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Year 2021, Volume: 42 Issue: 2, 413 - 421, 30.06.2021

Gokcen Altun

https://doi.org/10.17776/csj.835165

Abstract

References

  • [1] Altun G., Karesel Olumsallık Tablolarında Model Uyumunun Sapma Ölçüsü ile Belirlenmesi, PD Thesis, Hacettepe Üniversitesi, Fen Bilimleri Enstitüsü, (2018).
  • [2] Goodman L. A., Multiplicative models for square contingency tables with ordered categories, Biometrika., 66(3) (1979) 413-418.
  • [3] Fienberg S. E., The use of chi-squared statistics for categorical data problems, Journal of the Royal Statistical Society. Series B (Methodological.), 41(1) (1979) 54-64.
  • [4] Horn S. D., Goodness-of-fit tests for discrete data: a review and an application to a health impairment scale, Biometrics., 33(1) (1977) 237-247.
  • [5] Watson G. S., Some recent results in chi-square goodness-of-fit tests, Biometrics., 15(3) (1959) 440-468.
  • [6] Cochran W. G., The χ2 test of goodness of fit, The Annals of Mathematical Statistics., 23(3) (1952) 315-345.
  • [7] Tate M. W., Hyer L. A., Inaccuracy of the X2 test of goodness of fit when expected frequencies are small, Journal of the American Statistical Association., 68(344) (1973) 836-841.
  • [8] Yarnold J. K., The minimum expectation in X2 goodness of fit tests and the accuracy of approximations for the null distribution, Journal of the American Statistical Association., 65 (330) (1970) 864-886.
  • [9] Fisher R. A., Statistical methods for research workers,13th ed. New York: Hafner Publishing Co, (1958) 356.
  • [10] Roscoe J. T., Byars J. A., An investigation of the restraints with respect to sample size commonly imposed on the use of the chi-square statistic, Journal of the American Statistical Association., 66(336) (1971) 755-759.
  • [11] Agresti A, Categorical data analysis, 2nd ed. New Jersey: John Wiley & Sons, (2003) 482.
  • [12] Koehler K. J., Goodness-of-fit tests for log-linear models in sparse contingency tables, Journal of the American Statistical Association., 81 (394) (1986) 483-493.
  • [13] Koehler, K. J., Larntz, K., An empirical investigation of goodness-of-fit statistics for sparse multinomials. Journal of the American Statistical Association., 75(370) (1980) 336- 344.
  • [14] Larntz K., Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics, Journal of the American Statistical Association., 73(362) (1978) 253-263.
  • [15] Haberman S. J., A warning on the use of chi-squared statistics with frequency tables with small expected cell counts, Journal of the American Statistical Association., 83(402) (1988) 555-560.
  • [16] Cressie N., Read T. R., Pearson's X2 and the loglikelihood ratio statistic G2: a comparative review, International Statistical Review/Revue Internationale de Statistique., 57(1) (1989) 19-43.
  • [17] Lawal H. B., Comparisons of the X2, Y2, Freeman-Tukey and Williams's improved G2 test statistics in small samples of one-way multinomials, Biometrika, 71(2) (1984) 415-418.
  • [18] Baglivo J., Olivier D., Pagano M., Methods for exact goodness-of-fit tests, Journal of the American Statistical Association., 87(418) (1992) 464-469.
  • [19] Bishop Y. M. M., Fienberg S. E., Holland P. W., Discrete multivariate analysis: Theory and practice, Cambridge: The Massachusetts Institute of Technology Press Google Scholar., (1975).
  • [20] Aitchison J., Aitken C. G., Multivariate binary discrimination by the kernel method, Biometrika., 63(3) (1976) 413-420.
  • [21] Simonoff J. S., A penalty function approach to smoothing large sparse contingency tables, The Annals of Statistics., 11(1) (1983) 208-218
  • [22] Simonoff J. S., Probability estimation via smoothing in sparse contingency tables with ordered categories, Statistics & Probability Letters., 5(1) (1987) 55-63.
  • [23] Burman P., Central limit theorem for quadratic forms for sparse tables, Journal of Multivariate Analysis., 22(2) (1987) 258-277.
  • [24] Kim S. H., Choi H., Lee S., Estimate-based goodness-of-fit test for large sparse multinomial distributions, Computational Statistics & Data Analysis., 53(4) (2009) 1122-1131.
  • [25] Zelterman D., Goodness-of-fit tests for large sparse multinomial distributions, Journal of the American Statistical Association., 82(398) (1987) 624-629.
  • [26] AKTAŞ S., Power Divergence Statistics under Quasi Independence Model for Square Contingency Tables, Sains Malaysiana., 45(10) (2016) 1573-1578.
  • [27] Cressie N., Read T. R., Multinomial goodness-of-fit tests, Journal of the Royal Statistical Society, Series B (Methodological)., 46(3) (1984) 440-464.

A study on Freeman-Tukey test statistic under the symmetry model for square contingency tables

Year 2021, Volume: 42 Issue: 2, 413 - 421, 30.06.2021

Gokcen Altun

https://doi.org/10.17776/csj.835165

Abstract

The symmetry model is the basic model in the analysis of square contingency tables. Multiple test statistics have been developed for the goodness of fit test. Freeman-Tukey test statistics is appropriate to be used in large samples. However, the required sample size to use the Freeman-Tukey test statistics is not clear. In this paper, the asymptotic properties of Freeman-Tukey test statistic are discussed via extensive Monte-Carlo simulation study. The Freeman-Tukey test statistic is compared with members of power-divergence family test statistic under the symmetry model. The results of simulation study are evaluated based on the Type-I error and power of a test. The results of simulation study and artificial data study show that Freeman-Tukey’s T^2 test statistic does not converge to chi-squared distribution for both sparse and non-sparse square contingency tables.

Keywords

Freeman-Tukey test statistic Power-divergence family Goodness-of-fit test statistics Square contingency tables Symmetry model

References

  • [1] Altun G., Karesel Olumsallık Tablolarında Model Uyumunun Sapma Ölçüsü ile Belirlenmesi, PD Thesis, Hacettepe Üniversitesi, Fen Bilimleri Enstitüsü, (2018).
  • [2] Goodman L. A., Multiplicative models for square contingency tables with ordered categories, Biometrika., 66(3) (1979) 413-418.
  • [3] Fienberg S. E., The use of chi-squared statistics for categorical data problems, Journal of the Royal Statistical Society. Series B (Methodological.), 41(1) (1979) 54-64.
  • [4] Horn S. D., Goodness-of-fit tests for discrete data: a review and an application to a health impairment scale, Biometrics., 33(1) (1977) 237-247.
  • [5] Watson G. S., Some recent results in chi-square goodness-of-fit tests, Biometrics., 15(3) (1959) 440-468.
  • [6] Cochran W. G., The χ2 test of goodness of fit, The Annals of Mathematical Statistics., 23(3) (1952) 315-345.
  • [7] Tate M. W., Hyer L. A., Inaccuracy of the X2 test of goodness of fit when expected frequencies are small, Journal of the American Statistical Association., 68(344) (1973) 836-841.
  • [8] Yarnold J. K., The minimum expectation in X2 goodness of fit tests and the accuracy of approximations for the null distribution, Journal of the American Statistical Association., 65 (330) (1970) 864-886.
  • [9] Fisher R. A., Statistical methods for research workers,13th ed. New York: Hafner Publishing Co, (1958) 356.
  • [10] Roscoe J. T., Byars J. A., An investigation of the restraints with respect to sample size commonly imposed on the use of the chi-square statistic, Journal of the American Statistical Association., 66(336) (1971) 755-759.
  • [11] Agresti A, Categorical data analysis, 2nd ed. New Jersey: John Wiley & Sons, (2003) 482.
  • [12] Koehler K. J., Goodness-of-fit tests for log-linear models in sparse contingency tables, Journal of the American Statistical Association., 81 (394) (1986) 483-493.
  • [13] Koehler, K. J., Larntz, K., An empirical investigation of goodness-of-fit statistics for sparse multinomials. Journal of the American Statistical Association., 75(370) (1980) 336- 344.
  • [14] Larntz K., Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics, Journal of the American Statistical Association., 73(362) (1978) 253-263.
  • [15] Haberman S. J., A warning on the use of chi-squared statistics with frequency tables with small expected cell counts, Journal of the American Statistical Association., 83(402) (1988) 555-560.
  • [16] Cressie N., Read T. R., Pearson's X2 and the loglikelihood ratio statistic G2: a comparative review, International Statistical Review/Revue Internationale de Statistique., 57(1) (1989) 19-43.
  • [17] Lawal H. B., Comparisons of the X2, Y2, Freeman-Tukey and Williams's improved G2 test statistics in small samples of one-way multinomials, Biometrika, 71(2) (1984) 415-418.
  • [18] Baglivo J., Olivier D., Pagano M., Methods for exact goodness-of-fit tests, Journal of the American Statistical Association., 87(418) (1992) 464-469.
  • [19] Bishop Y. M. M., Fienberg S. E., Holland P. W., Discrete multivariate analysis: Theory and practice, Cambridge: The Massachusetts Institute of Technology Press Google Scholar., (1975).
  • [20] Aitchison J., Aitken C. G., Multivariate binary discrimination by the kernel method, Biometrika., 63(3) (1976) 413-420.
  • [21] Simonoff J. S., A penalty function approach to smoothing large sparse contingency tables, The Annals of Statistics., 11(1) (1983) 208-218
  • [22] Simonoff J. S., Probability estimation via smoothing in sparse contingency tables with ordered categories, Statistics & Probability Letters., 5(1) (1987) 55-63.
  • [23] Burman P., Central limit theorem for quadratic forms for sparse tables, Journal of Multivariate Analysis., 22(2) (1987) 258-277.
  • [24] Kim S. H., Choi H., Lee S., Estimate-based goodness-of-fit test for large sparse multinomial distributions, Computational Statistics & Data Analysis., 53(4) (2009) 1122-1131.
  • [25] Zelterman D., Goodness-of-fit tests for large sparse multinomial distributions, Journal of the American Statistical Association., 82(398) (1987) 624-629.
  • [26] AKTAŞ S., Power Divergence Statistics under Quasi Independence Model for Square Contingency Tables, Sains Malaysiana., 45(10) (2016) 1573-1578.
  • [27] Cressie N., Read T. R., Multinomial goodness-of-fit tests, Journal of the Royal Statistical Society, Series B (Methodological)., 46(3) (1984) 440-464.
There are 27 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Natural Sciences
Authors

Gokcen Altun 0000-0003-4311-6508

Publication Date June 30, 2021
Submission Date December 2, 2020
Acceptance Date March 31, 2021
Published in Issue Year 2021Volume: 42 Issue: 2

Cite

APA Altun, G. (2021). A study on Freeman-Tukey test statistic under the symmetry model for square contingency tables. Cumhuriyet Science Journal, 42(2), 413-421. https://doi.org/10.17776/csj.835165