2,-62,-20,-52) and expectation -42. Equation [34] has values (-37,-25,-22,-22,-25,-37) and expectation -28, equal to the true covariance and its population bound; the true treatment-total variance and its population bound are both 28. Every marginal is 1/2 and all distinct-cluster joint probabilities are positive (TT=CC=1/6, TC=1/3). Thus the printed display is unbiased for neither of the quantities named in the text and on its left side. Usual measure-theoretic conditions are assumed and hold automatically on this finite probability space.</p>",
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     "body": "Appendix E, p. 152 (PDF p. 18). The printed claim is false: for four singleton clusters with two treated uniformly, and sharp-null outcomes (1,2,4,7), the literal Appendix E display has the six values (-68,-52,2,-62,-20,-52) and expectation -42. Equation [34] has values (-37,-25,-22,-22,-25,-37) and expectation -28, equal to the true covariance and its population bound; the true treatment-total variance and its population bound are both 28. Every marginal is 1/2 and all distinct-cluster joint probabilities are positive (TT=CC=1/6, TC=1/3). Thus the printed display is unbiased for neither of the quantities named in the text and on its left side. Usual measure-theoretic conditions are assumed and hold automatically on this finite probability space.",
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     "body": "<p>p. 144 (PDF p. 10); p. 145 (PDF p. 11). Grouping by the fixed cluster partition preserves each displayed HT total identity and the resulting ATE contrast.</p>",
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     "body": "<p>p. 146 (PDF p. 12); Appendix D, pp. 151-152 (PDF pp. 17-18). The restored cluster-indicator formulas give the exact arm variance and conservative bounds, unbiased estimation of those bounds, and unbiased equation [31] under same-arm support.</p>",
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     "body": "repaired package help, man/estate.Rd lines 8, 30 and R/estate.R lines 4–6, 15, 29, 31–36. The fixed-prediction residual estimator is unbiased, while the all-unit intercept-inclusive OLS residual estimator has the covariance bias formula, a biased example, and zero-covariance and sharp-null sufficient conditions for unbiasedness.",
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     "body": "The printed claim is false: P026-T17, pp. 145–147, footnote 9 p. 147, introduction p. 137, abstract p. 135 (PDF pp. 11–13, 3, 1). The universal support and variance-conservativeness assertion fails, and a sharp-null example with positive same-arm joints separately violates footnote 9's unbiasedness assertion.",
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     "body": "The printed claim is false: P026-T18, Section 5.1 and footnote 5, p. 142 (PDF p. 8). The XOR design supplies individually independent fitting inputs whose deterministic fit has nonzero covariance with the focal treatment indicator, refuting the printed general rule.",
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     "body": "Section 4.1, equations [11]–[12], pp. 139–140 (PDF pp. 5–6). With exactly n of N units treated and equal inclusion probabilities, HT equals the actual difference in unit means on support, and both the displayed contrast and that difference are unbiased. For any partition into M nonempty clusters whose distinct treatment indicators disagree with positive probability, with treatment and control marginals m/M and (M-m)/M, 0 < m < M, HT equals M/N times the difference of observed unit arm totals divided by m and M-m, at every allocation; this contrast is unbiased, without a fixed cluster count. There is a real-valued four-unit example with clusters {1}, {2}, {3,4}, one uniformly treated cluster, distinct cluster indicators, and both unit arms nonempty on support: for sharp-null outcomes (1,1,0,0), the actual difference in means has expectation 1/9, whereas the ATE and expected HT are zero.",
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     "body": "p. 139, Appendix A pp. 149–150 and Appendix C p. 151 (PDF pp. 5, 15–17). HT obeys the affine formula and exact balance characterization with a supported failure example, while identically transforming outcomes and predictions scales the generalized estimator.",
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    "html": {
     "body": "<p>equations [25]–[30], pp. 144–145 (PDF pp. 10–11). The generalized estimator is residual HT, its fixed-prediction variance has the displayed formula, and actual cluster totals obey both arm-variance formulas and the full, split and off-diagonal cross-covariance formulas without a directed cross-support assumption.</p>",
     "headline": "Eqs. [25]-[26] (the generalized difference estimator as the HT estimator applied to U_i, and its sampling variance for constant f) and eqs. [27]-[30] (variance and covariance of the HT totals in terms of cluster totals)",
     "location": "p. 144 (PDF p. 10); p. 145 (PDF p. 11)"
    },
    "id": "P026-T24",
    "text": {
     "body": "equations [25]–[30], pp. 144–145 (PDF pp. 10–11). The generalized estimator is residual HT, its fixed-prediction variance has the displayed formula, and actual cluster totals obey both arm-variance formulas and the full, split and off-diagonal cross-covariance formulas without a directed cross-support assumption.",
     "headline": "Eqs. [25]-[26] (the generalized difference estimator as the HT estimator applied to U_i, and its sampling variance for constant f) and eqs. [27]-[30] (variance and covariance of the HT totals in terms of cluster totals)",
     "location": "p. 144 (PDF p. 10); p. 145 (PDF p. 11)"
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 "missing_sources": [],
 "reproduction": {
  "agreement": "agrees_in_part",
  "status": "selected_recalculation",
  "targets": [
   {
    "agreement": "agrees_in_part",
    "location": "p. 149 (PDF p. 15); estimator definitions p. 148 (PDF p. 14)",
    "method": "I enumerated all 90 assignments of the population in Table 1. All point-estimator entries for IPW, FE, HT and G (Prior) agree to three decimals; FE (Cov.), G (Linear) and G (Logit) differ (P026-E003). RE (Cov.) remains unsettled because the cited lme4 version was unavailable. Changing four covariates nearly reproduces selected printed entries but is not established as the original input.",
    "status": "recalculated",
    "target": "Table 2: ATE estimator properties"
   },
   {
    "agreement": "agrees_in_part",
    "location": "p. 149 (PDF p. 15); variance formula Eq. (35) p. 147 (PDF p. 13)",
    "method": "I enumerated all 90 assignments of the population in Table 1 and computed equation [35] directly. IPW, FE, HT and G (Prior) agree to three decimals; FE (Cov.), G (Linear) and G (Logit) have discrepancies (P026-E003). Regression variance rows use the stated cluster correction, whose finite-sample factor the paper leaves unspecified. RE (Cov.) remains unsettled because the cited lme4 version was unavailable. Alternative covariates leave an FE (Cov.) RMSE of 0.01255, rounding to 0.013 rather than 0.012.",
    "status": "recalculated",
    "target": "Table 2: variance estimator properties"
   }
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 },
 "summary": "The paper asks how to estimate the average treatment effect without bias in randomized experiments of any design, including clustered designs and designs with unequal probabilities. It proposes Horvitz--Thompson estimators, a generalized difference estimator that adjusts for covariates without bias when the predictions are uncorrelated with assignment, and conservative variance estimators. 8 entries are classified as errors (3 Moderate, 5 Minor), and 2 are clarity notes; 3 Moderate and 4 Minor errors are in the article, and 1 Minor error is in the help page of the companion \\texttt{ri} package (\\fid{P026-N004}). One Moderate error is \\fid{P026-E002}: the variance estimator [35] is not conservative for every design, and becomes so when one term is added.",
 "what_was_checked": "I checked the following points, which held up under the conditions stated.\n\\begin{itemize}\n\\item Equations [1]--[8]: the definitions, the decomposition into totals, and the unbiasedness of the Horvitz--Thompson estimators of each total and of the average treatment effect when the assignment probabilities are known and positive.\n\\item Equation [11], that the Horvitz--Thompson estimator is the difference in means under complete randomization of units, and equation [12], that it need not be under complete randomization of clusters. In an exact example with unequal clusters the difference in means is biased and the Horvitz--Thompson estimator is not.\n\\item Equations [13]--[16]: the generalized difference estimator equals $\\widehat{\\Delta}_{HT}$ minus the adjustment term.\n\\item Equations [17]--[19] and Appendix B: the bias identity and the sufficiency of $\\operatorname{Cov}(T_i,f(\\mathbf{X}_i,\\theta_i))=0$, apart from the bracket of \\fid{P026-N001}. The identity [19] agrees with exact enumeration on 1150 random designs in which $f$ depends on the randomization.\n\\item Equations [20]--[30]: the variance expressions, apart from the misprints of \\fid{P026-E005}. Equation [24] equals the enumerated variance on 1150 random designs in exact arithmetic, and [30] holds for every design (see \\fid{P026-E002}).\n\\item Equation [31] is unbiased when all $\\pi'_{1k1l}>0$. The Young bounds behind [32], [33] and Appendix D are correct, and these two estimators are conservative for the arm variances in every design.\n\\item Equation [35] equals $N^{-2}$ times ([32] $+$ [33] $-2\\,$[34]). It is conservative, and exactly unbiased under the sharp null when every pair of clusters has positive probability of joint treatment and of joint control, in each of 936 random designs with $\\pi'_{1k0l}>0$ for all $k\\ne l$, and in the example of Section 7.\n\\item Appendix A (equation [38]) and Appendix C (equation [42]), line by line, under the assumption [41].\n\\item Table 1: 16 units, 10 clusters, two blocks, 90 equally likely randomizations, and probabilities $2/4$ and $2/6$. For distinct clusters, the smallest joint probabilities are $\\pi'_{1k1l}=1/15$, $\\pi'_{0k0l}=1/6$ and $\\pi'_{1k0l}=1/6$.\n\\item Table 2, columns IPW, FE, HT and G (Prior): every entry agrees with a recalculation from Table 1 to the third decimal, with exact variances $353/1920$ for HT and $233/2560$ for G (Prior). Within each column of Table 2 the rows are consistent with one another: the variance is the square of the standard error, and the squared RMSE is the variance plus the squared bias, to rounding.\n\\item For the population of Table 1, G (Linear) and G (Logit) have mean zero, their variance estimators are unbiased, and their RMSE is lower than that of IPW, FE and FE (Cov.) (see \\fid{P026-E003}).\n\\item The cross-references to sections, equations and appendices, and the citation numbers other than the one in footnote 8.\n\\item In the \\texttt{ri} package, \\texttt{genperms} and \\texttt{genprobexact} return the 90 randomizations and the probabilities of Table 1; \\texttt{estate} with \\texttt{HT=TRUE} and no \\texttt{X} computes equation [8], and with \\texttt{Ypre} it computes the difference estimator with fixed $f$; and \\texttt{gendist} reproduces the expectation and SE rows of the IPW, HT and G (Prior) columns of Table 2.\n\\end{itemize}\nThe following were not checked or not settled. I did not read the cited works, among them Aronow and Samii [46], Wood [45], Williams [44] and Middleton and Aronow [5]. Statements attributed to them were checked by derivation or enumeration, and what the paper says about consistency and about the outside literature was not compared with the cited works. The records of Godambe's and of Horvitz and Thompson's articles were consulted for their bibliographic details only. The RE (Cov.) column of Table 2 could not be settled, because the version of \\texttt{lme4} that the paper cites was unavailable for comparison. The available replication files do not include the commands and inputs behind Table 2, and the \\texttt{ri} package does not implement the variance estimator [35], which was computed directly from the equation. That this audit records no further corrections is not a finding that the rest of the paper is correct.",
 "work": {
  "arxiv_id": null,
  "audit_complete": true,
  "authors": [
   "P. M. Aronow",
   "Joel A. Middleton"
  ],
  "category": "Articles - Theory and Methodology",
  "citation": "P. M. Aronow and Joel A. Middleton (2013). A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments. Journal of Causal Inference 1(1):135–154.",
  "doi": "10.1515/jci-2012-0009",
  "edition": {
   "pages_in_audited_file": 20,
   "sha256": "0e4291aa2052b86d09b912b88fb5990d95a2eb50746686ebf30de0b12c3c1e89",
   "source_url": "https://doi.org/10.1515/jci-2012-0009",
   "venue": "Journal of Causal Inference",
   "year": "2013"
  },
  "id": "P026",
  "later_materials": null,
  "title": "A Class of Unbiased Estimators of the Average Treatment Effect in Randomized Experiments",
  "type": "Journal article",
  "url": "https://pmaronow.github.io/llm-audit/works/P026/",
  "venue": "Journal of Causal Inference",
  "year": "2013"
 }
}