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Caning of Charles Sumner
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Representative Preston Brooks (left) brutally beat Senator Charles Sumner after Sumner gave a fiery speech attacking slavery and its practitioners.
Events leading to
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Manifest destiny
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Compromise of 1850
Fugitive Slave Act of 1850
Uncle Tom's Cabin
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Caning of Charles Sumner
Dred Scott v. Sandford
The Impending Crisis of the South
Brown's raid on Harpers Ferry
1860 presidential election
Crittenden Compromise
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The Caning of Charles Sumner, or the Brooks–Sumner Affair, occurred on May 22, 1856, in the United States Senate when Representative Preston Brooks (D-SC) attacked Senator Charles Sumner (R-MA), an abolitionist, with a walking cane in retaliation for a speech given by Sumner two days earlier in which he fiercely criticized slaveholders including a relative of Brooks. The beating nearly killed Sumner and it drew a sharply polarized response from the American public on the subject of the expansion of slavery in the United States. It has been considered symbolic of the "breakdown of reasoned discourse"[1] that eventually led to the American Civil War.
Contents
[hide]
1 Background
2 Day of the attack
3 Aftermath
4 References
5 External links
Background[edit]
Wikisource has original text related to this article:
The Crime against Kansas
In 1856, during the "Bleeding Kansas" crisis, Sumner denounced the Kansas–Nebraska Act in his "Crime against Kansas" speech, delivered on May 19 and May 20. The long speech argued for the immediate admission of Kansas as a free state and went on to denounce the "Slave Power"—the political arm of the slave owners:
Not in any common lust for power did this uncommon tragedy have its origin. It is the rape of a virgin Territory, compelling it to the hateful embrace of slavery; and it may be clearly traced to a depraved desire for a new Slave State, hideous offspring of such a crime, in the hope of adding to the power of slavery in the National Government.[2]
Sumner then attacked the authors of the Act, Senators Stephen A. Douglas of Illinois and Andrew Butler of South Carolina, saying,
The senator from South Carolina has read many books of chivalry, and believes himself a chivalrous knight with sentiments of honor and courage. Of course he has chosen a mistress to whom he has made his vows, and who, though ugly to others, is always lovely to him; though polluted in the sight of the world, is chaste in his sight—I mean the harlot, slavery. For her his tongue is always profuse in words. Let her be impeached in character, or any proposition made to shut her out from the extension of her wantonness, and no extravagance of manner or hardihood of assertion is then too great for this senator.
In addition, Sumner mocked Butler's speaking ability, which had been impeded by a recent stroke:
[He] touches nothing which he does not disfigure with error, sometimes of principle, sometimes of fact. He cannot open his mouth, but out there flies a blunder.[3]
According to Manisha Sinha (2003), Sumner had been ridiculed and insulted by both Douglas and Butler for his opposition to the Fugitive Slave Law and the Kansas-Nebraska Act earlier, with Butler crudely race-baiting Sumner by making sexual allusions to black women, like many slaveholders who accused abolitionists of promoting interracial marriage.[4]
According to Hoffer (2010), "It is also important to note the sexual imagery that recurred throughout the oration, which was neither accidental nor without precedent. Abolitionists routinely accused slaveholders of maintaining slavery so that they could engage in forcible sexual relations with their slaves."[5] Douglas said during the speech that "this damn fool is going to get himself killed by some other damn fool."[6]
Representative Preston Brooks, Butler's cousin, was infuriated. He later said that he intended to challenge Sumner to a duel, and consulted with fellow South Carolina Representative Laurence M. Keitt on dueling etiquette. Keitt told him that dueling was for gentlemen of equal social standing, and that Sumner was no better than a drunkard, due to the supposedly coarse language he had used during his speech. Brooks said that he concluded that since Sumner was no gentleman, he did not merit honorable treatment; to Keitt and Brooks, it was more appropriate to humiliate Sumner by beating him with a cane in a public setting.[7]
Day of the attack[edit]
Congressman Laurence Keitt advised Brooks and was with him when he assaulted Sumner.
Two days later, on the afternoon of May 22, Brooks entered the Senate chamber with Keitt and another ally, Congressman Henry A. Edmundson. They waited for the galleries to clear, especially concerned that there be no ladies present to witness what Brooks intended to do.[8] He confronted Sumner as he sat writing at his desk in the almost empty Senate chamber. "Mr. Sumner, I have read your speech twice over carefully. It is a libel on South Carolina, and Mr. Butler, who is a relative of mine," Brooks calmly announced in a low voice. As Sumner began to stand up, Brooks beat Sumner severely on the head before he could reach his feet, using a thick gutta-percha cane with a gold head. The force of the blows so shocked Sumner that he lost his sight immediately. "I no longer saw my assailant, nor any other person or object in the room. What I did afterwards was done almost unconsciously, acting under the instincts of self-defense," he recalled later.[9]
Congressman Henry A. Edmundson also advised Brooks and was with him during the assault on Sumner.
Lithograph of Preston Brooks' 1856 attack on Sumner; the artist depicts the faceless assailant bludgeoning the learned martyr
Sumner was knocked down and trapped under the heavy desk, which was bolted to the floor. His chair, which was pulled up to his desk, moved back and forth on a track; Sumner either could not or did not think to slide his chair back to escape, so it pinned him under his desk. Brooks continued to strike Sumner until Sumner rose to his feet and ripped the desk from the floor in an effort to get away from Brooks.[10] By this time, Sumner was blinded by his own blood. He staggered up the aisle and, arms outstretched, vainly attempted to defend himself. But then he was an even larger and easier target for Brooks, who continued to beat him across the head, face, and shoulders "to the full extent of [my] power." Brooks didn't stop when his cane snapped; he continued thrashing Sumner with the piece which held the gold head. Sumner stumbled and reeled convulsively, "Oh Lord," he gasped "Oh! Oh!" Near the end of the attack, Sumner collapsed unconscious, although shortly before he succumbed he "bellowed like a calf" according to Brooks. Brooks grabbed the falling Sumner, held him up by the lapel with one hand, and continued to lash out at him with the cane in the other.[11][12] Several other Senators and Representatives attempted to help Sumner, but were blocked by Edmundson, who yelled at the spectators to leave Brooks and Sumner alone,[13] and Keitt, who brandished his own cane and a pistol, and shouted, "Let them be!" and "Let them alone, God damn you, let them alone!"[14][15][16]
Senator John J. Crittenden attempted to intervene, and pleaded with Brooks not to kill Sumner. Senator Robert Toombs then interceded for Crittenden, telling Keitt not to attack someone who was not a party to the dispute, though Toombs also indicated later that he had no issue with Brooks beating Sumner, and in fact approved of it.[17]
Representatives Ambrose S. Murray and Edwin B. Morgan were finally able to intervene and restrain Brooks, at which point he quietly left the chamber.[18] Murray obtained the aid of a Senate page and the Sergeant at Arms, Dunning R. McNair.[19] As Sumner regained consciousness they were able to assist him to walk to a cloakroom.[20] Sumner received medical attention, including several stitches.[21] With the aid of Nathaniel P. Banks, the Speaker of the House, and Senator Henry Wilson, Sumner was able to travel by carriage to his lodgings, where he received further medical treatment.[22]
Brooks also required medical attention before leaving the Capitol; he had hit himself above his right eye with one of his backswings.[23] Brooks was arrested for the assault.[24]
The cane Brooks used was broken into several pieces, which he left on the blood soaked floor of the Senate chamber. Some, including the cane's gold head, were recovered by Edmundson, who gave the portion with the head to Adam John Glossbrenner, the House Sergeant at Arms.[25][26] This portion of the cane eventually ended up at the Old State House Museum in Boston; it was worked to smooth the edges and finish, and then put on display.[27] Southern lawmakers made rings out of the other pieces Edmundson recovered from the Senate floor, which they wore on neck chains to show their solidarity with Brooks, who boasted "[The pieces of my cane] are begged for as sacred relics." [28]
Aftermath[edit]
The walking cane used to attack Charles Sumner on exhibit at the Old State House in Boston.
The episode revealed the polarization in America, as Sumner became a martyr in the North and Brooks a hero in the South. Northerners were outraged. The Cincinnati Gazette said, "The South cannot tolerate free speech anywhere, and would stifle it in Washington with the bludgeon and the bowie-knife, as they are now trying to stifle it in Kansas by massacre, rapine, and murder."[29] William Cullen Bryant of the New York Evening Post, asked, "Has it come to this, that we must speak with bated breath in the presence of our Southern masters?... Are we to be chastised as they chastise their slaves? Are we too, slaves, slaves for life, a target for their brutal blows, when we do not comport ourselves to please them?"[30] Thousands attended rallies in support of Sumner in Boston, Albany, Cleveland, Detroit, New Haven, New York, and Providence. More than a million copies of Sumner's speech were distributed. Two weeks after the caning, Ralph Waldo Emerson described the divide the incident represented: "I do not see how a barbarous community and a civilized community can constitute one state. I think we must get rid of slavery, or we must get rid of freedom."[31]
Conversely, Brooks was praised by Southern newspapers. The Richmond Enquirer editorialized that Sumner should be caned "every morning", praising the attack as "good in conception, better in execution, and best of all in consequences" and denounced "these vulgar abolitionists in the Senate" who "have been suffered to run too long without collars. They must be lashed into submission." Southerners sent Brooks hundreds of new canes in endorsement of his assault. One was inscribed "Hit him again."[32]
Representative Anson Burlingame publicly humiliated Brooks by goading him into challenging Burlingame to a duel, only to set conditions designed to intimidate Brooks into backing down. (As the challenged party, Burlingame, who was a crack shot, had the choice of weapons and dueling ground. He selected rifles on the Canada side of Niagara Falls, where U.S. anti-dueling laws would not apply. Brooks withdrew his challenge, claiming that he did not want to expose himself to the risk of violence by traveling through northern states to get to Niagara Falls.)[33]
Brooks also threatened Senator Henry Wilson, Sumner's colleague from Massachusetts. Wilson called the beating by Brooks "brutal, murderous, and cowardly," and in response Brooks challenged Wilson to a duel.[34] Wilson declined, saying that he could not legally or by personal conviction participate, and calling dueling "the lingering relic of a barbarous civilization."[35] In reference to a rumor that Brooks might attack him in the Senate, Wilson told the press "I have sought no controversy, and I seek none, but I shall go where duty requires, uninfluenced by threats of any kind."[36] Wilson continued to perform his Senate duties, and Brooks did not make good on his threat.
Historian William Gienapp has concluded that Brooks' "assault was of critical importance in transforming the struggling Republican party into a major political force."[37]
Southerners mocked Sumner, claiming he was faking his injuries.[38] They argued that the cane Brooks used was not heavy enough to inflict severe injuries.[39] They also claimed that Brooks had not hit Sumner more than a few times, and had not hit him hard enough to cause serious health concerns.[40] In fact, Sumner suffered head trauma that caused him chronic, debilitating pain for the rest of his life and symptoms consistent with what is now called traumatic brain injury and post-traumatic stress disorder; he spent three years convalescing before returning to his Senate seat.[41]
Brooks claimed that he had not intended to kill Sumner, or else he would have used a different weapon.[42] In a speech to the House defending his actions, Brooks stated that he "meant no disrespect to the Senate of the United States" or the House by his attack on Sumner.[43] He was tried in a District of Columbia court, convicted for assault, and fined $300 ($8,000 in today's dollars), but received no prison sentence.[44] A motion to expel Brooks from the House failed, but he resigned on July 15 in order to permit his constituents to ratify or condemn his conduct via a special election.[45] They approved; Brooks was quickly returned to office in the August 1 special election,[46] and then re-elected to a new term of office later in 1856,[47] but he died of croup before the new term began.[48]
Keitt, who facilitated Brooks' attack, was censured by the House.[49] He resigned in protest, but his constituents ratified his conduct by overwhelmingly reelecting him to his seat within a month.[50] In 1858, he attempted to choke Representative Galusha Grow of Pennsylvania (Republican) for calling him a "negro driver" during an argument on the House floor.[51]
An effort to censure Edmundson failed to obtain a majority of votes in the House.[52]
During the 1856 lame duck session of Congress, Brooks made a speech calling for the admission of Kansas "even with a constitution rejecting slavery". His conciliatory tone impressed Northerners and disappointed slavery's supporters.[53]
References[edit]
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Jump up ^ Hendrix, Pat (2006). Murder and Mayhem in the Holy City. Charleston, SC: History Press. p. 50. ISBN 978-1-59629-162-1 – via Google Books.
Jump up ^ Sinha, Manisha (Summer 2003). "The Caning of Charles Sumner: Slavery, Race and Ideology in the Age of the Civil War". Journal of the Early Republic. 23: 233–262. doi:10.2307/3125037 – via EBSCO Host.
Jump up ^ Hoffer, Williamjames Hull (2010). The Caning of Charles Sumner: Honor, Idealism, and the Origins of the Civil War. Baltimore: Johns Hopkins University Press. p. 62. ISBN 978-0-8018-9468-8 – via Google Books.
Jump up ^ Walther, Eric H. (2004). The Shattering of the Union: America in the 1850s. Lanham, MD: SR Books. p. 97. ISBN 978-0-8420-2799-1 – via Google Books.
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Jump up ^ Puleo, Stephen (2013). The Caning: The Assault That Drove America to Civil War. Yardley, PA: Westholme Publishing. p. 112. ISBN 978-1-59416-187-2 – via Google Books.
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Jump up ^ Campbell, L. D. (June 2, 1856). U.S. House of Representatives Report 182, 34th Congress, 1st Session: Select Committee Report, Alleged Assault upon Senator Sumner. Washington, DC: U.S. Government Printing Office. pp. 64–65 – via Google Books.
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Jump up ^ "Outrage in the United States Senate: Senator Sumner, of Massachusetts, Knocked Down and Beaten till Insensible by Mr. Brooks, of South Carolina". The Baltimore Sun. May 23, 1856. p. 1 – via ProQuest Archiver.
Jump up ^ Dickey, J.D. (2014). Empire of Mud: The Secret History of Washington, DC. Guilford, CT: Globe Pequot Press. p. 141. ISBN 978-0-7627-8701-2 – via Google Books.
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Jump up ^ "#7 Raising Cane". Military History in 100 Objects: A Farewell to Arms (and Legs). MHN: Military History Now. May 5, 2015.
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Jump up ^ Puleo (2013), pp. 36-7.
Jump up ^ Palmer, John J. (June 3, 1856). "The Caning Affair". The Western Democrat. Charlotte, NC. p. 3. (Subscription required (help)).
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Jump up ^ "Senator Wilson and Mr. Brooks". Daily Courier. Louisville, KY. June 4, 1856. p. 3 – via Newspapers.com. (Subscription required (help)).
Jump up ^ Gienapp, William E. (1979). "The Crime Against Sumner: The Caning of Charles Sumner and the Rise of the Republican Party". Civil War History. 25 (25): 218–45.
Jump up ^ Mitchell, Thomas G. (2007). Antislavery Politics in Antebellum and Civil War America. Westport, CT: Praeger. p. 95. ISBN 978-0-275-99168-5.
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Jump up ^ Hoffer (2010), p. 83.
Jump up ^ Wilson, James Grant; Fiske, John (1898). Appletons' Cyclopaedia of American Biography. vol. V, Pickering–Sumter. New York: D. Appleton and Company. p. 747 – via Google Books.
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Jump up ^ Transactions of the Kansas State Historical Society. vol. 7. Topeka, KS: W. Y. Morgan. 1902. p. 424 – via Google Books.
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External links[edit]
The Caning of Senator Charles Sumner (U.S. Senate website)
C-SPAN Q&A interview with Stephen Puleo about his book The Caning: The Assault that Drove America to Civil War, June 21, 2015
Retrieved from "https://en.wikipedia.org/w/index.php?title=Caning_of_Charles_Sumner&oldid=799399149"
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text:
Fisher's exact test
From Wikipedia, the free encyclopedia
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Biologist and statistician Ronald Fisher
Fisher's exact test is a statistical significance test used in the analysis of contingency tables.[1][2][3] Although in practice it is employed when sample sizes are small, it is valid for all sample sizes. It is named after its inventor, Ronald Fisher, and is one of a class of exact tests, so called because the significance of the deviation from a null hypothesis (e.g., P-value) can be calculated exactly, rather than relying on an approximation that becomes exact in the limit as the sample size grows to infinity, as with many statistical tests.
Fisher is said to have devised the test following a comment from Muriel Bristol, who claimed to be able to detect whether the tea or the milk was added first to her cup. He tested her claim in the "lady tasting tea" experiment.[4]
Contents
[hide]
1 Purpose and scope
2 Example
3 Controversies
4 Alternatives
5 See also
6 References
7 External links
Purpose and scope[edit]
A teapot, a creamer and teacup full of tea with milk—can a taster tell if the milk went in first?
The test is useful for categorical data that result from classifying objects in two different ways; it is used to examine the significance of the association (contingency) between the two kinds of classification. So in Fisher's original example, one criterion of classification could be whether milk or tea was put in the cup first; the other could be whether Bristol thinks that the milk or tea was put in first. We want to know whether these two classifications are associated—that is, whether Bristol really can tell whether milk or tea was poured in first. Most uses of the Fisher test involve, like this example, a 2 × 2 contingency table. The p-value from the test is computed as if the margins of the table are fixed, i.e. as if, in the tea-tasting example, Bristol knows the number of cups with each treatment (milk or tea first) and will therefore provide guesses with the correct number in each category. As pointed out by Fisher, this leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table.
With large samples, a chi-squared test (or better yet, a G-test) can be used in this situation. However, the significance value it provides is only an approximation, because the sampling distribution of the test statistic that is calculated is only approximately equal to the theoretical chi-squared distribution. The approximation is inadequate when sample sizes are small, or the data are very unequally distributed among the cells of the table, resulting in the cell counts predicted on the null hypothesis (the “expected values”) being low. The usual rule of thumb for deciding whether the chi-squared approximation is good enough is that the chi-squared test is not suitable when the expected values in any of the cells of a contingency table are below 5, or below 10 when there is only one degree of freedom (this rule is now known to be overly conservative[5]). In fact, for small, sparse, or unbalanced data, the exact and asymptotic p-values can be quite different and may lead to opposite conclusions concerning the hypothesis of interest.[6][7] In contrast the Fisher exact test is, as its name states, exact as long as the experimental procedure keeps the row and column totals fixed, and it can therefore be used regardless of the sample characteristics. It becomes difficult to calculate with large samples or well-balanced tables, but fortunately these are exactly the conditions where the chi-squared test is appropriate.
For hand calculations, the test is only feasible in the case of a 2 × 2 contingency table. However the principle of the test can be extended to the general case of an m × n table,[8][9] and some statistical packages provide a calculation (sometimes using a Monte Carlo method to obtain an approximation) for the more general case.[10]
Example[edit]
For example, a sample of teenagers might be divided into male and female on the one hand, and those that are and are not currently studying for a statistics exam on the other. We hypothesize, for example, that the proportion of studying individuals is higher among the women than among the men, and we want to test whether any difference of proportions that we observe is significant. The data might look like this:
Men
Women
Row total
Studying
1 9 10
Not-studying
11 3 14
Column total 12 12 24
The question we ask about these data is: knowing that 10 of these 24 teenagers are studiers, and that 12 of the 24 are female, and assuming the null hypothesis that men and women are equally likely to study, what is the probability that these 10 studiers would be so unevenly distributed between the women and the men? If we were to choose 10 of the teenagers at random, what is the probability that 9 or more of them would be among the 12 women, and only 1 or fewer from among the 12 men?
Before we proceed with the Fisher test, we first introduce some notations. We represent the cells by the letters a, b, c and d, call the totals across rows and columns marginal totals, and represent the grand total by n. So the table now looks like this:
Men
Women
Row Total
Studying
a b a + b
Non-studying
c d c + d
Column Total a + c b + d a + b + c + d (=n)
Fisher showed that the probability of obtaining any such set of values was given by the hypergeometric distribution:
p = ( a + b a ) ( c + d c ) ( n a + c ) = ( a + b b ) ( c + d d ) ( n b + d ) = ( a + b ) ! ( c + d ) ! ( a + c ) ! ( b + d ) ! a ! b ! c ! d ! n ! {\displaystyle p={\frac {\displaystyle {{a+b} \choose {a}}\displaystyle {{c+d} \choose {c}}}{\displaystyle {{n} \choose {a+c}}}}={\frac {\displaystyle {{a+b} \choose {b}}\displaystyle {{c+d} \choose {d}}}{\displaystyle {{n} \choose {b+d}}}}={\frac {(a+b)!~(c+d)!~(a+c)!~(b+d)!}{a!~~b!~~c!~~d!~~n!}}}
where ( n k ) {\displaystyle {\tbinom {n}{k}}} is the binomial coefficient and the symbol ! indicates the factorial operator. With the data above (using the first of the equivalent forms), this gives:
p = ( 10 1 ) ( 14 11 ) / ( 24 12 ) = 10 ! 14 ! 12 ! 12 ! 1 ! 9 ! 11 ! 3 ! 24 ! ≈ 0.001346076 {\displaystyle p={{\tbinom {10}{1}}{\tbinom {14}{11}}}/{\tbinom {24}{12}}={\tfrac {10!~14!~12!~12!}{1!~9!~11!~3!~24!}}\approx 0.001346076}
The formula above gives the exact hypergeometric probability of observing this particular arrangement of the data, assuming the given marginal totals, on the null hypothesis that men and women are equally likely to be studiers. To put it another way, if we assume that the probability that a man is a studier is p {\displaystyle {\mathfrak {p}}} , the probability that a woman is a studier is also p {\displaystyle {\mathfrak {p}}} , and we assume that both men and women enter our sample independently of whether or not they are studiers, then this hypergeometric formula gives the conditional probability of observing the values a, b, c, d in the four cells, conditionally on the observed marginals (i.e., assuming the row and column totals shown in the margins of the table are given). This remains true even if men enter our sample with different probabilities than women. The requirement is merely that the two classification characteristics—gender, and studier (or not)—are not associated.
For example, suppose we knew probabilities P , Q , p , q {\displaystyle P,Q,{\mathfrak {p,q}}} with P + Q = p + q = 1 {\displaystyle P+Q={\mathfrak {p}}+{\mathfrak {q}}=1} such that (male studier, male non-studier, female studier, female non-studier) had respective probabilities ( P p , P q , Q p , Q q ) {\displaystyle (P{\mathfrak {p}},P{\mathfrak {q}},Q{\mathfrak {p}},Q{\mathfrak {q}})} for each individual encountered under our sampling procedure. Then still, were we to calculate the distribution of cell entries conditional given marginals, we would obtain the above formula in which neither p {\displaystyle {\mathfrak {p}}} nor P {\displaystyle P} occurs. Thus, we can calculate the exact probability of any arrangement of the 24 teenagers into the four cells of the table, but Fisher showed that to generate a significance level, we need consider only the cases where the marginal totals are the same as in the observed table, and among those, only the cases where the arrangement is as extreme as the observed arrangement, or more so. (Barnard's test relaxes this constraint on one set of the marginal totals.) In the example, there are 11 such cases. Of these only one is more extreme in the same direction as our data; it looks like this:
Men
Women
Row Total
Studying
0 10 10
Non-studying
12 2 14
Column Total 12 12 24
For this table (with extremely unequal studying proportions) the probability is p = ( 10 0 ) ( 14 12 ) / ( 24 12 ) ≈ 0.000033652 {\displaystyle {p={\tbinom {10}{0}}{\tbinom {14}{12}}}/{\tbinom {24}{12}}\approx 0.000033652} .
In order to calculate the significance of the observed data, i.e. the total probability of observing data as extreme or more extreme if the null hypothesis is true, we have to calculate the values of p for both these tables, and add them together. This gives a one-tailed test, with p approximately 0.001346076 + 0.000033652 = 0.001379728. For example, in the R statistical computing environment, this value can be obtained as fisher.test(rbind(c(1,9),c(11,3)), alternative="less")$p.value. This value can be interpreted as the sum of evidence provided by the observed data—or any more extreme table—for the null hypothesis (that there is no difference in the proportions of studiers between men and women). The smaller the value of p, the greater the evidence for rejecting the null hypothesis; so here the evidence is strong that men and women are not equally likely to be studiers.
For a two-tailed test we must also consider tables that are equally extreme, but in the opposite direction. Unfortunately, classification of the tables according to whether or not they are 'as extreme' is problematic. An approach used by the fisher.test function in R is to compute the p-value by summing the probabilities for all tables with probabilities less than or equal to that of the observed table. In the example here, the 2-sided p-value is twice the 1-sided value—but in general these can differ substantially for tables with small counts, unlike the case with test statistics that have a symmetric sampling distribution.
As noted above, most modern statistical packages will calculate the significance of Fisher tests, in some cases even where the chi-squared approximation would also be acceptable. The actual computations as performed by statistical software packages will as a rule differ from those described above, because numerical difficulties may result from the large values taken by the factorials. A simple, somewhat better computational approach relies on a gamma function or log-gamma function, but methods for accurate computation of hypergeometric and binomial probabilities remains an active research area.
Controversies[edit]
Despite the fact that Fisher's test gives exact p-values, some authors have argued that it is conservative, i.e. that its actual rejection rate is below the nominal significance level.[11][12][13] The apparent contradiction stems from the combination of a discrete statistic with fixed significance levels.[14][15] To be more precise, consider the following proposal for a significance test at the 5%-level: reject the null hypothesis for each table to which Fisher's test assigns a p-value equal to or smaller than 5%. Because the set of all tables is discrete, there may not be a table for which equality is achieved. If α e {\displaystyle \alpha _{e}} is the largest p-value smaller than 5% which can actually occur for some table, then the proposed test effectively tests at the α e {\displaystyle \alpha _{e}} -level. For small sample sizes, α e {\displaystyle \alpha _{e}} might be significantly lower than 5%.[11][12][13] While this effect occurs for any discrete statistic (not just in contingency tables, or for Fisher's test), it has been argued that the problem is compounded by the fact that Fisher's test conditions on the marginals.[16] To avoid the problem, many authors discourage the use of fixed significance levels when dealing with discrete problems.[14][15]
The decision to condition on the margins of the table is also controversial.[17][18] The p-values derived from Fisher's test come from the distribution that conditions on the margin totals. In this sense, the test is exact only for the conditional distribution and not the original table where the margin totals may change from experiment to experiment. It is possible to obtain an exact p-value for the 2x2 table when the margins are not held fixed. Barnard's test, for example, allows for random margins. However, some authors[14][15][18] (including, later, Barnard himself)[14] have criticized Barnard's test based on this property. They argue that the marginal success total is an (almost[15]) ancillary statistic, containing (almost) no information about the tested property.
The act of conditioning on the marginal success rate from a 2x2 table can be shown to ignore some information in the data about the unknown odds ratio.[19] The argument that the marginal totals are (almost) ancillary implies that the appropriate likelihood function for making inferences about this odds ratio should be conditioned on the marginal success rate.[19] Whether this lost information is important for inferential purposes is the essence of the controversy.[19]
Alternatives[edit]
An alternative exact test, Barnard's exact test, has been developed and proponents of it suggest that this method is more powerful, particularly in 2 × 2 tables. Another alternative is to use maximum likelihood estimates to calculate a p-value from the exact binomial or multinomial distributions and reject or fail to reject based on the p-value.[citation needed]
For stratified categorical data the Cochran–Mantel–Haenszel test must be used instead of Fisher's test.
Choi et al.[19] propose a p-value derived from the likelihood ratio test based on the conditional distribution of the odds ratio given the marginal success rate. This p-value is inferentially consistent with classical tests of normally distributed data as well as with likelihood ratios and support intervals based on this conditional likelihood function. It is also readily computable.[20]
See also[edit]
Barnard's exact test
Chi squared test
References[edit]
Jump up ^ Fisher, R. A. (1922). "On the interpretation of χ2 from contingency tables, and the calculation of P". Journal of the Royal Statistical Society. 85 (1): 87–94. doi:10.2307/2340521. JSTOR 2340521.
Jump up ^ Fisher, R.A. (1954). Statistical Methods for Research Workers. Oliver and Boyd. ISBN 0-05-002170-2.
Jump up ^ Agresti, Alan (1992). "A Survey of Exact Inference for Contingency Tables". Statistical Science. 7 (1): 131–153. CiteSeerX 10.1.1.296.874  . doi:10.1214/ss/1177011454. JSTOR 2246001.
Jump up ^ Fisher, Sir Ronald A. (1956) [The Design of Experiments (1935)]. "Mathematics of a Lady Tasting Tea". In James Roy Newman. The World of Mathematics, volume 3. Courier Dover Publications. ISBN 978-0-486-41151-4.
Jump up ^ Larntz, Kinley (1978). "Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics". Journal of the American Statistical Association. 73 (362): 253–263. doi:10.2307/2286650. JSTOR 2286650.
Jump up ^ Mehta, Cyrus R; Patel, Nitin R; Tsiatis, Anastasios A (1984). "Exact significance testing to establish treatment equivalence with ordered categorical data". Biometrics. 40 (3): 819–825. doi:10.2307/2530927. JSTOR 2530927. PMID 6518249.
Jump up ^ Mehta, C. R. 1995. SPSS 6.1 Exact test for Windows. Englewood Cliffs, NJ: Prentice Hall.
Jump up ^ Mehta C.R.; Patel N.R. (1983). "A Network Algorithm for Performing Fisher's Exact Test in r Xc Contingency Tables". Journal of the American Statistical Association. 78 (382): 427–434. doi:10.2307/2288652.
Jump up ^ mathworld.wolfram.com Page giving the formula for the general form of Fisher's exact test for m × n contingency tables
Jump up ^ Cyrus R. Mehta; Nitin R. Patel (1986). "ALGORITHM 643: FEXACT: a FORTRAN subroutine for Fisher's exact test on unordered r×c contingency tables". ACM Trans. Math. Softw. 12 (2): 154–161. doi:10.1145/6497.214326.
^ Jump up to: a b Liddell, Douglas (1976). "Practical tests of 2x2 contingency tables". The Statistician. 25 (4): 295–304. doi:10.2307/2988087. JSTOR 2988087.
^ Jump up to: a b Berkson, Joseph (1978). "In dispraise of the exact test". Journal of Statistic Planning and Inference. 2: 27–42. doi:10.1016/0378-3758(78)90019-8.
^ Jump up to: a b D'Agostino, R. B.; Chase, W. & Belanger, A. (1988). "The Appropriateness of Some Common Procedures for Testing Equality of Two Independent Binomial Proportions". The American Statistician. 42 (3): 198–202. doi:10.2307/2685002. JSTOR 2685002.
^ Jump up to: a b c d Yates, F. (1984). "Tests of Significance for 2 x 2 Contingency Tables (with discussion)". Journal of the Royal Statistical Society, Series A. 147 (3): 426–463. doi:10.2307/2981577. JSTOR 2981577.
^ Jump up to: a b c d Roderick J. A. Little (1989). "Testing the Equality of Two Independent Binomial Proportions". The American Statistician. 43 (4): 283–288. doi:10.2307/2685390. JSTOR 2685390.
Jump up ^ Cyrus R. Mehta and Pralay Senchaudhuri (4 September 2003). "Conditional versus Unconditional Exact Tests for Comparing Two Binomials". Retrieved 20 November 2009.
Jump up ^ Barnard, G.A (1945). "A New Test for 2×2 Tables". Nature. 156 (3954): 177. doi:10.1038/156177a0.
^ Jump up to: a b Fisher (1945). "A New Test for 2 × 2 Tables". Nature. 156 (3961): 388. doi:10.1038/156388a0. ; Barnard, G.A (1945). "A New Test for 2×2 Tables". Nature. 156 (3974): 783. doi:10.1038/156783b0.
^ Jump up to: a b c d Choi L, Blume JD, Dupont WD (2015). "Elucidating the Foundations of Statistical Inference with 2 x 2 Tables". PLoS ONE. 10 (4): e0121263. doi:10.1371/journal.pone.0121263.
Jump up ^ Choi, Leena (2011). "ProfileLikelihood: profile likelihood for a parameter in commonly used statistical models; 2011. R package version 1.1". See also: Likelihood Ratio Statistics for 2 x 2 Tables (Online calculator).
External links[edit]
Calculate Fisher's Exact Test Online
Likelihood Ratio Statistics for 2X2 Tables
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