Изворни код странице Hi-kvadratni test

[[File:Chi-square distributionCDF-English.png|thumb|right|300px|[[Hi-kvadratna distribucija]], sa prikazanim {{math|''χ''<sup>2</sup>}} na ''x''-osi i -{''p''}--vrednošću na ''y''-osi.<ref>{{cite web | url=http://www.planetmathematics.com/CentralChiDistr.pdf | title=Characteristic function of the central chi-square distribution | author=M.A. Sanders | accessdate=6. 3. 2009 | archive-url=https://web.archive.org/web/20110715091705/http://www.planetmathematics.com/CentralChiDistr.pdf# | archive-date=15. 7. 2011 |url-status=dead | df= }}</ref>]]

'''Hi-kvadratni test''', takođe napisan kao '''{{math|''χ''<sup>2</sup>}} test''', jeste [[Statistical hypothesis testing|test statističke hipoteze]] gde je [[sampling distribution|distribucija uzorka]] testirane statistike [[hi-kvadratna distribucija]] kad je [[null hypothesis|nulta hipoteza]] istinita. Bez druge kvalifikacije, hi-kvadratni test često se koristi kao zamena za [[Pearson's chi-squared test|Pirsonov hi-kvadratni test]].<ref>{{Cite journal | last = Pearson | first = Karl | authorlink = Karl Pearson | title = On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling | doi = 10.1080/14786440009463897 | journal = Philosophical Magazine |series=Series 5 | volume = 50 | issue = 302 | pages = 157–175 | year = 1900 | url = http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf | pmid =  | pmc = }}</ref> Hi-kvadratni test se koristi da se utvrdi da li postoji značajna razlika između očekivanih frekvencija i posmatranih frekvencija u jednoj ili više kategorija.

U standardnim primenama ovog testa, zapažanja su svrstana u međusobno isključive klase, i postoji neka teorija, ili nulta hipoteza, koja daje verovatnoću da bilo koje opažanje padne u odgovarajuću klasu. Svrha testa je da se proceni koliko su verovatne opservacije, pod pretpostavkom da je nulta hipoteza tačna.

Hi-kvadratni testovi se obično konstruišu iz [[Lack-of-fit sum of squares#Sums of squares|sume kvadriranih grešaka]], ili pomoću [[Варијанса#Distribucija varijanse uzorka|varijanse uzorka]]. Statistika testa koja sledi hi-kvadratnu distribuciju proizilazi iz pretpostavke nezavisnih normalno distribuiranih podataka,<ref name="abramowitz">{{Abramowitz_Stegun_ref|26|940}}</ref><ref>NIST (2006).  [http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm Engineering Statistics Handbook – Chi-Squared Distribution]</ref><ref name="Johnson_et_al">{{cite book
  | last = Johnson
  | first = N. L.
  | first2 = S. |last2=Kotz |first3=N. |last3=Balakrishnan
  | title = Continuous Univariate Distributions |edition=Second |volume=1 |chapter=Chi-Square Distributions including Chi and Rayleigh |pages=415–493
  | publisher = John Wiley and Sons
  | year = 1994
  | isbn = 978-0-471-58495-7
}}</ref><ref>{{cite book
  | last = Mood
  | first = Alexander
  | first2=Franklin A. |last2=Graybill |first3=Duane C. |last3=Boes
  | title = Introduction to the Theory of Statistics | url = https://archive.org/details/introductiontoth00mood_706
  |edition=Third |pages=[https://archive.org/details/introductiontoth00mood_706/page/241 241]–246
  | publisher = McGraw-Hill
  | year = 1974
  | isbn = 978-0-07-042864-5
}}</ref> koja je u mnogim slučajevima validna zbog [[Централна гранична теорема|centralne granične teoreme]]. Hi-kvadratni test može se koristiti za pokušaj odbacivanja nulte hipoteze da su podaci nezavisni.

== Istorija ==

U 19. veku, statističke analitičke metode uglavnom su primenjivane u analizi bioloških podataka i bilo je uobičajeno da istraživači pretpostavljaju da su zapažanja pratila [[normal distribution|normalnu distribuciju]], kao što su to činili [[Џорџ Бидел Ери|Ser Džordž Eri]] i [[Mansfield Merriman|profesor Meriman]], čije je radove kritikovao [[Karl Pearson|Karl Pirson]] u svom radu iz 1900. godine.<ref name = Pearson1900>
{{cite journal
 | last = Pearson | first = Karl
 | author-link = Karl Pearson
 | title = On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling
 | journal = Philosophical Magazine |series=Series 5
 | volume = 50
 | issue = 302
 | year = 1900
 | pages = 157–175
 | url = http://www.economics.soton.ac.uk/staff/aldrich/1900.pdf
 | doi = 10.1080/14786440009463897
}}</ref>

Do kraja 19. veka, Pirson je uočio postojanje značajne [[skewness|asimetrije]] unutar nekih bioloških posmatranja. Da bi modelovao zapažanja nezavisno od toga da li su normalna ili asimetrična, Pirson je u nizu članaka objavljenih od 1893 do 1916,<ref name = Pearson1893>
{{cite journal
 | last = Pearson | first = Karl
 | author-link = Karl Pearson
 | title = Contributions to the mathematical theory of evolution [abstract]
 | journal = Proceedings of the Royal Society
 | volume = 54
 | year = 1893
 | pages = 329–333
 | jstor = 115538
 | doi = 10.1098/rspl.1893.0079
}}</ref><ref name = Pearson1895>
{{cite journal
 | last = Pearson | first = Karl
 | author-link = Karl Pearson
 | title = Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material
 | journal = Philosophical Transactions of the Royal Society
 | volume = 186
 | year = 1895
 | pages = 343–414
 | bibcode = 1895RSPTA.186..343P
 | jstor = 90649
 | doi = 10.1098/rsta.1895.0010
}}</ref><ref name = Pearson1901>
{{cite journal
 | last = Pearson | first = Karl
 | author-link = Karl Pearson
 | title = Mathematical contributions to the theory of evolution, X: Supplement to a memoir on skew variation
 | journal = Philosophical Transactions of the Royal Society A
 | volume = 197
 | issue = 287–299
 | year = 1901
 | pages = 443–459
 | bibcode = 1901RSPTA.197..443P
 | jstor = 90841
 | doi = 10.1098/rsta.1901.0023
}}</ref><ref name = Pearson1916>
{{cite journal
 | last = Pearson | first = Karl
 | author-link = Karl Pearson
 | title = Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation
 | journal = Philosophical Transactions of the Royal Society A
 | volume = 216
 | issue = 538–548
 | year = 1916
 | pages = 429–457
 | bibcode = 1916RSPTA.216..429P
 | jstor = 91092
 | doi = 10.1098/rsta.1916.0009
}}</ref> osmislio [[Pearson distribution|Pirsonovu distribuciju]], porodicu neprekidnih raspodela verovatnoće, koja uključuje normalnu distribuciju i mnoge asimetrične distribucije. On je predložio metod statističke analize koji se sastoji od upotrebe Pirsonove distribucije za modelovanje posmatranja i vršenja testa dobrog uklapanja kako bi se utvrdilo koliko se model i posmatranje zaista uklapaju.

=== Pirsonov hi-kvadratni test ===
{{see also-lat|Pirsonov hi-kvadratni test}}

Godine 1900, Pirson je objavio publikaciju<ref name = Pearson1900 /> o {{math|''χ''<sup>2</sup>}} testu, koja se smatra jednim od fundamentalnih radova u modernoj statistici.<ref name = Cochran1952>
{{cite journal
 | last = Cochran | first = William G.
 | author-link = William G. Cochran
 | title = The Chi-square Test of Goodness of Fit
 | url = https://archive.org/details/sim_annals-of-mathematical-statistics_1952-09_23_3/page/315 | journal = The Annals of Mathematical Statistics
 | volume = 23
 | issue = 3
 | year = 1952
 | pages = 315–345
 | jstor = 2236678
 | doi=10.1214/aoms/1177729380
}}
</ref> U tom radu, Pirson je istražio test ''adekvatnosti uklapanja''.

Ako se pretpostavi da je {{mvar|n}} opažanja slučajnog uzorka iz populacije klasifikovano u {{mvar|k}} međusobno isključujućih klasa sa odgovarajućim posmatranim brojevima {{mvar|x<sub>i</sub>}} (za {{math|''i'' {{=}} 1,2,…,''k''}}), a da nulta hipoteza daje verovatnoću {{mvar|p<sub>i</sub>}} da će opažanje pasti u {{mvar|i}}-tu klasu. Dakle, dostupni su očekivani brojevi {{math|''m<sub>i</sub>'' {{=}} ''np<sub>i</sub>''}} za svako {{mvar|i}}, gde

:<math>\begin{align} \sum^k_{i=1}{p_i} &= 1 \\[8pt] \sum^k_{i=1}{m_i}&=n\sum^k_{i=1}{p_i} = \sum_{i=1}^{k}x_i \end{align}</math>

Pirson je predložio da u slučaju da je nulta hipoteza tačna kao {{math|''n'' → ∞}}, ograničavajuća distribucija navedene količine je raspodela {{math|''χ''<sup>2</sup>}}.

:<math>X^2=\sum^k_{i=1}{\frac{(x_i-m_i)^2}{m_i}}=\sum^k_{i=1}{\frac{x_i^2}{m_i}-n}</math>

Pirson se prvo bavio slučajem u kojem su očekivani brojevi {{mvar|m<sub>i</sub>}} dovoljno veliki poznati brojevi u svim ćelijama pod pretpostavkom da se svaki {{mvar|x<sub>i</sub>}} može uzeti kao [[normal distribution|normalno raspodeljen]], i postigao je rezultat da, na granici kada {{mvar|n}} postane veliko, {{math|''X''{{isup|2}}}} sledi {{math|''χ''<sup>2</sup>}} raspodelu sa {{math|''k'' − 1}} stepeni slobode.

Pirson je zatim razmotrio slučaj u kojem očekivani brojevi zavise od parametara koji se moraju proceniti iz uzorka. On je predložio da kad {{mvar|m<sub>i</sub>}} označava prave očekivane brojeve, a {{math|''m''′<sub>''i''</sub>}} procenjene očekivane brojeva, razlika

:<math>X^2-{X'}^2=\sum^k_{i=1}{\frac{x_i^2}{m_i}}-\sum^k_{i=1}{\frac{x_i^2}{m'_i}}</math>

obično biva pozitivna i dovoljno mala da se može zanemariti. U zaključku, Pirson je tvrdio da ako se smatra da je {{math|''X''′{{isup|2}}}} takođe raspodeljena kao {{math|''χ''<sup>2</sup>}} distribucija sa {{math|''k'' − 1}} stepeni slobode, greška u ovoj aproksimaciji neće uticati na praktične odluke. Ovaj zaključak izazvao je određene kontroverze u praktičnim primenama i to nije bilo rarešeno tokom 20 godina, do objavljivanja Fišerovih publikacija iz 1922. i 1924. godine.<ref name = Fisher1922>{{cite journal
 | last = Fisher | first = Ronald A.
 | author-link = Ronald A. Fisher
 | title = On the Interpretation of chi-squared from Contingency Tables, and the Calculation of P
 | journal = Journal of the Royal Statistical Society
 | volume = 85
 | issue = 1
 | year = 1922
 | pages = 87–94
 | jstor = 2340521
 | doi=10.2307/2340521
 | url = https://zenodo.org/record/1449484
 }}
</ref><ref name = Fisher1924>
{{cite journal
 | last = Fisher | first = Ronald A.
 | author-link = Ronald A. Fisher
 | title = The Conditions Under Which chi-squared Measures the Discrepancey Between Observation and Hypothesis
 | journal = Journal of the Royal Statistical Society
 | volume = 87
 | issue = 3
 | year = 1924
 | pages = 442–450
 | jstor = 2341149
}}</ref>

== Aplikacije ==

U [[Криптоанализа|kriptoanalizi]], hi-kvadratni test se koristi za upoređivanje distribucije [[Отворени текст|otvorenog teksta]] i (mogućeg) dekriptovanja [[ciphertext|šifroteksta]]. Najniža vrednost testa znači da je dešifrovanje bilo uspešno sa velikom verovatnoćom.<ref name="practicalcrypto">{{cite web|title=Chi-squared Statistic|url=http://practicalcryptography.com/cryptanalysis/text-characterisation/chi-squared-statistic/|website=Practical Cryptography|accessdate=18. 2. 2015|archive-url=https://web.archive.org/web/20150218203349/http://practicalcryptography.com/cryptanalysis/text-characterisation/chi-squared-statistic/|archive-date=18. 02. 2015|url-status=dead}}</ref><ref name="ibmath">{{cite web|title=Using Chi Squared to Crack Codes|url=http://ibmathsresources.com/2014/06/15/using-chi-squared-to-crack-codes/|website=IB Maths Resources|publisher=British International School Phuket}}</ref> Ova metoda se može generalizovati za rešavanje savremenih kriptografskih problema.<ref name="elsevier">{{cite journal|last1=Ryabko|first1=B. Ya.|last2=Stognienko|first2=V. S.|last3=Shokin|first3=Yu. I.|title=A new test for randomness and its application to some cryptographic problems|journal=Journal of Statistical Planning and Inference |date=2004|volume=123 |issue=2|pages=365–376|url=http://boris.ryabko.net/jspi.pdf|accessdate=18. 2. 2015|doi=10.1016/s0378-3758(03)00149-6}}</ref>

U [[bioinformatics|bioinformatici]], hi-kvadratni test se koristi za upoređivanje raspodele određenih svojstava gena (npr. genomskog sadržaja, brzine mutacije, interakcione mreže klasterovanja itd.) koje pripadaju različitim kategorijama (npr. geni bolesti, osnovni geni, geni na određenom hromozomu itd.).<ref name="pnas-bics">{{cite journal|last1=Feldman|first1=I.|last2=Rzhetsky|first2=A.|last3=Vitkup|first3=D.|title=Network properties of genes harboring inherited disease mutations |journal=PNAS|date=2008 |volume=105|issue=11 |pages=4323–432 |doi=10.1073/pnas.0701722105 |bibcode=2008PNAS..105.4323F|pmc=2393821}}</ref><ref name="chi-bics">{{cite web|title=chi-square-tests|url=https://visa.pharmacy.wsu.edu/bioinformatics/documents/chi-square-tests.pdf|accessdate=29. 6. 2018|archive-url=https://web.archive.org/web/20180629131548/https://visa.pharmacy.wsu.edu/bioinformatics/documents/chi-square-tests.pdf|archive-date=29. 06. 2018|url-status=dead}}</ref>

== Reference ==
{{Reflist}}

== Literatura ==
{{Литература|30em}}
* {{MathWorld | urlname=Chi-SquaredTest | title=Chi-Squared Test}}
* {{citation |last1=Corder |first1=G. W. |last2=Foreman |first2=D. I. |title=Nonparametric Statistics: A Step-by-Step Approach |year=2014 |publisher=Wiley |location=New York |isbn=978-1118840313 |ref=none|url=https://books.google.com/books?id=hYVYAwAAQBAJ&printsec=frontcover&dq=isbn:9781118840313&hl=en&sa=X&ved=0ahUKEwimtaSyo9PjAhUIQq0KHXT2A6UQ6AEIKjAA#v=snippet&q=%22Chi-square%22&f=false}}
* {{citation |last1=Greenwood |first1=Cindy |authorlink1=Cindy Greenwood |last2=Nikulin |first2=M. S. |title=A guide to chi-squared testing |year=1996 |publisher=Wiley |location=New York |isbn=0-471-55779-X |ref=none}}
* {{citation |last=Nikulin |first=M. S. |title=Chi-squared test for normality |year=1973 |work=Proceedings of the International Vilnius Conference on Probability Theory and Mathematical Statistics |volume=2 |pages=119–122 |ref=none}}
* {{citation |last1=Bagdonavicius |first1=V. |last2=Nikulin |first2=M. S. |title=Chi-squared goodness-of-fit test for right censored data |date=2011 |work=The International Journal of Applied Mathematics and Statistics |pages=30–50 |ref=none|url=https://www.researchgate.net/profile/Vilijandas_Bagdonavicius/publication/325193712_Chi-squared_goodness-of-fit_test_for_right_censored_data_International_Journal_of_Applied_Mathematics_and_Statistics_January_2011/links/5afd2d85aca272b5d8708548/Chi-squared-goodness-of-fit-test-for-right-censored-data-International-Journal-of-Applied-Mathematics-and-Statistics-January-2011.pdf}}{{full citation needed|date=januar 2013}}
* {{cite book |title=A history of mathematical statistics from 1750 to 1930 |last=Hald |first=Anders |authorlink=Anders Hald |year=1998 |publisher=Wiley |location=New York |isbn=978-0-471-17912-2 |ref=harv }}
* {{Cite journal | last = Elderton | first = William Palin | authorlink = William Palin Elderton| title = Tables for Testing the Goodness of Fit of Theory to Observation | doi = 10.1093/biomet/1.2.155 | journal = Biometrika | volume = 1 | issue = 2 | pages = 155–163 | year = 1902 | pmid =  | pmc = |ref=harv| url = https://zenodo.org/record/1431595 }}
* {{springer|title=Chi-squared distribution|id=Chi-squared_distribution}}
* {{cite journal|last1=Ramsey|first1=PH|title=Evaluating the Normal Approximation to the Binomial Test|journal=Journal of Educational Statistics|date=1988|volume=13|issue=2|pages=173–82|doi=10.2307/1164752|jstor=1164752}}
* {{Citation |last=Lancaster |first=H.O. |title=The Chi-squared Distribution |year=1969 |publisher=Wiley }}
* {{cite journal |last1=Dasgupta |first1=Sanjoy D. A. |last2=Gupta |first2=Anupam K. |date=januar 2003 |title=An Elementary Proof of a Theorem of Johnson and Lindenstrauss |journal=Random Structures and Algorithms |volume=22 |issue=1 |pages=60–65 |doi=10.1002/rsa.10073 |url=http://cseweb.ucsd.edu/~dasgupta/papers/jl.pdf |accessdate=1. 5. 2012 }}
* M. K. Simon, ''Probability Distributions Involving Gaussian Random Variables'', New York: Springer, 2002, eq. (2.35), {{ISBN|978-0-387-34657-1}}
* {{cite book |title=Statistics for experimenters|url=https://archive.org/details/statisticsforexp00geor|author=Box, Hunter and Hunter|publisher=Wiley|year=1978|isbn=978-0471093152|page=[https://archive.org/details/statisticsforexp00geor/page/118 118]}}
* {{cite journal |first=M. S. |last=Bartlett |first2=D. G. |last2=Kendall |title=The Statistical Analysis of Variance-Heterogeneity and the Logarithmic Transformation |journal=Supplement to the Journal of the Royal Statistical Society |volume=8 |issue=1 |year=1946 |pages=128–138 |jstor=2983618 |doi=10.2307/2983618 }}
* {{Cite journal|last=Pillai|first=Natesh S.|year=2016|title=An unexpected encounter with Cauchy and Lévy|journal=[[Annals of Statistics]]|volume=44|issue=5|pages=2089–2097|doi=10.1214/15-aos1407|jstor=|arxiv=1505.01957}}
{{Литература крај}}

== Spoljašnje veze ==
{{Commons category-lat|Chi-squared test}}
* -{[https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/chi-square/ Chi-squared test] {{Wayback|url=https://www.statisticshowto.datasciencecentral.com/probability-and-statistics/chi-square/ |date=20190802201102 }}}-

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{{СОРТИРАЊЕ:Хи-квадратни тест}}
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