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   "claim": "The FRT with the difference in means is asymptotically a Rao-type score test under constant effects, and its reference variance exceeds that of the pooled Wald-type analogue $\\widehat V(\\mathrm{OLS})$ by $\\tau^2/N$ to first order. It can therefore be less powerful under nonlocal constant-effect alternatives.",
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   "tied_to": "Section 2, p. 347: the displays for $\\widehat V(\\mathrm{Fisher})$, $\\widehat V(\\mathrm{OLS})$ and $\\widehat V(\\mathrm{Fisher})-\\widehat V(\\mathrm{OLS})=\\tau^2/N+o_p(N^{-1})$; Introduction, item (i), p. 346"
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   "claim": "A Wald-type analogue of the FRT, using homoskedastic OLS or the pooled-variance two-sample $z$-test, is asymptotically valid under Fisher's sharp null and asymptotically equivalent to the difference-in-means FRT under local alternatives $\\tau_N=o(1)$.",
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   "tied_to": "Section 2, p. 347: the display for $\\widehat V(\\mathrm{OLS})$ and the paragraph beginning “Under Pitman-type local alternatives,” with $\\widehat V(\\mathrm{Fisher})-\\widehat V(\\mathrm{OLS})=o_p(N^{-1})$; Introduction, item (i), p. 346"
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   "claim": "The Wald-type variance estimates $\\widehat V(\\mathrm{OLS})$ and $\\widehat V(\\mathrm{Neyman})$ differ asymptotically only if $S_1\\ne S_0$. Neyman's estimate is conservative for $V(\\hat\\tau)$ and asymptotically exact under constant effects; under heterogeneous effects the pooled OLS test can be invalid for Neyman's null. The OLS-based test can be more or less powerful, and the stated direction of the paradox occurs when $(p>1/2\\text{ and }S_1>S_0)$ or $(p<1/2\\text{ and }S_1<S_0)$.",
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   "tied_to": "Section 3, pp. 347–348: the display for $\\widehat V(\\mathrm{OLS})-\\widehat V(\\mathrm{Neyman})$, Neyman’s upper bound, and the inequality $S_1^2/p+S_0^2/(1-p)<S_0^2/p+S_1^2/(1-p)$ with its bracketed equivalent; Introduction, item (ii), p. 346"
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   "claim": "The comment explains Ding’s apparent paradox by combining the first-order reference-variance inflation of $\\tau^2/N$ for the Rao-type test under nonlocal alternatives with the Behrens–Fisher-type behavior of tests that assume $S_1=S_0$ when that equality fails.",
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   "tied_to": "Introduction, “Combining (i) and (ii),” p. 346; Sections 2 and 3 together; Conclusion, p. 348"
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   "claim": "The comment argues that Ding's Theorem 7, concerning equivalence of test statistics for binary outcomes, is too general. It proposes a weaker claim: in a completely randomized experiment with binary outcomes, under the sharp null every test statistic can be written as a function of the difference in means.",
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   "tied_to": "Section 4, 'A Note on Theorem 7' (p. 348), especially the replacement claim attributed to Copas (1973), equation (9); Introduction item (iii) (p. 346)"
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     "alt": "Crop of five lines at the top of the right column of page 348. They read: for completely randomized experiments, if outcomes are binary, then under the sharp null hypothesis of no treatment effect, all test statistics may be written as a function of the difference-in-means statistic. Stronger statements are likely available. A red outline surrounds the words all test statistics.",
     "caption": "Section 4, p. 348 (PDF p. 3), top of the right column: the claim, which ends the sentence. The outline marks “all test statistics”.",
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     "alt": "Crop of four lines at the foot of the left column of page 348. They read: ference in means (e.g., the absolute value of tau-hat). For such test statistics, the resulting inferences may substantively differ. Here, we state a weaker claim verifiable at full generality, which has been known since Copas [(1973), equation (9)]: A red outline surrounds the words at full generality and the comma after them.",
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     "alt": "Crop of five lines in the right column of page 346. They read: one of the constituent hypotheses is false. (iii) We also note an error in Ding's (2017) Theorem 7 and suggest a refinement that is correct at full generality. Note, we consider only asymptotic results, and do not discuss the finite N differences between exact. A red outline surrounds the words at full generality and the full stop after them.",
     "caption": "Introduction, item (iii), p. 346 (PDF p. 1), right column. The outline marks “at full generality”.",
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   "headline": "In the replacement for Ding’s Theorem 7, “all test statistics” should read “all test statistics based on the two-by-two treatment-by-outcome table”, which is the class that the claim concerns and for which it is correct.",
   "id": "P053-E001",
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    "as_printed": "Section 4, p.~348: Here, we state a weaker claim verifiable \\hl{at full generality}, which has been known since Copas [(1973), equation (9)]: for completely randomized experiments, if outcomes are binary, then under the sharp null hypothesis of no treatment effect, \\hl{all test statistics} may be written as a function of the difference-in-means statistic. Stronger statements are likely available.\n\nIntroduction, item (iii), p.~346: We also note an error in Ding's (2017) Theorem 7 and suggest a refinement that is correct \\hl{at full generality}.",
    "corrected": "Section 4, p.~348: \\ldots for completely randomized experiments, if outcomes are binary, then under the sharp null hypothesis of no treatment effect, all test statistics \\hl{based on the two-by-two treatment-by-outcome table} may be written as a function of the difference-in-means statistic.\n\nIntroduction, item (iii), p.~346: \\ldots a refinement that is correct at full generality \\hl{for statistics based on the two-by-two treatment-by-outcome table}.\n\nThe statement in full: in a completely randomized experiment with fixed positive treatment and control counts $N_1$ and $N_0$ and binary outcomes, under the sharp null hypothesis of no treatment effect, every test statistic that depends on the data only through the two-by-two treatment-by-outcome table is a function of the difference-in-means statistic $\\widehat\\tau$. With $K$ total successes and $a$ treated successes,\n\\[\na = \\frac{N_1 N_0 \\widehat\\tau + N_1 K}{N_1 + N_0},\n\\]\nwhich determines all four cells of the table.",
    "intended_meaning": "The sentence takes the words ``all test statistics'' from Ding's Theorem 7, which the comment quotes a few lines above, and it replaces ``are equivalent to'' by ``may be written as a function of''. In the preprint of Ding's article (arXiv:1402.0142v4, Supplementary Material, p.~38), the proof of Theorem 7 begins ``All test statistics are functions of the two by two table''. The preprint was not compared with the published article and its supplement. Read with that proof, ``all test statistics'' ranges over the statistics based on the two-by-two treatment-by-outcome table, and ``at full generality'' covers all of them, including those that are not monotone in the difference in means $\\widehat\\tau$. The comment's own example, $|\\widehat\\tau|$, is such a statistic. The article of Copas that the sentence cites is titled ``Randomization models for the matched and unmatched $2 \\times 2$ tables''; its equation (9) was not read. With this reading the claim is correct.\n\nTo see this, let $N_1$ and $N_0$ be the fixed positive numbers of treated and control units, and let $N = N_1 + N_0$. Under the sharp null hypothesis, unit $i$ has one binary outcome $y_i = Y_i(1) = Y_i(0)$, and the number of successes $K = \\sum_i y_i$ is the same for every assignment. If $a$ of the treated units are successes, then the table has cells $a$, $N_1 - a$, $K - a$ and $N_0 - K + a$, and\n\\[\n\\widehat\\tau = \\frac{a}{N_1} - \\frac{K - a}{N_0} = \\frac{N a - N_1 K}{N_1 N_0}.\n\\]\nSolving for $a$, I obtain\n\\[\na = \\frac{N_1 N_0 \\widehat\\tau + N_1 K}{N_1 + N_0}.\n\\]\nThen $\\widehat\\tau$ determines $a$, and $a$ determines the four cells. This implies that every statistic based on the table is a function of $\\widehat\\tau$, whether or not it is monotone in $\\widehat\\tau$. A statistic that depends on the treated outcomes and on the control outcomes, and not on their order within either sample, is based on the table, since a binary sample is determined up to order by its number of successes.\n\nThe sentence does not name this class. If ``all test statistics'' is taken to include statistics that use the labels of the units or fixed covariates, then the claim is false. For example, suppose that there are $N = 4$ units with $N_1 = N_0 = 2$, that the outcomes are $y = (1,1,0,0)$ under both treatments, and that $X = (0,1,2,3)$ is a fixed covariate. Let $Z$ be the vector of treatment indicators, and let $\\widehat\\beta$ be the coefficient on $Z$ in the least-squares regression of the outcome on a constant, $Z$ and $X$. The six assignments are equally likely.\n\\begin{audittable}{llrrr}{The six assignments in the example with four units. The cells of the two-by-two table are listed as (treated successes, treated failures, control successes, control failures).}\nAssignment $Z$ & Cells & $\\widehat\\tau$ & $\\widehat\\beta$ & $Z_1 y_1$ \\\\\n\\hline\n$(1,1,0,0)$ & $(2,0,0,2)$ & $1$ & $1$ & $1$ \\\\\n$(1,0,1,0)$ & $(1,1,1,1)$ & $0$ & $-1/2$ & $1$ \\\\\n$(1,0,0,1)$ & $(1,1,1,1)$ & $0$ & $0$ & $1$ \\\\\n$(0,1,1,0)$ & $(1,1,1,1)$ & $0$ & $0$ & $0$ \\\\\n$(0,1,0,1)$ & $(1,1,1,1)$ & $0$ & $1/2$ & $0$ \\\\\n$(0,0,1,1)$ & $(0,2,2,0)$ & $-1$ & $-1$ & $0$ \\\\\n\\end{audittable}\nThe assignments $(1,0,1,0)$ and $(0,1,0,1)$ have the same cells and $\\widehat\\tau = 0$, but $\\widehat\\beta$ is $-1/2$ at the first and $1/2$ at the second. The statistic $Z_1 y_1$, which uses no covariate, is $1$ at the first and $0$ at the second. Neither statistic is a function of $\\widehat\\tau$. Repeating the four units $m$ times gives the same values at every $N = 4m$, so the asymptotic conditions on p.~347 do not change this.\n\nNothing else in the comment uses the sentence. The distinction between functional dependence and identical inference, the variance comparisons of Sections 2 and 3, and the conclusion are unaffected. In the example, the assignment with $\\widehat\\tau = 1$ has upper-tail p-value $1/6$ for $\\widehat\\tau$ and $1/3$ for $|\\widehat\\tau|$ in the randomization test. This is the comment's own point: a function of $\\widehat\\tau$ need not give the same inference as $\\widehat\\tau$."
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    "title": "Understanding Ding's Apparent Paradox",
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     "alt": "Crop of six lines in the right column of page 346. They read: ear model; Samii and Aronow (2012) give the implied variance of this test a randomization basis, by establishing that the implied variance is equivalent to the randomization distribution of the difference-in-means estimator if the treatment effect is assumed to be constant and equal to the observed estimate. Gerber and. Two red outlines surround the words equivalent to the, at the end of the third line, and randomization distribution, at the start of the fourth line.",
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    "corrected": "Samii and Aronow (2012) give the implied variance of this test a randomization basis, by establishing that the implied variance is equivalent to \\hl{the variance of} the randomization distribution of the difference-in-means estimator if the treatment effect is assumed to be constant and equal to the observed estimate.",
    "intended_meaning": "As printed, the sentence compares a variance with a distribution. The intended comparison is with the variance of that distribution. The equivalence is asymptotic, which is the scope that the Introduction (p.~346) sets for the whole comment, and with this reading the statement is correct. Let $\\text{SSR}$ be the residual sum of squares from the regression of the outcome on a constant and the treatment indicator. When the outcomes are imputed under a constant effect equal to the observed estimate, the randomization distribution of the difference in means has variance $N\\,\\text{SSR}/(N_1 N_0 (N-1))$. This is $(N-2)/(N-1)$ times the usual estimate $\\widehat{V}(\\text{OLS}) = N\\,\\text{SSR}/(N_1 N_0 (N-2))$, and the factor tends to $1$. No result in the comment depends on the sentence. Samii and Aronow (2012) was not consulted."
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class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Fisher</span></span><span class=\"mclose\">)</span></span></span></span>, <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>OLS</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{OLS})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">OLS</span></span><span class=\"mclose\">)</span></span></span></span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>Fisher</mtext><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>OLS</mtext><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>τ</mi><mn>2</mn></msup><mi mathvariant=\"normal\">/</mi><mi>N</mi><mo>+</mo><msub><mi>o</mi><mi>p</mi></msub><mo stretchy=\"false\">(</mo><msup><mi>N</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{Fisher})-\\widehat{V}(\\text{OLS})=\\tau^2/N+o_p(N^{-1})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Fisher</span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">OLS</span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mord\">/</span><span class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1002em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">o</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" 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stretchy=\"false\">)</mo><mo>=</mo><msub><mi>o</mi><mi>p</mi></msub><mo stretchy=\"false\">(</mo><msup><mi>N</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{Fisher})-\\widehat{V}(\\text{OLS})=o_p(N^{-1})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Fisher</span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">OLS</span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1002em;vertical-align:-0.2861em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">o</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.1514em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span> under local alternatives <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>τ</mi><mi>N</mi></msub><mo>=</mo><mi>o</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\tau_N=o(1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">o</span><span class=\"mopen\">(</span><span class=\"mord\">1</span><span class=\"mclose\">)</span></span></span></span>; Introduction, item (i)",
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     "headline": "Section 3, unnumbered statements: Neyman&#x27;s bound <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>N</mi><mo>→</mo><mi mathvariant=\"normal\">∞</mi></mrow></msub><mi>N</mi><mi>V</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>τ</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">)</mo><mo>≤</mo><msubsup><mi>S</mi><mn>1</mn><mn>2</mn></msubsup><mi mathvariant=\"normal\">/</mi><mi>p</mi><mo>+</mo><msubsup><mi>S</mi><mn>0</mn><mn>2</mn></msubsup><mi mathvariant=\"normal\">/</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\lim_{N\\to\\infty}NV(\\widehat{\\tau})\\le S_1^2/p+S_0^2/(1-p)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mop\"><span class=\"mop\">lim</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3283em;\"><span style=\"top:-2.55em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N</span><span class=\"mrel mtight\">→</span><span class=\"mord mtight\">∞</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N</span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span><span class=\"mopen\">(</span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6706em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ</span></span><span class=\"svg-align\" style=\"width:calc(100% - 0.0556em);margin-left:0.0556em;top:-3.4306em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">≤</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-2.4519em;margin-left:-0.0576em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2481em;\"><span></span></span></span></span></span></span><span class=\"mord\">/</span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.0641em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-2.4519em;margin-left:-0.0576em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0</span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2481em;\"><span></span></span></span></span></span></span><span class=\"mord\">/</span><span class=\"mopen\">(</span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mclose\">)</span></span></span></span>, equality under constant effects, agreement of <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>Neyman</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{Neyman})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Neyman</span></span><span class=\"mclose\">)</span></span></span></span> and <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>OLS</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{OLS})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">OLS</span></span><span class=\"mclose\">)</span></span></span></span> under constant effects, and the existence of limit distributions for which <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>OLS</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{OLS})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">OLS</span></span><span class=\"mclose\">)</span></span></span></span> is anticonservative",
     "location": "p. 347 (PDF p. 2), Section 3, foot of the right column, continuing at the top of the left column of p. 348 (PDF p. 3)"
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    "id": "P053-T8",
    "text": {
     "body": "Section 3, pp. 347–348 (PDF pp. 2–3). The actual variance obeys Neyman's bound, constant effects give equality and agreement, and the specified heterogeneous Neyman-null sequence has all stated finite formulas and the actual/OLS/Neyman limiting coefficients 81/4, 10 and 85/4.",
     "headline": "Section 3, unnumbered statements: Neyman's bound $\\lim_{N\\to\\infty}NV(\\widehat{\\tau})\\le S_1^2/p+S_0^2/(1-p)$, equality under constant effects, agreement of $\\widehat{V}(\\text{Neyman})$ and $\\widehat{V}(\\text{OLS})$ under constant effects, and the existence of limit distributions for which $\\widehat{V}(\\text{OLS})$ is anticonservative",
     "location": "p. 347 (PDF p. 2), Section 3, foot of the right column, continuing at the top of the left column of p. 348 (PDF p. 3)"
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    "url": "https://pmaronow.github.io/llm-audit/works/P053/#P053-T8"
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     "body": "<p>Sections 2–3, p. 347 (PDF p. 2), with the sentence beginning on p. 346. The Neyman and OLS expansions and limits satisfy the stated difference identity and necessary conditions, and the local Fisher comparison has the same expansion and limits.</p>",
     "headline": "Section 3, unnumbered display for <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>OLS</mtext><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>Neyman</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{OLS})-\\widehat{V}(\\text{Neyman})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">OLS</span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Neyman</span></span><span class=\"mclose\">)</span></span></span></span> and the sentence that <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>S</mi><mn>1</mn></msub><mo mathvariant=\"normal\">≠</mo><msub><mi>S</mi><mn>0</mn></msub></mrow><annotation encoding=\"application/x-tex\">S_1\\neq S_0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\"><span class=\"mrel\"><span class=\"mord katex-vbox\"><span class=\"katex-thinbox\"><span class=\"rlap\"><span class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\"></span><span class=\"katex-inner\"><span class=\"mord\"><span class=\"mrel\"></span></span></span><span class=\"katex-fix\"></span></span></span></span></span><span class=\"mspace nobreak\"></span><span class=\"mrel\">=</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span> is a necessary condition for the two probability limits to differ; with the display for <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>Neyman</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{Neyman})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Neyman</span></span><span class=\"mclose\">)</span></span></span></span> of Section 2",
     "location": "p. 347 (PDF p. 2), Section 3, right column; the display for <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>V</mi><mo stretchy=\"true\">^</mo></mover><mo stretchy=\"false\">(</mo><mtext>Neyman</mtext><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\widehat{V}(\\text{Neyman})</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:1.1733em;vertical-align:-0.25em;\"></span><span class=\"mord katex-accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9233em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V</span></span><span class=\"svg-align\" style=\"top:-3.6833em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span style=\"height:0.24em;\"><svg xmlns=\"http://www.w3.org/2000/svg\" width=\"100%\" height=\"0.24em\" viewBox=\"0 0 1062 239\" preserveAspectRatio=\"none\"><path d=\"M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z\"/></svg></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord text\"><span class=\"mord\">Neyman</span></span><span class=\"mclose\">)</span></span></span></span> is at the top of the left column of p. 347, in a sentence that begins on p. 346 (PDF p. 1)"
    },
    "id": "P053-T7",
    "text": {
     "body": "Sections 2–3, p. 347 (PDF p. 2), with the sentence beginning on p. 346. The Neyman and OLS expansions and limits satisfy the stated difference identity and necessary conditions, and the local Fisher comparison has the same expansion and limits.",
     "headline": "Section 3, unnumbered display for $\\widehat{V}(\\text{OLS})-\\widehat{V}(\\text{Neyman})$ and the sentence that $S_1\\neq S_0$ is a necessary condition for the two probability limits to differ; with the display for $\\widehat{V}(\\text{Neyman})$ of Section 2",
     "location": "p. 347 (PDF p. 2), Section 3, right column; the display for $\\widehat{V}(\\text{Neyman})$ is at the top of the left column of p. 347, in a sentence that begins on p. 346 (PDF p. 1)"
    },
    "url": "https://pmaronow.github.io/llm-audit/works/P053/#P053-T7"
   }
  ]
 },
 "missing_sources": [],
 "reproduction": {
  "agreement": "not_applicable",
  "status": "not_applicable",
  "targets": []
 },
 "summary": "This three-page comment explains the apparent paradox of Ding (2017), in which Neyman's null of no average effect can be rejected while Fisher's sharp null is not. It combines two known results: the Fisher randomization test is asymptotically a Rao score test, whose reference variance exceeds that of its Wald-type analogue by $\\tau^2/N$ to first order, and a test that assumes equal variances behaves as in the Behrens--Fisher problem. No entries are classified as errors, and 2 are clarity notes. One possible error about a cited source was not confirmed, and one remains undetermined.",
 "what_was_checked": "The comment is three pages, with no table, figure, numerical example or appendix, and I read all of it. The following algebra and asymptotic comparisons held up under the comment's regularity conditions and, for the tests, a positive limiting variance.\n\\begin{itemize}\n\\item p.~347: the expansions $\\widehat{V}(\\text{Neyman}) = S_1^2/N_1 + S_0^2/N_0 + o_p(N^{-1})$, $\\widehat{V}(\\text{Fisher}) = S_1^2/N_0 + S_0^2/N_1 + \\tau^2/N + o_p(N^{-1})$ and $\\widehat{V}(\\text{OLS}) = S_1^2/N_0 + S_0^2/N_1 + o_p(N^{-1})$, and the difference $\\widehat{V}(\\text{Fisher}) - \\widehat{V}(\\text{OLS}) = \\tau^2/N + o_p(N^{-1})$, which is $o_p(N^{-1})$ when $\\tau_N = o(1)$.\n\\item p.~347: the Wald-type test with $\\widehat{V}(\\text{OLS})$ is asymptotically valid under the sharp null hypothesis and under constant effects. Under constant nonlocal effects, $\\widehat{V}(\\text{OLS})$ is consistent for $V(\\widehat\\tau)$, and $\\widehat{V}(\\text{Fisher})$ overstates it by $\\tau^2/N$ to first order.\n\\item p.~347: the display for $\\widehat{V}(\\text{OLS}) - \\widehat{V}(\\text{Neyman})$. Its limit after multiplication by $N$ is $(S_1^2 - S_0^2)(2p-1)/(p(1-p))$, so $S_1 \\ne S_0$ is necessary for a difference, as the comment states.\n\\item p.~347: $\\lim_{N\\to\\infty} N V(\\widehat\\tau) \\le S_1^2/p + S_0^2/(1-p)$, with equality under constant effects. The exact randomization variance $S_1^2/N_1 + S_0^2/N_0 - S_\\tau^2/N$ agreed with complete enumeration of small populations.\n\\item p.~348: there exist limit distributions such that $\\lim_{N\\to\\infty} N V(\\widehat\\tau) > S_0^2/p + S_1^2/(1-p)$. With $p = 1/5$, $S_1^2 = 4$, $S_0^2 = 1$ and $S_\\tau^2 = 1$, the limits after multiplication by $N$ are $81/4$ for the true variance, $10$ for $\\widehat{V}(\\text{OLS})$ and $85/4$ for $\\widehat{V}(\\text{Neyman})$. At nominal size $0.05$, the test with $\\widehat{V}(\\text{OLS})$ then has asymptotic size about $0.168$.\n\\item p.~348: the test with $\\widehat{V}(\\text{OLS})$ may be more or less powerful than the test with $\\widehat{V}(\\text{Neyman})$. There exist exact examples in which the limit for $\\widehat{V}(\\text{OLS})$ is no smaller than the true variance and is below the limit for $\\widehat{V}(\\text{Neyman})$, and examples in which it is above that limit. The inequality $S_1^2/p + S_0^2/(1-p) < S_0^2/p + S_1^2/(1-p)$ is equivalent to the bracketed condition, ($p > 1/2$ and $S_1 > S_0$) or ($p < 1/2$ and $S_1 < S_0$).\n\\item Sections 2 and 3 together: $\\widehat{V}(\\text{Fisher}) - \\widehat{V}(\\text{Neyman}) = \\tau^2/N + (1/N_0 - 1/N_1)(S_1^2 - S_0^2) + o_p(N^{-1})$, so the two parts of the explanation account for the whole difference to first order.\n\\item p.~348, Section 4: the signed-versus-absolute-value comparison. The statistic $|\\widehat\\tau|$ is not strictly increasing in $\\widehat\\tau$, and exact examples show that the two statistics can have different upper-tail p-values in the randomization test. The replacement claim holds for statistics based on the two-by-two treatment-by-outcome table; see \\fid{P053-E001}.\n\\item The citation of Engle (1984) for the possible suboptimality of Rao-type tests, and the quotation from p.~786, agree with the chapter.\n\\item The cross-references within the comment point to the right places. The pages cited for Engle (786) and for Reichardt and Gollob (124--125) lie within the page ranges in the reference list.\n\\end{itemize}\nThese checks concern the comparison of asymptotic normal approximations. They do not establish a uniform approximation of the exponentially small tails of the randomization test under fixed alternatives.\n\nThe following were not checked. Of Ding (2017), only the preprint was read, and only for Theorem 7, its proof and the definition of the test. The published article, its equation (7), its Section A.3.1 and Ding's rejoinder were not read, and neither was Copas (1973). Whether the published Theorem 7 is in error as the comment says, and whether Copas's equation (9) states the claim attributed to it, were not settled. The other cited works were not compared with their sources: Reichardt and Gollob (1999), Lin (2013), Samii and Aronow (2012), Freedman (2008), Pratt (1964), Romano (1990), Neyman (1990), Aronow (2013), Mosteller and Rourke (1973), and Gerber and Green (2012).\n\nThe comment has no simulation and no empirical application, and I found no supplement, erratum or replication files for it, so there were no published numerical results to reproduce from code. The absence of further corrections is not a finding that the comment is correct.",
 "work": {
  "arxiv_id": null,
  "audit_complete": true,
  "authors": [
   "P. M. Aronow",
   "Molly R. Offer-Westort"
  ],
  "category": "Essays, Commentary, Reports",
  "citation": "P. M. Aronow and Molly R. Offer-Westort (2017). Understanding Ding's Apparent Paradox. Statistical Science 32(3):346–348.",
  "doi": "10.1214/16-STS582",
  "edition": {
   "pages_in_audited_file": 3,
   "sha256": "15c35664f0b88617333122565ee62b23f8611e25af1fcdb06b9923f9c678be2f",
   "source_url": "https://doi.org/10.1214/16-STS582",
   "venue": "Statistical Science",
   "year": "2017"
  },
  "id": "P053",
  "later_materials": null,
  "title": "Understanding Ding's Apparent Paradox",
  "type": "Commentary/article",
  "url": "https://pmaronow.github.io/llm-audit/works/P053/",
  "venue": "Statistical Science",
  "year": "2017"
 }
}