{
 "claims": [
  {
   "claim": "Under random allocation of clusters with unequal cluster sizes the difference-in-means estimator is generally biased even though every individual has the same probability of treatment; the bias is a difference of two ratio-estimator covariance terms.",
   "claim_id": "P020-H1",
   "related_corrections": [
    "P020-E002"
   ],
   "tied_to": "Equations (6)-(8)"
  },
  {
   "claim": "Under the paper's replication scheme, the difference-in-means estimator is consistent when independent copies of the original clusters are added, but its bias is unchanged when individuals are copied within a fixed number of clusters.",
   "claim_id": "P020-H2",
   "related_corrections": [
    "P020-E010"
   ],
   "tied_to": "Equations (9)-(10); Figure 1; Section 3.3"
  },
  {
   "claim": "An asymptotically biased estimator is guaranteed to be relatively inefficient for a sufficiently large sample size.",
   "claim_id": "P020-H3",
   "related_corrections": [
    "P020-E001"
   ],
   "tied_to": "Introduction, p. 40; Section 3 opening, p. 42; Section 3.3, equation (10)"
  },
  {
   "claim": "With pair randomization of clusters, the block-fixed-effects regression estimator is a weighted average of block-level differences in means; the paper argues that its bias can persist as pairs are added.",
   "claim_id": "P020-H4",
   "related_corrections": [
    "P020-N008"
   ],
   "tied_to": "Section 3.4; Section 6.5; Conclusion; Figure 4, 'Reg. w/FE' panels"
  },
  {
   "claim": "The Horvitz-Thompson estimator built from cluster totals is unbiased for the ATE but is not location invariant and can be very imprecise.",
   "claim_id": "P020-H5",
   "related_corrections": [],
   "tied_to": "Equations (11)-(15); Appendix A"
  },
  {
   "claim": "The Des Raj difference estimator with a fixed, prespecified $k$, and its covariate extension, are unbiased, including in small samples; estimating $k$ from the same data can introduce bias. For the Des Raj estimator, location invariance uses $k^*=b_0+b_1k$ when outcomes are transformed as $Y^*=b_0+b_1Y$.",
   "claim_id": "P020-H6",
   "related_corrections": [
    "P020-E005",
    "P020-E010"
   ],
   "tied_to": "Equations (20)-(25), (29)-(31); Section 5.4; Appendix B"
  },
  {
   "claim": "The variance-minimizing single coefficient is $k_{\\mathit{optim}*}=(m_t/M)k_{\\mathit{optim}_c}+(m_c/M)k_{\\mathit{optim}_t}$, and the Des Raj estimator is more precise than the Horvitz–Thompson estimator exactly when $k^2<2k\\,k_{\\mathit{optim}*}$; adjustment for cluster size and covariates can improve precision.",
   "claim_id": "P020-H7",
   "related_corrections": [
    "P020-E007",
    "P020-E010"
   ],
   "tied_to": "Equation (26); Section 5.3; Appendix C, equations (32)-(35) and the final inequality; Section 5.4"
  },
  {
   "claim": "Two variance estimators are available: a Neyman-type estimator that is unbiased for an upper bound on the variance, and a 'sharp null' estimator that is exact under no effect and conservative when cluster-level effects are constant or the design is balanced; both extend to independent blocks.",
   "claim_id": "P020-H8",
   "related_corrections": [
    "P020-E009",
    "P020-E010",
    "P020-N001"
   ],
   "tied_to": "Equations (16)–(19), (23); the displays for $V_{\\mathrm{apx}}$, $\\widehat V$ and $V^N$ in Sections 4.2 and 5.1."
  },
  {
   "claim": "In the Green-Vavreck reanalysis the proposed estimators are unbiased and have precision comparable, if not superior, to the regression estimators: under the sharp null the Des Raj estimators are competitive with random-effects regression, and under heterogeneous effects they have the smallest RMSE.",
   "claim_id": "P020-H9",
   "related_corrections": [
    "P020-N008",
    "P020-E003"
   ],
   "tied_to": "Figures 2 and 4; Sections 6.2-6.5"
  },
  {
   "claim": "In the heterogeneous-effect application, the paper reports biases of $-0.2$, $0.9$, $0.9$ and $2.0$ percentage points for the difference-in-means, two fixed-effects regressions and random-effects regression, respectively, against a stated true effect of $0.7$ percentage points.",
   "claim_id": "P020-H10",
   "related_corrections": [
    "P020-N008"
   ],
   "tied_to": "Introduction, p. 41; Section 6.5, p. 66; Figure 4, top row"
  },
  {
   "claim": "Cluster-robust standard errors for the fixed-effects regressions are badly biased downward, the random-effects standard error performs well, and the sharp-null standard errors for the HT and Des Raj estimators are exact under the sharp null and slightly conservative under heterogeneous effects.",
   "claim_id": "P020-H11",
   "related_corrections": [
    "P020-N005"
   ],
   "tied_to": "Figures 3 and 5; Section 6.4, p. 66; Section 6.5, p. 68"
  }
 ],
 "date_published": "2026-10-25",
 "findings": [
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the Introduction paragraph on efficiency. A red box marks the words 'are guaranteed to be relatively inefficient' in the sentence 'Estimators that are asymptotically biased are guaranteed to be relatively inefficient for a sufficiently large sample size'. The lines above read 'researchers often consider efficiency (typically mean square error) to be of paramount importance'.",
     "caption": "Introduction, p. 40 (PDF p. 2). The marked words state the guarantee; two sentences earlier the paragraph takes efficiency to be “typically mean square error”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E001.png"
    },
    {
     "alt": "Crop of the first paragraph of Section 3. A red box marks the words 'hence, asymptotically inefficient' at the end of the sentence 'Third, the derivations will help us to identify conditions under which common estimators are not consistent and, hence, asymptotically inefficient.'",
     "caption": "Section 3, first paragraph, p. 42 (PDF p. 4). The marked words draw the same inference from inconsistency.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E001-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T1"
    },
    {
     "id": "P020-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T2"
    }
   ],
   "headline": "Persistent bias does not guarantee that an estimator eventually has the larger mean squared error; the guarantee holds against a comparator whose mean squared error tends to zero.",
   "id": "P020-E001",
   "location": "Introduction, paragraph beginning “Bias is not the only statistical property”, p. 40 (PDF p. 2); Section 3, first paragraph, p. 42 (PDF p. 4)",
   "pdf_page": 2,
   "printed_page": "40",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P020-T1",
    "P020-T2"
   ],
   "text": {
    "as_printed": "p.~40: ``Estimators that are asymptotically biased \\hl{are guaranteed to be relatively inefficient} for a sufficiently large sample size, and we provide an empirical example where bias is critical in undermining the relative efficiency of common estimators.''\n\np.~42: ``Third, the derivations will help us to identify conditions under which common estimators are not consistent and, \\hl{hence, asymptotically inefficient}.''",
    "corrected": "p.~40: ``\\hl{Persistent asymptotic bias contributes a nonvanishing squared-bias term to mean squared error and can undermine relative efficiency. It guarantees eventual mean-squared-error inferiority to a specified comparator when that comparator's mean squared error tends to zero}.'' The clause on the empirical example is unchanged.\n\np.~42: ``\\ldots\\ conditions under which common estimators are not consistent; \\hl{whether they are relatively inefficient depends on comparative mean squared errors}.''\n\nIn full: let $T_h$ and $U_h$ be estimators of $\\Delta_h$ along a sequence of finite populations indexed by $h$, each with finite second moment, such that the bias of $T_h$ converges to a limit $b\\neq0$ and $\\operatorname{MSE}(U_h)\\to0$ as $h\\to\\infty$. Then $\\operatorname{MSE}(U_h)<\\operatorname{MSE}(T_h)$ for all sufficiently large $h$.",
    "evidence": "The evidence is an exact counterexample, computed by enumerating every assignment in rational arithmetic. Let $M=3$, $m_t=1$ and $m_c=2$, and let the clusters have sizes $1$, $2$ and $4$. The control outcomes of the individuals are $(0)$, $(1,0)$ and $(1,0,0,0)$ in the three clusters, and every treatment outcome is $0$. Then $N=7$ and $\\Delta=-2/7$. Each cluster is the treated one with probability $1/3$. Enumerating the three assignments, I obtain the following values of the estimators.\n\\begin{audittable}{lrrr}{The three-cluster example: the estimate under each of the three equally likely assignments, and the bias and mean squared error of each estimator. The average treatment effect is $-2/7$.}\nTreated cluster & Difference in means & Horvitz--Thompson & Des Raj, $k=2/21$ \\\\\n\\hline\n1 & $-1/3$ & $-3/7$ & $-17/49$ \\\\\n2 & $-1/5$ & $-3/14$ & $-19/98$ \\\\\n3 & $-1/3$ & $-3/14$ & $-31/98$ \\\\\n\\hline\nBias & $-1/315$ & $0$ & $0$ \\\\\nMean squared error & $131/33075\\approx0.00396$ & $1/98\\approx0.01020$ & $3/686\\approx0.00437$ \\\\\n\\end{audittable}\nFor a fixed $k$ the Des Raj estimator of Section 5.1 is unbiased, and I find that its mean squared error is $3/686+(9/14)(k-2/21)^2$. This implies that no fixed $k$ does better than $3/686$. The choice $k=0$ is the Horvitz--Thompson estimator. Now suppose that each individual is copied $h$ times within its cluster, as in the construction on p.~47. The cluster sizes and totals are multiplied by $h$, and $N=7h$. Every entry of the table is then the same for every $h$. I conclude that the difference in means has bias $-1/315$ at every sample size. It is inconsistent, since its error is at least $1/21$ in absolute value under every assignment. Its mean squared error is nevertheless smaller than that of the Horvitz--Thompson estimator and of the Des Raj estimator with any fixed $k$.\n\nThe corrected statement holds because mean squared error is the sum of the variance and the squared bias. Suppose that the bias $b_h$ of $T_h$ tends to $b\\neq0$. Then $\\operatorname{MSE}(T_h)\\ge b_h^2>b^2/4$ for all large $h$. If also $\\operatorname{MSE}(U_h)\\to0$, then $\\operatorname{MSE}(U_h)<b^2/4$ for all large $h$.",
    "problem": "The paragraph on p.~40 measures efficiency by mean squared error. The marked sentence says that an estimator whose bias does not vanish must, in a large enough sample, have a larger mean squared error than the estimator with which it is compared, and the sentence on p.~42 draws the same conclusion from inconsistency. One sufficient condition is that the mean squared error of that comparator tend to zero, which neither sentence assumes. If asymptotic efficiency is defined so as to require consistency, the phrase on p.~42 holds by definition, but that reading gives no ranking by mean squared error and does not support the sentence on p.~40. The asymptotic regimes of the paper include the one in which the number of clusters is fixed and the clusters grow (p.~40 and Section 3.3, p.~47). In that regime the variance of an unbiased estimator need not vanish, and a biased estimator can keep the smaller mean squared error at every sample size.",
    "what_survives": "I found no use of the guarantee in the derivations checked. The bias formula and the replication arguments of Section 3 are unaffected, and so are the results on the Horvitz--Thompson estimator in Section 4. The same holds for the results of Section 5 on the Des Raj estimator: that it is unbiased, and the explicit comparison of its variance with that of the Horvitz--Thompson estimator. The rankings by root mean squared error in Sections 6.4 and 6.5 are computed for the application and do not rest on these sentences. The motivation survives in the weaker form of the next sentence of the paper: bias cannot always be ignored when efficiency is the concern. The counterexample does not show that biased estimators are better in general."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E001",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of equation (8), which gives the bias as minus M over N times the difference of two covariance terms divided by m_t and m_c, and of the lines after it. A red box marks the words 'is biased.' at the end of the sentence 'if the size of the cluster is correlated with the potential outcomes in the cluster, the difference-in-means estimator is biased.' The next paragraph begins 'In some special cases, there will be no bias'.",
     "caption": "Equation (8) and the sentences after it, p. 45 (PDF p. 7). The marked words state as certain a bias that equation (8) makes possible.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E002.png"
    },
    {
     "alt": "Crop of the first lines of the page. A red box marks the words 'the bias arises whenever' in the sentence 'As our derivation will show, the bias arises whenever outcomes are related to cluster size.'",
     "caption": "First lines of p. 44 (PDF p. 6). The marked words announce the same condition before the derivation.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E002-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T11",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T11"
    }
   ],
   "headline": "Read literally, two sentences say that the difference in means is biased whenever cluster size is related to outcomes; the meaning is that such a relation can bias it, and by equation (8) the terms of the two arms can cancel.",
   "id": "P020-E002",
   "location": "Section 3.2, the sentence after equation (8) and the paragraph that follows, p. 45 (PDF p. 7); first sentence of p. 44 (PDF p. 6)",
   "pdf_page": 7,
   "printed_page": "45",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [
    "P020-T11"
   ],
   "text": {
    "as_printed": "p.~44: ``As our derivation will show, \\hl{the bias arises whenever} outcomes are related to cluster size.''\n\np.~45: ``Inspection of this term reveals that, if the size of the cluster is correlated with the potential outcomes in the cluster, the difference-in-means estimator \\hl{is biased}. Moreover, the presence of the terms $1/m_t$ and $1/m_c$ shows that the magnitude (and even the direction) of the bias can depend on the relative number of clusters allocated to treatment and control.\n\nIn some special cases, there will be no bias, such as when the cluster size does not vary or when there is no covariance between cluster size and outcomes.''",
    "corrected": "p.~44: ``\\ldots\\ \\hl{bias can arise when} outcomes are related to cluster size.''\n\np.~45: ``Inspection of this term reveals that, if the size of the cluster is correlated with the potential outcomes in the cluster, the difference-in-means estimator \\hl{can be biased}. \\hl{The terms of the two arms in equation (8) can cancel, and the estimator is unbiased exactly when they are equal.} Moreover, the presence of the terms $1/m_t$ and $1/m_c$ shows that the magnitude (and even the direction) of the bias can depend on the relative number of clusters allocated to treatment and control.\n\nIn some special cases, there will be no bias, such as when the cluster size does not vary or when \\hl{both covariances in equation (8) are zero}.''",
    "intended_meaning": "The passage warns that the difference in means is not unbiased in general when cluster size is related to outcomes. The warning is correct, and equation (8) and the paragraph on special cases state the exact condition. I write $R_1$ and $R_0$ for the mean outcomes of the treated and the control individuals under a given assignment, and $n_t$ and $n_c$ for the numbers of treated and control individuals. Then equation (8) reads\n\\[\n\\mathrm{E}[\\widehat{\\Delta_S}]-\\Delta=-\\frac{M}{N}\\left[\\frac{\\operatorname{Cov}(R_1,n_t)}{m_t}-\\frac{\\operatorname{Cov}(R_0,n_c)}{m_c}\\right],\n\\]\nwhich implies that the bias is zero exactly when the two terms in the bracket are equal. The words ``whenever'' and ``is biased'' claim more than this. For example, let $M=2$ and $m_t=m_c=1$, and suppose that the first cluster has one individual with $Y_0=Y_1=0$ and the second has two individuals with $Y_0=Y_1=1$. The outcomes are perfectly correlated with cluster size, and both covariances equal $1/4$. The estimator takes the values $-1$ and $1$ with probability $1/2$ each, and its expectation is $0=\\Delta$. In a second example the arms have unequal numbers of clusters ($M=3$, $m_t=1$, sizes $1$, $2$, $3$), and the effects are positive and vary with cluster size. There $\\Delta=44/45$, each arm mean has bias $-7/90$, and the difference in means is unbiased. The balanced design under the sharp null in Section 6.4, for which the paper reports no bias, is another case of cancellation.\n\nNote that equation (8) is correct as printed, including its sign, and that the estimator can be biased: with sizes $(1,2)$, control means $(0,1)$ and treatment means $(0,2)$ in the two clusters, its bias is $-1/6$. In the clause ``no covariance between cluster size and outcomes'', the covariances are the two in equation (8). No estimator, derivation or reported number depends on the stronger wording."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E002",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "historical_lean_attribution",
   "exhibits": [
    {
     "alt": "Crop of the logistic regression model P(Y_ij = 1) = 1 - (1 + exp(alpha + tau t_j + beta n_j + phi n_j t_j + sum of gamma_f Gamma_f)) to the power -1 and of the paragraph below it. Red boxes mark 'n_j is the cluster size,' and the sentence 'We estimate tau as 0.4, beta as 1.4 and phi as -0.9.'",
     "caption": "The logistic model of Section 6.2, p. 61 (PDF p. 23). The marked passages define $n_j$ as the cluster size and report the coefficients; with $n_j$ a count of individuals, $0.4-0.9\\,n_j$ is negative for every cluster.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E003.png"
    },
    {
     "alt": "Crop of the last three lines of Section 6.2. A red box marks '= 0.007,' in 'Marginalizing over the imputation process, the ATE is in expectation the sum over i of (p_ti - p_ci) divided by N = 0.007, or 0.7 percentage points.'",
     "caption": "Last sentence of Section 6.2, p. 61 (PDF p. 23). The marked value is the positive average effect that the model is reported to give.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E003-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T3"
    },
    {
     "id": "P020-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T4"
    }
   ],
   "headline": "With $n_j$ the number of individuals in a cluster, the printed coefficients $\\tau=0.4$ and $\\phi=-0.9$ make the modelled effect negative for every individual, against the reported average effect of $+0.007$; the predictor’s units and coefficients should be reconciled.",
   "historical_evidence_type": "lean_proof",
   "id": "P020-E003",
   "location": "Section 6.2, p. 61 (PDF p. 23): the logistic regression model, the sentence that reports the coefficients, and the last sentence of the section",
   "pdf_page": 23,
   "printed_page": "61",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P020-T3",
    "P020-T4"
   ],
   "text": {
    "as_printed": "\\[\nP(Y_{ij}=1)=1-\\left(1+\\exp\\left(\\alpha+\\tau t_j+\\beta \\hl{n_j}+\\phi \\hl{n_j}t_j+\\sum_{f=1}^{F-1}\\gamma_f\\Gamma_f\\right)\\right)^{-1}\n\\]\n``where $t_j$ is a treatment indicator for cluster $j$, $\\hl{n_j}$ \\hl{is the cluster size}, $F$ is the number of blocks (in this case, 40), $\\Gamma_f$ is an indicator variable indicating whether cluster $j$ is in block $f$. The terms $\\alpha$, $\\tau$, $\\beta$, $\\phi$ and $\\gamma_f$ are coefficients estimated from the data using maximum likelihood methods. Note the coefficient $\\phi$ is responsible for the heterogeneous treatment effects. We estimate $\\tau$ as \\hl{0.4}, $\\beta$ as 1.4 and $\\phi$ as $\\hl{-0.9}$.''\n\n``Marginalizing over the imputation process, the ATE is in expectation\n\\[\n\\frac{\\sum_i(p_{ti}-p_{ci})}{N}\\hl{=0.007},\n\\]\nor 0.7 percentage points.''",
    "corrected": "The passage should state the units of the cluster-size predictor and give the coefficients in those units. If the predictor was cluster size in thousands of individuals, the passage reads:\n\\[\nP(Y_{ij}=1)=1-\\left(1+\\exp\\left(\\alpha+\\tau t_j+\\beta \\hl{s_j}+\\phi \\hl{s_j}t_j+\\sum_{f=1}^{F-1}\\gamma_f\\Gamma_f\\right)\\right)^{-1}\n\\]\n``where $t_j$ is a treatment indicator for cluster $j$, $\\hl{s_j=n_j/1000}$ \\hl{is the cluster size in thousands of individuals}, $F$ is the number of blocks (in this case, 40), $\\Gamma_f$ is an indicator variable indicating whether cluster $j$ is in block $f$.'' The coefficients are then as printed: $\\tau=0.4$, $\\beta=1.4$ and $\\phi=-0.9$.\n\nEquivalently, the passage can keep $n_j$ and report $\\beta$ as $0.0014$ and $\\phi$ as $-0.0009$. With this scale the difference $0.4-0.0009\\,n_j$ is positive for clusters of fewer than $4000/9\\approx444.4$ individuals and negative for larger ones, and the contradiction in sign disappears. The scale in thousands is an inference; the specification that was used has not been recovered. The reported value $0.007$ should be checked on the sample of 80 clusters.",
    "evidence": "The evidence is the derivation above, together with a Lean 4 proof of the same statement for the model as printed, which covers every finite nonempty population. Rounding does not change the sign: coefficients that round to $0.4$ and $-0.9$ are at most $0.45$ and $-0.85$, so that $\\tau+\\phi n_j\\le-0.40$ when $n_j\\ge1$. As a check of the scale only, I fitted the same model to all 85 clusters of the public file with cluster size in thousands. The fit gives $\\tau=0.43$, $\\beta=1.43$ and $\\phi=-0.99$, with an average of $p_{ti}-p_{ci}$ equal to $0.010$. This is not the fit to 80 clusters that the paper reports, and it does not reproduce $0.007$.",
    "problem": "The paper defines $n_j$ as the number of individuals in cluster $j$ (p.~44), and Section 6.2 does not rescale it. For an individual in cluster $j$, the difference between the linear predictors under treatment and under control is $\\tau+\\phi n_j=0.4-0.9\\,n_j$, which is at most $-0.5$ when $n_j\\ge1$. Since the printed probability is an increasing function of the linear predictor, I have $p_{ti}<p_{ci}$ for every individual. The average of $p_{ti}-p_{ci}$ is then negative for all values of $\\alpha$, $\\beta$ and $\\gamma_f$, and it cannot equal $0.007$. In the public file of the source study the clusters hold between 30 and 990 voters. A plausible explanation is that cluster size entered the fit in other units, such as thousands of voters, but the paper does not say so, and the scale cannot be recovered from the text.",
    "what_survives": "The results on the estimators and their variances in Sections 2--5 and the appendices do not use the fitted coefficients. The scenario under the sharp null in Section 6.4 and Figures 2 and 3 uses a separate imputation and is unaffected. Section 6.5 and Figures 4 and 5 use the scenario with heterogeneous effects. No result there is shown to change, but the scenario cannot be reproduced from the model as printed. The units that were used and the coefficients of the fit to 80 clusters are not recovered."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E003",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the end of Appendix B: equation (30), the sentence 'So the expected value of the estimator will be', equation (31) and the paragraph that follows. A red box marks the second argument of the second covariance in equation (31), n bar with subscript c j. The first covariance in the same equation has the same argument, unmarked.",
     "caption": "Appendix B, p. 71 (PDF p. 33), equations (30) and (31). The marked term in (31) repeats the control-arm mean $\\overline{n_{cj}}$ of the first covariance.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E005.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T5"
    },
    {
     "id": "P020-T6",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T6"
    }
   ],
   "headline": "In equation (31) the second covariance is with the mean cluster size in the treatment arm, $\\overline{n_{tj}}$, as in the same formula in Section 5.3 (p. 57).",
   "id": "P020-E005",
   "location": "Appendix B, equation (31), p. 71 (PDF p. 33)",
   "pdf_page": 33,
   "printed_page": "71",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P020-T5",
    "P020-T6"
   ],
   "text": {
    "as_printed": "\\[\n\\mathrm{E}\\left[\\frac{\\widehat{Y^T_{1,R1}}-\\widehat{Y^T_{0,R1}}}{N}\\right]=\\Delta+\\frac{M}{N}\\left(\\operatorname{Cov}(\\hat k,\\overline{n_{cj}})-\\operatorname{Cov}(\\hat k,\\overline{n_{\\hl{c}j}})\\right).\\qquad(31)\n\\]",
    "corrected": "Let $M$ clusters of sizes $n_j$, with $N=\\sum_j n_j$, be completely randomized, $m_t$ to treatment and $m_c=M-m_t$ to control. Let $\\hat k$ be an estimator of $k$ computed from the data of the same experiment, so that it is constant within a randomization and can vary across randomizations. Let $\\overline{n_{tj}}$ and $\\overline{n_{cj}}$ be the mean sizes of the clusters in treatment and in control in a given randomization. Then the Des Raj difference estimator of equations (20) and (21), with $\\hat k$ in place of $k$, satisfies\n\\[\n\\mathrm{E}\\left[\\frac{\\widehat{Y^T_{1,R1}}-\\widehat{Y^T_{0,R1}}}{N}\\right]=\\Delta+\\frac{M}{N}\\left(\\operatorname{Cov}(\\hat k,\\overline{n_{cj}})-\\operatorname{Cov}(\\hat k,\\overline{n_{\\hl{t}j}})\\right).\\qquad(31)\n\\]",
    "evidence": "The evidence is a derivation and an exact example. Let $\\overline{n_{tj}}$ and $\\overline{n_{cj}}$ be the mean sizes of the clusters assigned to treatment and to control in a given randomization, so that $\\mathrm{E}[\\overline{n_{tj}}]=\\mathrm{E}[\\overline{n_{cj}}]=N/M$. The estimator $\\hat k$ is constant within a randomization. By equation (21) with $\\hat k$ in place of $k$, I have\n\\[\n\\widehat{Y^T_{1,R1}}=\\frac{M}{m_t}\\sum_{j\\in J_1}Y_j^T-M\\hat k\\Bigl(\\overline{n_{tj}}-\\frac{N}{M}\\Bigr).\n\\]\nTaking expectations, I obtain\n\\[\n\\mathrm{E}\\bigl[\\widehat{Y^T_{1,R1}}\\bigr]=Y_1^T-M\\bigl(\\mathrm{E}[\\hat k\\,\\overline{n_{tj}}]-\\mathrm{E}[\\hat k]\\,\\mathrm{E}[\\overline{n_{tj}}]\\bigr)=Y_1^T-M\\operatorname{Cov}(\\hat k,\\overline{n_{tj}}),\n\\]\nwhich is equation (29). By the same argument for the control arm, I have equation (30), $\\mathrm{E}\\bigl[\\widehat{Y^T_{0,R1}}\\bigr]=Y_0^T-M\\operatorname{Cov}(\\hat k,\\overline{n_{cj}})$. Subtracting (30) from (29) and dividing by $N$, I obtain the corrected (31), since $\\Delta=(Y_1^T-Y_0^T)/N$.\n\nFor the example, take $M=3$ clusters of sizes $(1,2,3)$, so that $N=6$, with one cluster treated. Let the cluster totals be $(0,0,0)$ under control and $(0,0,3)$ under treatment, so that $\\Delta=1/2$, and let $\\hat k$ be the least-squares slope of the observed cluster totals on $n_j$, with an intercept. Over the three assignments, $\\hat k$ takes the values $(0,0,3/2)$ and the estimator takes the values $(0,0,3/8)$. Its expectation is $1/8$, and its bias is $-3/8$. Here $\\operatorname{Cov}(\\hat k,\\overline{n_{cj}})=-1/4$ and $\\operatorname{Cov}(\\hat k,\\overline{n_{tj}})=1/2$, and the bias term of the corrected formula is $(3/6)(-1/4-1/2)=-3/8$. The printed formula gives $0$.",
    "problem": "As printed, the two covariances in (31) are the same, and the term that the next sentence calls the bias is identically zero. Subtracting equation (30) from equation (29), I obtain the treatment-arm mean $\\overline{n_{tj}}$ in the second covariance, and Section 5.3 prints the formula in that form on p.~57. The second subscript $c$ is a misprint.",
    "what_survives": "Equations (29) and (30), their derivation and the bias formula of Section 5.3 are correct as printed. The unbiasedness of the Des Raj estimator for a coefficient $k$ fixed before the analysis is not affected. Only the last display of Appendix B changes."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E005",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "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 last four lines of the page. A red box marks '2 sigma squared of n_j, must be positive,' in the sentence 'Since the second derivative with respect to k, 2 sigma squared of n_j, must be positive, we may set the first derivative equal to zero and solve for k, so that'.",
     "caption": "Section 5.3, last lines of p. 56 (PDF p. 18). The marked words assert that $2\\sigma^2(n_j)$ is positive, which holds when the cluster sizes are not all equal.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E007.png"
    },
    {
     "alt": "Crop of the paragraph beginning 'The chosen value of k will reduce the variability of the Des Raj estimator'. Red boxes mark 'will have better precision than' and, on the next line, 'the HT estimator unless' in the sentence 'In other words, the Des Raj estimator will have better precision than the HT estimator unless the researcher picks a k with the wrong sign or chooses a k that is more than twice the magnitude of k_optim*.' The sentence before gives the intervals 0 < k < 2 k_optim* and 0 > k > 2 k_optim*.",
     "caption": "Section 5.3, p. 57 (PDF p. 19). The strict intervals are printed in the sentence before; the marked summary omits the boundary cases.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E007-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T14",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T14"
    }
   ],
   "headline": "The formulas for the optimal $k$ presuppose that the cluster sizes are not all equal, $\\sigma^2(n_j)>0$, and the informal summary on p. 57 leaves out the cases $k=0$ and $k=2k_{\\mathit{optim}*}$, in which the two variances are equal.",
   "id": "P020-E007",
   "location": "Section 5.3: last paragraph of p. 56 (PDF p. 18), leading to equation (26); paragraph beginning “The chosen value of $k$”, p. 57 (PDF p. 19)",
   "pdf_page": 18,
   "printed_page": "56",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [
    "P020-T14"
   ],
   "text": {
    "as_printed": "pp.~56--57: ``Since the second derivative with respect to $k$, $2\\sigma^2(n_j)$, \\hl{must be positive}, we may set the first derivative equal to zero and solve for $k$, so that'' and then equation (26),\n\\[\nk_{\\mathit{optim}_c}=\\frac{\\sigma(n_j,Y^T_{0j})}{\\sigma^2(n_j)}.\n\\]\n\np.~57: ``The chosen value of $k$ will reduce the variability of the Des Raj estimator, $\\widehat{\\Delta_{R1}}$, relative to the HT estimator when, for $k_{\\mathit{optim}*}>0$, $0<k<2k_{\\mathit{optim}*}$ and, for $k_{\\mathit{optim}*}<0$, $0>k>2k_{\\mathit{optim}*}$. In other words, the Des Raj estimator \\hl{will have better precision than the HT estimator unless} the researcher picks a $k$ with the wrong sign or chooses a $k$ that is more than twice the magnitude of $k_{\\mathit{optim}*}$.''",
    "corrected": "pp.~56--57: ``\\hl{Assume that the cluster sizes are not all equal, so that $\\sigma^2(n_j)>0$.} Since the second derivative with respect to $k$, $2\\sigma^2(n_j)$, \\hl{is then positive}, \\ldots\\ \\hl{If all clusters have the same size, the adjustment vanishes and every $k$ gives the HT estimator.}''\n\np.~57: ``In other words, the Des Raj estimator will have better precision than the HT estimator \\hl{when $k$ lies strictly between $0$ and $2k_{\\mathit{optim}*}$. The two estimators have the same precision at $k=0$ and at $k=2k_{\\mathit{optim}*}$, and when $k_{\\mathit{optim}*}=0$ no nonzero $k$ improves on the HT estimator.}''",
    "intended_meaning": "Section 5 concerns clusters of unequal size, and for these the formulas are correct. I write $\\mathrm{V}(k)$ for the variance of the Des Raj estimator with a fixed $k$, so that $\\mathrm{V}(0)$ is the variance of the Horvitz--Thompson estimator. Let $c=(M-m_c)/m_c$, $t=(M-m_t)/m_t$ and $v=(M-1)N^2/M^2$, as in Appendix C. By the expansion in Appendix C, I have\n\\[\nv\\,[\\mathrm{V}(k)-\\mathrm{V}(0)]=(c+t+2)\\,\\sigma^2(n_j)\\,k\\,(k-2k_{\\mathit{optim}*}),\\qquad k_{\\mathit{optim}*}=\\frac{m_t}{M}k_{\\mathit{optim}_c}+\\frac{m_c}{M}k_{\\mathit{optim}_t}.\n\\]\nWhen $\\sigma^2(n_j)>0$ the variance has a unique minimum at $k_{\\mathit{optim}*}$, and the Des Raj estimator has the smaller variance exactly when $k(k-2k_{\\mathit{optim}*})<0$. These are the strict intervals that the paper prints before the marked sentence. The two variances are equal at $k=0$ and at $k=2k_{\\mathit{optim}*}$, and when $k_{\\mathit{optim}*}=0$ no nonzero $k$ improves on the Horvitz--Thompson estimator. For example, let $M=4$ and $m_t=2$, with sizes $(1,2,3,4)$, control totals $(3,3,5,9)$ and a constant effect of $1$ on the totals. Then $k_{\\mathit{optim}*}=2$, and the variance is $1.28$ at $k=0$, $0.2133$ at $k=2$ and $1.28$ at $k=4$.\n\nWhen all clusters have the same size, $n_j-N/M=0$ for every $j$. The adjustment then vanishes, the Des Raj estimator coincides with the Horvitz--Thompson estimator for every $k$, and the quotients in equation (26) and in Appendix C are $0/0$. The Des Raj estimator remains unbiased in that case. Nothing in the paper depends on the two omissions."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E007",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "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 paragraph beginning 'Recent theoretical results suggest'. Red boxes mark the words 'will have a higher value' and, on the next line, 'than that of the true variance' in the sentence 'These results indicate V-hat-N of Delta-hat-HT will have a higher value than that of the true variance if treatment effects are in fact constant at the cluster scale.' The sentence before says the estimator 'will be conservative relative to the true variance if effects are constant (at the cluster scale) or if the number of clusters is balanced'.",
     "caption": "Section 4.2, p. 52 (PDF p. 14). The marked words describe the estimate itself as higher than the true variance; the property that holds is the one in the sentence before, that the estimator is conservative.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E009.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T16",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T16"
    }
   ],
   "headline": "“Will have a higher value than that of the true variance” is meant in expectation: under the stated conditions the sharp-null variance estimator has nonnegative bias, and the estimate from one assignment can fall below the true variance.",
   "id": "P020-E009",
   "location": "Section 4.2, paragraph beginning “Recent theoretical results suggest”, p. 52 (PDF p. 14)",
   "pdf_page": 14,
   "printed_page": "52",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [
    "P020-T16"
   ],
   "text": {
    "as_printed": "``In general, $\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})$ will be conservative relative to the true variance if effects are constant (at the cluster scale) or if the number of clusters is balanced, in a result that directly follows from theorem 3 of Ding (2014) and Samii and Aronow (2012) (by way of the relationship between pooled and combined variance). These results indicate $\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})$ \\hl{will have a higher value than that of the true variance} if treatment effects are in fact constant at the cluster scale.''",
    "corrected": "``\\hl{Under constant cluster-total treatment effects or balanced cluster counts,} $\\hl{\\mathrm{E}[\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})]\\ge\\mathrm{V}(\\widehat{\\Delta_{HT}})}$; \\hl{the realized estimate need not exceed the true variance}.''",
    "intended_meaning": "On p.~51 the paper calls a variance estimator conservative when its bias is nonnegative. The marked sentence restates that property for the sharp-null variance estimator: its expectation over the assignments is at least the true variance. The property holds. Let $d_j=Y^T_{1j}-Y^T_{0j}$ denote the effect on the total of cluster $j$, with mean $\\overline{d}$ and variance $\\sigma^2(d)$ over the $M$ clusters. Using equation (17) and the closed form of the estimator on p.~52, I obtain\n\\[\n\\mathrm{E}[\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})]-\\mathrm{V}(\\widehat{\\Delta_{HT}})=\\frac{M^2}{N^2(M-1)}\\,\\tau^2\\ge0\n\\]\nwhen $d_j=\\tau$ for every cluster, and\n\\[\n\\mathrm{E}[\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})]-\\mathrm{V}(\\widehat{\\Delta_{HT}})=\\frac{M^2}{N^2(M-1)}\\left[\\overline{d}^{\\,2}+\\frac{M-2}{M-1}\\,\\sigma^2(d)\\right]\\ge0\n\\]\nwhen $m_t=m_c$. Note that the inequality is weak: equality holds under the sharp null. The estimate from a single assignment can be smaller than the true variance. For example, let $M=N=4$ and $m_t=m_c=2$, with control totals $(0,0,2,2)$ and a constant effect of $2$. Then the true variance is $4/3$, and the estimator has mean $8/3$ over the six assignments. When the first two clusters are treated, each of the four observed totals equals $2$ and the estimate is $0$. Here ``constant at the cluster scale'' means a constant effect on the cluster totals. Nothing in the paper depends on the wording."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E009",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the paragraph that sets up growth in cluster size. Red boxes mark 'M units' in 'the original population of M units is exactly copied (h-1) times' and 'equation 9' in 'is defined as the difference-in-means estimator as in equation 9, but now the inner summation takes place across all h n_j units in each cluster'.",
     "caption": "Section 3.3, p. 47 (PDF p. 9), items 1 and 2. The marked words call the $M$ clusters “units” and cite equation 9, a probability limit, for the definition of the estimator.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010.png"
    },
    {
     "alt": "Crop of the display for the variance estimator V-hat of Delta-hat-HT, whose first sum contains Y_j^T minus Y-bar_cj^T, and of the sentence below it. A red box marks the symbol Y-bar_mj^T in 'where Y-bar_mj^T equals the sum over j in J_0 of Y_j^T divided by m_c'.",
     "caption": "The variance estimator and its “where” clause, p. 51 (PDF p. 13), item 3. The display uses the control mean with subscript $cj$; the marked symbol in the definition has subscript $mj$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p51.png"
    },
    {
     "alt": "Crop of the lines that introduce equation (20). A red box marks 'Y_0j^T and Y_0j^T' in 'we first derive our estimates of the study population totals, Y_0j^T and Y_0j^T by differencing off some of the variability'. Equation (20) follows.",
     "caption": "Sentence before equation (20), p. 54 (PDF p. 16), item 4. The marked symbols name the control total of cluster $j$ twice where the two population totals are meant.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p54.png"
    },
    {
     "alt": "Crop of the first paragraph of Section 5.2 and equation (24). A red box marks 'Y_0ij^T and Y_1ij^T' in 'the same transformation must be applied to k as to Y_0ij^T and Y_1ij^T'. Two lines above, the text reads 'When Y_0ij and Y_1ij are linearly transformed'.",
     "caption": "Section 5.2, first paragraph, p. 55 (PDF p. 17), item 5. The marked individual outcomes carry the superscript $T$ of a total; the same outcomes are printed without it two lines above.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p55.png"
    },
    {
     "alt": "Crop of the first three lines of equation (25). In the second line a red box marks U_1j^T inside the sum over j in J_0 of (b_0 N/M + b_1 U_1j^T). The third line reads M over N times the bracket of one over m_t times b_1 times the sum over J_1 of U_1j^T minus one over m_c times b_1 times the sum over J_0 of U_0j^T.",
     "caption": "Equation (25), first three lines, p. 56 (PDF p. 18), item 6. In the second line the marked term in the sum over the control clusters $J_0$ is $U^T_{1j}$; the third line has $U^T_{0j}$ in the same sum.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p56.png"
    },
    {
     "alt": "Crop of the first lines of the page. A red box marks 'for units' in 'n-bar_cj is the mean value of n_j for units in the control condition in a given randomization'. The line above reads 'n-bar_tj is the mean value of n_j for clusters in the treatment condition'.",
     "caption": "First lines of p. 58 (PDF p. 20), item 9. The marked words take the mean of $n_j$ over “units” in the control condition; the parallel clause for the treatment condition says “clusters”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p58a.png"
    },
    {
     "alt": "Crop of the first paragraph of Section 5.4. Red boxes mark X_aij^T in 'A covariates for each individual i in cluster j, denoted by X_aij^T' and Y_j in 'a regression of Y_j on cluster size and cluster-level covariates'. The same paragraph defines the cluster total X_aj^T as the sum over i of X_aij.",
     "caption": "Section 5.4, first paragraph, p. 58 (PDF p. 20), items 7 and 8. The marked individual covariate carries the superscript $T$, and the marked outcome of the regression lacks it.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p58b.png"
    },
    {
     "alt": "Crop of the top of the page. A red box marks 'equation 20' in 'In this scenario, the expected value of equation 20 yields'. The display below begins with the expectation of Y-hat_1,R1^T, written with M over m_t and a sum over j in J_1.",
     "caption": "Appendix B, sentence before equation (29), p. 71 (PDF p. 33), item 10. The marked reference is to equation 20, the estimator of the control total; the display that follows is the expectation of the estimator of the treatment total, equation 21.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p71a.png"
    },
    {
     "alt": "Crop of equation (30) and the sentence after it. A red box marks 'for units' in 'where n-bar_cj is the mean value of n_j for units in the control condition in a given randomization'.",
     "caption": "Appendix B, sentence after equation (30), p. 71 (PDF p. 33), item 9. The marked words repeat “for units” from p. 58.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p71b.png"
    },
    {
     "alt": "Crop of the top of the page and equation (33). A red box marks the first of two identical terms sigma squared of U_j0^T in 'Now note that the terms sigma squared of U_j0^T, sigma squared of U_j0^T, and sigma of U_j0^T, U_j1^T in equation 32 can be written as follows'. Equation (33) below expands sigma squared of U_j1^T.",
     "caption": "Appendix C, sentence before equation (33), p. 72 (PDF p. 34), item 11. The list names $\\sigma^2(U^T_{j0})$ twice; the first, marked, should be $\\sigma^2(U^T_{j1})$, the term expanded in equation (33).",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p72a.png"
    },
    {
     "alt": "Crop of three lines of a display. The first line reads sigma of U_j0^T, U_j1^T equals E[U_j0^T U_j1^T] minus U-bar_0^T U-bar_1^T. In the second line a red box marks 'E[Y_j0^T - k delta_j](Y_j1^T - k delta_j)', where the square bracket closes after the first factor. The third line reads E[Y_j0^T Y_j1^T - Y_j0^T k delta_j - Y_j1^T k delta_j + k squared delta_j squared] minus Y-bar_0^T Y-bar_1^T.",
     "caption": "Appendix C, first three lines of the display that ends in equation (35), p. 72 (PDF p. 34), item 12. In the marked expression the bracket of the expectation closes before the second factor; the next line expands the product inside the expectation.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E010-p72b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T7"
    },
    {
     "id": "P020-T8",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T8"
    },
    {
     "id": "P020-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T9"
    },
    {
     "id": "P020-T15",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T15"
    }
   ],
   "headline": "Local misprints in labels, indices, cross-references and one bracket: among them, the control sum in the second line of equation (25) should contain $U^T_{0j}$, and the expectation in the second line of the display that ends in equation (35) should enclose both factors.",
   "id": "P020-E010",
   "location": "Section 3.3, p. 47 (PDF p. 9); p. 51 (PDF p. 13); p. 54 (PDF p. 16); p. 55 (PDF p. 17); equation (25), p. 56 (PDF p. 18); p. 58 (PDF p. 20); Appendix B, p. 71 (PDF p. 33); Appendix C, p. 72 (PDF p. 34)",
   "pdf_page": 9,
   "printed_page": "47",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P020-T15",
    "P020-T7",
    "P020-T8",
    "P020-T9"
   ],
   "text": {
    "as_printed": "\\begin{enumerate}\n\\item p.~47: ``(1) the original population of \\hl{$M$ units} is exactly copied $(h-1)$ times, but this time the $h$ copies of a cluster are considered part of one supercluster''\n\\item p.~47: ``(3) $\\widehat{\\Delta_S}$ is defined as the difference-in-means estimator as in \\hl{equation 9}, but now the inner summation takes place across all $hn_j$ units in each cluster''\n\\item p.~51: ``where $\\hl{\\overline{Y^T_{mj}}}=\\sum_{j\\in J_0}Y^T_j/m_c$, the mean value of $Y^T_j$ over all $j\\in J_0$''\n\\item p.~54: ``we first derive our estimates of the study population totals, $\\hl{Y^T_{0j}}$ and $\\hl{Y^T_{0j}}$''\n\\item p.~55: ``the same transformation must be applied to $k$ as to $\\hl{Y^T_{0ij}}$ and $\\hl{Y^T_{1ij}}$''\n\\item p.~56, equation (25), second line:\n\\[\n=\\frac{M}{N}\\left[\\frac{1}{m_t}\\sum_{j\\in J_1}(b_0\\cdot N/M+b_1\\cdot U^T_{1j})-\\frac{1}{m_c}\\sum_{j\\in J_0}(b_0\\cdot N/M+b_1\\cdot \\hl{U^T_{1j}})\\right]\n\\]\n\\item p.~58: ``$A$ covariates for each individual $i$ in cluster $j$, denoted by $\\hl{X^T_{aij}}$''\n\\item p.~58: ``with a regression of $\\hl{Y_j}$ on cluster size and cluster-level covariates''\n\\item pp.~58 and~71: ``$\\overline{n_{cj}}$ is the mean value of $n_j$ \\hl{for units} in the control condition in a given randomization''\n\\item p.~71: ``In this scenario, the expected value of \\hl{equation 20} yields'', followed by the expectation of $\\widehat{Y^T_{1,R1}}$\n\\item p.~72: ``Now note that the terms $\\hl{\\sigma^2(U^T_{j0})}$, $\\sigma^2(U^T_{j0})$, and $\\sigma(U^T_{j0},U^T_{j1})$ in equation 32 can be written as follows''\n\\item p.~72, second line of the display that ends in equation (35):\n\\[\n=\\mathrm{E}[Y^T_{j0}-k\\delta_j\\hl{]}(Y^T_{j1}-k\\delta_j)-\\overline{Y^T_0}\\,\\overline{Y^T_1}\n\\]\n\\end{enumerate}",
    "corrected": "\\begin{enumerate}\n\\item p.~47: ``(1) the original population of \\hl{$M$ clusters} is exactly copied $(h-1)$ times''\n\\item p.~47: ``(3) $\\widehat{\\Delta_S}$ is defined as the difference-in-means estimator as in \\hl{equation 6}''\n\\item p.~51: ``where $\\hl{\\overline{Y^T_{cj}}}=\\sum_{j\\in J_0}Y^T_j/m_c$''\n\\item p.~54: ``the study population totals, $\\hl{Y^T_0}$ and $\\hl{Y^T_1}$''\n\\item p.~55: ``the same transformation must be applied to $k$ as to $\\hl{Y_{0ij}}$ and $\\hl{Y_{1ij}}$''\n\\item p.~56, equation (25):\n\\[\n\\begin{aligned}\n\\widehat{\\Delta^*_{R1}}&=\\frac{M}{N}\\left[\\frac{1}{m_t}\\sum_{j\\in J_1}U^{T*}_j-\\frac{1}{m_c}\\sum_{j\\in J_0}U^{T*}_j\\right]\\\\\n&=\\frac{M}{N}\\left[\\frac{1}{m_t}\\sum_{j\\in J_1}(b_0\\cdot N/M+b_1\\cdot U^T_{1j})-\\frac{1}{m_c}\\sum_{j\\in J_0}(b_0\\cdot N/M+b_1\\cdot \\hl{U^T_{0j}})\\right]\\\\\n&=\\frac{M}{N}\\left[\\frac{1}{m_t}b_1\\sum_{j\\in J_1}U^T_{1j}-\\frac{1}{m_c}b_1\\sum_{j\\in J_0}U^T_{0j}\\right]\\\\\n&=b_1\\cdot\\widehat{\\Delta_{R1}}.\n\\end{aligned}\n\\]\n\\item p.~58: ``denoted by $\\hl{X_{aij}}$''\n\\item p.~58: ``with a regression of $\\hl{Y^T_j}$ on cluster size and cluster-level covariates''\n\\item pp.~58 and~71: ``$\\overline{n_{cj}}$ is the mean value of $n_j$ \\hl{for clusters} in the control condition in a given randomization''\n\\item p.~71: ``In this scenario, the expected value of \\hl{equation 21} yields''\n\\item p.~72: ``Now note that the terms $\\hl{\\sigma^2(U^T_{j1})}$, $\\sigma^2(U^T_{j0})$, and $\\sigma(U^T_{j0},U^T_{j1})$ in equation 32 can be written as follows''\n\\item p.~72, with $\\delta_j=n_j-N/M$ and $\\mathrm{E}$ the average over the $M$ clusters:\n\\[\n\\begin{aligned}\n\\sigma(U^T_{j0},U^T_{j1})&=\\mathrm{E}[U^T_{j0}U^T_{j1}]-\\overline{U^T_0}\\,\\overline{U^T_1}\\\\\n&=\\mathrm{E}[\\hl{(}Y^T_{j0}-k\\delta_j\\hl{)}(Y^T_{j1}-k\\delta_j)\\hl{]}-\\overline{Y^T_0}\\,\\overline{Y^T_1}\\\\\n&=\\sigma(Y^T_{j0},Y^T_{j1})-k\\sigma(Y^T_{j0},n_j)-k\\sigma(Y^T_{j1},n_j)+k^2\\sigma^2(n_j).\n\\end{aligned}\n\\]\n\\end{enumerate}",
    "evidence": "The evidence is a check of the pages against the definitions on pp.~44, 46, 48 and 54, with a derivation for the two displayed lines. In equation (25), with $U^T_{0j}$ in the control sum, the constants $b_0\\cdot N/M$ cancel between the two arm means, and $b_1$ times the original estimator remains. For example, let the cluster sizes be $(1,2)$, the control totals $(0,0)$ and the treatment totals $(1,2)$, with $k=0$, $b_0=0$ and $b_1=1$, and let the first cluster be treated. Then the estimate is $2/3$, and the second line as printed evaluates to $-2/3$. In the display that ends in equation (35), $\\mathrm{E}$ is the average over the $M$ clusters and $\\delta_j=n_j-N/M$ has average zero. Expanding $\\mathrm{E}[(Y^T_{j0}-k\\delta_j)(Y^T_{j1}-k\\delta_j)]-\\overline{Y^T_0}\\,\\overline{Y^T_1}$, I obtain $\\sigma(Y^T_{j0},Y^T_{j1})-k\\sigma(Y^T_{j0},n_j)-k\\sigma(Y^T_{j1},n_j)+k^2\\sigma^2(n_j)$, which is equation (35). With $k=0$ and both vectors of totals equal to $(0,2)$, the covariance is $1$. The line as printed instead equals $\\mathrm{E}[Y^T_{j0}]\\,Y^T_{j1}-1=Y^T_{j1}-1$, which is $-1$ for one cluster and $1$ for the other.",
    "problem": "Each item is a slip in a label, an index, a cross-reference or a bracket, and the neighbouring lines show the intended form. In items 6 and 12 a displayed line is false as printed.\n\\begin{enumerate}\n\\item $M$ is the number of clusters (p.~44), and the parallel construction on p.~46 says ``$M$ clusters''.\n\\item Equation 9 is a probability limit. The estimator with an inner summation is defined in equation 6.\n\\item The display above the clause writes the control mean as $\\overline{Y^T_{cj}}$.\n\\item Equations (20) and (21) estimate the population totals $Y^T_0$ and $Y^T_1$. The symbol $Y^T_{0j}$ is the control total of cluster $j$ (p.~48).\n\\item The superscript $T$ marks a total (p.~48). Individual outcomes do not carry it.\n\\item The sentence before the display defines $U^{T*}_{0j}=b_0\\cdot N/M+b_1\\cdot U^T_{0j}$ for the control clusters, and the third line of the display has $U^T_{0j}$ in the sum over $J_0$.\n\\item The individual covariate is $X_{aij}$. The same paragraph defines the total $X^T_{aj}=\\sum_{i=1}^{n_j}X_{aij}$.\n\\item The regression is of the cluster total $Y^T_j$, as on p.~54.\n\\item The mean $\\overline{n_{cj}}$ is taken over the clusters in the control condition. The clause for $\\overline{n_{tj}}$ says ``for clusters in the treatment condition''.\n\\item Equation 20 defines the estimator of the control total. The display that follows is the expectation of the estimator of the treatment total, equation 21.\n\\item Equation 32 has one variance for each arm, and the expansions that follow, ``respectively'', are (33) for $\\sigma^2(U^T_{j1})$, (34) for $\\sigma^2(U^T_{j0})$ and (35) for the covariance.\n\\item The expectation is of the product of the two factors. The next line of the display expands that product inside the expectation.\n\\end{enumerate}\nNote that Appendix C writes the indices in the order $j0$, $j1$, and the main text writes $0j$, $1j$. This is a relabelling and needs no correction.",
    "what_survives": "No result changes. Equation (10), the variance estimator on p.~51, the estimators (20) and (21), the invariance result (25), the bias formulas (29) and (30) and the optimal $k$ of Appendix C hold once the slips are corrected. The last lines of the displays (25) and (35) are correct as printed."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E010",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of the three design-based panels of Figure 2 and its caption. One red box marks the word 'upper-right' in the last sentence of the caption. A second red box marks the labels Bias = 0.0, SD = 9.2, RMSE = 9.2, which sit in the upper left corner of the Horvitz-Thompson panel. The other two panels carry their labels in the upper left as well.",
     "caption": "Figure 2, p. 64 (PDF p. 26), lower row of panels and caption, turned upright. The caption places the labels in the upper right of each plot (the word is marked). They are in the upper left, as the marked labels of the first panel show.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E011.png"
    },
    {
     "alt": "Crop of the three design-based panels of Figure 4 and its caption. One red box marks the word 'upper-right' in the caption. A second red box marks the labels Bias = 0.0, SD = 1.5, RMSE = 1.5 in the upper left corner of the panel 'Raj difference (n j)'.",
     "caption": "Figure 4, p. 67 (PDF p. 29), lower row of panels and caption, turned upright. The same sentence appears (the word “upper-right” is marked), and the labels are again in the upper left, as marked in the second panel.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E011-b.png"
    },
    {
     "alt": "Crop of the three design-based panels of Figure 5 and its caption. A red box marks the last sentence of the caption, 'Distributions for the SEs under the sharp null hypothesis of no treatment effect were too narrow to display.' Each of the three panels above it shows a density curve with a solid and a dotted vertical line.",
     "caption": "Figure 5, p. 69 (PDF p. 31), lower row of panels and caption, turned upright. The marked sentence says that distributions under the sharp null were too narrow to display; Figure 5 shows the scenario with heterogeneous effects, and every panel has its distribution.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-E011-c.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "In Figures 2 and 4 the labels are in the upper left of each panel, not the upper right, and the last sentence of the caption of Figure 5 belongs to Figure 3.",
   "id": "P020-E011",
   "location": "Captions of Figure 2, p. 64 (PDF p. 26), Figure 4, p. 67 (PDF p. 29), and Figure 5, p. 69 (PDF p. 31)",
   "pdf_page": 26,
   "printed_page": "64",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Captions of Figures 2 and 4: ``Bias and SE estimates in the \\hl{upper-right} of each plot are computed from each empirical distribution.''\n\nCaption of Figure 5, last sentence: ``\\hl{Distributions for the SEs under the sharp null hypothesis of no treatment effect were too narrow to display.}''",
    "corrected": "Captions of Figures 2 and 4: ``\\hl{Bias, SD, and RMSE summaries} in the \\hl{upper-left} of each plot are computed from each empirical distribution.''\n\nCaption of Figure 5: the caption ends with ``Bias and SD estimates are computed from each empirical distribution.'' \\del{Distributions for the SEs under the sharp null hypothesis of no treatment effect were too narrow to display.}",
    "evidence": "The evidence is a check of the pages: Figures 2 to 5 were inspected upright, panel by panel.",
    "problem": "Every panel of Figures 2 and 4 carries its labels (Bias, SD and RMSE) in the upper left. Figure 5 shows the scenario with heterogeneous effects and displays all seven distributions of standard errors. Its last sentence repeats a remark from the caption of Figure 3. That figure concerns the sharp-null scenario, and the distributions for its three design-based estimators are omitted.",
    "what_survives": "The panels and their numbers are unaffected. The sentence about the sharp null is true of Figure 3, whose caption already contains it. Naming the three labels in the corrected caption is optional, since ``SE'' is acceptable usage for the standard deviation of a point estimator."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-E011",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of the opening lines of the page. Red boxes mark 'Y_0 i = Y_1i = Y_i', in which the i of the first symbol sits on the baseline, and the two identities 'sigma squared of Y_0j = sigma squared of Y_1j = sigma squared of Y_j' and 'sigma of Y_0j, Y_1j = sigma squared of Y_j', none of which carries a superscript T.",
     "caption": "First lines of p. 52 (PDF p. 14). The marked symbols are the broken subscript in $Y_{0i}$ and the two facts, printed without the superscript $T$ of a cluster total.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N001.png"
    },
    {
     "alt": "Crop of the paragraph. A red box marks 'n_c or n_t' in 'it tends to be more stable than V-hat of Delta-hat-HT, particularly when either n_c or n_t is small, because it combines the variance of the treatment and control groups'.",
     "caption": "Paragraph beginning “The primary benefit of using”, p. 52 (PDF p. 14). The marked symbols are the numbers of individuals in the two arms; the numbers of clusters $m_c$ and $m_t$ are meant.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N001-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P020-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T10"
    }
   ],
   "headline": "On p. 52 the two facts under the sharp null should be written for the cluster totals $Y^T_{0j}$, $Y^T_{1j}$ and $Y^T_j$, the subscript of $Y_{0i}$ is broken, and “$n_c$ or $n_t$” should read “$m_c$ or $m_t$”.",
   "id": "P020-N001",
   "location": "Section 4.2, first two lines of p. 52 (PDF p. 14) and the paragraph beginning “The primary benefit of using”",
   "pdf_page": 14,
   "printed_page": "52",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P020-T10"
   ],
   "text": {
    "as_printed": "``identical: $\\hl{Y_0i}=Y_{1i}=Y_i$. When the sharp null hypothesis of no effect holds, we know two important facts: $\\hl{\\sigma^2(Y_{0j})=\\sigma^2(Y_{1j})=\\sigma^2(Y_j)}$ and $\\hl{\\sigma(Y_{0j},Y_{1j})=\\sigma^2(Y_j)}$.''\n\n``The primary benefit of using $\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})$ is that it tends to be more stable than $\\hat{\\mathrm{V}}(\\widehat{\\Delta_{HT}})$, particularly when either $\\hl{n_c}$ or $\\hl{n_t}$ is small, because it combines the variance of the treatment and control groups.''",
    "corrected": "``identical: $\\hl{Y_{0i}}=Y_{1i}=Y_i$. \\ldots\\ two important facts: $\\sigma^2(\\hl{Y^T_{0j}})=\\sigma^2(\\hl{Y^T_{1j}})=\\sigma^2(\\hl{Y^T_j})$ and $\\sigma(\\hl{Y^T_{0j}},\\hl{Y^T_{1j}})=\\sigma^2(\\hl{Y^T_j})$.''\n\n``The primary benefit of using $\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})$ is that it tends to be more stable than $\\hat{\\mathrm{V}}(\\widehat{\\Delta_{HT}})$, particularly when either $\\hl{m_c}$ or $\\hl{m_t}$ is small, because it combines the variance of the treatment and control groups.''",
    "evidence": "The evidence is a check of the page against the definitions on pp.~44, 48, 50 and 51. The display that follows the two facts on p.~52 writes $\\sigma^2(Y^T_j)$.",
    "problem": "The variances that enter equation (17), and the display that follows these lines on p.~52, are variances of cluster totals: $Y^T_{0j}$ and $Y^T_{1j}$ are defined on p.~48 and their variances on p.~50. Without the superscript the symbols $Y_{0j}$, $Y_{1j}$ and $Y_j$ are not defined. In $Y_0i$ the index $i$ is printed on the baseline. The symbols $n_c$ and $n_t$ denote the numbers of individuals in the control and treatment arms (p.~44), which are random under the assignment of clusters. The estimator $\\hat{\\mathrm{V}}(\\widehat{\\Delta_{HT}})$ has the denominators $m_c(m_c-1)$ and $m_t(m_t-1)$ and is undefined when $m_c$ or $m_t$ is $1$ (p.~51). Its stability therefore depends on the number of clusters in each arm.",
    "what_survives": "Equation (17), the display for $\\mathrm{V}^N(\\widehat{\\Delta_{HT}})$ and the remark on stability are correct with these readings. Under the sharp null the identities hold for the cluster totals as they do for the individual outcomes. Nothing else changes."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-N001",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of the last paragraph of Section 6.1. A red box marks the word 'three' in 'We use randomization inference to examine the behavior of our estimators and compare them with the behavior of three commonly used estimators.'",
     "caption": "Section 6.1, last paragraph, p. 60 (PDF p. 22). The marked word gives the number of comparison estimators as three; Section 6.3 defines four.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N003.png"
    },
    {
     "alt": "Crop of the fixed-effects regression with covariates and the sentence below it. A red box marks 'precinct-level' in 'where beta_2 is the coefficient on precinct-level voter turnout in 2000, beta_3 is the coefficient on age'.",
     "caption": "Section 6.3, p. 62 (PDF p. 24). The marked word places the turnout covariate at the level of the precinct; p. 59 defines it as the turnout rate in the cable district.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N003-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "“Three commonly used estimators” should read “four”, and the turnout covariate is measured for the cable district and is not “precinct-level”.",
   "id": "P020-N003",
   "location": "Section 6.1, last paragraph, p. 60 (PDF p. 22); Section 6.3, p. 62 (PDF p. 24)",
   "pdf_page": 22,
   "printed_page": "60",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "p.~60: ``We use randomization inference to examine the behavior of our estimators and compare them with the behavior of \\hl{three} commonly used estimators.''\n\np.~62: ``where $\\beta_2$ is the coefficient on \\hl{precinct-level} voter turnout in 2000, $\\beta_3$ is the coefficient on age, and the model is fitted with ordinary least squares.''",
    "corrected": "p.~60: ``\\ldots\\ compare them with the behavior of \\hl{four} commonly used estimators.''\n\np.~62: ``where $\\beta_2$ is the coefficient on \\hl{the cable-district turnout rate} in 2000, $\\beta_3$ is the coefficient on age, and the model is fitted with ordinary least squares.''",
    "evidence": "I checked pp.~59--63 and the panels of Figures 2--5.",
    "problem": "Section 6.3 opens with ``We will consider four regression-based estimators as well as the three design-based estimators proposed in this paper'' (p.~61). It defines four regression estimators, and each of Figures 2--5 has four regression panels. The turnout covariate is defined on p.~59 as the ``turnout rate in the cable district in the 2000 election''. The clusters are cable districts, and the turnout covariate is defined at that level.",
    "what_survives": "The estimators that are compared are those of Section 6.3 and the figures, and no number changes. The two covariates are indexed $X_2$, $X_3$ in the regressions (p.~62) and $X^T_1$, $X^T_2$ in the Des Raj model (p.~63). Each model defines its own indices, and this is not an error."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-N003",
   "work": {
    "arxiv_id": null,
    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
   },
   "work_id": "P020"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "recalculation_from_source",
   "exhibits": [
    {
     "alt": "Crop of a paragraph of Section 6.3. Red boxes mark the sentence 'For both fixed effects models, Huber-White robust cluster standard errors are estimated', which runs over two lines. The sentences before it discuss bias from regression adjustment, and the sentence after it says that these standard errors may be unreliable in finite samples.",
     "caption": "Section 6.3, p. 62 (PDF p. 24). The marked sentence is the only statement of a standard-error procedure for the least-squares regressions, and it covers the two fixed-effects models only.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N005.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Section 6.3 states a standard-error procedure for every estimator except the regression without covariates of equation (27); the procedure used for that regression needs to be stated, and ordinary least-squares standard errors cannot explain the figure on whole-cluster subsets of the public data.",
   "id": "P020-N005",
   "location": "Section 6.3, p. 62 (PDF p. 24), paragraph beginning “As Freedman (2008a) notes”; equation (27), p. 61 (PDF p. 23); Figure 3, p. 65 (PDF p. 27)",
   "pdf_page": 24,
   "printed_page": "62",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Section 6.3, p.~61, after equation (27): ``Our estimate of $\\beta_1$ follows from fitting the model with ordinary least squares.''\n\nSection 6.3, p.~62: ``For \\hl{both fixed effects models}, Huber-White ``robust'' cluster standard errors are estimated.''\n\nSection 6.4, pp.~63 and 66: ``In the case of the regression with no covariates (difference-in-means) the standard errors are biased upwards due to the failure of this model to account for blocking.''",
    "corrected": "``For both fixed effects models, Huber-White ``robust'' cluster standard errors are estimated. \\hl{For the regression without covariates, the standard errors are computed by [the procedure that was used].}''\n\nThe added sentence cannot be completed from the article or from the replication files of the source study. For each robust standard error, the description should also give the clustering variable, the covariance formula, the finite-sample correction and the treatment of the blocks.",
    "evidence": "The evidence is a check of the pages and a recalculation from the public data of the source study. Figure 3 reports a bias of 0.53 percentage points for this panel, measured from the standard deviation of the estimator, which Figure 2 reports as 2.4. Then the mean of the plotted standard errors is about 2.93 points. On whole-cluster subsets of the public data, the standard error from ordinary least squares cannot be that large. For a binary outcome regressed on a constant and a treatment indicator, the residual sum of squares is at most $N/4$, so that the residual variance satisfies $s^2\\le N/(4(N-2))$. With $n_t$ and $n_c$ individuals in the two arms, the squared standard error of $\\hat\\beta_1$ is $s^2(1/n_t+1/n_c)$. Each arm consists of 40 whole clusters, and in the public data the 40 smallest of the 85 clusters contain $L=5{,}654$ individuals together, so that each arm has at least $L$ individuals. I have\n\\[\ns^2\\Bigl(\\frac{1}{n_t}+\\frac{1}{n_c}\\Bigr)\\le\\frac{N}{4(N-2)}\\Bigl(\\frac{1}{L}+\\frac{1}{N-L}\\Bigr).\n\\]\nWith $N=22{,}733$ the square root of the right side gives a bound of about $0.767$ percentage points, for every assignment under these assumptions. This rules out ordinary least-squares standard errors for that sample construction. It does not identify the procedure or establish that the authors used the public data unchanged.",
    "problem": "Section 6.3 specifies the standard errors of the two fixed-effects regressions (cluster-robust), of the random-effects regression (produced by \\texttt{lmer()}) and of the three design-based estimators (the square root of the estimated variance under the sharp null). For the first regression, equation (27), it says only that the coefficient is fitted by ordinary least squares. Figures 3 and 5 nevertheless show a distribution of standard errors for this regression, and Section 6.4 interprets its upward bias. The paper does not say which standard error was computed, and the procedure cannot be recovered from the article or from the scripts of the source study, which serve a different analysis.",
    "what_survives": "The point estimators as displayed and the labels printed in the figures stand. No standard-error implementation and no design-based formula is shown to be wrong. The distributions of the regression standard errors and the comparisons of their biases cannot be reproduced exactly."
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   "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-N005",
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    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
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   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
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   "exhibits": [
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     "alt": "Crop of the four regression panels of Figure 4: Reg. (Diff-in-means), Reg. w/FE, Reg. w/FE, cov., and Reg. w/FE, cov, RE. Each panel shows a density curve over an axis from minus 10 to 10 percentage points, a solid vertical line at the mean and a dotted vertical line just right of zero. Red boxes mark the labels Bias = -0.2, Bias = 0.9, Bias = 0.9 and Bias = 2.0. Below them, unmarked, are SD = 2.3, 1.5, 1.5, 1.3 and RMSE = 2.3, 1.8, 1.8, 2.4.",
     "caption": "Figure 4, p. 67 (PDF p. 29), upper row of panels, turned upright. The four marked bias labels agree, to one decimal place, with the solid line minus the dotted line, and the dotted lines stand near 1.0 on the axis, not at 0.7.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N008.png"
    },
    {
     "alt": "Crop of eight lines of Section 6.5. One red box marks '(0.7 percentage points)' at the end of the sentence 'Dotted vertical lines indicate the true treatment effect (0.7 percentage points).' A second red box marks 'is only slightly biased downward' in the sentence about the regression without covariates.",
     "caption": "Section 6.5, p. 66 (PDF p. 28), first two paragraphs. The marked phrases are the true effect that the dotted lines are said to indicate and the description of the difference-in-means that rests on the printed bias of $-0.2$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N008-b.png"
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    {
     "alt": "Chart with one row for each of the seven panels of Figure 4 and a horizontal axis from 0 to 3.3 percentage points. A blue vertical line marks 0.7, the true effect stated in the text. Orange diamonds mark the dotted lines of the four regression panels, all at 1.0. Black circles mark the solid lines: 0.75 for the difference-in-means, 1.90 and 1.92 for the two fixed-effects regressions, 3.02 for the random-effects regression, 0.82 for Horvitz-Thompson and 0.77 and 0.75 for the two Des Raj estimators. Two columns of numbers at the right give the printed bias (minus 0.2, 0.9, 0.9, 2.0, 0.0, 0.0, 0.0) and the mean minus 0.7 (0.05, 1.20, 1.22, 2.32, 0.12, 0.07, 0.05).",
     "caption": "The lines of Figure 4 as read from the published PDF, for all seven panels. In the four regression panels the dotted line (diamond) stands at 1.0 and not at the stated true effect of 0.7 (vertical line). The columns at the right give the printed bias and the plotted mean minus 0.7, in percentage points.",
     "url": "https://pmaronow.github.io/llm-audit/works/P020/exhibits/P020-N008-reference.png"
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   "headline": "In the four regression panels of Figure 4, bias is measured from a dotted line at 1.0 percentage point and not from the stated true effect of 0.7; measured from 0.7, the biases of the fixed-effects and random-effects regressions are about 1.2 and 2.3 points instead of 0.9 and 2.0, and the figure does not establish downward bias of the difference-in-means.",
   "id": "P020-N008",
   "location": "Figure 4, p. 67 (PDF p. 29), the four regression panels; Section 6.5, p. 66 (PDF p. 28), first two paragraphs; Section 6.2, p. 61 (PDF p. 23), last sentence",
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    "as_printed": "Section 6.2, p.~61, last sentence: ``Marginalizing over the imputation process, the ATE is in expectation $\\frac{\\sum_i(p_{ti}-p_{ci})}{N}=0.007$, or 0.7 percentage points.''\n\nSection 6.5, p.~66, first paragraph: ``Figure 4 displays results under treatment effect heterogeneity. Solid vertical lines indicate the mean of the sampling distributions. Dotted vertical lines indicate the true treatment effect (0.7 percentage points).''\n\nSection 6.5, p.~66, second paragraph: ``Interestingly, the regression without covariates (difference-in-means) is only slightly biased \\hl{downward}.''\n\nFigure 4, p.~67, labels of the four regression panels, in percentage points. In each of these panels the dotted line is drawn near 1.0 on the horizontal axis.\n\n\\begin{audittable}{lrrr}{Figure 4 as printed: labels of the regression panels.}\nPanel & Bias & SD & RMSE \\\\\nReg. (Diff-in-means) & $\\hl{-0.2}$ & $2.3$ & $2.3$ \\\\\nReg. w/FE & $\\hl{0.9}$ & $1.5$ & $\\hl{1.8}$ \\\\\nReg. w/FE, cov. & $\\hl{0.9}$ & $1.5$ & $\\hl{1.8}$ \\\\\nReg. w/FE, cov, RE & $\\hl{2.0}$ & $1.3$ & $\\hl{2.4}$ \\\\\n\\end{audittable}",
    "corrected": "In Figure 4, the dotted lines \\hl{should be drawn at the common, unrounded true effect of the scenario}, and \\hl{all empirical bias and RMSE labels should be computed from it}. If that effect is 0.7 percentage points, as stated on pp.~61 and 66, then the published means and standard deviations give the following approximate labels.\n\n\\begin{audittable}{lrrr}{Figure 4 with bias measured from 0.7: approximate labels of the regression panels, in percentage points.}\nPanel & Bias & SD & RMSE \\\\\nReg. (Diff-in-means) & $\\hl{+0.05}$ & $2.3$ & $2.30$ \\\\\nReg. w/FE & $\\hl{1.20}$ & $1.5$ & $\\hl{1.92}$ \\\\\nReg. w/FE, cov. & $\\hl{1.22}$ & $1.5$ & $\\hl{1.93}$ \\\\\nReg. w/FE, cov, RE & $\\hl{2.32}$ & $1.3$ & $\\hl{2.66}$ \\\\\n\\end{audittable}\n\nSection 6.5, second paragraph: ``Interestingly, the regression without covariates (difference-in-means) \\hl{has a mean close to the true effect}.''\n\nExact labels require the unrounded true effect and the original draws. Changing the text to 1.0 has not been checked against the imputation model, and it would leave the labels ``Bias = 0.0'' of the three design-based panels in conflict with their plotted means of 0.82, 0.77 and 0.75.",
    "evidence": "The evidence is a recalculation from the published PDF, in which Figure 4 is stored as vector artwork. The coordinates of the axis ticks, the solid lines and the dotted lines were read from the page and converted to the scale of the axis in exact rational arithmetic. The remaining ticks fit this calibration to within 0.0015 points, and the incompatibility with the stated effect persists with a tolerance of 0.005. A second, independent reading of the coordinates agrees with the first to within 0.002 points.\n\n\\begin{audittable}{lrrrrrr}{Figure 4, regression panels: positions of the solid line (the mean) and of the dotted line, read from the published PDF, and the bias and RMSE that follow if the true effect is 0.7. All entries are in percentage points.}\nPanel & Solid & Dotted & Printed bias & Solid minus 0.7 & Printed RMSE & RMSE at 0.7 \\\\\nReg. (Diff-in-means) & $0.753$ & $1.001$ & $-0.2$ & $+0.053$ & $2.3$ & $2.25$--$2.35$ \\\\\nReg. w/FE & $1.904$ & $1.001$ & $0.9$ & $+1.204$ & $1.8$ & $1.88$--$1.96$ \\\\\nReg. w/FE, cov. & $1.915$ & $0.999$ & $0.9$ & $+1.215$ & $1.8$ & $1.89$--$1.97$ \\\\\nReg. w/FE, cov, RE & $3.024$ & $0.999$ & $2.0$ & $+2.324$ & $2.4$ & $2.64$--$2.69$ \\\\\n\\end{audittable}\n\nThe solid line minus the dotted line is $-0.248$, $0.903$, $0.916$ and $2.025$ in the four panels, and these values round to the printed biases. Suppose that a printed bias $b$ is rounded to one decimal and that the solid line is at $m$. Then a target compatible with the label lies in $[m-b-0.05,\\,m-b+0.05]$. For the four panels these intervals are $[0.903,1.003]$, $[0.954,1.054]$, $[0.965,1.065]$ and $[0.974,1.074]$, and each of them lies above $0.75$. For the RMSE, I use the identity\n\\[\n\\mathrm{E}\\bigl[(T-\\theta)^2\\bigr]=\\operatorname{Var}(T)+\\bigl(\\mathrm{E}[T]-\\theta\\bigr)^2\n\\]\napplied to the empirical distribution of the 5,000 estimates, with $\\theta=0.7$, the plotted mean and a standard deviation within $0.05$ of the printed one. If the standard deviation uses divisor 4,999, its square is multiplied by $4{,}999/5{,}000$ to give the empirical variance. The last column of the table shows the resulting approximate range. The printed $2.3$ of the difference-in-means panel lies in its range, and the printed $1.8$, $1.8$ and $2.4$ lie outside theirs. The last column gives guides rather than exact replacement labels: the plotted positions and the printed standard deviations are rounded, and the original draws are not available.",
    "problem": "Section 6.2 states that the true effect in the scenario with heterogeneous effects is 0.7 percentage points, and Section 6.5 says that the dotted lines of Figure 4 mark it. In the published figure, the dotted lines stand at 1.00 on the axis in the four regression panels, and each printed bias equals the position of the solid line minus the position of the dotted line, to the printed decimal. The three design-based panels show no separate dotted line. The text and the figure cannot both be right, and rounding does not reconcile them: a value printed as 0.007 lies between 0.65 and 0.75 percentage points, and every dotted line lies between 0.99 and 1.01. If the true effect is 0.7, as the text states twice, then the biases of the four regressions are about 0.3 points higher than printed: about $+0.05$ for the difference-in-means, $+1.20$ and $+1.22$ for the two fixed-effects regressions, and $+2.32$ for the random-effects regression. The RMSE labels of the fixed-effects and random-effects panels ($1.8$, $1.8$ and $2.4$) are then too small. From the published means and standard deviations, I obtain about $1.9$, $1.9$ and a value between $2.6$ and $2.7$. The sentence that the difference-in-means is ``only slightly biased downward'' rests on the printed $-0.2$. Measured from 0.7, the plotted mean of that estimator is higher by about 0.05, with a Monte Carlo standard error of about 0.03, so the figure does not establish downward bias.\n\nThe label ``Bias = 0.0'' of the Horvitz--Thompson panel also needs to be recomputed. The mean drawn in that panel is 0.82, which exceeds every target between 0.65 and 0.75 by more than 0.05. If the target is exactly 0.7, the same holds for the Des Raj panel with $n_j$, whose mean is 0.77. These are disagreements between labels and plotted means in a simulation of 5,000 draws, and they do not imply that either design-based estimator is biased. The cause of the mismatch is not established, since the original simulation code and draws are not available.",
    "what_survives": "The inconsistency concerns the reporting of Figure 4 and the downward-bias sentence of Section 6.5. With the published means and standard deviations and a target near 0.7, the biases of the fixed-effects and random-effects regressions remain positive, the random-effects regression still has the smallest standard deviation, and the Horvitz--Thompson estimator remains imprecise. The grouping by RMSE is unchanged: the smallest values belong to the Des Raj estimators, and then come the fixed-effects regressions, the difference-in-means, the random-effects regression and the Horvitz--Thompson estimator. On these numbers, the statement that the Des Raj estimators are the most precise in terms of RMSE stands. The arguments on bias and consistency in Section 3 and the estimators and unbiasedness results of Sections 4 and 5 do not use Figure 4. Figures 2 and 3 come from the separate calculation under the sharp null. Figure 5 concerns standard errors and does not depend on the placement of Figure 4's reference lines. What changes is the size of the regression biases, the RMSE labels of three panels, and the statement that the difference-in-means shows a downward bias in this example; the empirical bias labels of the design-based panels also require checking. The theoretical result that the difference-in-means can be biased is not affected."
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    "doi": "10.1515/spp-2013-0002",
    "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
    "url": "https://pmaronow.github.io/llm-audit/works/P020/"
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     "body": "<p>p. 40 (PDF p. 2); p. 42 (PDF p. 4). The explicit three-cluster population has persistent bias and probability-one error, yet its difference in means beats HT and every fixed-coefficient Des Raj estimator at every positive within-cluster replication.</p>",
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     "body": "p. 40 (PDF p. 2); p. 42 (PDF p. 4). The explicit three-cluster population has persistent bias and probability-one error, yet its difference in means beats HT and every fixed-coefficient Des Raj estimator at every positive within-cluster replication.",
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     "body": "<p>pp. 51-52 (PDF pp. 13-14). The individual sharp null identifies observations with both potentials, gives all corrected total moment identities, and makes actual HT variance equal to its exact observed-total formula.</p>",
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     "body": "<p>pp. 44-45 (PDF pp. 6-7). For every finite-support probability law with denominator positive on its support, the ratio identity holds, and complete cluster randomization gives the exact difference-in-means expectation and bias.</p>",
     "headline": "Equation (7) (ratio identity, Hartley and Ross 1954) and equation (8) (bias of the difference-in-means estimator under random allocation of clusters)",
     "location": "pp. 44-45 (PDF pp. 6-7)"
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    "text": {
     "body": "pp. 44-45 (PDF pp. 6-7). For every finite-support probability law with denominator positive on its support, the ratio identity holds, and complete cluster randomization gives the exact difference-in-means expectation and bias.",
     "headline": "Equation (7) (ratio identity, Hartley and Ross 1954) and equation (8) (bias of the difference-in-means estimator under random allocation of clusters)",
     "location": "pp. 44-45 (PDF pp. 6-7)"
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      "status": "proved",
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     "body": "<p>pp. 48-49 (PDF pp. 10-11); Appendix A, p. 70 (PDF p. 32). HT estimates both potential totals and the ATE without bias, obeys the affine shift identity, and fails location invariance in the stated nonzero-shift cases; an admissible counterexample exists.</p>",
     "headline": "Equations (11)-(13) (Horvitz-Thompson estimator, unbiased) and equations (14)-(15) with Appendix A (not invariant to location shifts)",
     "location": "pp. 48-49 (PDF pp. 10-11); Appendix A, p. 70 (PDF p. 32)"
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    "text": {
     "body": "pp. 48-49 (PDF pp. 10-11); Appendix A, p. 70 (PDF p. 32). HT estimates both potential totals and the ATE without bias, obeys the affine shift identity, and fails location invariance in the stated nonzero-shift cases; an admissible counterexample exists.",
     "headline": "Equations (11)-(13) (Horvitz-Thompson estimator, unbiased) and equations (14)-(15) with Appendix A (not invariant to location shifts)",
     "location": "pp. 48-49 (PDF pp. 10-11); Appendix A, p. 70 (PDF p. 32)"
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     "body": "<p>p. 54 (PDF p. 16), equations (20)-(22) and the following display; pp. 58-59 (PDF pp. 20-21), Section 5.4. Under complete random allocation of fixed positive numbers of clusters to both arms, the Des Raj estimators with any fixed common size coefficient, and with any finite collection of fixed covariates and coefficients, unbiasedly estimate both potential-outcome totals and their difference divided by the population size. Equation (22) averages the selected cluster or arm mean: the fixed coefficient factors out of expectation, the centered arm mean has expectation zero, and so does its fixed multiple. The combined size and covariate arm-mean adjustment also has expectation zero. Both adjusted arm-total estimators have the same expectations as HT. Covariate totals and all coefficients are fixed before analysis; zero covariates and zero size or covariate variance are allowed. The usual measure-theoretic conditions are assumed; here all expectations are finite design sums.</p>",
     "headline": "Equations (20)-(22) and the display that follows (Des Raj difference estimator, unbiased for fixed k); Section 5.4 estimators <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><msubsup><mi>Y</mi><mrow><mn>0</mn><mo separator=\"true\">,</mo><mi>R</mi><mn>2</mn></mrow><mi>T</mi></msubsup><mo stretchy=\"true\">^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\widehat{Y^T_{0,R2}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.4928em;vertical-align:-0.4296em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0631em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8231em;\"><span style=\"top:-2.4065em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">0</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0077em;\">R</span><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.0448em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4296em;\"><span></span></span></span></span></span></span></span><span class=\"svg-align\" style=\"top:-3.8231em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4296em;\"><span></span></span></span></span></span></span></span></span>, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><msubsup><mi>Y</mi><mrow><mn>1</mn><mo separator=\"true\">,</mo><mi>R</mi><mn>2</mn></mrow><mi>T</mi></msubsup><mo stretchy=\"true\">^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\widehat{Y^T_{1,R2}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.4928em;vertical-align:-0.4296em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0631em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8231em;\"><span style=\"top:-2.4065em;margin-left:-0.2222em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0077em;\">R</span><span class=\"mord mtight\">2</span></span></span></span><span style=\"top:-3.0448em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4296em;\"><span></span></span></span></span></span></span></span><span class=\"svg-align\" style=\"top:-3.8231em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.4296em;\"><span></span></span></span></span></span></span></span></span>, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><msub><mi mathvariant=\"normal\">Δ</mi><mrow><mi>R</mi><mn>2</mn></mrow></msub><mo stretchy=\"true\">^</mo></mover></mrow><annotation encoding=\"application/x-tex\">\\widehat{\\Delta_{R2}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.0733em;vertical-align:-0.15em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">Δ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0077em;\">R</span><span class=\"mord mtight\">2</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span> (covariate version, unbiased)",
     "location": "p. 54 (PDF p. 16); pp. 58-59 (PDF pp. 20-21)"
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    "text": {
     "body": "p. 54 (PDF p. 16), equations (20)-(22) and the following display; pp. 58-59 (PDF pp. 20-21), Section 5.4. Under complete random allocation of fixed positive numbers of clusters to both arms, the Des Raj estimators with any fixed common size coefficient, and with any finite collection of fixed covariates and coefficients, unbiasedly estimate both potential-outcome totals and their difference divided by the population size. Equation (22) averages the selected cluster or arm mean: the fixed coefficient factors out of expectation, the centered arm mean has expectation zero, and so does its fixed multiple. The combined size and covariate arm-mean adjustment also has expectation zero. Both adjusted arm-total estimators have the same expectations as HT. Covariate totals and all coefficients are fixed before analysis; zero covariates and zero size or covariate variance are allowed. The usual measure-theoretic conditions are assumed; here all expectations are finite design sums.",
     "headline": "Equations (20)-(22) and the display that follows (Des Raj difference estimator, unbiased for fixed k); Section 5.4 estimators $\\widehat{Y^T_{0,R2}}$, $\\widehat{Y^T_{1,R2}}$, $\\widehat{\\Delta_{R2}}$ (covariate version, unbiased)",
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     "headline": "Equations (16)-(17), <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi mathvariant=\"normal\">V</mi><mrow><mi>a</mi><mi>p</mi><mi>x</mi></mrow></msub></mrow><annotation encoding=\"application/x-tex\">\\mathrm{V}_{apx}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathrm\" style=\"margin-right:0.0139em;\">V</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:-0.0139em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">a</span><span class=\"mord mathnormal mtight\">p</span><span class=\"mord mathnormal mtight\">x</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span></span></span> and the estimator <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi mathvariant=\"normal\">V</mi><mo>^</mo></mover><mo stretchy=\"false\">(</mo><mover accent=\"true\"><msub><mi mathvariant=\"normal\">Δ</mi><mrow><mi>H</mi><mi>T</mi></mrow></msub><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\hat{\\mathrm{V}}(\\widehat{\\Delta_{HT}})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathrm\" style=\"margin-right:0.0139em;\">V</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord katex-accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">Δ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0813em;\">H</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span> (Section 4.2); equations (18)-(19) (block designs, Section 4.3); equation (23) (Des Raj version, Section 5.1)",
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     "body": "pp. 50-51 (PDF pp. 12-13); p. 53 (PDF p. 15); p. 55 (PDF p. 17). The exact HT variance components and upper bound, two-clusters-per-arm conservative estimators for HT and fixed-coefficient Des Raj, and independent-block unbiasedness, variance and conservative estimation formulas all hold.",
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     "body": "<p>p. 52 (PDF p. 14); explicit form on p. 52 and, for the Des Raj estimator, p. 55 (PDF p. 17). The sharp-null variance estimator is conservative in expectation when cluster-total effects are constant or arm cluster counts are balanced.</p>",
     "headline": "Section 4.2, unnumbered statement that the sharp-null variance estimator <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mover accent=\"true\"><mi mathvariant=\"normal\">V</mi><mo>^</mo></mover><mi>N</mi></msup><mo stretchy=\"false\">(</mo><mover accent=\"true\"><msub><mi mathvariant=\"normal\">Δ</mi><mrow><mi>H</mi><mi>T</mi></mrow></msub><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1968em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9468em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathrm\" style=\"margin-right:0.0139em;\">V</span></span><span style=\"top:-3.2523em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N</span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord katex-accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord\">Δ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.0813em;\">H</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span> is conservative if effects are constant at the cluster scale or if the number of clusters is balanced",
     "location": "p. 52 (PDF p. 14); explicit form on p. 52 and, for the Des Raj estimator, p. 55 (PDF p. 17)"
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     "headline": "Section 4.2, unnumbered statement that the sharp-null variance estimator $\\hat{\\mathrm{V}}^N(\\widehat{\\Delta_{HT}})$ is conservative if effects are constant at the cluster scale or if the number of clusters is balanced",
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     "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P020-T17",
     "decision": "whole",
     "release_id": "sap-2026-11-08-assured-release",
     "target_id": "P020-T17"
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    "html": {
     "body": "<p>pp. 45-47 (PDF pp. 7-9); Figure 1 on p. 46 (PDF p. 8). Consistency means convergence in probability along populations whose total and both realized arm counts strictly increase. Independent exact copies have a common path realization with the stipulated uniform prefix laws, all growth certificates, all four weak laws, positive denominators and pooled consistency. Within-cluster replication with a common draw has the same strict growth but preserves the target, estimates, law and bias; nonzero bias implies inconsistency. The sizes (1,2,4), control totals (0,1,1), treatment totals (0,0,0) give N_h=7h, ATE=-2/7, bias=-1/315 and error probability one at tolerance 1/25 for every h.</p>",
     "headline": "Section 3.3, equation (9) (consistency of the difference-in-means estimator as the number of clusters grows) and equation (10) (the estimate is unchanged when cluster sizes grow with the number of clusters fixed)",
     "location": "pp. 45-47 (PDF pp. 7-9); Figure 1 on p. 46 (PDF p. 8)"
    },
    "id": "P020-T17",
    "text": {
     "body": "pp. 45-47 (PDF pp. 7-9); Figure 1 on p. 46 (PDF p. 8). Consistency means convergence in probability along populations whose total and both realized arm counts strictly increase. Independent exact copies have a common path realization with the stipulated uniform prefix laws, all growth certificates, all four weak laws, positive denominators and pooled consistency. Within-cluster replication with a common draw has the same strict growth but preserves the target, estimates, law and bias; nonzero bias implies inconsistency. The sizes (1,2,4), control totals (0,1,1), treatment totals (0,0,0) give N_h=7h, ATE=-2/7, bias=-1/315 and error probability one at tolerance 1/25 for every h.",
     "headline": "Section 3.3, equation (9) (consistency of the difference-in-means estimator as the number of clusters grows) and equation (10) (the estimate is unchanged when cluster sizes grow with the number of clusters fixed)",
     "location": "pp. 45-47 (PDF pp. 7-9); Figure 1 on p. 46 (PDF p. 8)"
    },
    "url": "https://pmaronow.github.io/llm-audit/works/P020/#P020-T17"
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 "missing_sources": [],
 "reproduction": {
  "agreement": "agrees_in_part",
  "status": "selected_recalculation",
  "targets": [
   {
    "agreement": "not_compared",
    "location": "p. 59 (PDF p. 21)",
    "method": "I checked the public source-study file: its 23,869 voters, 85 clusters and 40 blocks agree exactly. The restricted analysis sample of 22,733 voters in 80 clusters remains unavailable and was not compared.",
    "status": "blocked",
    "target": "Section 6 analysis sample"
   },
   {
    "agreement": "not_compared",
    "location": "p. 61 (PDF p. 23)",
    "method": "I inspected the printed model, but I could not regenerate the fit or expected average effect for the paper's 80-cluster sample without the exact sample, prior-turnout covariate and simulation code. With cluster size counted in individuals, the printed coefficients make the modelled effect negative for every individual, so the model as printed cannot give the reported +0.007 (P020-E003). As a scale check, I fitted the model to all 85 public clusters with cluster size in thousands; the average modelled effect is 0.010. This does not reproduce the reported 80-cluster fit.",
    "status": "blocked",
    "target": "Section 6.2 heterogeneous-effect imputation"
   },
   {
    "agreement": "to_rounding",
    "location": "p. 64 (PDF p. 26)",
    "method": "I recalculated RMSE from the printed bias and SD in all seven panels. The labels are compatible after allowing each input its rounding interval. This compares printed arithmetic only.",
    "status": "recalculated",
    "target": "Figure 2"
   },
   {
    "agreement": "to_rounding",
    "location": "p. 65 (PDF p. 27)",
    "method": "I inspected the printed labels against the plotted lines; they agree at the resolution of the figure. This comparison concerns the figure itself, not regenerated sampling distributions. I could not identify the standard-error procedure for the regression without covariates. Ordinary least-squares standard errors cannot produce that panel on whole-cluster subsets of the public data (P020-N005); this does not establish which procedure was used.",
    "status": "inspected_only",
    "target": "Figure 3"
   },
   {
    "agreement": "disagrees",
    "location": "p. 67 (PDF p. 29)",
    "method": "I recalculated the regression reference-line positions, biases and RMSE ranges from the published PDF coordinates and rounded labels, using the stated effect of 0.7 percentage points. The reference lines near 1.0 and several labels disagree (P020-N008). Separately, all seven printed bias/SD/RMSE triples are arithmetically compatible after rounding.",
    "status": "recalculated",
    "target": "Figure 4"
   },
   {
    "agreement": "to_rounding",
    "location": "p. 69 (PDF p. 31)",
    "method": "I inspected the printed labels against the plotted lines; they agree at the resolution of the figure. This comparison concerns the figure itself, not regenerated sampling distributions.",
    "status": "inspected_only",
    "target": "Figure 5"
   }
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 "summary": "The paper studies unbiased estimation of the average treatment effect when clusters of unequal size are randomized. It shows that the difference in means can be biased even as clusters grow, proposes Horvitz--Thompson and Des Raj estimators with variance estimators, and reanalyzes a voter-turnout experiment. 9 entries are classified as errors (1 Moderate, 8 Minor), and 3 are clarity notes. The main finding is \\fid{P020-N008}: the regression panels of Figure 4 measure bias from 1.0 percentage point, not the stated 0.7.",
 "what_was_checked": "I checked the following results.\n\\begin{itemize}\n\\item Sections 2 and 3: equations (1)--(6). I derived the ratio identity (7) and the bias formula (8) again, and the formula agrees with the bias obtained by enumerating every assignment in small examples. The estimator is consistent when clusters are replicated, equation (9), and the estimate is unchanged when individuals are replicated within a fixed number of clusters, equation (10). The wording around equation (8) is the subject of \\fid{P020-E002}.\n\\item Sections 3.4, 6.5 and 7: with pairs of one treated and one control cluster, the weights of the fixed-effects estimator are fixed, and its bias can persist as pairs are added.\n\\item Section 4 and Appendix A: the Horvitz--Thompson estimators (11)--(13) are unbiased. Equations (14) and (15) and every step of Appendix A hold, as do equation (16), both lines of equation (17) and the expression $\\mathrm{V}_{\\mathrm{apx}}$, for which $\\hat{\\mathrm{V}}$ is unbiased. The closed form of the sharp-null variance on p.~52 and its Des Raj version on p.~55 are exact under the sharp null, and they are conservative in expectation under a constant effect on cluster totals or a balanced design. Equations (18) and (19) for block designs hold.\n\\item Section 5 and Appendices B and C: equations (20)--(23). The Des Raj estimator and its version with covariates are unbiased for fixed coefficients, and also with $k$ taken from other blocks when the blocks are randomized independently. The invariance results (24) and (25) hold when $k$ is mapped to $b_0+b_1k$, after the subscript corrected in \\fid{P020-E010}. Equation (26), equations (32)--(35), the optimal $k$ and the condition $k^2<2k\\,k_{\\mathit{optim}*}$ hold when $\\sigma^2(n_j)>0$ (see \\fid{P020-E007}), with both factors inside the expectation in (35), as corrected in \\fid{P020-E010}. Equations (29) and (30) hold and agree with the bias formula printed on p.~57. The display (31) is the subject of \\fid{P020-E005}.\n\\item Section 6: the description of the source data (23,869 voters, 85 cable systems, 40 blocks of two or three) agrees with the public file of Green and Vavreck (2008). The expected average effect equals the mean of $p_{ti}-p_{ci}$, as Section 6.2 states. The model that supplies these probabilities is the subject of \\fid{P020-E003}. The paper prints no test for its statement on p.~45 that ``cluster size is significantly related to treatment effect''. In the public file the interaction of treatment and cluster size has a cluster-robust $z$ statistic of about $-3.1$ to $-3.4$, which is consistent with the statement.\n\\item Figures 2--5: the printed bias, SD and root mean squared error are compatible with $\\text{RMSE}=\\sqrt{\\text{bias}^2+\\text{SD}^2}$ after separate rounding, in all 14 panels of Figures 2 and 4. The labels of Figures 3 and 5 agree with the plotted lines. Under the sharp null every estimator is centred at zero, as the symmetry of the balanced pair design requires. The rankings by precision stated in Sections 6.4 and 6.5 agree with the panels as printed. This agreement concerns the arithmetic of the printed labels. Figure 4's regression reference lines, biases and several RMSE labels disagree with the stated true effect, as detailed in \\fid{P020-N008}. On several subsets of 80 districts of the public file, I obtain standard deviations under the sharp null of 8.8--9.5 for the Horvitz--Thompson estimator, 2.38--2.53 for the difference in means and 1.85--1.99 for the fixed-effects estimator; the printed values are 9.2, 2.4 and 1.9.\n\\item The reference list: every work cited in the text appears, and no year disagrees.\n\\end{itemize}\nMy checks had the following limits. The authors' simulation code, their analysis file of 80 districts and the 2000 turnout covariate were not available. The full seven-estimator simulations behind Figures 2--5 remain blocked: I could not regenerate their sampling distributions, and I did not recompute the panels that need the turnout covariate or the random-effects fit. The reproduction rows for these figures describe only the completed comparisons of their printed numbers and lines. The replication files are those of the source study, and they do not implement the estimators or the simulations of this paper. I did not run the R script of the source study, which needs a retired package. Two questions about the application remain open, because the authors' analysis file is needed to answer them. The first is whether the sample of 22,733 voters can be obtained from the public file. The second is whether the standard deviations printed for the Des Raj estimators in Figures 2 and 4 can be reproduced from it. The works cited for particular results (Ding 2014, Samii and Aronow 2012, Cochran 1977, Hansen and Bowers 2008 and 2009, Imai et al.\\ 2009, Angrist and Pischke 2009, Middleton 2008) were not consulted, and I checked the statements attributed to them only for whether they hold. That this audit lists no further corrections is not a finding that the rest of the paper is correct.",
 "work": {
  "arxiv_id": null,
  "audit_complete": true,
  "authors": [
   "Joel A. Middleton",
   "P. M. Aronow"
  ],
  "category": "Articles - Theory and Methodology",
  "citation": "Joel A. Middleton and P. M. Aronow (2015). Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments. Statistics, Politics and Policy 6(1–2):39–75.",
  "doi": "10.1515/spp-2013-0002",
  "edition": {
   "pages_in_audited_file": 37,
   "sha256": "16068acdcbd7b1f84e285ba32dfd511d6b523855bd5cdaf9f0b6d5572824e434",
   "source_url": "https://doi.org/10.1515/spp-2013-0002",
   "venue": "Statistics, Politics and Policy",
   "year": "2015"
  },
  "id": "P020",
  "later_materials": null,
  "title": "Unbiased Estimation of the Average Treatment Effect in Cluster-Randomized Experiments",
  "type": "Journal article",
  "url": "https://pmaronow.github.io/llm-audit/works/P020/",
  "venue": "Statistics, Politics and Policy",
  "year": "2015"
 }
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