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The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables

Yıl 2024, , 169 - 185, 30.09.2024

Gokcen Altun

https://doi.org/10.33401/fujma.1432348

Öz

The family of power-divergence (PD) test statistic contains many well-known test statistics used in the analysis of the contingency tables under the independence model. In this work, we compare the various test statistics for the independence model. The type-I and type-II errors of the test statistics are obtained and compared via simulation study considering the different degree of freedoms and sample sizes. According to the simulation results, we recommend the PD(0.4) test statistic for the small sample size based on its power and type-I error rates. Two applications are given to demonstrate the usefulness of the PD(0.4) test statistic over the chi-square test statistic {contingency tables}.

Anahtar Kelimeler

Goodness-of-fit test statistics Independence model Power-divergence family Small sample

Kaynakça

  • [1] K. Pearson, 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, Lond. Edinb. Dubl., 50(302) (1900), 157-175. $ \href{https://doi.org/10.1080/14786440009463897}{\mbox{[CrossRef]}} $
  • [2] T.M. Franke, T. Ho and C.A. Christie, The chi-square test: Often used and more often misinterpreted. Am. J. Evaluation, 33(3)(2012), 448-458. $ \href{https://doi.org/10.1177/1098214011426594}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84864389048&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1177%2F1098214011426594%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000306732300009}{\mbox{[Web of Science]}} $
  • [3] J. Cohen, Statistical Power Analysis for the Behavioral Sciences, England, Routledge, (1988). $ \href{https://doi.org/10.4324/9780203771587}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1988T109700068}{\mbox{[Web of Science]}} $
  • [4] K.R. Murphy and B. Myors, Testing the hypothesis that treatments have negligible effects: Minimum-effect tests in the general linear model, J. Appl. Psychol., 84(2)(1999), 234. $ \href{https://doi.org/10.1037//0021-9010.84.2.234}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0033115552&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Testing+the+hypothesis+that+treatments+have+negligible+effects%3A+Minimum-effect+tests+in+the+general+linear+model%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000080562000007}{\mbox{[Web of Science]}} $
  • [5] G.M. Oyeyemi, A.A. Adewara, F.B. Adebola and S.I. Salau, On the estimation of power and sample size in test of independence, Asian J. Math. Stat., 3(3) (2010), 139-146. $\href{https://uilspace.unilorin.edu.ng/handle/20.500.12484/11569}{\mbox{[CrossRef]}} $
  • [6] A. Agresti, Categorical Data Analysis John Wiley & Sons, 482(2002). $\href{https://doi.org/10.1002/0471249688}{\mbox{[CrossRef]}} $
  • [7] A. Agresti, An Introduction to Categorical Data Analysis, John Wiley & Sons (2007). $ \href{https://doi.org/10.1002/0470114754}{\mbox{[CrossRef]}} $
  • [8] P. Burman, On some testing problems for sparse contingency tables. J. Multivariate Anal., 88(1)(2004), 1-18. $ \href{https://doi.org/10.1016/S0047-259X(02)00052-0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000187785400001}{\mbox{[Web of Science]}} $
  • [9] J.K. Yarnold, The minimum expectation in c2 goodness of fit tests and the accuracy of approximations for the null distribution, J. Amer. Statist. Assoc., 65(330) (1970), 864-886. $ \href{https://doi.org/10.2307/2284594}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0001105364&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22goodness+of+fit+tests+and+the+accuracy+of+approximations+for+the+null+distribution%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} $
  • [10] T. Rudas, A Monte Carlo comparison of the small sample behaviour of the Pearson, the likelihood ratio and the Cressie-Read statistics, J. Stat. Comput. Simul., 24(2)(1986), 107-120. $\href{https://doi.org/10.1080/00949658608810894}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0002389220&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1080%2F00949658608810894%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1986D071500003}{\mbox{[Web of Science]}} $
  • [11] N. Cressie and T.R. Read, Multinomial goodness-of-fit tests, J. R. Stat. Soc. Ser. B. Stat. Methodol., 46(3)(1984), 440-464. $ \href{https://doi.org/10.1111/j.2517-6161.1984.tb01318.x}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1984AFP6300010}{\mbox{[Web of Science]}} $
  • [12] S.E. Fienberg, The use of chi-squared statistics for categorical data problems, J. R. Stat. Soc. Ser. B. Stat. Methodol., 41(1)(1979), 54-64. $ \href{https://doi.org/10.1111/j.2517-6161.1979.tb01057.x}{\mbox{[CrossRef]}} $
  • [13] K. Larntz, Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics, J. Amer. Statist. Assoc., 73(362)(1978), 253- 263. $ \href{https://doi.org/10.2307/2286650}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84950665320&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Small-sample+comparisons+of+exact+levels+for+chi-squared+goodness-of-fit+statistics%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} $
  • [14] M.A. Garcia-Perez and V. Nunez-Anton, Accuracy of power-divergence statistics for testing independence and homogeneity in two-way contingency tables, Commun Stat-Simul C., 38(3)(2009), 503-512. $ \href{https://doi.org/10.1080/03610910802538351}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-60849127003&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1080%2F03610910802538351%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000208104000001}{\mbox{[Web of Science]}} $
  • [15] S. Aktaş, Power divergence statistics under quasi independence model for square contingency tables, Sains Malays., 45(10)(2016), 1573- 1578. $\href{http://ukm.my/jsm/pdf_files/SM-PDF-45-10-2016/20%20Serpil%20Aktas.pdf}{\mbox{[Web]}} $
  • [16] G. Altun, A study on Freeman-Tukey test statistic under the symmetry model for square contingency tables, Cumhuriyet Sci J., 42(2)(2021), 413-421. $ \href{https://doi.org/10.17776/csj.835165}{\mbox{[CrossRef]}} $
  • [17] S.D. Horn, Goodness-of-fit tests for discrete data: a review and an application to a health impairment scale, Biometrics, 33(1)(1977), 237-247. $ \href{https://doi.org/10.2307/2529319}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0017472349&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.2307%2F2529319%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} $
  • [18] G.S. Watson, Some recent results in chi-square goodness-of-fit tests, Biometrics, 15(3) (1959), 440-468. $ \href{https://doi.org/10.2307/2527749}{\mbox{[CrossRef]}} $
  • [19] M. Kendall and A. Stuart, Handbook of Statistics, (1979).
  • [20] K. Larntz, Small Sample Comparison of Likelihood-Ratio and Pearson Chi-Square Statistics for the Null Distribution, University of Minnesota, (1973). $ \href{https://doi.org/10.2307/2286650}{\mbox{[CrossRef]}} $
  • [21] R. Durrett, Probability: Theory and Examples, Cambridge university press, 49(2019). $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85097483110&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22probability+theory+and+examples%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} $
  • [22] G.H. Freeman and J.H. Halton, Note on an exact treatment of contingency, goodness of fit and other problems of significance, Biometrika, 38(1/2)(1951), 141-149. $ \href{https://doi.org/10.2307/2332323}{\mbox{[CrossRef]}} $
  • [23] M. L. McHugh, The chi-square test of independence, Biochem. Med., 23(2)(2013), 143-149. $ \href{https://doi.org/10.11613/bm.2013.018}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84878767384&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.11613%2Fbm.2013.018%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000320224300004}{\mbox{[Web of Science]}} $
  • [24] P.R. Nelson, P.S. Wludyka and K.A. Copeland, The Analysis of Means: A Graphical Method for Comparing Means, Rates, and Proportions, SIAM, (2005). $ \href{http://dx.doi.org/10.1137/1.9780898718362}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276913900002}{\mbox{[Web of Science]}} $

Yıl 2024, , 169 - 185, 30.09.2024

Gokcen Altun

https://doi.org/10.33401/fujma.1432348

Öz

Kaynakça

  • [1] K. Pearson, 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, Lond. Edinb. Dubl., 50(302) (1900), 157-175. $ \href{https://doi.org/10.1080/14786440009463897}{\mbox{[CrossRef]}} $
  • [2] T.M. Franke, T. Ho and C.A. Christie, The chi-square test: Often used and more often misinterpreted. Am. J. Evaluation, 33(3)(2012), 448-458. $ \href{https://doi.org/10.1177/1098214011426594}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84864389048&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1177%2F1098214011426594%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000306732300009}{\mbox{[Web of Science]}} $
  • [3] J. Cohen, Statistical Power Analysis for the Behavioral Sciences, England, Routledge, (1988). $ \href{https://doi.org/10.4324/9780203771587}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1988T109700068}{\mbox{[Web of Science]}} $
  • [4] K.R. Murphy and B. Myors, Testing the hypothesis that treatments have negligible effects: Minimum-effect tests in the general linear model, J. Appl. Psychol., 84(2)(1999), 234. $ \href{https://doi.org/10.1037//0021-9010.84.2.234}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0033115552&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Testing+the+hypothesis+that+treatments+have+negligible+effects%3A+Minimum-effect+tests+in+the+general+linear+model%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000080562000007}{\mbox{[Web of Science]}} $
  • [5] G.M. Oyeyemi, A.A. Adewara, F.B. Adebola and S.I. Salau, On the estimation of power and sample size in test of independence, Asian J. Math. Stat., 3(3) (2010), 139-146. $\href{https://uilspace.unilorin.edu.ng/handle/20.500.12484/11569}{\mbox{[CrossRef]}} $
  • [6] A. Agresti, Categorical Data Analysis John Wiley & Sons, 482(2002). $\href{https://doi.org/10.1002/0471249688}{\mbox{[CrossRef]}} $
  • [7] A. Agresti, An Introduction to Categorical Data Analysis, John Wiley & Sons (2007). $ \href{https://doi.org/10.1002/0470114754}{\mbox{[CrossRef]}} $
  • [8] P. Burman, On some testing problems for sparse contingency tables. J. Multivariate Anal., 88(1)(2004), 1-18. $ \href{https://doi.org/10.1016/S0047-259X(02)00052-0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000187785400001}{\mbox{[Web of Science]}} $
  • [9] J.K. Yarnold, The minimum expectation in c2 goodness of fit tests and the accuracy of approximations for the null distribution, J. Amer. Statist. Assoc., 65(330) (1970), 864-886. $ \href{https://doi.org/10.2307/2284594}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0001105364&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22goodness+of+fit+tests+and+the+accuracy+of+approximations+for+the+null+distribution%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} $
  • [10] T. Rudas, A Monte Carlo comparison of the small sample behaviour of the Pearson, the likelihood ratio and the Cressie-Read statistics, J. Stat. Comput. Simul., 24(2)(1986), 107-120. $\href{https://doi.org/10.1080/00949658608810894}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0002389220&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1080%2F00949658608810894%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1986D071500003}{\mbox{[Web of Science]}} $
  • [11] N. Cressie and T.R. Read, Multinomial goodness-of-fit tests, J. R. Stat. Soc. Ser. B. Stat. Methodol., 46(3)(1984), 440-464. $ \href{https://doi.org/10.1111/j.2517-6161.1984.tb01318.x}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1984AFP6300010}{\mbox{[Web of Science]}} $
  • [12] S.E. Fienberg, The use of chi-squared statistics for categorical data problems, J. R. Stat. Soc. Ser. B. Stat. Methodol., 41(1)(1979), 54-64. $ \href{https://doi.org/10.1111/j.2517-6161.1979.tb01057.x}{\mbox{[CrossRef]}} $
  • [13] K. Larntz, Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics, J. Amer. Statist. Assoc., 73(362)(1978), 253- 263. $ \href{https://doi.org/10.2307/2286650}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84950665320&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Small-sample+comparisons+of+exact+levels+for+chi-squared+goodness-of-fit+statistics%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} $
  • [14] M.A. Garcia-Perez and V. Nunez-Anton, Accuracy of power-divergence statistics for testing independence and homogeneity in two-way contingency tables, Commun Stat-Simul C., 38(3)(2009), 503-512. $ \href{https://doi.org/10.1080/03610910802538351}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-60849127003&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1080%2F03610910802538351%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000208104000001}{\mbox{[Web of Science]}} $
  • [15] S. Aktaş, Power divergence statistics under quasi independence model for square contingency tables, Sains Malays., 45(10)(2016), 1573- 1578. $\href{http://ukm.my/jsm/pdf_files/SM-PDF-45-10-2016/20%20Serpil%20Aktas.pdf}{\mbox{[Web]}} $
  • [16] G. Altun, A study on Freeman-Tukey test statistic under the symmetry model for square contingency tables, Cumhuriyet Sci J., 42(2)(2021), 413-421. $ \href{https://doi.org/10.17776/csj.835165}{\mbox{[CrossRef]}} $
  • [17] S.D. Horn, Goodness-of-fit tests for discrete data: a review and an application to a health impairment scale, Biometrics, 33(1)(1977), 237-247. $ \href{https://doi.org/10.2307/2529319}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0017472349&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.2307%2F2529319%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} $
  • [18] G.S. Watson, Some recent results in chi-square goodness-of-fit tests, Biometrics, 15(3) (1959), 440-468. $ \href{https://doi.org/10.2307/2527749}{\mbox{[CrossRef]}} $
  • [19] M. Kendall and A. Stuart, Handbook of Statistics, (1979).
  • [20] K. Larntz, Small Sample Comparison of Likelihood-Ratio and Pearson Chi-Square Statistics for the Null Distribution, University of Minnesota, (1973). $ \href{https://doi.org/10.2307/2286650}{\mbox{[CrossRef]}} $
  • [21] R. Durrett, Probability: Theory and Examples, Cambridge university press, 49(2019). $ \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85097483110&origin=resultslist&sort=plf-f&src=s&sid=3a9e791c833e2a7ccf70ee494ec6e7f7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22probability+theory+and+examples%22%29&sl=111&sessionSearchId=3a9e791c833e2a7ccf70ee494ec6e7f7&relpos=0}{\mbox{[Scopus]}} $
  • [22] G.H. Freeman and J.H. Halton, Note on an exact treatment of contingency, goodness of fit and other problems of significance, Biometrika, 38(1/2)(1951), 141-149. $ \href{https://doi.org/10.2307/2332323}{\mbox{[CrossRef]}} $
  • [23] M. L. McHugh, The chi-square test of independence, Biochem. Med., 23(2)(2013), 143-149. $ \href{https://doi.org/10.11613/bm.2013.018}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84878767384&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.11613%2Fbm.2013.018%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000320224300004}{\mbox{[Web of Science]}} $
  • [24] P.R. Nelson, P.S. Wludyka and K.A. Copeland, The Analysis of Means: A Graphical Method for Comparing Means, Rates, and Proportions, SIAM, (2005). $ \href{http://dx.doi.org/10.1137/1.9780898718362}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276913900002}{\mbox{[Web of Science]}} $
Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular İstatistik (Diğer)
Bölüm Makaleler
Yazarlar

Gokcen Altun 0000-0003-4311-6508

Erken Görünüm Tarihi 30 Eylül 2024
Yayımlanma Tarihi 30 Eylül 2024
Gönderilme Tarihi 5 Şubat 2024
Kabul Tarihi 24 Eylül 2024
Yayımlandığı Sayı Yıl 2024

Kaynak Göster

APA Altun, G. (2024). The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables. Fundamental Journal of Mathematics and Applications, 7(3), 169-185. https://doi.org/10.33401/fujma.1432348
AMA Altun G. The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables. Fundam. J. Math. Appl. Eylül 2024;7(3):169-185. doi:10.33401/fujma.1432348
Chicago Altun, Gokcen. “The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables”. Fundamental Journal of Mathematics and Applications 7, sy. 3 (Eylül 2024): 169-85. https://doi.org/10.33401/fujma.1432348.
EndNote Altun G (01 Eylül 2024) The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables. Fundamental Journal of Mathematics and Applications 7 3 169–185.
IEEE G. Altun, “The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables”, Fundam. J. Math. Appl., c. 7, sy. 3, ss. 169–185, 2024, doi: 10.33401/fujma.1432348.
ISNAD Altun, Gokcen. “The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables”. Fundamental Journal of Mathematics and Applications 7/3 (Eylül 2024), 169-185. https://doi.org/10.33401/fujma.1432348.
JAMA Altun G. The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables. Fundam. J. Math. Appl. 2024;7:169–185.
MLA Altun, Gokcen. “The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables”. Fundamental Journal of Mathematics and Applications, c. 7, sy. 3, 2024, ss. 169-85, doi:10.33401/fujma.1432348.
Vancouver Altun G. The Most Powerful Member of the Power-Divergence Family for the Independence Model in Contingency Tables. Fundam. J. Math. Appl. 2024;7(3):169-85.

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