{ "claims": [ { "claim": "Under unconfoundedness, unit-level effects linear in treatment, a linear conditional treatment mean and regularity conditions, the OLS coefficient on $D_i$ converges to $\\mathrm{E}[w_i\\tau_i]/\\mathrm{E}[w_i]$, where $w_i=(D_i-\\mathrm{E}[D_i\\mid X_i])^2$ and $\\mathrm{E}[w_i\\mid X_i]=\\operatorname{Var}[D_i\\mid X_i]$. Thus covariate-adjusted OLS can estimate a nonrepresentative average of heterogeneous effects even with a representative nominal sample.", "claim_id": "P015-H1", "related_corrections": [ "P015-N004" ], "tied_to": "Expressions (7) and (8); Theorem A.1 (Appendix A, pp. 262-263)" }, { "claim": "The effective sample can be studied by reweighting the nominal sample with estimated weights: the squared residuals from a regression of $D_i$ on $X_i$. Weighted covariate means, shares and distributions converge to their effective-sample counterparts.", "claim_id": "P015-H2", "related_corrections": [ "P015-E001", "P015-N008" ], "tied_to": "Expression (9); Figure 1; Table 1" }, { "claim": "If the treatment is randomly assigned with equal probabilities, the weights are independent of the effects and regression with covariates converges to the average causal effect.", "claim_id": "P015-H3", "related_corrections": [], "tied_to": "Corollary A.1 (p. 263); 'Randomized Experiments', pp. 255-256; footnote 15" }, { "claim": "The leverage statistic does not measure a unit's contribution to the causal effect estimate and can point the opposite way from the multiple regression weight.", "claim_id": "P015-H4", "related_corrections": [ "P015-E012" ], "tied_to": "Appendix B: Figure B1 and the binary example (pp. 263-264); Expression (8)" }, { "claim": "The same weighting carries over, exactly or to a first-order approximation, from OLS to nonlinear models fit by maximum likelihood (logit, probit, multinomial, Poisson, duration and other generalized linear models).", "claim_id": "P015-H5", "related_corrections": [ "P015-E008", "P015-E009", "P015-E010", "P015-E011", "P015-E013" ], "tied_to": "'Maximum Likelihood Estimation and Generalized Linear Models', pp. 256-257; Appendix C (pp. 264-265)" }, { "claim": "Random-coefficient models do not remove the nonrepresentative within-group weighting created by regression.", "claim_id": "P015-H6", "related_corrections": [ "P015-E005" ], "tied_to": "'Random Coefficient Models', p. 257; Expression (10); footnote 17" }, { "claim": "In Jensen (2003) the effective sample differs sharply from the nominal sample of 114 countries: 12 countries carry over half (51%) of the weight, 32 carry 90%, the lowest 14 carry under 0.05%, and Russia, China and Germany are essentially unrepresented.", "claim_id": "P015-H7", "related_corrections": [ "P015-E001", "P015-E002", "P015-E003", "P015-N008" ], "tied_to": "Figure 1; unnumbered results in the text of p. 258" }, { "claim": "In Gerber and Huber (2010) the nominal and effective samples are close on demographics, so the nominal sample describes the population for which the partisanship effect is estimated.", "claim_id": "P015-H8", "related_corrections": [ "P015-E004", "P015-N010" ], "tied_to": "Table 1 (p. 260); pp. 258-259" }, { "claim": "The paper states that representative average causal effects can be recovered under unconfoundedness and positivity by inverse probability weighting, imputation, interacted regression or inverse conditional-variance weighting. It argues that where positivity fails, as it describes in the Jensen example, recovery requires extrapolation and the requisite conditions cannot always be met.", "claim_id": "P015-H9", "related_corrections": [ "P015-E006", "P015-N011" ], "tied_to": "Expression (11); the positivity display and discussion on pp. 260-261; the remark on inverse propensity weighting on p. 255" }, { "claim": "The paper argues that there is no general external-validity basis for preferring multiple regression on representative samples over experimental or quasi-experimental methods, even under assumptions that are highly favorable to regression.", "claim_id": "P015-H10", "related_corrections": [ "P015-N003" ], "tied_to": "Abstract; p. 251; conclusion pp. 261-262; Theorem A.1; the two applications" } ], "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 last fourteen lines of the left column. Red boxes mark the words 'the sum of the weights' in the definition; the words '12 contribute over half (51 percent)', which run over two lines; and five names in the list of the top twelve countries: Niger, Madagascar, Poland, Lesotho and Belarus.", "caption": "P. 258 (PDF p. 9), left column. The first marked phrase defines the contribution by sums. The marked share and the five marked countries come from the index of country means.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E001.png" }, { "alt": "Crop of Figure 1, two world maps side by side with the note beneath them. The left map, 'Nominal Sample', shows the countries of the sample in black. A red box marks the right map, 'Effective Sample', in which a few countries are dark, among them Argentina, Niger and Pakistan, and most are white or pale gray.", "caption": "Figure 1, p. 258 (PDF p. 9). The marked right panel, “Effective Sample”, is shaded by the index of country means.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E001-b.png" }, { "alt": "Dot plot in three groups, with the share of the weight in percent on the horizontal axis. For each country an open orange diamond gives the printed index and a filled blue circle gives the share by sums, joined by a line. First group, named on p. 258 and among the twelve largest sums: Uruguay, Philippines, Pakistan, Hungary, Argentina, Peru, Zimbabwe; Uruguay's circle is at 8.9 and its diamond at 6.4. Second group, named on p. 258 but not among the twelve largest sums: Lesotho, Madagascar, Poland, Niger, Belarus; Niger's diamond is at 4.7 and its circle at 1.2. Third group, among the twelve largest sums but not named: Chile, Fiji, Zambia, Ecuador, Nepal, with circles between 3.2 and 4.0 and diamonds between 2.3 and 2.8. A note says that the twelve named countries hold 50.9 percent of the printed index and 47.6 percent of the summed weight, and that the twelve countries with the largest sums hold 56.3 percent.", "caption": "The 17 countries that are among the twelve largest under either index, with the number of observed country-years in parentheses. Diamonds give the index behind the printed list (country mean, normalized); circles give the share that the text defines (country sum over the total). Drawn from the exact recalculation.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E001-shares.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T23", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T23" } ], "headline": "The country contributions reported for the Jensen illustration were computed from each country’s mean weight and not from the sum that the text describes; with sums, the twelve largest contributors hold 56% of the weight, five of the twelve named countries are replaced, and 34 countries are needed to reach 90%.", "id": "P015-E001", "location": "Section “Effects of Regime Type on FDI”, p. 258 (PDF p. 9): left column, from “To measure the extent” to the list of twelve countries; right column, first paragraph; Figure 1, right panel", "pdf_page": 9, "printed_page": "258", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P015-T23" ], "text": { "as_printed": "To measure the extent to which a given country contributes to the effect estimate, we compute the analogue to the left-hand side of Expression (9), taking the sum of the weights for that country (summing over all time periods) and dividing it by the sum of all the weights in the sample. The right panel in Figure 1 shows the results. Darker shading means a larger contribution by that country, and white shading means a small or negligible contribution. From the sample of 114 countries, 12 contribute over half (\\hl{51\\%}) of the weight used to construct the estimate of the effect of regime type on FDI, and \\hl{32} contribute 90\\% of the weight. The top 12 contributing countries, in descending order of their weights, are Uruguay, Hungary, \\hl{Niger}, Philippines, Argentina, \\hl{Madagascar}, Pakistan, Zimbabwe, \\hl{Poland}, Peru, \\hl{Lesotho}, and \\hl{Belarus}.\n\nImportantly, the lowest-contributing 14 countries combine to contribute less than 0.05\\% of the weight used to construct the effect estimate. These lowest-contributing countries, in ascending order of their weights, are Central African Republic, Albania, Germany, Russia, Haiti, Benin, South Africa, Yemen, Honduras, \\hl{DR Congo}, Czech Republic, Lithuania, \\hl{China}, and Latvia.", "corrected": "To measure the extent to which a given country contributes to the effect estimate, we compute the analogue to the left-hand side of Expression (9), taking the sum of the weights for that country (summing over all time periods) and dividing it by the sum of all the weights in the sample. The right panel in Figure 1 shows the results. Darker shading means a larger contribution by that country, and white shading means a small or negligible contribution. From the sample of 114 countries, 12 contribute over half (\\hl{56\\%}) of the weight used to construct the estimate of the effect of regime type on FDI, and \\hl{34} contribute \\hl{at least} 90\\% of the weight. The top 12 contributing countries, in descending order of their weights, are Uruguay, Philippines, Pakistan, Hungary, Argentina, Peru, Zimbabwe, \\hl{Chile}, \\hl{Fiji}, \\hl{Zambia}, \\hl{Ecuador}, and \\hl{Nepal}.\n\nImportantly, the lowest-contributing 14 countries combine to contribute less than 0.05\\% of the weight used to construct the effect estimate. These lowest-contributing countries, in ascending order of their weights, are Central African Republic and Albania (both with weight zero), Russia, Germany, Haiti, Honduras, Benin, South Africa, Yemen, \\hl{Republic of the Congo}, Czech Republic, Latvia, Lithuania, and \\hl{Namibia}.\n\nIn both lists the order of the countries that remain also changes. The right panel of Figure~1 is to be redrawn with shading proportional to the country sums, and the regional summary and the remarks on case selection are to be rewritten from the corrected lists. The exact shares are 56.3347\\% for the twelve largest contributors, 90.3453\\% for the largest 34 (the largest 33 hold 89.8429\\%), and 0.0107\\% for the lowest 14.", "evidence": "I checked the exact recalculation from the authors' data. The regression of regime type on the stated controls was solved in rational arithmetic for all 1,630 country-years and 114 countries, and the residuals satisfy all 123 normal equations exactly. The index of country means reproduces the printed twelve in the printed order, with a share of 50.91\\%, and it reproduces the printed list of the 14 lowest once its tenth name is corrected (see \\fid{P015-E002}). The shares by country sums are in the last column of the table. Under sums, 10 countries hold 49.7\\% and 11 hold 53.1\\%. The printed 90\\% rounds the mean-based share of 89.71\\%; it does not assert a minimum of 32 countries to reach at least 90\\%.\n\n\\begin{audittable}{lrrr}{Shares of the weight in the Jensen illustration, p.~258: as printed, and recalculated from the authors' data. ``Means'' is each country's mean weight divided by the sum of the 114 country means; ``sums'' is each country's summed weight divided by the total, as the text defines the contribution.}\nQuantity & As printed & Means & Sums \\\\\nTwelve largest countries & 51\\% & 50.91\\% & 56.33\\% \\\\\nTwelve countries named in the text & 51\\% & 50.91\\% & 47.58\\% \\\\\nLargest 32 countries & 90\\% & 89.71\\% & 89.25\\% \\\\\nLargest 33 countries & -- & 90.25\\% & 89.84\\% \\\\\nLargest 34 countries & -- & 90.72\\% & 90.35\\% \\\\\nLowest 14 countries & $<0.05\\%$ & 0.0458\\% & 0.0107\\% \\\\\n\\end{audittable}", "problem": "The page defines the contribution of a country as the sum of its weights over all its years, divided by the sum of all the weights in the sample. This is the left side of Expression~(9) with $Z_i$ the indicator of the country. The numbers that follow were computed differently: from the mean weight of each country, divided by the sum of the 114 country means. The replication script takes the country mean (\\texttt{tapply(w, jensen.cc\\$country, mean)}, line 33) and shades the right panel of Figure~1 with it. Let $p_c$ be the share of country $c$ by sums and $n_c$ its number of observed years. Then the index of country means is\n\\[\nq_c=\\frac{p_c/n_c}{\\sum_d p_d/n_d},\n\\]\nand $q_c=p_c$ for every country only if all countries with positive weight are observed for the same number of years. In this panel $n_c$ ranges from 1 to 22, and the index of country means raises the countries that are observed in few years. Niger, with 4 years, is third on the printed list and 27th by its sum. Under the definition that the page gives, the printed statements change as follows. The twelve named countries hold 47.6\\% of the weight, which is less than half. Five of them (Niger, Madagascar, Poland, Lesotho and Belarus) are not among the twelve largest contributors, and the order of the other seven changes. The largest 32 countries hold 89.2\\% of the weight, and 34 are needed to pass 90\\%. The list of the 14 lowest contributors changes in order and in one member: Namibia replaces China (its tenth name is a separate error, \\fid{P015-E002}). The regional summary and the advice on case selection in the right column were written from the printed list and the map.", "what_survives": "The finding that few countries carry the estimate survives: 12 of 114 countries hold 56.3\\% of the weight, 11 pass one half, and 34 pass 90\\%. The 14 lowest contributors hold 0.011\\%, inside the printed bound of 0.05\\%, and Russia, China and Germany each hold less than 0.005\\%. The aggregation by country takes place after the regression is fitted, so the OLS coefficient and the weights $\\hat w_i$ do not change. The left panel of Figure~1, the definition of the weights and the application to Gerber and Huber (2010) are not touched by this error. What changes is the right panel of Figure~1, the two lists, the shares, and the regional summary. Niger, Madagascar, Poland, Lesotho and Belarus leave the twelve, and Chile, Fiji, Zambia, Ecuador and Nepal enter. If the printed numbers were kept, the index would have to be described as a country's average weight per observed year, which is a different quantity from the contribution that Expression~(9) defines." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E001", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of nine lines at the top of the right column. A red box marks the words 'DR Congo' in the list 'Central African Republic, Albania, Germany, Russia, Haiti, Benin, South Africa, Yemen, Honduras, DR Congo, Czech Republic, Lithuania, China, and Latvia'. The first line of the crop says that these countries contribute less than 0.05 percent of the weight.", "caption": "P. 258 (PDF p. 9), right column, list of the 14 lowest-contributing countries. The marked name is the tenth entry.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E002.png" } ], "formal_release": null, "formal_statements": [], "headline": "The tenth country in the list of the 14 lowest contributors is the Republic of the Congo, not DR Congo.", "id": "P015-E002", "location": "Section “Effects of Regime Type on FDI”, p. 258 (PDF p. 9), right column, list of the 14 lowest-contributing countries, tenth entry", "pdf_page": 9, "printed_page": "258", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "These lowest-contributing countries, in ascending order of their weights, are Central African Republic, Albania, Germany, Russia, Haiti, Benin, South Africa, Yemen, Honduras, \\hl{DR Congo}, Czech Republic, Lithuania, China, and Latvia.", "corrected": "\\hl{In the original ranking by country means}, these lowest-contributing countries, in ascending order of their weights, are Central African Republic, Albania, Germany, Russia, Haiti, Benin, South Africa, Yemen, Honduras, \\hl{Republic of the Congo}, Czech Republic, Lithuania, China, and Latvia.", "evidence": "The evidence is an exact recalculation from the authors' data, cross-checked against the authors' regression in R. With the Republic of the Congo in tenth place, the list is the list of the 14 lowest under the index of country means, which produced the printed figures, and these countries hold 0.0458\\% of it; the first two entries are tied at zero. The 14 names as printed hold 0.2225\\% by means and 0.2173\\% by sums. The authors' file that links the country names of the data to the names of the map also keeps the two countries apart.", "problem": "``DR Congo'' denotes the Democratic Republic of the Congo. In the data it is a separate country with 18 observed years, and it is not among the 14 lowest contributors: counted from the bottom it ranks 63rd by country means and 64th by country sums. The country in tenth place is the Republic of the Congo, with 4 observed years, under either way of aggregating the weights. With DR Congo in the list, the 14 named countries hold about 0.22\\% of the weight, which is above the bound of 0.05\\% stated in the preceding sentence.", "what_survives": "Only the name changes; no weight, coefficient or map does. The bound of 0.05\\% holds for the correct 14 countries. This correction repairs the list as it was computed. The list by country sums, which is the quantity the text describes, differs further, and \\fid{P015-E001} gives it." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E002", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of eight lines of the left column. One red box marks the number 0.020 in the words 'corresponds to a 0.020 increase in net FDI inflows'. A second red box marks '(p less than 0.001)' at the end of the same sentence.", "caption": "P. 258 (PDF p. 9), left column, first paragraph below Figure 1. The marked coefficient and significance level are those attributed to Model 10 of Jensen (2003).", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E003.png" } ], "formal_release": null, "formal_statements": [], "headline": "Jensen (2003), Table 4, Model 10, reports a coefficient of 0.021 that is significant at the .01 level; the paper quotes it as 0.020 with $p<0.001$.", "id": "P015-E003", "location": "Section “Effects of Regime Type on FDI”, p. 258 (PDF p. 9), left column, first paragraph below Figure 1", "pdf_page": 9, "printed_page": "258", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "The resulting estimate implies that a one-unit increase in the Polity score corresponds to a \\hl{0.020} increase in net FDI inflows as a percentage of gross domestic product ($p < \\hl{0.001}$).", "corrected": "The resulting estimate implies that a one-unit increase in the Polity score corresponds to a \\hl{0.021} increase in net FDI inflows as a percentage of gross domestic product ($p < \\hl{0.01}$).", "evidence": "This is a check of the cited source: Jensen (2003), \\emph{International Organization} 57(3), p.~604, Table~4, Model~10, in the published version; p.~602 of the same article says that the standard errors are panel-corrected. The replication files agree on the coefficient: fitting OLS with the stated controls on all 1,630 country-years, I find 0.0211. The conventional standard error of that fit gives a two-sided $p$ of 0.043, which follows a different convention from Jensen's and is not what the sentence quotes.", "problem": "The sentence summarizes Model~10 in Table~4 of Jensen (2003), which the preceding sentences name. In that table the coefficient on democracy is 0.021 with three stars, the test statistic beneath it is 2.606, and the legend reads ``***$p<.01$''. The coefficient is therefore misquoted in the third decimal, and the significance level is overstated: the source supports $p<.01$, and under a normal reference a statistic of 2.606 has a two-sided $p$ of about 0.009, which is above 0.001.", "what_survives": "The association remains positive and significant at the .01 level in the source. No weight, figure or result of the paper uses these two numbers. Whether the ``$p<.05$'' quoted on p.~259 for the estimate of Gerber and Huber (2010) matches their table could not be settled, because their Table~4A was not available to me; \\fid{P015-E004} reports what the replication data show." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E003", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of seven lines of the left column. A red box marks '20 percent', printed with a percent sign, in the words 'is associated with a 0.077 increase in log change in projected vacation spending (p less than .05), implying about a 20 percent increase'.", "caption": "P. 259 (PDF p. 10), left column, end of the paragraph beginning “We focus on regression Model (4)”. The marked percentage is the reading of the coefficient 0.077.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E004.png" }, { "alt": "Crop of five lines of the left column. A red box marks the words 'the natural log of the change' in the sentence 'We focus on regression Model (4) in Table 4A of their article, which uses a behavioral indicator of economic assessments: the natural log of the change in respondents' predicted vacation spending over the pre- and post-election waves of the survey'.", "caption": "P. 259 (PDF p. 10), left column, beginning of the same paragraph. The marked words describe the outcome.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E004-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T16", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T16" } ], "headline": "A coefficient of 0.077 on a natural-log outcome corresponds to an increase of about 8%, not 20%, and the outcome is the change in the natural log of one plus predicted vacation spending, not the natural log of the change.", "id": "P015-E004", "location": "Section “Effects of Partisanship on Economic Assessments”, p. 259 (PDF p. 10), left column, paragraph beginning “We focus on regression Model (4)”", "pdf_page": 10, "printed_page": "259", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P015-T16" ], "text": { "as_printed": "We focus on regression Model (4) in Table 4A of their article, which uses a behavioral indicator of economic assessments: \\hl{the natural log of the change} in respondents' predicted vacation spending over the pre- and post-election waves of the survey. [\\ldots] Gerber and Huber(2010) estimate that a unit change in a respondent's party identification scale in the direction of ``strong Democrat'' is associated with a 0.077 increase in log change in projected vacation spending ($p < .05$), implying about a \\hl{20\\%} increase.", "corrected": "We focus on regression Model (4) in Table 4A of their article, which uses a behavioral indicator of economic assessments: \\hl{the change in the natural log of one plus} respondents' predicted vacation spending over the pre- and post-election waves of the survey. [\\ldots] Gerber and Huber(2010) estimate that a unit change in a respondent's party identification scale in the direction of ``strong Democrat'' is associated with a 0.077 increase in log change in projected vacation spending ($p < .05$), implying about \\hl{an 8\\%} increase \\hl{in the exponentiated fitted outcome}.\n\nHere the outcome is $Y_i=\\log(1+S_{i,\\mathrm{post}})-\\log(1+S_{i,\\mathrm{pre}})$, and the exponentiated fitted outcome is the fitted geometric growth factor of $(1+S_{\\mathrm{post}})/(1+S_{\\mathrm{pre}})$. The 8\\% is not a percentage effect on arithmetic mean spending. Standard errors clustered by state (50 clusters, with a finite-sample adjustment and a $t_{49}$ reference distribution) give $p=0.0274<.05$.", "evidence": "The evidence is a derivation and a recalculation from the replication files. Let $\\hat m(d,x)$ be the fitted linear predictor at party identification $d$ and controls $x$, and let $\\hat\\beta$ be the coefficient on party identification. Then $\\hat m(d+1,x)-\\hat m(d,x)=\\hat\\beta$, which implies that\n\\[\n\\frac{\\exp\\{\\hat m(d+1,x)\\}}{\\exp\\{\\hat m(d,x)\\}}=e^{\\hat\\beta},\n\\]\nso the relative increase in the exponentiated fitted outcome is $e^{\\hat\\beta}-1$. For $x=77/1000$, bounding $e^x$ below by the partial sum $P_7=\\sum_{k=0}^{7}x^k/k!$ and above by $P_7+(x^8/8!)/(1-x/9)$ and comparing exact fractions, I obtain\n\\[\n1.0800420763 0, \\qquad \\Pr[D_i = d' \\mid X_i = x] > 0\n\\]\nfor all values of $x$ in the target population represented by the nominal sample.\n\n\\emph{The passage, p.~261.} We can draw a connection between the positivity assumption and the results that we obtained above for multiple regression. The positivity assumption implies that $\\operatorname{Var}[D_i \\mid X_i = x] > 0$, for all values of $x$. Thus, when a multiple regression places (either \\hl{approximately} or \\hl{exactly}) no weight on a set of units with a given covariate profile, this result implies that \\hl{positivity is violated} and representative causal effects cannot be estimated for the target population without assumption-based extrapolation or interpolation. Examination of the effective sample provides insight about what can be learned from any given study. In the Jensen (2003) study, many substantively important countries---including Russia and China---cannot be included in estimates of the causal effects of regime type on FDI. \\hl{No consistent, extrapolation-free estimator} of $\\mathrm{E}[\\tau_i(d,d')]$ is available for any $(d,d')$ if our interest is in the average over the entirety of the 114 countries in the nominal sample. This is not due to any fault on Jensen's part; it is an implication of the variation that nature has provided. However, in other studies, such as Gerber and Huber (2010), positivity appears to hold, and thus estimation of representative average causal effects is straightforward. Of course, the effective sample that multiple regression generates is one for which \\hl{positivity holds by construction}. In conjunction with the linearity assumptions embodied by Expressions (1) and (3) (assumptions that are with little loss of generality when $D_i$ is binary), regression facilitates estimation in a manner that reflects the regions of the data where causal inference is possible.", "corrected": "\\emph{p.~261, from ``The positivity assumption implies'' to ``holds by construction''.} Positivity for two distinct treatment values $d$ and $d'$ implies that $\\operatorname{Var}[D_i \\mid X_i = x] > 0$, for all values of $x$ in the target population. Thus, when the conditional variance of the treatment at a covariate profile is \\hl{exactly zero in the population}, positivity is violated at that profile for every pair $(d,d')$ with $d \\neq d'$, and \\hl{representative effects for that profile require} assumption-based extrapolation or interpolation. \\hl{A small positive conditional variance, or a small estimated regression weight, does not establish a violation of positivity.} \\hl{A small positive population conditional variance} shows that the profile supplies little identifying variation, and estimates for it may be imprecise.\n\nIn the Jensen (2003) study, the weights show that many substantively important countries---including Russia and China---\\hl{contribute very little to this regression estimate}. The statement that no consistent, extrapolation-free estimator of $\\mathrm{E}[\\tau_i(d,d')]$ is available for the 114 countries of the nominal sample \\hl{requires in addition a violation of positivity in the population} for the chosen pair, and the weights alone do not establish one.\n\nIn Gerber and Huber (2010), \\hl{the weighting diagnostics alone do not establish positivity for a chosen pair}.\n\nThe effective \\hl{population defined by the population weights $w_i$} is one in which \\hl{the conditional variance of the treatment is positive} at every represented covariate profile. For a binary treatment this is positivity for the pair $(0,1)$. For a treatment with more than two values, positivity for a chosen pair $(d,d')$ requires in addition that \\hl{both treatment values have positive conditional probability at each of these profiles in the effective population}.\n\nThe opening sentence, the sentence beginning ``Examination of the effective sample'', and the final sentence on the linearity assumptions are unchanged.", "evidence": "The evidence is three exact examples, each computed in rational arithmetic. By the effective population I mean the nominal population reweighted in proportion to $w_i$.\n\n\\emph{Example A: a small weight with positivity.} Let $X_i$ equal $0$ or $1$ with probability $1/2$ each, let $D_i$ be binary with $\\Pr[D_i = 1 \\mid X_i = 0] = \\varepsilon = 10^{-6}$ and $\\Pr[D_i = 1 \\mid X_i = 1] = 1/2$, and let $Y_i(d) = 2X_i + (1 + X_i)d$. Since the potential outcomes are functions of $X_i$, unconfoundedness holds. The effects are linear in $d$ with $\\tau_i = 1 + X_i$, and $\\mathrm{E}[D_i \\mid X_i] = \\varepsilon + (1/2 - \\varepsilon)X_i$ is linear in $X_i$. Both treatment values have positive probability at both profiles.\n\\begin{audittable}{rrrrrr}{Example A: the two covariate profiles.}\n$X_i$ & $\\Pr[D_i = 1 \\mid X_i]$ & $\\operatorname{Var}[D_i \\mid X_i]$ & Nominal share & Effective share & $\\tau_i$ \\\\\n$0$ & $1/10^{6}$ & $999999/10^{12}$ & $1/2$ & $999999/250000999999$ & $1$ \\\\\n$1$ & $1/2$ & $1/4$ & $1/2$ & the remainder & $2$ \\\\\n\\end{audittable}\nThe effective share of the profile $X_i = 0$ is about $4.0 \\times 10^{-6}$. The average effect in the nominal population is $3/2$. The population OLS coefficient on $D_i$ is $500000999999/250000999999$, which is the weighted average of Theorem~A.1. Let $e(x) = \\Pr[D_i = 1 \\mid X_i = x]$ and\n\\[\nZ_i = \\frac{D_i Y_i}{e(X_i)} - \\frac{(1 - D_i) Y_i}{1 - e(X_i)} .\n\\]\nComputing its first two moments, I find $\\mathrm{E}[Z_i] = 3/2$ and $\\operatorname{Var}(Z_i) = 2000071/4$. The mean of $n$ independent copies of $Z_i$ is then unbiased for $3/2$ with variance $2000071/(4n)$, and Chebyshev's inequality implies that it is consistent. Note that this estimator uses only outcomes that are observed under each treatment at each profile. The same construction has positivity for every $\\varepsilon$ in $(0, 1/2)$. The effective share of the profile $X_i = 0$ is $\\varepsilon(1-\\varepsilon)/[\\varepsilon(1-\\varepsilon) + 1/4]$, which tends to $0$ as $\\varepsilon \\to 0$. Hence no threshold for ``approximately no weight'' implies a violation of positivity.\n\n\\emph{Example B: positive weights, and a chosen pair without support.} Let $(X_i, D_i)$ take the values $(0,0)$, $(0,1)$, $(1,0)$ and $(1,2)$ with probability $1/4$ each. Then $\\mathrm{E}[D_i \\mid X_i] = 1/2 + X_i/2$ is linear in $X_i$.\n\\begin{audittable}{rrrrr}{Example B: the four cells, each with nominal probability $1/4$.}\n$X_i$ & $D_i$ & $\\mathrm{E}[D_i \\mid X_i]$ & $w_i$ & Effective probability \\\\\n$0$ & $0$ & $1/2$ & $1/4$ & $1/10$ \\\\\n$0$ & $1$ & $1/2$ & $1/4$ & $1/10$ \\\\\n$1$ & $0$ & $1$ & $1$ & $2/5$ \\\\\n$1$ & $2$ & $1$ & $1$ & $2/5$ \\\\\n\\end{audittable}\nEvery weight is positive and the profile $X_i = 1$ has effective share $4/5$, yet $\\Pr[D_i = 1 \\mid X_i = 1] = 0$ in the nominal and in the effective population. Positivity for the pair $(0,1)$ fails. I compare two sets of potential outcomes. In the first, $Y_i(d) = d$ for every unit. The second differs from the first only in that $Y_i(1) = 3$ when $X_i = 1$. Both satisfy unconfoundedness. Since $D_i = 1$ never occurs at $X_i = 1$, the observed $(X_i, D_i, Y_i)$ has the same distribution under both. The average of $\\tau_i(0,1)$ over the nominal population is $1$ in the first and $2$ in the second; over the effective population it is $1$ and $13/5$. I conclude that no estimator based on the observed data is consistent in both cases.\n\n\\emph{Example C: an estimated weight of zero.} Suppose that $X_i$ and $D_i$ are independent with $\\Pr[X_i = 1] = \\Pr[D_i = 1] = 1/2$, so that positivity holds and $\\operatorname{Var}[D_i \\mid X_i = x] = 1/4$ at both profiles. The ordered sample $(X_i, D_i) = (0,0), (0,0), (1,0), (1,1)$ has probability $1/256$. In this sample the regression of $D_i$ on a constant and $X_i$ has coefficients $(0, 1/2)$ and residuals $(0, 0, -1/2, 1/2)$. The two units with $X_i = 0$ receive an estimated weight of exactly zero.", "problem": "The paragraph draws three conclusions from the regression weights. Recall that $w_i = (D_i - \\mathrm{E}[D_i \\mid X_i])^2$, so that $\\mathrm{E}[w_i \\mid X_i = x] = \\operatorname{Var}[D_i \\mid X_i = x]$ by the definition of the conditional variance. One direction is correct. Positivity for two distinct values $d$ and $d'$ at $x$ implies that $\\operatorname{Var}[D_i \\mid X_i = x] > 0$. Hence positivity fails for every pair at any covariate profile such that this variance is zero in the population. The three conclusions of the paragraph go further, and none of them follows.\n\nFirst, the paragraph treats a weight of approximately zero as a violation of positivity. A conditional variance that is small and positive is compatible with positivity at every profile, and with a consistent estimator of the representative effect that uses only observed outcomes. A small positive population conditional variance shows that the profile supplies little variation in the treatment and that estimates for it may be imprecise. Second, the paragraph says that positivity holds in the effective sample by construction. At the population level, the effective sample assigns positive weight to the profiles at which the treatment varies. For a binary treatment this is positivity for the pair $(0,1)$. For a treatment with more than two values it means only that $D_i$ takes at least two values at $x$, and these need not be the chosen $d$ and $d'$. The positivity display on the same page is stated for the chosen pair. Both applications have such treatments: a regime score with 21 values in Jensen (2003), and party identification on a five-point scale in Gerber and Huber (2010). Third, the weights used in an application are estimated: they are squared residuals from a fitted regression. An estimated weight that is zero or small does not establish that the conditional variance is zero in the population.\n\nThe statement about Jensen (2003), that no consistent, extrapolation-free estimator exists for any $(d,d')$ over the 114 countries, therefore does not follow from the small weights of countries such as Russia and China. Whether the statement is true is a separate question, which I do not settle. In the replication data Russia has two observations, both with regime score 14, and China has seven, all with score 3. These observations are compatible with a violation of positivity, and they do not determine the conditional distribution of the regime score in the population.", "what_survives": "Theorem~A.1, the weights of Expressions (7) and (8), the effective-sample summaries of Expression (9), the simulations and the weighting diagnostics for both applications are unaffected. So is the conclusion that OLS estimates an average effect weighted by the conditional variance of the treatment, which can differ from the average effect in the nominal population. It remains true that zero conditional variance in the population rules out positivity, and that positive conditional variance is positivity for a binary treatment. The paragraph can no longer conclude from near-zero weights that positivity is violated, or that positivity holds in the effective sample when the treatment has more than two values. The statement about Jensen (2003) becomes conditional on a violation of positivity that is shown by other means. No reported number changes. The form of the positivity display for a continuous treatment is a separate point, \\fid{P015-N011}." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E006", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the heading Appendix C: Maximum Likelihood Estimation for Nonlinear Models and the fourteen lines of text below it, at the foot of the right column of page 264. Red outlines surround the sentence This holds for generalized linear models such as probit, Poisson regression, ordered logit or probit, multinomial logit or probit, and duration models, and the words To apply the result below for these other models, with which the page ends.", "caption": "Appendix C, first paragraph, p. 264 (PDF p. 15), right column. The outlines mark the sentence that lists the other models and the start of the instruction, which continues on p. 265.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E008.png" }, { "alt": "Crop of the top of the left column of page 265. The first two lines read: one substitutes the relevant predicted mean or inverse link function into the estimating equation. Red outlines surround these two lines. Below them are the logistic regression model, Pr[Y_i = 1 given D_i, X_i] = inverse logit of (alpha_ell + beta_ell D_i + X_i gamma_ell), the definition of W_i, and the estimating equation: the sum over i of W_i times (Y_i minus the inverse logit of W_i theta-hat) equals the zero vector of size 1 by (K+2).", "caption": "Appendix C, p. 265 (PDF p. 16), top of the left column: the end of the instruction, the logistic model and its estimating equation. The outlines mark “one substitutes the relevant predicted mean or inverse link function into the estimating equation.”", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E008-b.png" }, { "alt": "Crop of the section heading Maximum Likelihood Estimation and Generalized Linear Models and the five lines below it. Red outlines surround the first sentence: The results for linear regression carry over directly to nonlinear regression models fit via maximum likelihood. The text continues: This includes the class of generalized linear models, such as logit, probit, multinomial logit or probit, Poisson regression, and parametric or semiparametric (e.g., Cox); the sentence continues on the next page.", "caption": "Main text, p. 256 (PDF p. 7), right column, start of the section “Maximum Likelihood Estimation and Generalized Linear Models”. The outlines mark its first sentence.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E008-c.png" }, { "alt": "Crop of seven lines of text from the left column of page 257. The passage ends: (See Appendix C for the mathematical details. The logic extends to maximum likelihood estimates of any generalized linear model.) Red outlines surround the sentence The logic extends to maximum likelihood estimates of any generalized linear model.", "caption": "Main text, p. 257 (PDF p. 8), left column, end of the paragraph on logistic regression. The outlines mark “The logic extends to maximum likelihood estimates of any generalized linear model.”", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E008-d.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T5", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T5" }, { "id": "P015-T6", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T6" }, { "id": "P015-T7", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T7" }, { "id": "P015-T8", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T8" }, { "id": "P015-T18", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T18" } ], "headline": "Substituting another inverse link into the logistic estimating equation of Appendix C does not yield the maximum-likelihood equations of probit and the other listed models, and the argument that carries the OLS weights over to nonlinear models holds as printed only for models whose likelihood score is a constant multiple of the equation so obtained, such as logistic regression and Poisson regression with the log link.", "id": "P015-E008", "location": "Appendix C, first paragraph, p. 264 (PDF p. 15), continued at the top of p. 265 (PDF p. 16); the related statements of the main text are in the section “Maximum Likelihood Estimation and Generalized Linear Models”, pp. 256–257 (PDF pp. 7–8)", "pdf_page": 15, "printed_page": "264", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P015-T18", "P015-T5", "P015-T6", "P015-T7", "P015-T8" ], "text": { "as_printed": "\\emph{Appendix~C, pp.~264--265.} While our illustration is for logistic regression, the results hold for any nonlinear model that can be expressed in terms of estimating equations in the manner that follows (Liang and Zeger 1986). \\hl{This holds for generalized linear models such as probit, Poisson regression, ordered logit or probit, multinomial logit or probit, and duration models} (McCullagh and Nelder 1999). To apply the result below for these other models, \\hl{one substitutes the relevant predicted mean or inverse link function into the estimating equation}.\n\n\\emph{Appendix~C, p.~265.} The ML estimator for $\\hat\\theta$ solves the following estimating equation (Liang and Zeger 1986):\n\\[\n\\sum_{i=1}^{n} \\psi(Y_i, W_i, \\hat\\theta) = \\sum_{i=1}^{n} \\bigl( W_i\\{Y_i - \\operatorname{logit}^{-1}(W_i\\hat\\theta)\\} \\bigr) = \\mathbf{0}_{1\\times(K+2)},\n\\]\nwhere $K$ is the number of variables in $X_i$.\n\n\\emph{Main text, pp.~256--257.} The results for linear regression \\hl{carry over directly} to nonlinear regression models fit via maximum likelihood. This includes the class of generalized linear models, such as logit, probit, multinomial logit or probit, Poisson regression, and parametric or semiparametric (e.g., Cox) duration models (Long 1997; McCullagh and Nelder 1999). [\\ldots] (See Appendix C for the mathematical details. The logic extends to maximum likelihood estimates of \\hl{any generalized linear model}.)", "corrected": "\\emph{Appendix~C, first paragraph.} The illustration is for logistic regression. The same argument applies to any model \\hl{whose maximum-likelihood score is a common nonzero multiple of the estimating equation displayed below}, with the inverse link of the model in place of the inverse logit. A scalar generalized linear model \\hl{with its canonical link} and a common dispersion is of this kind; examples are logistic regression and Poisson regression with the log link, and for such a model one substitutes the inverse link function into the estimating equation. Other links and likelihoods, including probit, ordered and multinomial models and duration models, \\hl{require their own score equations and their own approximation arguments}. For a binary probit model the score is\n\\[\n\\sum_{i=1}^{n} W_i \\, \\hl{\\frac{\\phi(W_i\\theta)}{\\Phi(W_i\\theta)\\{1 - \\Phi(W_i\\theta)\\}}} \\, \\{Y_i - \\Phi(W_i\\theta)\\} ,\n\\]\nwhere $\\phi$ and $\\Phi$ are the standard normal density and distribution function. A first-order expansion of this score must also differentiate the multiplier $\\phi/[\\Phi(1-\\Phi)]$. It reduces to the OLS equations only if the resulting term cancels, or under a further local approximation that is stated and justified. The logistic calculation alone does not give the OLS weights for the other listed models.\n\n\\emph{Main text, p.~256.} The results for linear regression carry over, \\hl{as a first-order approximation}, to logistic regression and to other nonlinear regression models fit via maximum likelihood \\hl{whose score has the form of the estimating equation of Appendix~C}, such as Poisson regression with the log link.\n\n\\emph{Main text, p.~257.} (See Appendix C for the mathematical details. The logic extends to maximum likelihood estimates of \\hl{scalar} generalized linear models \\hl{with a canonical link and a common dispersion}; other models, including probit, \\hl{need a separate argument}.)\n\nThe sentence on p.~252 that the results ``also carry through to generalized linear models (logit, probit, etc.)'' and the last sentence of Appendix~C are limited in the same way.", "evidence": "The evidence is a derivation of the score and exact counterexamples.\n\n\\emph{The score.} For a binary outcome with $\\Pr[Y_i = 1 \\mid W_i] = \\mu(\\eta_i)$, the log likelihood of one observation is $Y_i \\log \\mu(\\eta_i) + (1 - Y_i)\\log\\{1 - \\mu(\\eta_i)\\}$. Differentiating in $\\eta_i$, I have\n\\[\n\\mu'(\\eta_i)\\left\\{ \\frac{Y_i}{\\mu(\\eta_i)} - \\frac{1 - Y_i}{1 - \\mu(\\eta_i)} \\right\\} = h(\\eta_i)\\{Y_i - \\mu(\\eta_i)\\}, \\qquad h = \\frac{\\mu'}{\\mu(1-\\mu)} .\n\\]\nFor $\\mu = \\operatorname{logit}^{-1}$, $\\mu' = \\mu(1-\\mu)$ and $h \\equiv 1$.\n\n\\emph{Two parameter values that solve the substituted equation and are not ML estimates.} In both examples $W_i = (1 \\;\\; D_i \\;\\; X_i)$ with $D_i$ binary, and the cells are the same for $D_i = 0$ and for $D_i = 1$. In the first example $\\mu(\\eta) = \\eta/(1+\\eta)$ on $\\eta>0$ and $\\theta = (1, 0, 1)'$, and the cells $X_i = 0, 1, 2$ have $12$ observations each, with $7$, $6$ and $10$ successes. In the second example the model is probit and $\\theta = (0, 0, a)'$ with $a = \\Phi^{-1}(3/4)$, and the cells $X_i = -1, 0, 1$ have $8$ observations each, with $3$, $2$ and $7$ successes.\n\\begin{audittable}{llll}{The substituted equation and the likelihood score at the same parameter value.}\nModel & $\\theta$ & $\\sum_i W_i'\\{Y_i - \\mu(W_i\\theta)\\}$ & Likelihood score \\\\\n$\\mu(\\eta) = \\eta/(1+\\eta)$ & $(1, 0, 1)'$ & $(0, 0, 0)'$ & $(2/3,\\; 1/3,\\; -2/3)'$ \\\\\nProbit & $(0, 0, a)'$ & $(0, 0, 0)'$ & $(4s,\\; 2s,\\; 0)'$ with $s = h(a) - h(0)$ \\\\\n\\end{audittable}\nIn the probit example the treatment coordinate $2s$ of the score exceeds $13/625$, by a bound obtained in rational arithmetic, and its value is about $0.198$. A point at which the score is not zero is not an ML estimate.\n\n\\emph{The first-order term.} Fix $p$ in $(0,1)$ and let $\\eta_0 = \\mu^{-1}(p)$, $c = \\mu'(\\eta_0)$, $h_0 = h(\\eta_0)$ and $h_1 = h'(\\eta_0)$. Writing $\\eta_i = \\eta_0 + \\delta_i$ and expanding both factors, I obtain\n\\[\nh(\\eta_i)\\{Y_i - \\mu(\\eta_i)\\} = h_0 (Y_i - p) - h_0 c\\, \\delta_i + h_1 (Y_i - p)\\, \\delta_i + O(\\delta_i^2) .\n\\]\nThe substitution keeps the first two terms and drops the third, which is of first order in $\\delta_i$. Note that $h_1 = 0$ for the logit.\n\n\\emph{Its effect on the coefficient.} Let $X_i$ take the values $-1$, $0$ and $1$ with probability $1/3$ each, let $D_i$ be binary with $\\Pr[D_i = 1 \\mid X_i] = 1/2 + X_i/4$, and let $\\mathrm{E}[Y_i \\mid D_i, X_i] = q_0(X_i) + \\varepsilon (D_i - 1/2 - X_i/4)$, where $q_0(-1) = q_0(1) = p + 1/16$ and $q_0(0) = p - 1/8$. The mean of $Y_i$ is $p$. Potential outcomes with this conditional mean can be chosen such that unconfoundedness holds; the treatment is binary, and $\\mathrm{E}[D_i \\mid X_i]$ is linear in $X_i$. The OLS coefficient on $D_i$ is exactly $\\varepsilon$. If the substitution were valid, the ML coefficient on $D_i$ would then be $\\varepsilon/c$ to first order. Differentiating the equation that sets the likelihood score to zero, I find that the derivative of the ML coefficient at $\\varepsilon = 0$ is $1/(c + k/80)$, where $k = h_1/h_0$.\n\\begin{audittable}{llll}{Derivative at $\\varepsilon = 0$ of the ML coefficient on $D_i$: the value implied by the substitution and the value from the likelihood score.}\nModel & $p$ & Substitution: $1/c$ & Likelihood score: $1/(c + k/80)$ \\\\\n$\\mu(\\eta) = \\eta^2$, $0<\\eta<1$ & $1/4$ & $1$ & $120/119$ \\\\\nProbit & $\\Phi(-1)$ & in $[4.132731354122,\\, 4.132731354123]$ & in $[4.184073577247,\\, 4.184073577248]$ \\\\\n\\end{audittable}\nHence the ML coefficient differs at first order in $\\varepsilon$ from the value implied by the substitution. With the same $p$ and with $q_0$ constant, the derivative equals $1/c$. For these models the first-order scale is therefore not a function of the mean of $Y_i$ alone. If $\\varepsilon$ is replaced by $\\varepsilon\\tau(X_i)$ in this example, the relative weights on the effects $\\tau(X_i)$ in the derivative still follow $\\operatorname{Var}[D_i \\mid X_i]$. The example shows that the scaling fails, and it does not show different relative weights.", "problem": "Appendix~C derives its result for logistic regression in two steps. The ML estimate solves the estimating equation displayed above. Replacing $\\operatorname{logit}^{-1}$ by its first-order Taylor polynomial about $\\operatorname{logit}(P_Y)$ then turns this equation into the OLS normal equations, up to the scale factor $c = P_Y(1-P_Y)$. The appendix says that the result holds for probit, Poisson, ordered, multinomial and duration models as well, and that it follows by substituting the predicted mean or inverse link of the model into the estimating equation.\n\nThe first step fails in general. For a binary outcome with mean function $\\mu$ and linear predictor $\\eta_i = W_i\\theta$, the likelihood score is $\\sum_i W_i\\, h(\\eta_i)\\{Y_i - \\mu(\\eta_i)\\}$ with $h = \\mu'/[\\mu(1-\\mu)]$. For the logit, $h$ is identically $1$ and the printed equation is the likelihood score. For probit, $h(\\eta) = \\phi(\\eta)/[\\Phi(\\eta)\\{1-\\Phi(\\eta)\\}]$ varies with $\\eta$ and cannot be taken out of the sum. The substituted equation is then not the probit score, and a root need not be the probit ML estimate. The scores of ordered, multinomial and duration models have other forms again. For example, the Cox partial-likelihood score involves the risk sets, and it cannot be obtained by substituting a scalar inverse link.\n\nThe second step changes as well. A first-order expansion of the probit score must expand $h$ along with $\\Phi$, which leaves a first-order term that involves the outcome and has no counterpart in the OLS equations. This term vanishes for probit when the expansion point is $P_Y = 1/2$. It can cancel in particular designs, and it can be of higher order under further local assumptions, none of which the appendix states. The appendix therefore establishes the first-order OLS weighting for logistic regression, and the same argument covers the models whose score is a constant multiple of the substituted equation, such as Poisson regression with the log link. For probit and the other listed models the statement is not established by this argument. I do not show that a local version with suitable qualifications fails for those models.", "what_survives": "Theorem~A.1 and Corollary~A.1, the formulas for the effective sample under OLS, the comparison with the leverage in Appendix~B and the argument about external validity do not use this extension. Both applications compute their weights from least-squares regressions of the treatment on the controls, and none of their reported numbers changes on this account. The logistic score and its first-order reduction to the OLS equations stand, apart from the misprint \\fid{P015-E010} and the notational point \\fid{P015-E011}. The extension narrows to the models whose score is a constant multiple of the substituted equation. For probit, ordered, multinomial and duration models the same weights are not established by Appendix~C and would need a separate argument. The erratum of 2017 does not address this passage." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E008", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of ten lines of the left column. Two red boxes mark the words 'approximation of arbitrary precision', which run over two lines, in the sentence 'We can obtain an approximation of arbitrary precision for the weights by using a Taylor expansion of the inverse link function'.", "caption": "P. 257 (PDF p. 8), left column. The marked words promise an approximation of arbitrary precision; the next sentence fixes the center of the expansion.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E009.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T11", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T11" }, { "id": "P015-T12", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T12" } ], "headline": "One Taylor expansion of the inverse logit about the logit of the mean outcome does not give an approximation of arbitrary precision for the weights, because the series has a finite radius of convergence and the fitted linear predictors of a logistic regression can lie outside it; what remains is the first-order, local approximation that the paper derives and uses.", "id": "P015-E009", "location": "Section “Maximum Likelihood Estimation and Generalized Linear Models”, p. 257 (PDF p. 8), left column, sentence beginning “We can obtain an approximation”", "pdf_page": 8, "printed_page": "257", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P015-T11", "P015-T12" ], "text": { "as_printed": "We can obtain \\hl{an approximation of arbitrary precision} for the weights by using a Taylor expansion of the inverse link function, $\\mathrm{logit}^{-1}(.)$. Suppose we center such an expansion on the logit of the mean of $Y_i$.", "corrected": "\\hl{A first-order} Taylor expansion of the inverse link function, $\\mathrm{logit}^{-1}(.)$, \\hl{provides a local approximation} for the weights. Suppose we center \\hl{this} expansion on the logit of the mean of $Y_i$; \\hl{its accuracy depends on how close the fitted linear predictors are to that center.}\n\nA statement about expansions of increasing order needs the following hypotheses. Let $f(a)=1/[1+\\exp(-a)]$, let $09/2>4>\\pi$. Both fitted linear predictors lie outside the disk of convergence.\n\nFurther, the error is bounded below at every degree. The Taylor series of $f$ at $0$ is\n\\[\nf(a)=\\frac12+\\sum_{m\\ge 1}(-1)^{m-1}b_m a^{2m-1},\\qquad b_m=2\\sum_{k\\ge 0}\\bigl((2k+1)\\pi\\bigr)^{-2m}>0,\n\\]\nwith $b_1=1/4$ and $b_2=1/48$. By the Cauchy--Schwarz inequality, $b_m^2\\le b_{m-1}b_{m+1}$ for $m\\ge 2$, which implies that $b_{m+1}/b_m\\ge b_2/b_1=1/12$ for all $m\\ge 1$. For $a\\ge 9/2$ the terms $t_m=b_ma^{2m-1}$ therefore satisfy $t_{m+1}/t_m\\ge 27/16$. Pairing consecutive terms, I find that at such $a$ every Taylor polynomial with an odd number of nonconstant terms is at least $1/2+a/4\\ge 13/8$, and every one with an even number is at most $1/2+a/4-a^3/48\\le -35/128$. The fitted probability at $a=\\log 99$ is $99/100$, and the constant polynomial equals $1/2$. Hence every Taylor polynomial $T_N$ of $f$ at $0$, of every degree $N\\ge 0$, satisfies\n\\[\n\\bigl|T_N(\\log 99)-\\tfrac{99}{100}\\bigr|\\ge \\tfrac{49}{100},\n\\]\nand by the symmetry $f(-a)=1-f(a)$ the same bound holds at $-\\log 99$. Repeating the data at each value of a binary control $X_i\\in\\{0,1\\}$, I obtain 400 observations, a control coefficient of zero, and the same failure.", "problem": "The sentence says that one Taylor expansion of $f(a)=1/[1+\\exp(-a)]$, centered at the logit of the mean of $Y_i$, can be carried to any accuracy by taking more terms. That holds only at linear predictors inside the radius of convergence of the series. Since $f$ has poles at the odd multiples of $i\\pi$, its Taylor series about a real center $c$ has radius $\\sqrt{c^2+\\pi^2}$, which equals $\\pi$ when $c=0$. A logistic regression can have a design of full rank and a maximum likelihood estimate that is unique and finite, and still have fitted linear predictors that lie farther than this from the center. At such predictors the Taylor polynomials do not approach $f$ as the degree grows. The argument that follows the sentence, and Appendix~C, use the first-order term only.", "what_survives": "The first-order step of Appendix~C is a formal linearization and does not require the series to converge; how accurate it is for given data depends on how far the fitted linear predictors are from the center, which I did not quantify. The OLS weighting result and its derivation in Appendix~A, the comparison with leverage in Appendix~B and Figure~B1, and the two applications, which are fitted by OLS, do not use the sentence. What is lost is the unrestricted promise of arbitrary precision for the nonlinear extension on pp.~256--257, to which the introduction refers on p.~252. The counterexample concerns the single expansion that the sentence specifies; it says nothing about approximations with other centers or by other methods." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E009", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of a short passage from the left column of page 265. The text reads: We may approximate the inverse logit function by taking a first-order Taylor expansion about logit(P_Y) = log[P_Y/(1 - P_Y)], where P_Y = Pr[Y_i = 1]. The display reads: inverse logit of a is approximately P_Y plus the fraction exp[logit(P_Y)] over [1 + logit(P_Y)] squared, times [a - logit(P_y)], which equals b + ca. Red outlines surround the denominator [1 + logit(P_Y)] squared and the symbol P with a lowercase subscript y in the last bracket. Below: where b = P_Y - P_Y(1 - P_Y) logit(P_Y) and the scale factor c = P_Y(1 - P_Y).", "caption": "Appendix C, p. 265 (PDF p. 16), left column: the Taylor expansion of $\\operatorname{logit}^{-1}(a)$ and the definitions of $b$ and $c$. The outlines mark the denominator $[1 + \\operatorname{logit}(P_Y)]^2$ and the symbol $P_y$.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E010.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T13", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T13" } ], "headline": "In the displayed Taylor expansion the denominator lacks an exponential and should read $[1+\\exp\\{\\operatorname{logit}(P_Y)\\}]^2$, and $P_y$ in the last bracket should read $P_Y$; the constants $b$ and $c$ printed below the display are correct.", "id": "P015-E010", "location": "Appendix C, display of the first-order Taylor expansion of $\\operatorname{logit}^{-1}(a)$, p. 265 (PDF p. 16), left column", "pdf_page": 16, "printed_page": "265", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P015-T13" ], "text": { "as_printed": "We may approximate the $\\operatorname{logit}^{-1}(.)$ function by taking a first-order Taylor expansion about $\\operatorname{logit}(P_Y) = \\log[P_Y/(1-P_Y)]$, where $P_Y = \\Pr[Y_i = 1]$:\n\\[\n\\begin{aligned}\n\\operatorname{logit}^{-1}(a) &\\approx P_Y + \\frac{\\exp[\\operatorname{logit}(P_Y)]}{[1 + \\hl{\\operatorname{logit}(P_Y)}]^2}\\,[a - \\operatorname{logit}(\\hl{P_y})]\\\\\n&= b + ca,\n\\end{aligned}\n\\]\nwhere $b = P_Y - P_Y(1-P_Y)\\operatorname{logit}(P_Y)$ and the scale factor $c = P_Y(1-P_Y)$.", "corrected": "With $\\operatorname{logit}^{-1}(a) = [1 + \\exp(-a)]^{-1}$ and $P_Y = \\Pr[Y_i = 1]$ in $(0,1)$, the first-order Taylor expansion about $\\operatorname{logit}(P_Y) = \\log[P_Y/(1-P_Y)]$ is\n\\[\n\\begin{aligned}\n\\operatorname{logit}^{-1}(a) &\\approx P_Y + \\frac{\\exp[\\operatorname{logit}(P_Y)]}{[1 + \\hl{\\exp\\{\\operatorname{logit}(P_Y)\\}}]^2}\\,[a - \\operatorname{logit}(\\hl{P_Y})]\\\\\n&= b + ca,\n\\end{aligned}\n\\]\nwhere $b = P_Y - P_Y(1-P_Y)\\operatorname{logit}(P_Y)$ and the scale factor $c = P_Y(1-P_Y)$.", "evidence": "The evidence is a derivation and a check of the page. Let $g(a) = \\operatorname{logit}^{-1}(a) = [1 + \\exp(-a)]^{-1}$. Differentiating, I have\n\\[\ng'(a) = \\frac{\\exp(-a)}{[1 + \\exp(-a)]^2} = \\frac{\\exp(a)}{[1 + \\exp(a)]^2} .\n\\]\nLet $L = \\operatorname{logit}(P_Y)$, so that $\\exp(L) = P_Y/(1-P_Y)$ and $1 + \\exp(L) = 1/(1-P_Y)$. Substituting these into the derivative, I obtain\n\\[\ng'(L) = \\frac{P_Y}{1-P_Y}\\,(1-P_Y)^2 = P_Y(1-P_Y) = c .\n\\]\nSince $g(L) = P_Y$, the expansion is $g(a) \\approx P_Y + c\\,(a - L) = (P_Y - cL) + ca$, which is $b + ca$ with $b$ as printed. At $P_Y = 1/2$, $L = 0$: the printed fraction is $\\exp(0)/[1+0]^2 = 1$, and $c = 1/4$.", "problem": "The coefficient of $a - \\operatorname{logit}(P_Y)$ in the expansion is the derivative of $\\operatorname{logit}^{-1}$ at $\\operatorname{logit}(P_Y)$. The printed fraction is not this derivative, because its denominator has $\\operatorname{logit}(P_Y)$ where $\\exp[\\operatorname{logit}(P_Y)]$ belongs. At $P_Y = 1/2$ the printed fraction equals $1$, and the derivative equals $1/4$. The last bracket prints $P_y$ for $P_Y$. The constants $b$ and $c$ that follow the display are those of the correct expansion, and both slips are confined to the display.", "what_survives": "The definitions of $b$ and $c$ and the calculation that follows them are unaffected; the notation of that calculation is the subject of \\fid{P015-E011}. Two symbols change in one display." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E010", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the lower part of the left column of page 265. It begins: Using the linearized inverse logit function, we may define approximate estimating equations. The display reads: the sum over i of psi(Y_i, W_i, theta-hat) is approximately the sum over i of psi-tilde(Y_i, W_i, theta-hat), which equals the sum over i of [W_i(Y_i - b - c W_i theta-hat)], which equals the zero vector of size 1 by (K+2). Red outlines surround the summand W_i(Y_i - b - c W_i theta-hat) and the final equality with the zero vector. Below: Without loss of generality, we may reframe the problem by defining an alternative coefficient vector theta-tilde = ((b + c alpha-hat), c beta-hat, c gamma-hat)', so that the sum of psi-tilde equals the sum of [W_i(Y_i - W_i theta-tilde)] equals zero, which are precisely the estimating equations for the OLS estimator.", "caption": "Appendix C, p. 265 (PDF p. 16), left column: the approximate estimating equations and the reparametrization. The outlines mark the linearized summand evaluated at $\\hat\\theta$ and the “$= \\mathbf{0}_{1\\times(K+2)}$” that follows it.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E011.png" } ], "formal_release": null, "formal_statements": [], "headline": "The symbol $\\hat\\theta$ is defined as the root of the exact logistic score and is then used for the root of the linearized score; the two roots differ, and the equality with the OLS estimating equations holds for the root of the linearized score.", "id": "P015-E011", "location": "Appendix C, the display introduced by “we may define approximate estimating equations” and the display that follows it, p. 265 (PDF p. 16), left column", "pdf_page": 16, "printed_page": "265", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "The ML estimator for $\\hat\\theta$ solves the following estimating equation (Liang and Zeger 1986):\n\\[\n\\sum_{i=1}^{n} \\psi(Y_i, W_i, \\hat\\theta) = \\sum_{i=1}^{n} \\bigl( W_i\\{Y_i - \\operatorname{logit}^{-1}(W_i\\hat\\theta)\\} \\bigr) = \\mathbf{0}_{1\\times(K+2)},\n\\]\n[\\ldots] Using the linearized $\\operatorname{logit}^{-1}(.)$ function, we may define approximate estimating equations:\n\\[\n\\begin{aligned}\n\\sum_{i=1}^{n} \\psi(Y_i, W_i, \\hat\\theta) &\\approx \\sum_{i=1}^{n} \\tilde\\psi(Y_i, W_i, \\hat\\theta)\\\\\n&= \\sum_{i=1}^{n} \\bigl[ W_i(Y_i - b - c W_i \\hl{\\hat\\theta}) \\bigr]\\\\\n&{}\\hl{= \\mathbf{0}_{1\\times(K+2)}}.\n\\end{aligned}\n\\]\nWithout loss of generality, we may reframe the problem by defining an alternative coefficient vector $\\tilde\\theta = ((b + c\\hat\\alpha_\\ell) \\;\\; c\\hat\\beta_\\ell \\;\\; c\\hat\\gamma_\\ell)'$, so that\n\\[\n\\sum_{i=1}^{n} \\tilde\\psi(Y_i, W_i, \\hat\\theta) = \\sum_{i=1}^{n} \\bigl[ W_i(Y_i - W_i\\tilde\\theta) \\bigr] = \\mathbf{0}_{1\\times(K+2)},\n\\]\nwhich are precisely the estimating equations for the OLS estimator.", "corrected": "Let $\\hl{\\hat\\theta_{\\mathrm{lin}}}$ solve the linearized estimating equations\n\\[\n\\sum_{i=1}^{n} \\bigl[ W_i(Y_i - b - c W_i \\hl{\\hat\\theta_{\\mathrm{lin}}}) \\bigr] = \\mathbf{0}_{1\\times(K+2)} ,\n\\]\nand define $\\tilde\\theta = b e_1 + c\\,\\hat\\theta_{\\mathrm{lin}}$, where $e_1 = (1 \\;\\; 0 \\;\\; \\cdots \\;\\; 0)'$. Then $\\sum_{i=1}^{n} [ W_i(Y_i - W_i\\tilde\\theta) ] = \\mathbf{0}_{1\\times(K+2)}$, which are the estimating equations for the OLS estimator. The vector $\\hat\\theta_{\\mathrm{lin}}$ is the first-order surrogate for the ML root $\\hat\\theta$, so that $\\hat\\beta_{\\mathrm{OLS}} \\approx c\\,\\hat\\beta_\\ell$. If the linearized score is evaluated at $\\hat\\theta$ itself, the last equality becomes approximate unless the Taylor remainder is included.", "intended_meaning": "I read the passage as defining a linearized score and solving it. If $\\hat\\theta_{\\mathrm{lin}}$ denotes the root of the linearized score and $e_1 = (1, 0, \\ldots, 0)'$, then $b e_1 + c\\,\\hat\\theta_{\\mathrm{lin}}$ is exactly the OLS estimate, and $\\hat\\theta_{\\mathrm{lin}}$ serves as a first-order surrogate for the ML root. This reading is correct, and the passage itself speaks of ``approximate estimating equations'' and of a ``first-order approximation''. Read literally, $\\hat\\theta$ keeps its definition as the ML root, and the ML root does not make the linearized score zero. For example, suppose that 2 of 10 controls and 8 of 10 treated units are successes, and that the regressors are a constant and $D_i$. Then the ML root is $(-\\log 4,\\, 2\\log 4)$ and the linearized score at this root is $(0,\\, 3 - \\log 32)$. The root of the linearized score is $(-6/5,\\, 12/5)$. The OLS slope is $3/5$, and $c$ times the ML slope is $(\\log 4)/2$, about $0.693$." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E011", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of the foot of the left column of page 264. It begins: Now we consider the leverage. The leverage for unit i is h_ii = W_i (W'W)^(-1) W_i', where W_i and W are defined as in Appendix A. After normalizing by n, straightforward mathematical calculations obtain. Four displayed lines follow: E[n h_ii given D_i = 0, X_i = 0] = 7.43; E[n h_ii given D_i = 1, X_i = 0] = 4.21; E[n h_ii given D_i = 0, X_i = 1] = 1.71; E[n h_ii given D_i = 1, X_i = 1] = 4.21. Then: Applying the law of iterated expectations, E[n h_ii given X_i = 0] = 4.86, E[n h_ii given X_i = 1] = 2.38. Red outlines surround the left-hand sides of the six displayed equations, that is, the six conditional expectations of n h_ii.", "caption": "Appendix B, p. 264 (PDF p. 15), left column: the definition of the leverage and the six displayed values. The outlines mark the six conditional expectations of $n \\cdot h_{ii}$.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E012.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T20", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T20" } ], "headline": "The six values printed as conditional expectations of $n \\cdot h_{ii}$ are population values of the scaled leverage, or exact values for a fixed design with the stated proportions; they are not expectations of the leverage in a random sample of size $n$, and all six values are correct.", "id": "P015-E012", "location": "Appendix B, the displays that follow “Now we consider the leverage”, p. 264 (PDF p. 15), left column", "pdf_page": 15, "printed_page": "264", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P015-T20" ], "text": { "as_printed": "The leverage for unit $i$ is\n\\[\nh_{ii} = W_i(\\mathbf{W}'\\mathbf{W})^{-1}W_i',\n\\]\nwhere $W_i$ and $\\mathbf{W}$ are defined as in Appendix A. After normalizing by $n$, straightforward mathematical calculations obtain\n\\[\n\\begin{aligned}\n{}\\hl{\\mathrm{E}[n \\cdot h_{ii} \\mid D_i = 0, X_i = 0]} &= 7.43,\\\\\n{}\\hl{\\mathrm{E}[n \\cdot h_{ii} \\mid D_i = 1, X_i = 0]} &= 4.21,\\\\\n{}\\hl{\\mathrm{E}[n \\cdot h_{ii} \\mid D_i = 0, X_i = 1]} &= 1.71,\\\\\n{}\\hl{\\mathrm{E}[n \\cdot h_{ii} \\mid D_i = 1, X_i = 1]} &= 4.21.\n\\end{aligned}\n\\]\nApplying the law of iterated expectations,\n\\[\n\\begin{aligned}\n{}\\hl{\\mathrm{E}[n \\cdot h_{ii} \\mid X_i = 0]} &= 4.86,\\\\\n{}\\hl{\\mathrm{E}[n \\cdot h_{ii} \\mid X_i = 1]} &= 2.38,\n\\end{aligned}\n\\]\nso that units with $X_i = 1$ have, on average, 51\\% \\emph{less} leverage than units with $X_i = 0$ have.", "corrected": "Let $Q = \\mathrm{E}[W_i'W_i]$ and define the population scaled leverage $\\hl{\\ell_i = W_i Q^{-1} W_i'}$, the limit of $n \\cdot h_{ii}$. Straightforward calculations give, to two decimal places,\n\\[\n\\begin{aligned}\n\\mathrm{E}[\\hl{\\ell_i} \\mid D_i = 0, X_i = 0] &= 7.43, & \\mathrm{E}[\\hl{\\ell_i} \\mid D_i = 1, X_i = 0] &= 4.21,\\\\\n\\mathrm{E}[\\hl{\\ell_i} \\mid D_i = 0, X_i = 1] &= 1.71, & \\mathrm{E}[\\hl{\\ell_i} \\mid D_i = 1, X_i = 1] &= 4.21,\n\\end{aligned}\n\\]\nand $\\mathrm{E}[\\hl{\\ell_i} \\mid X_i = 0] = 4.86$, $\\mathrm{E}[\\hl{\\ell_i} \\mid X_i = 1] = 2.38$. These are population values of the scaled leverage, or exact values of $n \\cdot h_{ii}$ for a fixed design with the stated proportions, and not exact expectations for a random design of finite size. The population values and the comparisons of 22\\% and 51\\% remain correct.", "intended_meaning": "The example compares the scaled leverage of the four cells of $(D_i, X_i)$ in the population whose joint distribution is printed on p.~263. With $Q = \\mathrm{E}[W_i'W_i]$, the population scaled leverage is $\\ell_i = W_i Q^{-1} W_i'$. I obtain the exact values $52/7$, $59/14$, $12/7$ and $59/14$ in the four cells, with means $34/7$ at $X_i = 0$ and $50/21$ at $X_i = 1$, and these round to the printed numbers. A fixed design of $n = 20$ units with cell counts $1$, $4$, $11$ and $4$ has the same values of $n \\cdot h_{ii}$ exactly. Since the printed values do not depend on $n$, this reading is clear. Read literally, the notation refers to the leverage in a random sample of size $n$, and the values are not its expectations. For example, for $n = 3$ every sample of full rank has $n \\cdot h_{ii} = 3$ in every cell. The comparisons of 22\\% and 51\\% are correct." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E012", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the display 'Pr[Y 1 = 1] = inverse logit of alpha plus beta D i plus X i gamma', with three lines of text above and three below. A red box marks the symbol Y with subscript 1 on the left side.", "caption": "Display of the logistic regression model, p. 257 (PDF p. 8), left column. The marked outcome carries the subscript $1$.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E013.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T14", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T14" } ], "headline": "In the logistic regression model the outcome carries the subscript $i$, not $1$, and the probability is conditional on $D_i$ and $X_i$.", "id": "P015-E013", "location": "Section “Maximum Likelihood Estimation and Generalized Linear Models”, p. 257 (PDF p. 8), left column, display of the logistic regression model", "pdf_page": 8, "printed_page": "257", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P015-T14" ], "text": { "as_printed": "\\[\n\\Pr[Y_{\\hl{1}} = 1] = \\mathrm{logit}^{-1}(\\alpha_\\ell + \\beta_\\ell D_i + X_i\\gamma_\\ell),\n\\]", "corrected": "\\[\n\\Pr[Y_{\\hl{i}} = 1 \\hl{\\mid D_i, X_i}] = \\mathrm{logit}^{-1}(\\alpha_\\ell + \\beta_\\ell D_i + X_i\\gamma_\\ell),\n\\]", "evidence": "This is a check of the page. At 300 dpi the subscript is the numeral $1$ and no conditioning is printed. Appendix~C, p.~265, prints $\\Pr[Y_i = 1 \\mid D_i, X_i]$ with the same right side.", "problem": "The left side names the outcome of unit $1$ and conditions on nothing, while the right side depends on the treatment and the covariates of unit $i$. One unconditional probability for unit $1$ cannot in general equal the different values that the right side takes as $i$ varies. The intended model is the conditional model for unit $i$, which Appendix~C prints on p.~265 with the same right side.", "what_survives": "This correction changes no fitted model or numerical result." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E013", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of a four-line reference entry: Angrist, Joshua D., and Alan B. Krueger. 1999. Empirical Strategies in Labor Economics. In Handbook of Labor Economics (Vol. 3), ed. Orley C. Ahsenfelter and David Card. Amsterdam: North Holland, 1277-366. A red outline surrounds the name Ahsenfelter.", "caption": "References, p. 265 (PDF p. 16): the entry Angrist and Krueger (1999). The outline marks “Ahsenfelter”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E014.png" }, { "alt": "Crop of a five-line reference entry: Barron, Reuben M., and David A. Kenny. 1986. The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations. Journal of Personality and Social Psychology 51(6): 1173-82. A red outline surrounds the name Barron at the start of the entry.", "caption": "References, p. 265 (PDF p. 16): the entry Barron and Kenny (1986). The outline marks “Barron”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E014-b.png" }, { "alt": "Crop of footnote 12, six lines: Background characteristics that modify the effect of a treatment are known in the methodology literature as moderators (Barron and Kenny 1986). In other words, they are covariates for which there are nonzero interaction effects with the causal factor. VanderWeele and Robins (2007) discuss challenges for causal inference when studying moderators (which they term effect modifiers). A red outline surrounds the name Barron at the end of the second line.", "caption": "Footnote 12, p. 255 (PDF p. 6). The outline marks “Barron”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E014-c.png" }, { "alt": "Crop of a three-line reference entry: Campbell, Donald T. 1957. Factors Relevant to the Validity of Experiments in Social Settings. Psychology Bulletin 45(4): 297-312. Red outlines surround the word Psychology and the number 45.", "caption": "References, p. 265 (PDF p. 16): the entry Campbell (1957). The outlines mark “Psychology” and the volume number “45”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E014-d.png" }, { "alt": "Crop of a three-line reference entry: Van der Laan, Mark, and Sherri Rose. 2011. Targeted Learning: Causal Inference for Obervational and Experimental Data. New York: Springer. A red outline surrounds the misspelt word Obervational.", "caption": "References, p. 267 (PDF p. 18): the entry Van der Laan and Rose (2011). The outline marks “Obervational”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-E014-e.png" } ], "formal_release": null, "formal_statements": [], "headline": "Four reference entries contain misprints: “Ahsenfelter” should read “Ashenfelter”, “Barron” should read “Baron” (also in footnote 12), Campbell (1957) appeared in Psychological Bulletin 54(4), and “Obervational” should read “Observational”.", "id": "P015-E014", "location": "References, p. 265 (PDF p. 16): the entries Angrist and Krueger (1999), Barron and Kenny (1986) and Campbell (1957); footnote 12, p. 255 (PDF p. 6); References, p. 267 (PDF p. 18): the entry Van der Laan and Rose (2011)", "pdf_page": 16, "printed_page": "265", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{itemize}\n\\item p.~265: Angrist, Joshua D., and Alan B. Krueger. 1999. ``Empirical Strategies in Labor Economics.'' In \\emph{Handbook of Labor Economics} (Vol. 3), ed. Orley C. \\hl{Ahsenfelter} and David Card. Amsterdam: North Holland, 1277--366.\n\\item p.~265: \\hl{Barron}, Reuben M., and David A. Kenny. 1986. ``The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations.'' \\emph{Journal of Personality and Social Psychology} 51(6): 1173--82.\n\\item p.~255, footnote~12: Background characteristics that modify the effect of a treatment are known in the methodology literature as ``moderators'' (\\hl{Barron} and Kenny 1986).\n\\item p.~265: Campbell, Donald T. 1957. ``Factors Relevant to the Validity of Experiments in Social Settings.'' \\emph{\\hl{Psychology} Bulletin} \\hl{45}(4): 297--312.\n\\item p.~267: Van der Laan, Mark, and Sherri Rose. 2011. \\emph{Targeted Learning: Causal Inference for \\hl{Obervational} and Experimental Data.} New York: Springer.\n\\end{itemize}", "corrected": "\\begin{itemize}\n\\item Angrist, Joshua D., and Alan B. Krueger. 1999. ``Empirical Strategies in Labor Economics.'' In \\emph{Handbook of Labor Economics} (Vol. 3), ed. Orley C. \\hl{Ashenfelter} and David Card. Amsterdam: North Holland, 1277--366.\n\\item \\hl{Baron}, Reuben M., and David A. Kenny. 1986. ``The Moderator-Mediator Variable Distinction in Social Psychological Research: Conceptual, Strategic, and Statistical Considerations.'' \\emph{Journal of Personality and Social Psychology} 51(6): 1173--82.\n\\item Footnote~12: \\ldots known in the methodology literature as ``moderators'' (\\hl{Baron} and Kenny 1986).\n\\item Campbell, Donald T. 1957. ``Factors Relevant to the Validity of Experiments in Social Settings.'' \\emph{\\hl{Psychological} Bulletin} \\hl{54}(4): 297--312.\n\\item Van der Laan, Mark, and Sherri Rose. 2011. \\emph{Targeted Learning: Causal Inference for \\hl{Observational} and Experimental Data.} New York: Springer.\n\\end{itemize}", "evidence": "I checked the pages against the publishers' pages and the entries of the National Library of Medicine. The PubMed entry for Campbell (1957) reads ``1957 Jul;54(4):297-312'' (\\href{https://pubmed.ncbi.nlm.nih.gov/13465924/}{PubMed 13465924}), and the NLM Catalog has the title of the journal as ``Psychological bulletin''. The PubMed entry for the article of 1986 lists the authors as R M Baron and D A Kenny (\\href{https://pubmed.ncbi.nlm.nih.gov/3806354/}{PubMed 3806354}). Elsevier lists the editors of volume~3 of the \\emph{Handbook of Labor Economics} as Orley Ashenfelter and David Card (\\href{https://shop.elsevier.com/books/handbook-of-labor-economics/ashenfelter/978-0-444-82289-5}{ISBN 978-0-444-82289-5}). On Springer's page the title is \\emph{Targeted Learning: Causal Inference for Observational and Experimental Data} (\\href{https://link.springer.com/book/10.1007/978-1-4419-9782-1}{doi:10.1007/978-1-4419-9782-1}), and the paper's own entry for Petersen et al.\\ (2011) on p.~266 prints the same title with ``Observational''.", "problem": "Two names, a journal title with its volume number, and a word of a book title are misprinted in the reference list, and one of the names recurs in footnote~12. Each cited work can still be located.", "what_survives": "Nothing in the argument, the formulas or the numbers depends on these entries. Five passages change: four reference entries and footnote~12." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-E014", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Nine lines of the left column of page 253. A red box marks the words The only way, which open the sentence: The only way that our assumptions could be more restrictive would be to assume either that causal effects are constant for all units or that all sources of effect heterogeneity have been modeled.", "caption": "Page 253 (PDF p. 4), left column. The marked words open the sentence that names two stronger assumptions.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N003.png" } ], "formal_release": null, "formal_statements": [], "headline": "The sentence that begins “The only way” names two stronger assumptions under which regression recovers the average effect; they are examples, and not the only such assumptions.", "id": "P015-N003", "location": "p. 253 (PDF p. 4), left column, first full paragraph", "pdf_page": 4, "printed_page": "253", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\hl{The only way} that our assumptions could be more restrictive would be to assume either that causal effects are constant for all units or that all sources of effect heterogeneity have been modeled.", "corrected": "\\hl{Two ways to make} our assumptions more restrictive would be to assume either that causal effects are constant for all units or that all sources of effect heterogeneity have been modeled.", "intended_meaning": "The paragraph argues that the assumptions of the paper are as favourable to regression as possible, and the sentence names two stronger assumptions under which the unequal weights no longer matter. Read in this way the sentence is correct, and no result uses it. Taken literally, ``the only way'' says too much. Under Conditions (1)--(3), Expressions (7) and (8) imply that the regression coefficient converges to $\\mathrm{E}[\\tau_i]+\\operatorname{Cov}(v(X_i),t(X_i))/\\mathrm{E}[v(X_i)]$, where $v(X_i)=\\operatorname{Var}[D_i\\mid X_i]$ and $t(X_i)=\\mathrm{E}[\\tau_i\\mid X_i]$. The limit is the average effect whenever this covariance is zero, for instance when the conditional variance of the treatment is constant, although the effects vary and are not modeled. In an exact example with $\\operatorname{Var}[D_i\\mid X_i]=2/3$ for every value of $X_i$, the regression coefficient and the average effect both equal $7/5$." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-N003", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Seven lines of the right column of page 254, followed by Expression (7). A red box marks the words Conditions (1) and (3) in the sentence: Applying results from partial regression (Greene 2008, 27 to 29) along with Conditions (1) and (3), multiple regression generates a weighted average of causal effects of the form. Expression (7) below states that beta hat converges in probability to the ratio of E of w i tau i to E of w i, where w i is the square of D i minus E of D i given X i.", "caption": "Page 254 (PDF p. 5), right column: the sentence that introduces Expression (7). The marked cross-reference omits Condition (2).", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N004.png" }, { "alt": "Seven lines of the left column of page 256. A red box marks the word density in the phrase: consistent estimation of each of the covariate cumulative density functions in the effective sample.", "caption": "Page 256 (PDF p. 7), left column. The marked word should be “distribution”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N004-b.png" }, { "alt": "The sentence The OLS estimates may be written as, followed by a display: a column vector with entries alpha hat, beta hat, gamma hat 1, vertical dots and gamma hat k equals the inverse of W prime W times W prime Y. A red box marks the last entry, gamma hat with a lower-case subscript k.", "caption": "Page 262 (PDF p. 13), right column, proof of Theorem A.1. The marked entry carries the subscript $k$; the number of covariates is $K$.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N004-c.png" }, { "alt": "A four-line display from the proof of Theorem A.1, which shows that E of D i Y 0 i minus E of V i omega Y 0 i equals zero. A red box marks its equation number, (15). The next sentence reads: We now consider the second term in the right-hand side of Expression (13), where.", "caption": "Page 263 (PDF p. 14), left column, proof of Theorem A.1. The marked number follows display (13), which is higher on the same page; no display is numbered (14).", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N004-d.png" } ], "formal_release": null, "formal_statements": [ { "id": "P015-T15", "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T15" } ], "headline": "Four small slips: the derivation of Expression (7) uses Conditions (1)–(3), not only Conditions (1) and (3); the last entry of the OLS coefficient vector on p. 262 is $\\hat\\gamma_K$; the display numbered (15) on p. 263 should be numbered (14); and the covariate functions on p. 256 are cumulative distribution functions.", "id": "P015-N004", "location": "p. 254 (PDF p. 5), sentence before Expression (7); p. 256 (PDF p. 7), left column; p. 262 (PDF p. 13), display after “The OLS estimates may be written as”; p. 263 (PDF p. 14), display numbered (15)", "pdf_page": 5, "printed_page": "254", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P015-T15" ], "text": { "as_printed": "Page 254: Applying results from partial regression (Greene 2008, 27--29) along with \\hl{Conditions (1) and (3)}, multiple regression generates a weighted average of causal effects of the form\n\\[\n\\hat\\beta\\xrightarrow{p}\\frac{\\mathrm{E}[w_i\\tau_i]}{\\mathrm{E}[w_i]},\\ \\text{where } w_i=(D_i-\\mathrm{E}[D_i\\mid X_i])^2.\\qquad(7)\n\\]\n\nPage 256: Similar calculations---redefining $Z_i$ appropriately---permit consistent estimation of each of the covariate cumulative \\hl{density} functions in the effective sample.$^{14}$\n\nPage 262: The OLS estimates may be written as\n\\[\n\\begin{pmatrix}\\hat\\alpha\\\\ \\hat\\beta\\\\ \\hat\\gamma_1\\\\ \\vdots\\\\ \\hat\\gamma_{\\hl{k}}\\end{pmatrix}=(\\mathbf{W}'\\mathbf{W})^{-1}\\mathbf{W}'Y.\n\\]\n\nPage 263:\n\\[\n\\begin{aligned}\n\\mathrm{E}[D_iY_{0i}]-\\mathrm{E}[(V_i\\omega)Y_{0i}]&=\\mathrm{E}_X[\\mathrm{E}[D_iY_{0i}\\mid X_i]]-\\mathrm{E}_X[(V_i\\omega)\\mathrm{E}[Y_{0i}\\mid X_i]]\\\\\n&=\\mathrm{E}_X[\\mathrm{E}[D_i\\mid X_i]\\mathrm{E}[Y_{0i}\\mid X_i]]-\\mathrm{E}_X[(V_i\\omega)\\mathrm{E}[Y_{0i}\\mid X_i]]=0.\\qquad\\hl{(15)}\n\\end{aligned}\n\\]", "corrected": "Page 254: Applying results from partial regression (Greene 2008, 27--29) along with \\hl{Conditions (1)--(3)}, multiple regression generates a weighted average of causal effects of the form\n\\[\n\\hat\\beta\\xrightarrow{p}\\frac{\\mathrm{E}[w_i\\tau_i]}{\\mathrm{E}[w_i]},\\ \\text{where } w_i=(D_i-\\mathrm{E}[D_i\\mid X_i])^2.\\qquad(7)\n\\]\nHere Condition (1) is $Y_i(d)=Y_i(0)+\\tau_i\\cdot d$, Condition (2) is $(Y_i(0),\\tau_i)\\perp\\!\\!\\!\\perp D_i\\mid X_i$, and Condition (3) is $\\mathrm{E}[D_i\\mid X_i]=V_i\\omega$ with $V_i=(1\\ \\ X_i)$. Theorem A.1 states the result under these three conditions and the regularity condition that the OLS solutions converge to a limit.\n\nPage 256: Similar calculations---redefining $Z_i$ appropriately---permit consistent estimation of each of the covariate cumulative \\hl{distribution} functions in the effective sample.$^{14}$\n\nPage 262: The OLS estimates may be written as\n\\[\n\\begin{pmatrix}\\hat\\alpha\\\\ \\hat\\beta\\\\ \\hat\\gamma_1\\\\ \\vdots\\\\ \\hat\\gamma_{\\hl{K}}\\end{pmatrix}=(\\mathbf{W}'\\mathbf{W})^{-1}\\mathbf{W}'Y.\n\\]\n\nPage 263: the display that ends with ``$=0.$'' is numbered \\hl{(14)}.", "evidence": "The evidence is a check of the pages and, for the cross-reference, a derivation. In the proof on p.~263, the term of Expression (13) that involves the baseline outcome has the limit $(\\mathrm{E}[D_iY_{0i}]-\\mathrm{E}[(V_i\\omega)Y_{0i}])/\\mathrm{E}[w_i]$. By Condition (2) and then Condition (3), I have\n\\[\n\\mathrm{E}[D_iY_{0i}\\mid X_i]=\\mathrm{E}[D_i\\mid X_i]\\,\\mathrm{E}[Y_{0i}\\mid X_i]=(V_i\\omega)\\,\\mathrm{E}[Y_{0i}\\mid X_i],\n\\]\nand the numerator of this limit is zero. For the second term of Expression (13), the same two conditions imply that\n\\[\n\\mathrm{E}[(D_i-V_i\\omega)\\tau_i\\mid X_i]=\\mathrm{E}[D_i-V_i\\omega\\mid X_i]\\,\\mathrm{E}[\\tau_i\\mid X_i]=0,\n\\]\nand this term vanishes in the limit. Conditions (1) and (3) alone do not give either factorization. The proof in the paper names unconfoundedness at both steps.", "problem": "The sentence before Expression (7) cites Conditions (1) and (3) as the conditions that the result needs, but the derivation also uses Condition (2), unconfoundedness, which the proof of Theorem A.1 on p.~263 invokes twice. The three other slips are typographical. In the OLS coefficient vector on p.~262 the last entry carries the subscript $k$, while the number of covariates is $K$ everywhere else; the numbered displays of Appendix A run (12), (13), (15), with no (14); and p.~256 has ``cumulative density functions'' for cumulative distribution functions.", "what_survives": "Nothing else changes. Conditions (1)--(3) are stated on p.~253 and Theorem A.1 lists unconfoundedness among its hypotheses. The correction adds no assumption, and Expression (7) stands as printed. Two neighbouring points of notation are not errors. The symbol $\\gamma$ denotes the intercept in Model (5) and the coefficient vector on $X_i$ in Model (6), and each use is defined where it appears. The symbol $Y_{0i}$ in Theorem A.1 denotes the baseline outcome $Y_i(0)$ of the main text." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-N004", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of seven lines of the left column. A red box marks the words 'FDI variable' in the sentence 'We first regress the FDI variable on the various control variables (including the country and decade fixed effects) and then extract the residuals'.", "caption": "P. 258 (PDF p. 9), left column, paragraph beginning “Of course”. The marked words name the outcome where the treatment is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N008.png" } ], "formal_release": null, "formal_statements": [], "headline": "The weights come from regressing the regime type variable, not the FDI variable, on the control variables; the published erratum of 2017 already makes this correction.", "id": "P015-N008", "location": "Section “Effects of Regime Type on FDI”, p. 258 (PDF p. 9), left column, paragraph beginning “Of course”", "pdf_page": 9, "printed_page": "258", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "We first regress the \\hl{FDI} variable on the various control variables (including the country and decade fixed effects) and then extract the residuals.", "corrected": "We first regress the \\hl{regime type} variable on the various control variables (including the country and decade fixed effects) and then extract the residuals.", "evidence": "This is a check of the page against Expression~(7), against the replication script, which regresses \\texttt{regime} on the controls and squares the residuals, and against the erratum (\\emph{American Journal of Political Science} 61(2), April 2017, p.~504), which prints the replacement sentence.", "problem": "FDI is the outcome of the regression. The multiple regression weights are the squared residuals of the treatment, which here is regime type: Expression~(7) on p.~254 defines $w_i=(D_i-\\mathrm{E}[D_i\\mid X_i])^2$, and the sentences before this one speak of the unexplained variation in regime type.", "what_survives": "Since the weights were computed from the regime type variable, no number changes. The correction does not bear on how the weights are then aggregated by country, which is the subject of \\fid{P015-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-N008", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the top of page 260 with the title of Table 1 and its column headings. The title reads: Table 1, Summary Statistics for the Nominal and Effective Samples Relevant to Model (4) in Table 4A of Gerber and Huber Gerber and Huber (2010). A red outline surrounds the second occurrence of the words Gerber and Huber. Below the title are the headings Nominal Sample and Effective Sample, each with the columns Mean and S.D., and the heading Variable.", "caption": "Table 1, p. 260 (PDF p. 11): the title and the column headings. The outline marks the second “Gerber and Huber”.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N010.png" } ], "formal_release": null, "formal_statements": [], "headline": "The title of Table 1 prints “Gerber and Huber” twice; it should end “of Gerber and Huber (2010)”, as the published erratum of 2017 states.", "id": "P015-N010", "location": "Table 1, title, p. 260 (PDF p. 11)", "pdf_page": 11, "printed_page": "260", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "TABLE 1 Summary Statistics for the Nominal and Effective Samples Relevant to Model (4) in Table 4A of Gerber and Huber \\hl{Gerber and Huber} (2010)", "corrected": "TABLE 1 Summary Statistics for the Nominal and Effective Samples Relevant to Model (4) in Table 4A \\hl{of Gerber and Huber (2010)}", "evidence": "I checked the page and the published erratum (\\emph{American Journal of Political Science} 61(2), April 2017, p.~504), which notes an error in the title for Table~1 and says that it should simply read ``\\ldots of Gerber and Huber (2010).''", "problem": "The names of the authors of the study are printed twice in succession. The table concerns Model (4) in Table 4A of one study, Gerber and Huber (2010).", "what_survives": "No entry of Table~1 changes, and the erratum corrects none. Nothing else in the paper depends on the title." }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-N010", "work": { "arxiv_id": null, "doi": "10.1111/ajps.12185", "title": "Does Regression Produce Representative Estimates of Causal Effects?", "url": "https://pmaronow.github.io/llm-audit/works/P015/" }, "work_id": "P015" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the first eight lines of text and one displayed line at the top of the left column of page 261. The text before the display ends: Formally, the positivity assumption is as follows. A displayed line follows, reading Pr[D_i = d given X_i = x] greater than 0, Pr[D_i = d-prime given X_i = x] greater than 0, and a red outline surrounds this display. Below it: for all values of x in the target population represented by the nominal sample.", "caption": "p. 261 (PDF p. 12), top of the left column. The outline marks the display of the positivity assumption.", "url": "https://pmaronow.github.io/llm-audit/works/P015/exhibits/P015-N011.png" } ], "formal_release": null, "formal_statements": [], "headline": "The positivity display uses point probabilities for a discrete treatment; for a continuous treatment, both chosen values must lie in the support of the conditional treatment distribution.", "id": "P015-N011", "location": "Section “Estimating Representative Causal Effects”, display of the positivity assumption, p. 261 (PDF p. 12), left column", "pdf_page": 12, "printed_page": "261", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Formally, the positivity assumption is as follows:\n\\[\n\\hl{\\Pr[D_i = d \\mid X_i = x]} > 0, \\qquad \\hl{\\Pr[D_i = d' \\mid X_i = x]} > 0\n\\]\nfor all values of $x$ in the target population represented by the nominal sample.", "corrected": "For a discrete treatment, $\\Pr[D_i = d \\mid X_i = x] > 0$ and $\\Pr[D_i = d' \\mid X_i = x] > 0$ for all values of $x$ in the target population represented by the nominal sample. \\hl{For a continuous treatment, require both $d$ and $d'$ to be in the support of the conditional treatment distribution: for every $\\varepsilon>0$, $\\Pr[|D_i-d|<\\varepsilon\\mid X_i=x]>0$ and $\\Pr[|D_i-d'|<\\varepsilon\\mid X_i=x]>0$ for all such $x$.} A conditional density version continuous and positive at both target values is sufficient for this support condition. Pointwise positivity of an unspecified density version is insufficient. Identification at the two points also requires suitable regularity of the response in the treatment value.", "intended_meaning": "I read the condition as requiring both chosen treatment values to be available at every covariate profile of the target population, as the display states for a discrete treatment. The paper also allows a continuous treatment (pp.~252--253), and then every point probability is zero even when both values are available. For example, if $D_i$ is uniform on $(0,1)$ and independent of $X_i$, then $\\Pr[D_i = d \\mid X_i = x] = 0$ for every $d\\in(0,1)$, while the usual conditional density equals $1$.\n\nFor a continuous treatment, availability means that every open neighborhood of each chosen value has positive conditional probability at the relevant covariate profile. This is a condition on the conditional treatment law. Pointwise positivity of an unspecified density version is insufficient: changing a density at isolated points leaves the law unchanged. In the uniform example, a density can be set to $1$ at the isolated value $2$ without changing the law, although $2$ remains outside the support. A conditional density version continuous and positive at both target values is sufficient for the neighborhood condition. Identification at the two points also requires suitable regularity of the response in the treatment value. 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p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6). The full canonical rare-overlap counterexample holds: at epsilon = 1/1000000, the observed-outcome IPW estimator is unbiased and consistent with the stated exact moments and deviation bounds, and the actual nonsingular population OLS coefficient is the effective causal mean. The family refutes treating approximately no weight as a positive threshold for failure of strict endpoint positivity. The estimator moments, every-sample-size bounds, and pointwise consistency are at epsilon = 1/1000000; positivity and arbitrarily small effective-profile mass are asserted separately for every 0 < epsilon < 1/2.

", "headline": "Section ‘Estimating Representative Causal Effects’: ‘when a multiple regression places (either approximately or exactly) no weight on a set of units … positivity is violated and representative causal effects cannot be estimated …’, as printed (the ‘approximately’ case)", "location": "p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6)" }, "id": "P015-T1", "text": { "body": "p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6). The full canonical rare-overlap counterexample holds: at epsilon = 1/1000000, the observed-outcome IPW estimator is unbiased and consistent with the stated exact moments and deviation bounds, and the actual nonsingular population OLS coefficient is the effective causal mean. The family refutes treating approximately no weight as a positive threshold for failure of strict endpoint positivity. The estimator moments, every-sample-size bounds, and pointwise consistency are at epsilon = 1/1000000; positivity and arbitrarily small effective-profile mass are asserted separately for every 0 < epsilon < 1/2.", "headline": "Section ‘Estimating Representative Causal Effects’: ‘when a multiple regression places (either approximately or exactly) no weight on a set of units … positivity is violated and representative causal effects cannot be estimated …’, as printed (the ‘approximately’ case)", "location": "p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T1" }, { "declarations": [ { "claim_revision_id": "catalog:dec49b4e856c46016236ae79870cb70e7ecc1d99549e3aef0e6272e1b742e5af", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": 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p. 261 (PDF p. 12), right column; positivity display on p. 261 (PDF p. 12); τi(d,d′)\\tau_i(d, d')τi​(d,d′) and (11) on p. 260 (PDF p. 11). The full canonical four-atom counterexample holds: every regression weight is positive, treatment 1 is absent at profile 1, and two observationally identical outcome worlds have different nominal and effective effects and no common consistent observed-data estimator. The counterexample concerns positivity for a fixed distinct treatment pair at every effective-population profile. It retains positive within-profile variance, so it does not refute positivity understood only as some within-profile treatment variation.

", "headline": "Section ‘Estimating Representative Causal Effects’: ‘the effective sample that multiple regression generates is one for which positivity holds by construction’, as printed", "location": "p. 261 (PDF p. 12), right column; positivity display on p. 261 (PDF p. 12); τi(d,d′)\\tau_i(d, d')τi​(d,d′) and (11) on p. 260 (PDF p. 11)" }, "id": "P015-T2", "text": { "body": "p. 261 (PDF p. 12), right column; positivity display on p. 261 (PDF p. 12); $\\tau_i(d, d')$ and (11) on p. 260 (PDF p. 11). The full canonical four-atom counterexample holds: every regression weight is positive, treatment 1 is absent at profile 1, and two observationally identical outcome worlds have different nominal and effective effects and no common consistent observed-data estimator. The counterexample concerns positivity for a fixed distinct treatment pair at every effective-population profile. It retains positive within-profile variance, so it does not refute positivity understood only as some within-profile treatment variation.", "headline": "Section ‘Estimating Representative Causal Effects’: ‘the effective sample that multiple regression generates is one for which positivity holds by construction’, as printed", "location": "p. 261 (PDF p. 12), right column; positivity display on p. 261 (PDF p. 12); $\\tau_i(d, d')$ and (11) on p. 260 (PDF p. 11)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T2" }, { "declarations": [ { "claim_revision_id": "catalog:60acca1ed65fd3f7335c5f1040f3d4b903dcc73b61285e029c15046ed3a3108f", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.fitted_zero_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L285", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:60acca1ed65fd3f7335c5f1040f3d4b903dcc73b61285e029c15046ed3a3108f", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.fitted_zero_actual_ols", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L3080", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:60acca1ed65fd3f7335c5f1040f3d4b903dcc73b61285e029c15046ed3a3108f", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.PaperSpecializations.zero_sample_ols_arithmetic", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperSpecializations.lean#L145", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:60acca1ed65fd3f7335c5f1040f3d4b903dcc73b61285e029c15046ed3a3108f", "conditional_on": [], "credit_reason": "The target explicitly concerns estimated sample weights. 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p. 261 (PDF p. 12); estimated weights w^i=D~i2\\hat w_i = \\tilde D_i^2w^i​=D~i2​ on p. 255 (PDF p. 6). The full canonical finite-sample counterexample holds: a sample of probability 1/256 from a binary population with positive overlap has fitted weights (0,0,1/4,1/4), nonsingular controls Gram matrix, and full-regression Gram determinant 2. The counterexample concerns inferring population positivity failure from exactly zero estimated sample-profile weights. Its population conditional variances remain 1/4; it does not contradict the exact-zero implication for population profile weights.

", "headline": "Section ‘Estimating Representative Causal Effects’: ‘places (either approximately or exactly) no weight … this result implies that positivity is violated’, as printed (an exactly zero estimated weight in a sample)", "location": "p. 261 (PDF p. 12); estimated weights w^i=D~i2\\hat w_i = \\tilde D_i^2w^i​=D~i2​ on p. 255 (PDF p. 6)" }, "id": "P015-T3", "text": { "body": "p. 261 (PDF p. 12); estimated weights $\\hat w_i = \\tilde D_i^2$ on p. 255 (PDF p. 6). The full canonical finite-sample counterexample holds: a sample of probability 1/256 from a binary population with positive overlap has fitted weights (0,0,1/4,1/4), nonsingular controls Gram matrix, and full-regression Gram determinant 2. The counterexample concerns inferring population positivity failure from exactly zero estimated sample-profile weights. Its population conditional variances remain 1/4; it does not contradict the exact-zero implication for population profile weights.", "headline": "Section ‘Estimating Representative Causal Effects’: ‘places (either approximately or exactly) no weight … this result implies that positivity is violated’, as printed (an exactly zero estimated weight in a sample)", "location": "p. 261 (PDF p. 12); estimated weights $\\hat w_i = \\tilde D_i^2$ on p. 255 (PDF p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T3" }, { "declarations": [ { "claim_revision_id": "catalog:c0acde11d7d113c08bbbe271a061301f7b9c0cbeb295e6b4c88c2fdc7f046a19", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.logistic_exp", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L315", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c0acde11d7d113c08bbbe271a061301f7b9c0cbeb295e6b4c88c2fdc7f046a19", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.logistic_logit", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L323", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c0acde11d7d113c08bbbe271a061301f7b9c0cbeb295e6b4c88c2fdc7f046a19", "conditional_on": [], "credit_reason": "All four formal components, including the P=1/2 evaluation of the misprinted fraction, are exported.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.corrected_logistic_taylor", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L331", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:c0acde11d7d113c08bbbe271a061301f7b9c0cbeb295e6b4c88c2fdc7f046a19", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T13", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T13" }, "html": { "body": "

p. 265 (PDF p. 16), left column. The corrected inverse-logit derivative and affine Taylor polynomial hold, and the printed fraction differs from the derivative at probability one half.

", "headline": "Appendix C: the first-order Taylor expansion of logit−1\\mathrm{logit}^{-1}logit−1 about logit(PY)\\mathrm{logit}(P_Y)logit(PY​), corrected", "location": "p. 265 (PDF p. 16), left column" }, "id": "P015-T13", "text": { "body": "p. 265 (PDF p. 16), left column. The corrected inverse-logit derivative and affine Taylor polynomial hold, and the printed fraction differs from the derivative at probability one half.", "headline": "Appendix C: the first-order Taylor expansion of $\\mathrm{logit}^{-1}$ about $\\mathrm{logit}(P_Y)$, corrected", "location": "p. 265 (PDF p. 16), left column" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T13" }, { "declarations": [ { "claim_revision_id": "catalog:e2d950aa90fe4494249aa922a832ecd1ee17c65b1a324f05f7afe2aeaf0d8f71", "conditional_on": [], "credit_reason": "This is the local remainder clause; no resulting estimator or weight accuracy is claimed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.local_logistic_remainder", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L364", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e2d950aa90fe4494249aa922a832ecd1ee17c65b1a324f05f7afe2aeaf0d8f71", "conditional_on": [], "credit_reason": "The pointwise bound for every point in the closed real interval is equivalent to the stated supremum bound. The supremum defining M is over a nonempty compact circle strictly inside the nearest poles, so it is finite and attained.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.restricted_taylor_convergence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1795", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e2d950aa90fe4494249aa922a832ecd1ee17c65b1a324f05f7afe2aeaf0d8f71", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T12", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T12" }, "html": { "body": "

p. 257 (PDF p. 8), left column. At logit(p), for 0 < p < 1, the inverse-logit's first Taylor polynomial is p + p(1-p)(a-logit(p)), with a local quadratic remainder. The complex Taylor radius is exactly sqrt(logit(p)^2+pi^2). On every smaller real closed interval the Taylor errors have the stated uniform Cauchy bound, using the attained finite maximum on an intermediate complex circle, and this bound tends to zero. The explicit 200-observation, full-rank binary data have a unique finite logistic ML estimate with both predictors outside the disk; every Taylor degree has probability error at least 49/100 there. No accuracy of coefficients or weights is asserted.

", "headline": "p. 257: the Taylor approximation of the inverse logit for the weights, corrected (local first-order approximation)", "location": "p. 257 (PDF p. 8), left column" }, "id": "P015-T12", "text": { "body": "p. 257 (PDF p. 8), left column. At logit(p), for 0 < p < 1, the inverse-logit's first Taylor polynomial is p + p(1-p)(a-logit(p)), with a local quadratic remainder. The complex Taylor radius is exactly sqrt(logit(p)^2+pi^2). On every smaller real closed interval the Taylor errors have the stated uniform Cauchy bound, using the attained finite maximum on an intermediate complex circle, and this bound tends to zero. The explicit 200-observation, full-rank binary data have a unique finite logistic ML estimate with both predictors outside the disk; every Taylor degree has probability error at least 49/100 there. No accuracy of coefficients or weights is asserted.", "headline": "p. 257: the Taylor approximation of the inverse logit for the weights, corrected (local first-order approximation)", "location": "p. 257 (PDF p. 8), left column" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T12" }, { "declarations": [ { "claim_revision_id": "CT-01:P015-T8:accepted", "conditional_on": [], "credit_reason": "Assessed as the single-observation Bernoulli derivative used to sum the gradient.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.bernoulli_score_derivative", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L2497", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-01:P015-T8:accepted", "conditional_on": [], "credit_reason": "This row covers the constant-multiplier clause only.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.logistic_multiplier", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L409", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-01:P015-T8:accepted", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.canonical_score_derivative", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L2513", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "CT-01:P015-T8:accepted", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P015_Regression.lean:2646 concludes the vector sum of exact first Taylor polynomials, using firstTaylor as defined at P015_Regression.lean:313, whereas T8(iv) asserts an expansion of the actual nonlinear score with an O(‖δ‖²) remainder. The flagged missing conclusion is real under the claim-text standard; the record's permission to substitute a polynomial identity does not change that claim. However, the corrected target itself overstates what its hypotheses imply: twice differentiability of m at the center yields a differentiable score and a first-order little-o remainder, without guaranteeing a quadratic remainder. For example, on (-1/4,1/4), m(t)=1/2+t/4+|t|^(5/2)/16 is admissible and twice differentiable at zero, but the y=1 score m'(t)/m(t) has remainder (5/16)t^(3/2)+O(t²) for t>0 after its first-order polynomial. Thus this needs correction of the target's expansion, rather than simply demanding a proof of its present remainder under the existing hypotheses.\", 'suggested_fix': \"For this credited component, replace T8(iv)'s actual-score quadratic-remainder assertion with the exact first-Taylor-polynomial identity already authorized by the target note and proved by this declaration. 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pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8). The Bernoulli gradient is the residual score multiplied by m'/(m(1-m)); an admissible ML fit satisfies its zero equation. Logistic has multiplier one. Common-dispersion scalar canonical families have gradient s^-1 times the residual score, with mean κ' for their normalized response laws and locally analytic cumulants, including Bernoulli logit and Poisson log link. Probit has the nonconstant normal-density multiplier and the explicit residual-root counterexample. The exact first Taylor polynomial of the full score includes h' B; division by a nonzero h gives its linearized equation, while mean-only linearization gives the OLS normal equation after an invertible affine reparameterization. The equations coincide when h'=0 or Bδ=0; logistic and symmetric probit give the first case, and the square-link causal model exhibits a discrepancy in general. These are Taylor-polynomial identities. No nonlinear-score quadratic remainder or blanket extension to ordered, multinomial, or duration models is asserted. The Cox score illustrates the additional risk-set terms and their dependence on the risk sets.

", "headline": "Appendix C and pp. 256–257: the extension of the logistic argument to other maximum likelihood models, corrected", "location": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8)" }, "id": "P015-T8", "text": { "body": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8). The Bernoulli gradient is the residual score multiplied by m'/(m(1-m)); an admissible ML fit satisfies its zero equation. Logistic has multiplier one. Common-dispersion scalar canonical families have gradient s^-1 times the residual score, with mean κ' for their normalized response laws and locally analytic cumulants, including Bernoulli logit and Poisson log link. Probit has the nonconstant normal-density multiplier and the explicit residual-root counterexample. The exact first Taylor polynomial of the full score includes h' B; division by a nonzero h gives its linearized equation, while mean-only linearization gives the OLS normal equation after an invertible affine reparameterization. The equations coincide when h'=0 or Bδ=0; logistic and symmetric probit give the first case, and the square-link causal model exhibits a discrepancy in general. These are Taylor-polynomial identities. No nonlinear-score quadratic remainder or blanket extension to ordered, multinomial, or duration models is asserted. The Cox score illustrates the additional risk-set terms and their dependence on the risk sets.", "headline": "Appendix C and pp. 256–257: the extension of the logistic argument to other maximum likelihood models, corrected", "location": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T8" }, { "declarations": [ { "claim_revision_id": "catalog:d4d6a9b6dd26bc77334f8015503d1e053c9c1e52f7e98fe6874ff4d8df8cb82b", "conditional_on": [], "credit_reason": "Judged as the advertised auxiliary/computational component, not as a complete refutation of the printed ML necessity implication.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.odds_multiplier", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L469", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d4d6a9b6dd26bc77334f8015503d1e053c9c1e52f7e98fe6874ff4d8df8cb82b", "conditional_on": [], "credit_reason": "Judged as the advertised auxiliary/computational component, not as a complete refutation of the printed ML necessity implication. 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pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8). The specified 72-observation odds-link data have admissible means, full rank, a zero substituted residual equation, multiplier 1/eta, a genuine nonzero score, likelihood directional derivatives, and nonmaximality at the displayed parameter. This realizes the whole canonical 72-observation residual-root/nonmaximum example. It refutes the reading that every root of the inverse-link-substituted residual equation is an ML estimate. It does not contradict the printed necessary implication that an ML estimate solves that equation.

", "headline": "Appendix C: ‘To apply the result below for these other models, one substitutes the relevant predicted mean or inverse link function into the estimating equation’, as printed (exact rational counterexample)", "location": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8)" }, "id": "P015-T5", "text": { "body": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8). The specified 72-observation odds-link data have admissible means, full rank, a zero substituted residual equation, multiplier 1/eta, a genuine nonzero score, likelihood directional derivatives, and nonmaximality at the displayed parameter. This realizes the whole canonical 72-observation residual-root/nonmaximum example. It refutes the reading that every root of the inverse-link-substituted residual equation is an ML estimate. 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pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8). The specified 48-observation probit data have full rank, a well-defined normal quantile, a zero substituted residual equation, the exact genuine score with treatment coordinate strictly above 13/625, likelihood directional derivatives, and nonmaximality at the displayed parameter. This realizes the whole canonical 48-observation probit residual-root/nonmaximum example. It refutes the reading that every substituted residual root is an ML estimate. It does not contradict the printed necessary implication that an ML estimate solves the substituted equation.

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p. 257 (PDF p. 8); pp. 264–265 (PDF pp. 15–16). At the common outcome mean 1/4, the a = 1/16 and a = 0 laws have unique admissible expected-likelihood maximizing branches near epsilon = 0 with treatment derivatives 120/119 and 1; the first law has OLS treatment coefficient epsilon. The theorem includes the moment and score derivatives and the treatment-preserving change of basis.

", "headline": "p. 257 and Appendix C: ‘to a first approximation, the only difference … is a scaling factor that depends solely on the mean of YiY_iYi​’, extended to ‘any generalized linear model’, as printed", "location": "p. 257 (PDF p. 8); pp. 264–265 (PDF pp. 15–16)" }, "id": "P015-T7", "text": { "body": "p. 257 (PDF p. 8); pp. 264–265 (PDF pp. 15–16). At the common outcome mean 1/4, the a = 1/16 and a = 0 laws have unique admissible expected-likelihood maximizing branches near epsilon = 0 with treatment derivatives 120/119 and 1; the first law has OLS treatment coefficient epsilon. The theorem includes the moment and score derivatives and the treatment-preserving change of basis.", "headline": "p. 257 and Appendix C: ‘to a first approximation, the only difference … is a scaling factor that depends solely on the mean of $Y_i$’, extended to ‘any generalized linear model’, as printed", "location": "p. 257 (PDF p. 8); pp. 264–265 (PDF pp. 15–16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T7" }, { "declarations": [ { "claim_revision_id": "catalog:09db7438df5283baed8ba5a8d7781e44fe602601374c29dc9b72f0acc40926d2", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.log_points_numeric_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L612", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:09db7438df5283baed8ba5a8d7781e44fe602601374c29dc9b72f0acc40926d2", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.exponentiated_fitted_contrast", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L631", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:09db7438df5283baed8ba5a8d7781e44fe602601374c29dc9b72f0acc40926d2", "conditional_on": [], "credit_reason": "The log quotient equals the corrected difference of logs because 1+S_post and 1+S_pre are positive. The theorem includes the actual OLS coefficient, rather than only cancellation of two logarithms.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.spending_mean_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L2563", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:09db7438df5283baed8ba5a8d7781e44fe602601374c29dc9b72f0acc40926d2", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.spending_ols_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1133", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:09db7438df5283baed8ba5a8d7781e44fe602601374c29dc9b72f0acc40926d2", "conditional_on": [], "credit_reason": "Redundant component or alias: the selected declarations already supply the retained target statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.PaperSpecializations.spending_mean_arithmetic", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperSpecializations.lean#L66", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:09db7438df5283baed8ba5a8d7781e44fe602601374c29dc9b72f0acc40926d2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T16", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T16" }, "html": { "body": "

p. 259 (PDF p. 10), left column. The 0.077 log-point conversion is about eight percent, exponentiated fitted contrasts equal the exponentiated coefficient, and the four-observation OLS example separates this factor from arithmetic mean spending.

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The integral/integrability premises in the effective-law branch implement the target 0p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6). For discrete covariates and finite second moments, endpoint positivity implies positive conditional variance, zero variance is degenerate, and the binary equivalences and effective-law reversal hold, with the explicit multivalued-support and fitted-zero examples.

", "headline": "Section ‘Estimating Representative Causal Effects’: the connection between positivity and the multiple regression weights, corrected", "location": "p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6)" }, "id": "P015-T4", "text": { "body": "p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6). For discrete covariates and finite second moments, endpoint positivity implies positive conditional variance, zero variance is degenerate, and the binary equivalences and effective-law reversal hold, with the explicit multivalued-support and fitted-zero examples.", "headline": "Section ‘Estimating Representative Causal Effects’: the connection between positivity and the multiple regression weights, corrected", "location": "p. 261 (PDF p. 12); positivity display on p. 261 (PDF p. 12); Expression (8) on p. 255 (PDF p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T4" }, { "declarations": [ { "claim_revision_id": "catalog:1a686e7c43c2e278fea4dd184fd1ef28ca333bc1cfd9685ebcd8e3907e460d34", "conditional_on": [], "credit_reason": "The countable positive-mass-cell companion is redundant and does not cover arbitrary covariates.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P015.binary_variance_and_reversal", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L3450", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:1a686e7c43c2e278fea4dd184fd1ef28ca333bc1cfd9685ebcd8e3907e460d34", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'AronowAudit/P015/P015_Regression.lean:1342 identifies the conditional expectation of the squared treatment residual with conditionalVariance, defined at lines 1331–1338 by precisely that standard conditional expectation. This is the same definition used by Mathlib/Probability/CondVar.lean:40 and matches the claim\\'s explicit phrase, \"By the definition of conditional variance\"; reflexivity here is faithful to the mathematical claim, rather than an artificial zero or undefined-case convention. The probability, square-integrability and measurable-covariate assumptions give the intended conditional-variance domain, and the binary formula is a separate component at line 1349 rather than a conclusion advertised by this row.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.conditional_weight_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1323", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1a686e7c43c2e278fea4dd184fd1ef28ca333bc1cfd9685ebcd8e3907e460d34", "conditional_on": [], "credit_reason": "No proof audit was performed; the record’s historical sorryAx sentence is inconsistent with its current standard-axiom list and does not affect this statement assessment. No change from the saved claim-stage verdict.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.binary_conditional_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1330", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1a686e7c43c2e278fea4dd184fd1ef28ca333bc1cfd9685ebcd8e3907e460d34", "conditional_on": [], "credit_reason": "The countable positive-mass-cell companion is redundant and does not cover arbitrary covariates.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P015.PaperSpecializations.binary_variance_reversal_arithmetic", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperSpecializations.lean#L75", "status": "proved", "support_scope": "limited", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:1a686e7c43c2e278fea4dd184fd1ef28ca333bc1cfd9685ebcd8e3907e460d34", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T19", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T19" }, "html": { "body": "

p. 255 (PDF p. 6), left column. For general measurable covariates and square-integrable treatment, the conditional mean of the squared residual is conditional variance and has the usual binary propensity formula.

", "headline": "Expression (8), with its binary-treatment form", "location": "p. 255 (PDF p. 6), left column" }, "id": "P015-T19", "text": { "body": "p. 255 (PDF p. 6), left column. For general measurable covariates and square-integrable treatment, the conditional mean of the squared residual is conditional variance and has the usual binary propensity formula.", "headline": "Expression (8), with its binary-treatment form", "location": "p. 255 (PDF p. 6), left column" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T19" }, { "declarations": [ { "claim_revision_id": "catalog:45548b8c1f562edea362956971fc1de2ac2c148ed63c84e15a46f0bd2acca665", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'AronowAudit/P015/P015_Regression.lean:887 gives the two nonsingular Gram matrices, actual group OLS vectors, inverse identities defining the stated precision matrices and pooled vector, and the universal impossibility of obtaining the pooled slope from two zero slopes. It also states the arbitrary-vector treatment-coordinate formula in T9, including the intercept contributions that explain the failure of scalar pooling. T9 itself expressly says to read the passage as scalar weighting of treatment coefficients; the theorem matches that specified counterexample and makes no claim to refute matrix weighting, bias, consistency or an asymptotic approximation.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.group_pooling_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L880", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:45548b8c1f562edea362956971fc1de2ac2c148ed63c84e15a46f0bd2acca665", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'AronowAudit/P015/P015_Regression.lean:2298 constructs V_g = Z_g I Z_g′ + I and the raw GLS estimator directly, then concludes that its vector is (68/61, -10/61), both actual group OLS slopes vanish, and every scalar combination of those slopes differs from the pooled slope. These are exactly the raw-covariance GLS component and scalar-pooling failure specified in T9. The fixed full-rank designs and positive error variances make these genuine covariance inverses, and restricting the interpretation to scalar treatment-coefficient pooling comes from the claim itself rather than an extra Lean hypothesis.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.group_raw_gls_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L2256", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:45548b8c1f562edea362956971fc1de2ac2c148ed63c84e15a46f0bd2acca665", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T9", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T9" }, "html": { "body": "

p. 257 (PDF p. 8), Random Coefficient Models and expression (10). The two full-rank binary-treatment groups have zero within-group slopes but pooled GLS slope -10/61, with the exact inverse certificates, raw-data GLS value, and arbitrary-vector coordinate formula. This is a counterexample to interpreting the pooled treatment coefficient as a scalar precision-weighted average of the group-specific OLS treatment coefficients.

", "headline": "Section ‘Random Coefficient Models’: the estimator ‘computes a series of quantities along the lines of Expression (7) and aggregates them on the basis of their precision’, as printed", "location": "p. 257 (PDF p. 8), left column to right column; Expression (10) on p. 257 (PDF p. 8)" }, "id": "P015-T9", "text": { "body": "p. 257 (PDF p. 8), Random Coefficient Models and expression (10). The two full-rank binary-treatment groups have zero within-group slopes but pooled GLS slope -10/61, with the exact inverse certificates, raw-data GLS value, and arbitrary-vector coordinate formula. This is a counterexample to interpreting the pooled treatment coefficient as a scalar precision-weighted average of the group-specific OLS treatment coefficients.", "headline": "Section ‘Random Coefficient Models’: the estimator ‘computes a series of quantities along the lines of Expression (7) and aggregates them on the basis of their precision’, as printed", "location": "p. 257 (PDF p. 8), left column to right column; Expression (10) on p. 257 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T9" }, { "declarations": [ { "claim_revision_id": "source-claim-import-2026-11-08:P015-T10:1", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.corrected_matrix_pooling", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L922", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "source-claim-import-2026-11-08:P015-T10:1", "conditional_on": [], "credit_reason": "The exact criterion fixes a before quantifying every t and includes A_g[b,b]=a_g; the target's final abbreviated gloss about only off-treatment entries omits this fixed-diagonal requirement. Lean matches the preceding exact formula.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.scalar_pooling_iff", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L2549", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "source-claim-import-2026-11-08:P015-T10:1", "conditional_on": [], "credit_reason": "Duplicate specialization of the scalar-row criterion already supplied by scalar_pooling_iff.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.PaperSpecializations.scalar_pooling_iff_algebra", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperSpecializations.lean#L89", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "source-claim-import-2026-11-08:P015-T10:1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T10", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T10" }, "html": { "body": "

p. 257 (PDF p. 8), corrected random-coefficient aggregation description; expression (7), p. 254 (PDF p. 5), under its assumptions within each group. Usual measure-theoretic conditions are assumed. For nonempty finite groups and coordinates, full-rank group designs, positive error variances and a symmetric positive semidefinite coefficient covariance, all covariance inverses exist. Precision pooling uniquely minimizes the GLS loss, equals raw-observation GLS and the sum of the matrix-weighted group OLS vectors, and its matrix weights sum to the identity. Its treatment coordinate can include other coefficients. For prescribed scalar weights, pooling all coefficient vectors is equivalent to the full treatment-row identity, hence to off-treatment zeros AND the prescribed diagonal entries. Treatment-block diagonal precisions suffice, with weights p_g/sum_h p_h. Along any sequence with fixed finite dimensions, vanishing off-treatment entries give a vanishing error for uniformly bounded coefficient inputs, using the treatment diagonal entries as weights. Expression (7) applies separately in each group satisfying its causal and regularity assumptions. This qualification is compatible with consistency under correctly specified random-coefficient and exogeneity assumptions; it makes no additional consistency assertion.

", "headline": "Section ‘Random Coefficient Models’: the description of the estimator as an aggregation of within-group Expression (7) quantities, corrected", "location": "p. 257 (PDF p. 8)" }, "id": "P015-T10", "text": { "body": "p. 257 (PDF p. 8), corrected random-coefficient aggregation description; expression (7), p. 254 (PDF p. 5), under its assumptions within each group. Usual measure-theoretic conditions are assumed. For nonempty finite groups and coordinates, full-rank group designs, positive error variances and a symmetric positive semidefinite coefficient covariance, all covariance inverses exist. Precision pooling uniquely minimizes the GLS loss, equals raw-observation GLS and the sum of the matrix-weighted group OLS vectors, and its matrix weights sum to the identity. Its treatment coordinate can include other coefficients. For prescribed scalar weights, pooling all coefficient vectors is equivalent to the full treatment-row identity, hence to off-treatment zeros AND the prescribed diagonal entries. Treatment-block diagonal precisions suffice, with weights p_g/sum_h p_h. Along any sequence with fixed finite dimensions, vanishing off-treatment entries give a vanishing error for uniformly bounded coefficient inputs, using the treatment diagonal entries as weights. Expression (7) applies separately in each group satisfying its causal and regularity assumptions. This qualification is compatible with consistency under correctly specified random-coefficient and exogeneity assumptions; it makes no additional consistency assertion.", "headline": "Section ‘Random Coefficient Models’: the description of the estimator as an aggregation of within-group Expression (7) quantities, corrected", "location": "p. 257 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T10" }, { "declarations": [ { "claim_revision_id": "catalog:700256a6427669d305e8033f5852400ed6eb1d45ab21ccdc0e30f503d3859289", "conditional_on": [], "credit_reason": "The target’s definitions_needed explicitly fixes the population interpretation of scaled leverage and the intended rounding, so no finite-iid expectation assertion is required. No change from the saved claim-stage verdict.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.leverage_weight_comparison", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L979", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:700256a6427669d305e8033f5852400ed6eb1d45ab21ccdc0e30f503d3859289", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T20", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T20" }, "html": { "body": "

pp. 263-264 (PDF pp. 14-15), Appendix B. The specified population has the exact regression weights and scaled leverages, their conditional averages, and all printed decimal and percentage roundings, with opposite profile comparisons.

", "headline": "Appendix B, the example with a binary treatment and a binary covariate (leverage against multiple regression weights)", "location": "pp. 263–264 (PDF pp. 14–15)" }, "id": "P015-T20", "text": { "body": "pp. 263-264 (PDF pp. 14-15), Appendix B. The specified population has the exact regression weights and scaled leverages, their conditional averages, and all printed decimal and percentage roundings, with opposite profile comparisons.", "headline": "Appendix B, the example with a binary treatment and a binary covariate (leverage against multiple regression weights)", "location": "pp. 263–264 (PDF pp. 14–15)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T20" }, { "declarations": [ { "claim_revision_id": "catalog:6b6eaad8d2bbf10383243fd4dfab635e3f58e6fdc0433708a860c5ee2f8fcc17", "conditional_on": [], "credit_reason": "This row establishes only the target's misprint observation. The separately defined LogisticModel has the correct conditional-expectation formulation, but that definition is not a credited declaration in this chunk.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.unconditional_logistic_misprint", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1009", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:6b6eaad8d2bbf10383243fd4dfab635e3f58e6fdc0433708a860c5ee2f8fcc17", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T14", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T14" }, "html": { "body": "

p. 257 (PDF p. 8), logistic display; Appendix C, p. 265 (PDF p. 16). The binary logistic model specifies common real intercept, treatment and control coefficients for which the conditional success probability given treatment and controls is 1/(1+exp(-predictor)). Model membership supplies those parameters. The printed unconditional probability cannot equal both predictions at distinct treatments with fixed controls and a nonzero treatment coefficient. The usual measure-theoretic conditions are assumed; the conditional-probability equalities hold almost surely.

", "headline": "Section ‘Maximum Likelihood Estimation and Generalized Linear Models’: the displayed logistic regression model, corrected", "location": "p. 257 (PDF p. 8), left column; the same model in Appendix C, p. 265 (PDF p. 16)" }, "id": "P015-T14", "text": { "body": "p. 257 (PDF p. 8), logistic display; Appendix C, p. 265 (PDF p. 16). The binary logistic model specifies common real intercept, treatment and control coefficients for which the conditional success probability given treatment and controls is 1/(1+exp(-predictor)). Model membership supplies those parameters. The printed unconditional probability cannot equal both predictions at distinct treatments with fixed controls and a nonzero treatment coefficient. The usual measure-theoretic conditions are assumed; the conditional-probability equalities hold almost surely.", "headline": "Section ‘Maximum Likelihood Estimation and Generalized Linear Models’: the displayed logistic regression model, corrected", "location": "p. 257 (PDF p. 8), left column; the same model in Appendix C, p. 265 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T14" }, { "declarations": [ { "claim_revision_id": "catalog:50b30c082c45fdbc33295c5cb8e14bbbbad95b643e7f622b356aadc452d69f56", "conditional_on": [], "credit_reason": "This is the exact auxiliary T21(ii) component. Allowing arbitrary responses, c=0 and deficient rank loses none of its cases; no unique-root or likelihood claim is attached to this row. 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For example, W=(1,D,X) with D=X=(0,0,1,1) and Y=(0,1,0,1) has a varying, non-collinear control column and finite logistic fits, but singular WᵀW; its meaningful score identity is excluded here. The unique affine root and coefficient scaling are correctly covered on the nonsingular domain.', 'suggested_fix': 'Export the score-derivative conjunct without hQ, and make det(WᵀW)≠0 an antecedent only of the unique-root and OLS-scaling conjuncts.', 'second_review': 'P015_Regression.lean:1195 assumes det(WᵀW)≠0 before the entire conjunction, including the logistic likelihood derivative at :1196. T21(i) requires this score identity for binary observations and an intercept without a rank condition; invertibility is required only for the unique-root assertion in T21(iii). Designs with D=X=(0,0,1,1) and Y=(0,1,0,1) have a varying control and a meaningful logistic score but fail the extra determinant premise. 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p. 265 (PDF p. 16), Appendix C; main text p. 257 (PDF p. 8). For every binary sample the logistic log likelihood is Frechet differentiable with score sum W_i'(Y_i - sigmoid(W_i theta)), without a rank assumption. Affine coefficients give identical observation predictors and residual equations. If W'W is nonsingular and the affine scale is nonzero, the surrogate equations have a unique root whose affine transform is OLS; its treatment coefficient multiplied by the scale equals the OLS treatment coefficient.

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p. 257 (PDF p. 8); expansion point also on p. 265 (PDF p. 16). The full-rank 200-observation binary example has the unique logistic fit at predictors plus or minus log 99, outside the exact Taylor radius pi, and every degree has probability error at least 49/100 and treatment-equation error at least 49.

", "headline": "p. 257: ‘We can obtain an approximation of arbitrary precision for the weights by using a Taylor expansion of the inverse link function’, as printed", "location": "p. 257 (PDF p. 8), left column; the expansion point is also in Appendix C, p. 265 (PDF p. 16)" }, "id": "P015-T11", "text": { "body": "p. 257 (PDF p. 8); expansion point also on p. 265 (PDF p. 16). 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p. 254 (PDF p. 5), corrected expression (7); Appendix A, p. 262 (PDF p. 13). Under unconfoundedness, affine potential outcomes, the linear conditional treatment mean, and the stated eight-family regularity, the actual OLS treatment coefficient converges in probability to E[w tau]/E[w], with w=(D-E[D|X])^2, on any ambient probability space. The two conditional factorizations and their zero-moment consequences hold; the corrected OLS coefficient column has K+2 coordinates. A fair-bit confounding example shows that conditions (1) and (3) alone do not imply the factorizations.

", "headline": "Expression (7) and the sentence that introduces it, corrected (Conditions (1)–(3)); the OLS coefficient column of Appendix A, corrected (γ^K\\hat\\gamma_Kγ^​K​)", "location": "p. 254 (PDF p. 5), right column; Appendix A, p. 262 (PDF p. 13)" }, "id": "P015-T15", "text": { "body": "p. 254 (PDF p. 5), corrected expression (7); Appendix A, p. 262 (PDF p. 13). Under unconfoundedness, affine potential outcomes, the linear conditional treatment mean, and the stated eight-family regularity, the actual OLS treatment coefficient converges in probability to E[w tau]/E[w], with w=(D-E[D|X])^2, on any ambient probability space. The two conditional factorizations and their zero-moment consequences hold; the corrected OLS coefficient column has K+2 coordinates. A fair-bit confounding example shows that conditions (1) and (3) alone do not imply the factorizations.", "headline": "Expression (7) and the sentence that introduces it, corrected (Conditions (1)–(3)); the OLS coefficient column of Appendix A, corrected ($\\hat\\gamma_K$)", "location": "p. 254 (PDF p. 5), right column; Appendix A, p. 262 (PDF p. 13)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T15" }, { "declarations": [ { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P015_Regression.lean:5029 requires a standard Borel carrier, but this loses no statistical case of T22's OLS probability-limit claim. For any model on an arbitrary probability space, push forward by the entire sequence ((D_i, Y0_i, tau_i, X_i)) to (ℝ³ × ℝ^K)^ℕ; that canonical space is standard Borel. The joint law preserves iid sampling, conditional unconfoundedness, the conditional-mean relation, all eight moment/LLN families, and the probabilities defining convergence of betaHat to E[w tau]/E[w]. The reviewer's enlargement by unused independent coins changes the carrier without changing any of these quantities, so the difference falls under the permitted representation convention.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.ols_weighted_effect_limit", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L4974", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "Finite partial-regression auxiliary already incorporated in the exported actual-estimator limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.partial_regression", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1406", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "Finite decomposition auxiliary already incorporated in the exported actual-estimator limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.regression_decomposition", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L2576", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"The standard Borel assumption at P015_Regression.lean:1749 is a representation condition for the conditional-independence kernel, and every joint law of (X, D, Y0, tau) is represented on the standard Borel space R^(K+3). The conditional-expectation factorizations and the two zero population terms at P015_Regression.lean:1764-1770 are almost-sure or integral identities preserved by this representation, together with conditional unconfoundedness, the linear conditional mean and the required integrability. The displayed moment assumptions are supplied by T22's regularity families when this component is used there, so no further substantive hypothesis is introduced. Thus arbitrary additional randomness on the original carrier excludes no case of the claimed zero-moment argument.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.unconfounded_zero_terms", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1715", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "Finite decomposition auxiliary; the Challenge states the causal probability limit rather than a separate decomposition lemma.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.PaperSpecializations.regression_decomposition_algebra", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperSpecializations.lean#L106", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "Coefficient assembly auxiliary; the SAP regressor vector already has the corrected K+2 coordinates.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.combined_predictor", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1382", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "conditional_on": [], "credit_reason": "Residual-orthogonality auxiliary; no separate orthogonality claim is required for Theorem A.1.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.residual_orthogonal_predictor", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L1391", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:54864e8dd9231ce0343f89b9fa3f63db44785e31ca1e4f9af8bdd6bf791b909d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T22", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T22" }, "html": { "body": "

p. 262 (PDF p. 13), Theorem A.1; proof pp. 262-263 (PDF pp. 13-14); expression (7), p. 254 (PDF p. 5). For measurable iid unit data on any probability space, unconfoundedness, affine potential outcomes, a linear conditional treatment mean, and the stated eight-family regularity imply convergence in probability of the actual OLS treatment coefficient to E[w tau]/E[w], where w=(D-E[D|X])^2. The numerator is integrable and the denominator has positive finite expectation. The inverse formula completes the estimator on singular finite samples.

", "headline": "Theorem A.1", "location": "p. 262 (PDF p. 13); proof on pp. 262–263 (PDF pp. 13–14); main-text form, Expression (7), on p. 254 (PDF p. 5)" }, "id": "P015-T22", "text": { "body": "p. 262 (PDF p. 13), Theorem A.1; proof pp. 262-263 (PDF pp. 13-14); expression (7), p. 254 (PDF p. 5). For measurable iid unit data on any probability space, unconfoundedness, affine potential outcomes, a linear conditional treatment mean, and the stated eight-family regularity imply convergence in probability of the actual OLS treatment coefficient to E[w tau]/E[w], where w=(D-E[D|X])^2. The numerator is integrable and the denominator has positive finite expectation. The inverse formula completes the estimator on singular finite samples.", "headline": "Theorem A.1", "location": "p. 262 (PDF p. 13); proof on pp. 262–263 (PDF pp. 13–14); main-text form, Expression (7), on p. 254 (PDF p. 5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T22" }, { "declarations": [ { "claim_revision_id": "catalog:91f401dcfcf75c03552ccdc3843a07cde3a7bf0b970f094ee6d834dec94f7314", "conditional_on": [], "credit_reason": "Finite fitted-residual expansion auxiliary; expression (9) is packaged as the actual fitted-weight mean limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.PaperDomains.residual_weight_expansion", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperDomains.lean#L62", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:91f401dcfcf75c03552ccdc3843a07cde3a7bf0b970f094ee6d834dec94f7314", "conditional_on": [], "credit_reason": "The seven moment families are exactly those in the target’s sufficient reading. Zero extensions at singular finite samples or zero fitted denominators are compatible with a positive limiting residual variance and do not change the asserted convergence in probability. No change from the saved claim-stage verdict.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.effective_covariate_mean_limit", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L4955", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:91f401dcfcf75c03552ccdc3843a07cde3a7bf0b970f094ee6d834dec94f7314", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T23", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T23" }, "html": { "body": "

pp. 255–256 (PDF pp. 6–7), expression (9). For iid measurable observations with a linear conditional treatment mean, the seven integrable moment families and nonsingular population control Gram in the claim's sufficient regularity reading, and positive mean population weight, the mean formed with fitted squared treatment residuals converges in probability to E[w Z]/E[w], the effective population mean. Finite singular samples and zero fitted denominators use the package's totalized conventions.

", "headline": "Expression (9): the weighted mean of a covariate with the estimated weights converges to its effective-sample mean", "location": "p. 255 (PDF p. 6), right column, continued on p. 256 (PDF p. 7)" }, "id": "P015-T23", "text": { "body": "pp. 255–256 (PDF pp. 6–7), expression (9). For iid measurable observations with a linear conditional treatment mean, the seven integrable moment families and nonsingular population control Gram in the claim's sufficient regularity reading, and positive mean population weight, the mean formed with fitted squared treatment residuals converges in probability to E[w Z]/E[w], the effective population mean. Finite singular samples and zero fitted denominators use the package's totalized conventions.", "headline": "Expression (9): the weighted mean of a covariate with the estimated weights converges to its effective-sample mean", "location": "p. 255 (PDF p. 6), right column, continued on p. 256 (PDF p. 7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T23" }, { "declarations": [ { "claim_revision_id": "catalog:da899491c1891aa7caa95fd94d658b03b1de2ff92c9abd2ea630782d5b008a38", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P015_Regression.lean:4592 concludes conditional-mean constancy, an almost-everywhere centered-square weight identity, mean factorization and ratio cancellation at lines 4598-4602, but does not conclude IndepFun (populationWeight P D X) tau P. T24's printed proof explicitly asserts that w_i is independent of tau_i, and this row also expressly credits that population independence conclusion. Mean factorization does not assert independence, so the credited component omits a promised conclusion even though it follows mathematically from random assignment and the centered-square identity.\", 'suggested_fix': 'Add IndepFun (populationWeight P D X) τ P to the conclusion, using measurable transformation of the independent treatment variable and almost-everywhere congruence of the weight; keep the existing assumptions and cancellation conclusions.', 'second_review': \"P015_Regression.lean:4592 concludes conditional-mean constancy, the almost-everywhere centered-square weight identity, factorization and cancellation at P015_Regression.lean:4598, but omits the weight-effect independence explicitly asserted in T24's printed proof and main-text passage. Add that conclusion for this credited population component. The repair is mathematically trivial: apply measurable transformations to hrandom at P015_Regression.lean:4596 to obtain independence of (D-E[D])^2 and tau, then transfer it through the exported almost-everywhere identity using IndepFun.comp and IndepFun.congr (Mathlib/Probability/Independence/Basic.lean:756 and Mathlib/Probability/Independence/Basic.lean:714). Thus I agree about the omission and repair, but disagree with the major severity.\", 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P015.ProbabilityCompletion.randomization_population_weight_cancels", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L4532", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:da899491c1891aa7caa95fd94d658b03b1de2ff92c9abd2ea630782d5b008a38", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P015_Regression.lean:5048 retains Corollary A.1's actual OLS convergence-in-probability conclusion, common random assignment and the supplied OLSRegularity reading. Its standard Borel carrier represents every joint law of the countable sequence of real-valued treatment, baseline, effect and covariate vectors: use the pushforward law on (R^3 x R^K) indexed by the natural numbers. This preserves the full random-assignment independence, all sample moments and LLNs, and both betaHat's convergence probabilities and E[tau]. Consequently the reviewer's nonstandard carrier example supplies no omitted statistical model, and the extra typeclass is a harmless representation difference under the stated standard.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.randomized_ols_limit", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_Regression.lean#L4996", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:da899491c1891aa7caa95fd94d658b03b1de2ff92c9abd2ea630782d5b008a38", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T24", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T24" }, "html": { "body": "

p. 263 (PDF p. 14), Corollary A.1; p. 256 (PDF p. 7). Common random assignment makes the population treatment mean conditional on controls constant and the squared residual weight independent of the effect. Its mean is the positive treatment variance; the product expectation factors and the weighted-effect ratio cancels to the average effect. Under iid sampling and the eight-family regularity reading, both the actual OLS treatment estimate and the intermediate fitted-weight effect ratio converge in probability to that average effect, on every measurable sampling space.

", "headline": "Corollary A.1", "location": "p. 263 (PDF p. 14); main-text form in section ‘Randomized Experiments’, p. 256 (PDF p. 7)" }, "id": "P015-T24", "text": { "body": "p. 263 (PDF p. 14), Corollary A.1; p. 256 (PDF p. 7). Common random assignment makes the population treatment mean conditional on controls constant and the squared residual weight independent of the effect. Its mean is the positive treatment variance; the product expectation factors and the weighted-effect ratio cancels to the average effect. Under iid sampling and the eight-family regularity reading, both the actual OLS treatment estimate and the intermediate fitted-weight effect ratio converge in probability to that average effect, on every measurable sampling space.", "headline": "Corollary A.1", "location": "p. 263 (PDF p. 14); main-text form in section ‘Randomized Experiments’, p. 256 (PDF p. 7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T24" }, { "declarations": [ { "claim_revision_id": "catalog:a759a400dcebb3a368bbf3403839bcde649e30cb00edbd763648d400152ca234", "conditional_on": [], "credit_reason": "Final status wrong_credit: its identity-link witness does not refute score substitution for any specifically listed model family.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P015.InverseLinkMLCounterexample.printed_inverse_link_claim_false", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_InverseLinkMLCounterexample.lean#L106", "status": "counterexample_proved", "support_scope": "withdrawn", "used_in_solution": true }, { "claim_revision_id": "catalog:a759a400dcebb3a368bbf3403839bcde649e30cb00edbd763648d400152ca234", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P015_PaperAssessments.lean:19 conjoins the logistic ML score result, the arbitrary-Bernoulli-link negation from P015_InverseLinkMLCounterexample.lean:104, and RootLinkCounterexample. The logistic conjunct covers T18(A), and the square-link first-order counterexample defined at P015_Regression.lean:3003 addresses the any-GLM mean-only scaling assertion in T18(C). Neither supplies a score-substitution failure for one of the models specified in T18(B), and the middle conjunct has no such model restriction. Consequently the advertised complete assessment omits the listed-model component, even though its counterexample to the blanket first-order extension is substantive.', 'suggested_fix': 'Add a named-model score-substitution failure to the conjunction, for example the explicit probit genuine-score/residual mismatch packaged by ProbitCounterexample at P015_Regression.lean:2980. Keep the existing logistic and first-order counterexample conjuncts.', 'second_review': 'P015_PaperAssessments.lean:19 conjoins the logistic ML score, an arbitrary-link negation, and RootLinkCounterexample. T18 separately requires assessment of (A) the logistic equation, (B) substitution for the listed other models, and (C) mean-only first-order scaling for any GLM; the square inverse link defined at P015_Regression.lean:552 and packaged at :3003 addresses (C). The arbitrary-link conjunct uses the identity-link witness of P015_InverseLinkMLCounterexample.lean:90, so neither counterexample supplies a failure for one of the models in (B). Add a listed-model score mismatch to the exported conjunction to complete the credited assessment.', 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P015.PaperAssessments.nonlinear_extension_assessment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperAssessments.lean#L18", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:a759a400dcebb3a368bbf3403839bcde649e30cb00edbd763648d400152ca234", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T18", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T18" }, "html": { "body": "

The printed claim is false: P015-T18; pp. 264-265 (PDF pp. 15-16), Appendix C; pp. 256-257 (PDF pp. 7-8). Finite logistic ML fits solve their residual equations, but a genuine square-link ML branch in the stated causal model refutes universal mean-only first-order OLS scaling. Scope: Retains clause (A) and an explicit counterexample to universal clause (C); omits an assessment of score substitution for every specifically listed nonlogistic family in (B), and does not assert a separate counterexample for each named family.

", "headline": "Appendix C (opening paragraph, estimating equation and closing sentences) and pp. 256–257: the extension of the OLS weighting result to nonlinear models fit by maximum likelihood", "location": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8)" }, "id": "P015-T18", "text": { "body": "The printed claim is false: P015-T18; pp. 264-265 (PDF pp. 15-16), Appendix C; pp. 256-257 (PDF pp. 7-8). Finite logistic ML fits solve their residual equations, but a genuine square-link ML branch in the stated causal model refutes universal mean-only first-order OLS scaling. Scope: Retains clause (A) and an explicit counterexample to universal clause (C); omits an assessment of score substitution for every specifically listed nonlogistic family in (B), and does not assert a separate counterexample for each named family.", "headline": "Appendix C (opening paragraph, estimating equation and closing sentences) and pp. 256–257: the extension of the OLS weighting result to nonlinear models fit by maximum likelihood", "location": "pp. 264–265 (PDF pp. 15–16); main text pp. 256–257 (PDF pp. 7–8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T18" }, { "declarations": [ { "claim_revision_id": "catalog:db3fce31add82388774e0a533e3b43cbfc51e93972b31708292278bc9edb081a", "conditional_on": [], "credit_reason": "A concrete witness to failure of T17(C) under its specified pair-specific reading: globally possible endpoints 0 and 1 fail conditional positivity at the effectively represented profile X=1. This row does not claim to establish all of T17(A)–(C).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P015.PositivityRefutation.printed_effective_positivity_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PositivityRefutation.lean#L71", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:db3fce31add82388774e0a533e3b43cbfc51e93972b31708292278bc9edb081a", "conditional_on": [], "credit_reason": "Duplicate correction bundle; the used passage-assessment row supplies the same restricted positive implication and concrete refutations.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P015.PaperAssessments.positivity_implies_variance", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperAssessments.lean#L13", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:db3fce31add82388774e0a533e3b43cbfc51e93972b31708292278bc9edb081a", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P015_PaperAssessments.lean:12 exports PositivityCorrection, whose general positive clauses at P015_Regression.lean:3094 require [Countable B] and at P015_Regression.lean:3096 require MemLp D 2 P. T17(A) says positivity implies positive conditional variance at every represented profile, without restricting covariates to a countable range or requiring a global treatment second moment; the standing assumptions do not supply those restrictions for this positivity passage. Finite conditional variances, or a finite positive mean residual weight, do not imply a finite unconditional second moment, because conditional treatment means can have an infinite second moment. The three closed counterexamples remain valid witnesses against the false readings of (B) and (C), but they do not extend the restricted general positive assessment to the omitted cases.', 'suggested_fix': 'State the general positive implications for conditional treatment laws at represented profiles without a countable covariate-range hypothesis or global MemLp D 2 P. Use extended conditional variance where finiteness is not assumed, and retain finite positive mean weight only where the normalized effective law is needed; keep the existing concrete refutations as separate conjuncts.', 'second_review': \"P015_PaperAssessments.lean:12 states PositivityCorrection, and its definition at P015_Regression.lean:3093 puts both countability of the covariate type and MemLp D 2 P ahead of the positive variance implication. T17(A)'s endpoint-positivity implication and the standing assumptions for this passage do not require a global finite treatment second moment; finite conditional variances and even finite mean residual weight allow conditional means with infinite second moment. Consequently this complete-assessment row omits legitimate cases of the target's positive component, despite supplying concrete counterexamples to the false components at P015_Regression.lean:3114. The repair belongs in the exported statement.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P015.PaperAssessments.positivity_passage_assessment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P015/P015_PaperAssessments.lean#L11", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:db3fce31add82388774e0a533e3b43cbfc51e93972b31708292278bc9edb081a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P015-T17", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P015-T17" }, "html": { "body": "

The printed claim is false: P015-T17; p. 261 (PDF p. 12), positivity passage. Positive endpoint probabilities imply positive variance on the exported domain, while small effective mass, zero fitted sample weights, and a literal effective-law example refute the printed weight and pair-positivity implications. Scope: The positive implication (A) requires countable covariates with measurable singletons and a global finite treatment second moment; omits its remaining meaningful cases. Clauses (B) and (C) are refuted by explicit examples, without a universal characterization of identification or an applied country-specific conclusion.

", "headline": "Section ‘Estimating Representative Causal Effects’: the paragraph connecting positivity and the multiple regression weights", "location": "p. 261 (PDF p. 12), last paragraph of the left column to the right column; positivity display on p. 261 (PDF p. 12)" }, "id": "P015-T17", "text": { "body": "The printed claim is false: P015-T17; p. 261 (PDF p. 12), positivity passage. Positive endpoint probabilities imply positive variance on the exported domain, while small effective mass, zero fitted sample weights, and a literal effective-law example refute the printed weight and pair-positivity implications. Scope: The positive implication (A) requires countable covariates with measurable singletons and a global finite treatment second moment; omits its remaining meaningful cases. Clauses (B) and (C) are refuted by explicit examples, without a universal characterization of identification or an applied country-specific conclusion.", "headline": "Section ‘Estimating Representative Causal Effects’: the paragraph connecting positivity and the multiple regression weights", "location": "p. 261 (PDF p. 12), last paragraph of the left column to the right column; positivity display on p. 261 (PDF p. 12)" }, "url": "https://pmaronow.github.io/llm-audit/works/P015/#P015-T17" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "partial", "targets": [ { "agreement": "agrees_in_part", "location": "p. 258 (PDF p. 9)", "method": "I regenerated both maps with the authors' code and a replacement map reader. Country-mean weights reproduce the plotted shares to rounding; summing weights as the text describes changes the shares and rankings, and the bottom-country list names the wrong Congo (P015-E001, P015-E002).", "status": "reproduced", "target": "Figure 1 and country-weight concentration" }, { "agreement": "disagrees", "location": "p. 258 (PDF p. 9), left column", "method": "I ran the supplied OLS regression on all 1,630 country-years: the coefficient is 0.0211 and its conventional two-sided p-value is 0.043. The paper quotes 0.020 and p < 0.001; Jensen's source table reports 0.021 with p < 0.01 under panel-corrected inference (P015-E003).", "status": "reproduced", "target": "Jensen regression effect" }, { "agreement": "to_rounding", "location": "p. 260 (PDF p. 11)", "method": "I ran the authors' code on 1,469 observations and compared all 76 cells at two decimals. I independently refitted the treatment regression and verified the stored squared-residual weights to storage precision.", "status": "reproduced", "target": "Table 1" }, { "agreement": "agrees_in_part", "location": "p. 259 (PDF p. 10), left column", "method": "I reproduced the coefficient as 0.0768, agreeing with 0.077 to rounding. Its exponentiated one-point contrast is about 8%, rather than the stated 20% (P015-E004). State-clustered inference supports p < 0.05; I could not compare the cited source table.", "status": "reproduced", "target": "Gerber–Huber regression effect" } ] }, "summary": "The paper asks whether regression on a representative sample estimates a representative causal effect. It shows that, under unconfoundedness, two linearity conditions and the regularity conditions used in its proof, the OLS coefficient converges to an average of unit effects weighted by squared treatment residuals, and it examines two published studies. 14 entries are classified as errors (3 Moderate, 11 Minor), and 4 are clarity notes; 8 allegations were not confirmed, 1 is undetermined and 4 are out of scope. The main finding is \\fid{P015-E001}: the country shares, lists and map for Jensen (2003) use country means of the weights where the text describes sums.", "what_was_checked": "The following were checked and held up.\n\\begin{itemize}\n\\item Theorem A.1 and its proof (pp.~262--263), using Conditions (2) and (3) for the zero-term identities and the corrections in \\fid{P015-N004}: each step follows, and the limit $\\mathrm{E}[w_i\\tau_i]/\\mathrm{E}[w_i]$ agrees with the population OLS coefficient, computed exactly, on small discrete designs, for example a design with two strata in which the coefficient is $15/7$ and the average effect is $2$. The cross-reference before Expression (7) is corrected in \\fid{P015-N004}.\n\\item Expression (8), $\\mathrm{E}[w_i\\mid X_i]=\\operatorname{Var}[D_i\\mid X_i]$, and its form $\\pi(X_i)[1-\\pi(X_i)]$ for a binary treatment.\n\\item Expression (9): the means weighted by squared residuals converge to $\\mathrm{E}[w_iZ_i]/\\mathrm{E}[w_i]$, and on both data sets $\\sum_i\\tilde D_iY_i/\\sum_i\\tilde D_i^2$ equals the OLS coefficient exactly. In the illustration of a sample of 50\\% women and 50\\% men in which women receive three times the weight of men, the effective sample is 75\\% women and 25\\% men.\n\\item Corollary A.1 and its proof for the experiment of the main text, in which the assignment probabilities are equal, and the warning of footnote 15 for assignment probabilities that vary by block.\n\\item Two statements on weighting. On p.~255, for a binary treatment with positivity, a regression with ordinary inverse propensity score weighting and correct propensity scores has squared treatment residuals that are constant, and it converges to the average effect under the paper's assumptions. On p.~259, weighting by the inverse of the conditional variance of the treatment recovers the average effect under the assumptions of the paper, provided that variance is positive.\n\\item The binary example of Appendix B: $0.160$, $0.196$ and 22\\%; $7.43$, $4.21$, $1.71$ and $4.21$; $4.86$, $2.38$ and 51\\%. The notation for these values is the subject of \\fid{P015-E012}. Figure B1 agrees with its note and with the text.\n\\item Appendix C for the logistic model: the constants $b$ and $c$, the reparametrization, and the first-order relation between the OLS and logistic coefficients. The misprint in the derivative is \\fid{P015-E010}. The extension to other models, the promise of arbitrary precision and two points of notation are the subjects of \\fid{P015-E008}, \\fid{P015-E009}, \\fid{P015-E011} and \\fid{P015-E013}.\n\\item Table 1: all 76 cells are reproduced to two decimals from the replication data, with $n=1{,}469$, and the stored weights agree to storage precision with the squared residuals of party identification on the nine controls. The Gerber--Huber coefficient $0.077$ is reproduced as $0.0768$, and $p<.05$ holds with standard errors clustered by state. The reading of that coefficient is corrected in \\fid{P015-E004}.\n\\item Jensen: the sample has 114 countries, and the weights come from the regression of regime type on the controls, as the 2017 erratum states (\\fid{P015-N008}). Russia, Germany and China hold weight near zero, and the lowest 14 countries hold less than 0.05\\% of the weight, whether a country is scored by the sum or by the mean of its weights. The shares, lists and map are the subject of \\fid{P015-E001} and \\fid{P015-E002}.\n\\item The 2017 erratum: both of its corrections describe the original page accurately and agree with the replication code (\\fid{P015-N008}, \\fid{P015-N010}).\n\\item Footnote 11: the studentized Breusch--Pagan test with all controls rejects at $p<.001$ in both examples.\n\\item The map data: all 114 Jensen countries match the file that maps country names, all 211 map names match the attribute table of the authors' map, and the records of that table correspond to the polygons.\n\\item The conclusion that regression on a representative sample carries no general guarantee of a representative effect follows from Theorem A.1. The side remark on which further assumptions would remove the problem is the subject of \\fid{P015-N003}, and the statements on positivity are corrected in \\fid{P015-E006}.\n\\end{itemize}\nMy checks had the following limits. The simulation behind Figure B1 is not in the replication files. I compared the figure with its own description and did not regenerate it. The files hold no code for the numbers of Appendix B or for the country shares on p.~258, and I recomputed these independently. Both panels of Figure 1 were regenerated from the replication data with a substitute for the retired package that reads the map, and I compared them with the published figure by eye. Only the analysis data deposited by the authors were available, not the processing of the source studies. Among the cited works, I read Table 4 of Jensen (2003) for \\fid{P015-E003} and consulted publisher and library records for \\fid{P015-E014}. I could not read Table 4A of Gerber and Huber (2010) or p.~297 of Campbell (1957), and I did not consult the other cited works. I read the third-party Stata package and did not run it. That this audit records no further corrections is not a finding that the rest of the paper is correct.", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "P. M. Aronow", "Cyrus Samii" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow and Cyrus Samii (2016). Does Regression Produce Representative Estimates of Causal Effects?. American Journal of Political Science 60(1):250–267.", "doi": "10.1111/ajps.12185", "edition": { "pages_in_audited_file": 18, "sha256": "71736913b122a572563c084bde32697d83d36277e835dd13e7cd7e34ee8e209b", "source_url": "https://doi.org/10.1111/ajps.12185", "venue": "American Journal of Political Science", "year": "2016" }, "id": "P015", "later_materials": null, "title": "Does Regression Produce Representative Estimates of Causal Effects?", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P015/", "venue": "American Journal of Political Science", "year": "2016" } }