The analysis of 2 × 1 and 2 × 2 contingency tables: an historical review

Abstract

This paper describes the historical development of analytic techniques for frequency data presented as 2 x 1 or 2 x 2 contingency tables. The issues raised include: the need for a continuity correction when estimating the exceedance probability of a discrete statistic; the vulnerability of such statistics to a conservative bias when used in hypothesis testing; and the different statistical models that may underlie contingency tables. In the case of 2 x 1 contingency tables the recommended technique is the binomial test with a modified decision procedure for hypothesis testing. In the case of 2 x 2 contingency tables there is a single statistic approximating the chi-square distribution which can be used to test the hypothesis of an association between the two relevant variables and which in most practical situations is robust with respect to the occurrence of small expected cell frequencies.

References

Kempthorne O. In dispraise of the exact test: Reactions. Journal of Statistical Planning and Inference 1979; 3: 199-213.

Pearson K. 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. Philosophical Magazine 1900; Series 5, 50: 157-75.

Greenwood, M. Jr, Yule GU The statistics of anti-typhoid and anti-cholera inoculations and the interpretation of such statistics in general. Proceedings of the Royal Society of Medicine: Section on Epidemiology and State Medicine 1915; 8: 113-94.

Fisher RA On the interpretation of χ2 from contingency tables and the calculation of P. Journal of the Royal Statistical Society 1922; 85: 87-94.

Cochran WG The χ2 test of goodness of fit. Annals of Mathematical Statistics 1952; 23: 315-45.

Fienberg SE Fisher's contributions to the analysis of categorical data. In Fienberg SE , Hinkley DV eds, R.A. Fisher: An appreciation. New York: Springer-Verlag, 1980: 75-84.

Fisher RA Statistical methods for research workers, 1st ed. Edinburgh: Oliver and Boyd, 1925.

Croxton FE Elementary statistics with applications in medicine and the biological sciences . New York: Dover, 1953.

Ferguson GA , Takane Y. Statistical analysis in psychology and education, 6th ed. New York: McGraw-Hill, 1989.

10.

Howell DC Statistical methods for psychology, 2nd ed. Boston, MA: Duxbury Press, 1987.

11.

Kendall MG , Stuart A. (1967). The advanced theory of statistics, Vol. 2, 2nd ed. London : Griffin, 1967.

12.

Wilks SS The likelihood test of independence in contingency tables. Annals of Mathematical Statistics 1935; 6: 190-96.

13.

Kullback S. Information theory and statistics. New York: Wiley, 1959.

14.

Akaike H. Information theory and an extension of the maximum likelihood principle. In Petrov BN , Csaki F eds, Second international symposium on information theory. Budapest: Akademiai Kiado , 1973: 267-81.

15.

Sakamoto Y. , Ishiguro M. , Kitagawa G. Akaike information criterion statistics. Dordrecht: Reidel, 1986.

16.

Bulmer MG Principles of statistics, 2nd ed. Edinburgh: Oliver and Boyd, 1967 .

17.

Barnard GA Significance tests for 2 x 2 tables. Biometrika 1947; 34: 123-38.

18.

Pearson ES The choice of statistical tests illustrated on the interpretation of data classed in a 2 x 2 table. Biometrika 1947; 34: 139-67.

19.

Rhoades HM , Overall JE

A sample size correction for Pearson chi-square in 2 x 2 contingency tables. Psychological

Bulletin 1982; 91: 418-23.

20.

Upton Gjg. Cross-classified data. In Lovie AD ed. New developments in statistics for psychology and the social sciences . London: The British Psychological Society and Methuen, 1986: 70-92.

21.

Fisher RA The design of experiments. Edinburgh: Oliver and Boyd, 1935.

22.

Gridgeman NT The lady tasting tea and allied topics. Journal of the American Statistical Association 1959; 54: 776-83.

23.

Holschuh N. Randomization and design: 1. In Fienberg SE , Hinkley DV eds, R.A. Fisher: An appreciation. New York : Springer-Verlag, 1980: 35-45.

24.

Phillips DS Basic statistics for health science students. San Francisco : WH Freeman, 1978.

25.

Siegel S. Nonparametric statistics for the behavioral sciences. New York: McGraw-Hill, 1956.

26.

Haviland MG Yates's correction for continuity and the analysis of 2 x 2 contingency tables. Statistics in Medicine 1990; 9: 363-67.

27.

Kendall MG , Buckland WR A dictionary of statistical terms, 3rd ed. Edinburgh: Oliver and Boyd, 1971 .

28.

Yates F. Contingency tables involving small numbers and the χ2 test. Journal of the Royal Statistical Society 1934; Suppl. 1: 217-35.

29.

Fisher RA Statistical methods for research workers, 5th ed. Edinburgh: Oliver and Boyd, 1934.

30.

Irwin JO Tests of significance for differences between percentages based on small numbers . Metron 1935; 12: 83-94.

31.

Rohatgi VK Statistical inference. New York: Wiley, 1984.

32.

Basu D. Discussion of Joseph Berkson's paper 'In dispraise of the exact test'. Journal of Statistical Planning and Inference 1979; 3: 189-92.

33.

Richardson Jte. Variants of chi-square for 2 x 2 contingency tables. British Journal of Mathematical and Statistical Psychology 1990; 43: 309-26.

34.

Baker RJ Exact distributions derived from two-way tables. Applied Statistics 1977; 26: 199-206.

35.

Conover WJ Some reasons for not using the Yates continuity correction on 2 x 2 contingency tables. Journal of the American Statistical Association 1974; 69: 374-76.

36.

Haber, M. A comparison of some continuity corrections for the chi-squared test on 2 x 2 tables. Journal of the American Statistical Association 1980; 75: 510-15.

37.

Lawal HB , Upton Gjg. On the use of X2 as a test of independence in contingency tables with small cell expectations. Australian Journal of Statistics 1984; 26: 75-85.

38.

Mantel N. , Greenhouse SW

What is the continuity correction?

The American Statistician 1968; 22(5): 27-30.

39.

Maxwell EA Analysis of contingency tables and further reasons for not using Yates correction in 2 x 2 tables. Canadian Journal of Statistics 1976; 4: 277-90.

40.

Camilli G. , Hopkins KD

Applicability of chi-square to 2 x 2 contingency tables with small expected cell frequencies. Psychological

Bulletin 1978 ; 85: 163-67.

41.

Yates F. Tests of significance for 2 x 2 contingency tables. Journal of the Royal Statistical Society Series A 1984; 147: 426-63.

42.

Lewith GT , Machin D. A randomised trial to evaluate the effect of infra-red stimulation of local trigger points, versus placebo, on the pain caused by cervical osteoarthrosis . International Journal of Acupuncture and Electro-Therapeutic Research 1981; 6: 277-84.

43.

Daly LE , Bourke GJ , McGilvray J. Interpretation and uses of medical statistics, 4th ed. Oxford: Blackwell, 1991.

44.

Fisher RA The logic of inductive inference. Journal of the Royal Statistical Society 1935; 98: 39-82.

45.

Fisher RA The interpretation of experimental four-fold tables. Science 1941; 94: 210-11.

46.

Berkson J. In dispraise of the exact test: Do the marginal totals of the 2 x 2 table contain relevant information respecting the table proportions? Journal of statistical Planning and Inference 1978; 2: 27-42.

47.

Barnard GA A new test for 2 x 2 tables. Nature 1945;156: 177.

48.

Fisher RA Statistical methods and scientific inference. Edinburgh : Oliver and Boyd, 1956.

49.

Fisher RA A new test for 2 x 2 tables. Nature 1945; 156: 388.

50.

Berkson J. Do the marginal totals of the 2 x 2 table contain relevant information respecting the table proportions? Journal of Statistical Planning and Inference 1978; 2: 43-44.

51.

Camilli G. The test for homogeneity for 2 x 2 contingency tables: A review of and some personal opinions on the controversy. Psychological Bulletin 1990; 108: 135-45.

52.

Fisher RA The logical inversion of the notion of the random variable. Sankhyä 1945; 7: 129-32.

53.

Barnard GA Statistical inference. Journal of the Royal Statistical Society Series B 1949; 11: 115-49.

54.

Barnard GA Discussion of Dr Yates's paper. Journal of the Royal Statistical Society Series A 1984; 147:449-50.

55.

Upton Gjg. Fisher's exact test. Journal of the Royal Statistical Society Series A 1992 ; 155: 395-402.

56.

Haber M.

Comments on 'The test of homogeneity for 2 x 2 contingency tables: A review of and some personal opinions on the controversy' by G Camilli. Psychological

Bulletin 1990; 108: 146-49.

57.

Plackett RL The continuity correction in 2 x 2 tables. Biometrika 1964; 51: 327-37.

58.

Haber M. An exact unconditional test for the 2 x 2 comparative trial. Psychological Bulletin 1986; 99: 129-32.

59.

Barnard GA 2 x 2 tables: A note on ES Pearson's paper. Biometrika 1947; 34: 168-69.

60.

Kendall MG , Stuart A. The advanced theory of statistics, Vol. 1, 2nd ed. London: Griffin , 1963.

61.

Patnaik PB The power function of the test for the difference between two proportions in a 2 x 2 table. Biometrika 1948; 35: 157-75.

62.

Upton Gjg. A comparison of alternative tests for the 2 x 2 comparative trial. Journal of the Royal Statistical Society Series A 1982; 145: 86-105.

63.

Haldane Jbs. The mean and variance of χ2, when used as a test of homogeneity, when expectations are small. Biometrika 1939; 31: 346-55.

64.

Garside GR , Mack C. Actual Type I error probabilities for various tests in the homogeneity case of the 2 x 2 contingency table. The American Statistician 1976; 30: 18-21.

65.

D'Agostino RB , Chase W. , Belanger A. The appropriateness of some common procedures for testing the equality of two independent binomial populations. The American Statistician 1988; 42: 198-202.

66.

Haber M. The continuity correction and statistical testing. International Statistical Review 1982; 50: 135-44.

67.

Liddell D. Practical tests of 2 x 2 contingency tables. The Statistician 1976; 25: 295-304.

68.

Overall JE Continuity correction for Fisher's exact probability test. Journal of Educational Statistics 1980; 5: 177-90.

69.

Overall JE , Rhoades HM , Starbuck RR

Small-sample tests for homogeneity of response probabilities in 2 x 2 contingency tables. Psychological

Bulletin 1987; 102: 307-14.

70.

Overall JE , Starbuck RR F-test alternatives to Fisher's exact test and to the chi-square test of homogeneity in 2 x 2 tables. Journal of Educational Statistics 1983; 8: 59-73.

71.

D'Agostino RB , Chase W. , Belanger A. The appropriateness of some common procedures for testing the equality of two independent binomical populations. The American Statistician 1988; 42: 198-202.

72.

Camilli G. , Hopkins KD Testing for association in 2 x 2 contingency tables with very small sample sizes. Psychological Bulletin 1979; 86: 1011-14.

73.

Grizzle JE Continuity correction in the χ2-test for 2 x 2 tables. The American Statistician 1967; 21(4): 28-32.

74.

Roscoe JT , Byars JA 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 1971; 66: 755-59.

75.

Starmer CF , Grizzle JE , Sen PK

Comment. Journal of the

American Statistical Association 1974;69:376-78.

76.

Streitberg B. , Roehmel J. Alternatives to Fisher's exact test? Biometrie und Informatik in Medizin und Biologie 1991; 22: 139-46.

77.

Cox DR , Snell EJ Analysis of binary data, 2nd ed. London: Chapman and Hall, 1989.

78.

Bradley DR , Bradley TD , McGrath SG , Cutcomb SD Type I error rate of the chi-square test of independence in R x C tables that have small expected frequencies. Psychological Bulletin 1979; 86: 1290-97.

79.

Holland WW , Bailey P. , Bland JM Long-term consequences of respiratory disease in infancy. Journal of Epidemiology and Community Health 1978;32:256-59.

80.

Centerwall BS , Armstrong CW , Funkhouser LS , Elzay RP Erosion of dental enamel among competitive swimmers at a gas-chlorinated swimming pool. American Journal of Epidemiology 1986;123: 641-47.

81.

Pirie WR , Hamdan MA Some revised continuity corrections for discrete distributions. Biometrics 1972; 28: 693-701.

82.

Bradley DR , Cutcomb S. Monte Carlo simulations and the chi-square test of independence. Behavior Research Methods and Instrumentation 1977; 9: 193-201.

83.

Cox DR The continuity correction. Biometrika 1970 ;57:217-19.

84.

Siegel S. , Castellan NJ Jr. Nonparametric statistics for the behavioral sciences, 2nd ed. New York: McGraw-Hill , 1988.

85.

Kendall MG , Stuart A. The advanced theory of statistics, Vol. 2, 4th ed. London: Griffin , 1979.

86.

Mantel N. Comment and a suggestion. Journal of the American Statistical Association 1974; 69: 378-80.

87.

Mantel N. The continuity correction. The American Statistician 1976; 30: 103-104.

88.

Tocher KD Extension of the Neyman-Pearson theory of tests to discontinuous variates . Biometrika 1950; 37: 130-44.

89.

Lewis D. , Burke CJ The use and misuse of the chi-square test. Psychological Bulletin 1949; 46: 433-89.

90.

Hays WL Statistics. New York: Holt, Rinehart and Winston, 1963.

91.

Overall JE Comment. Statistics in Medicine 1990; 9:379-82.

92.

Corsten Lca , de Kroon Jpm. Comment on J Berkson's paper `In dispraise of the exact test'. Journal of Statistical Planning and Inference 1979; 3: 193-97.

93.

Bradley DR , Seely DL Empirical determination of the power of the chi-square test of independence in 2 x 2 tables. Proceedings of the Statistical Computing Section of the American Statistical Association 1977; 138-44.

94.

Comrey AL A first course in factor analysis. New York: Academic Press, 1973.

95.

Mehta CR , Hilton JF Exact power of conditional and unconditional tests: Going beyond the 2 x 2 contingency table. The American Statistician 1993; 47: 91-98.

96.

Overall JE Power of chi-square tests for 2 x 2 contingency tables with small expected frequencies. Psychological Bulletin 1980; 87: 132-35.