A critical review of asymptotic methods for comparing two proportions by means of independent samples

The test for comparing two proportions (based on the use of independent samples) is one of the best-known in statistics, and at the same time is the one that generates most disagreement, both in the asymptotic and non-asymptotic versions. The present article offers a critical review of the former, grouping them by "families", points out their theoretical relation to the latter, and details the main errors that have given rise to lop-sided arguments or have resulted in unsound conclusions. It also selects the optimal known versions (from the point of view mentioned) and their conditions of validity. Finally, it shows the main gaps that exist at present, and suggests how to solve them.

[1]  M. G. Haviland.  Yates's correction for continuity and the analysis of 2 x 2 contingency tables. , 1990, Statistics in medicine.

[2]  Choongrak Kim,et al.  Exact Properties of Some Exact Test Statistics for Comparing Two Binomial Proportions , 1990 .

[3]  G. Barnard,et al.  On alleged gains in power from lower P-values. , 1989, Statistics in medicine.

[4]  Timothy R. C. Read,et al.  Pearsons-X2 and the loglikelihood ratio statistic-G2: a comparative review , 1989 .

[5]  N. Kroll.  Testing Independence in 2 × 2 Contingency Tables , 1989 .

[6]  C. Lloyd.  Doubling the one-sided P-value in testing independence in 2 x 2 tables against a two-sided alternative. , 1988, Statistics in medicine.

[7]  Ralph B. D'Agostino,et al.  The Appropriateness of Some Common Procedures for Testing the Equality of Two Independent Binomial Populations , 1988 .

[8]  A. M. Andrés,et al.  Tablas 2×2 y test exacto de Fisher , 1987 .

[9]  M. Aitkin,et al.  Canonical likelihoods: A new likelihood treatment of nuisance parameters , 1987 .

[10]  W. Dupont.  Sensitivity of Fisher's exact test to minor perturbations in 2 x 2 contingency tables. , 1986, Statistics in medicine.

[11]  C. P. Cox,et al.  Analytic Results on the Difference Between the G 2 and χ 2 Test Statistics in One Degree of Freedom Cases , 1986 .

[12]  J. Shuster,et al.  Are Uniformly Most Powerful Unbiased Tests Really Best , 1984 .

[13]  F. Yates,et al.  Tests of Significance for 2 × 2 Contingency Tables , 1984 .

[14]  M. Haber.  The Continuity Correction and Statistical Testing , 1982 .

[15]  J. Overall,et al.  A sample size correction for Pearson chi-square in 2 × 2 contingency tables. , 1982 .

[16]  Michael Haber,et al.  A Comparison of Some Continuity Corrections for the Chi-Squared Test on 2×2 Tables , 1980 .

[17]  Michael A. Fligner,et al.  A Comparison of Two Tests for Equality of Two Proportions , 1977 .

[18]  Debabrata Basu,et al.  On the Elimination of Nuisance Parameters , 1977 .

[19]  George A. Milliken,et al.  A Nonrandomized Unconditional Test for Comparing Two Proportions in 2×2 Contingency Tables , 1977 .

[20]  H. Schouten.  On the Continuity Correction , 1976 .

[21]  C. Mack,et al.  Actual Type 1 Error Probabilities for Various Tests in the Homogeneity Case of the 2 × 2 Contingency Table , 1976 .

[22]  M. A. Hamdan.  On The Continuity Correction , 1974 .

[23]  N. Mantel.  Comment and a Suggestion , 1974 .

[24]  P. Sen,et al.  Some Reasons for Not Using the Yates Continuity Correction on 2 × 2 Contingency Tables: Comment , 1974 .

[25]  W. Conover.  Some Reasons for Not Using the Yates Continuity Correction on 2×2 Contingency Tables , 1974 .

[26]  W. Pirie,et al.  Some Revised Continuity Corrections for Discrete Distributions , 1972 .

[27]  H. Berchtold.  Vertrauensgrenzen und Vergleich zweier Wahrscheinlichkeiten , 1972 .

[28]  David R. Cox,et al.  The continuity correction , 1970 .

[29]  R. D. Boschloo.  Raised conditional level of significance for the 2 × 2‐table when testing the equality of two probabilities , 1970 .

[30]  S. Greenhouse,et al.  What is the Continuity Correction , 1968 .

[31]  R. Plackett.  The continuity correction in 2×2 tables , 1964 .

[32]  L. A. Goodman.  Simultaneous Confidence Intervals for Contrasts Among Multinomial Populations , 1964 .

[33]  D. G. Beech,et al.  Statistical Theory and Methodology in Science and Engineering. , 1961 .

[34]  W. G. Cochran.  Some Methods for Strengthening the Common χ 2 Tests , 1954 .

[35]  David Lindley,et al.  Advanced Statistical Methods in Biometric Research. , 1953 .

[36]  J. Tukey,et al.  Transformations Related to the Angular and the Square Root , 1950 .

[37]  K. D. Tocher.  Extension of the Neyman-Pearson theory of tests to discontinuous variates. , 1950, Biometrika.

[38]  Maurice G. Kendall,et al.  The advanced theory of statistics , 1945 .

[39]  R Fisher,et al.  Design of Experiments , 1936 .

[40]  S. S. Wilks.  The Likelihood Test of Independence in Contingency Tables , 1935 .

[41]  F. Yates.  Contingency Tables Involving Small Numbers and the χ2 Test , 1934 .

[42]  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 , 1900 .

[43]  A. Martín Andrés,et al.  A review of classic non-asymptotic methods for comparing two proportions by means of independent sampLES , 1991 .

[44]  Roderick J. A. Little,et al.  Testing the Equality of Two Independent Binomial Proportions , 1989 .

[45]  J. Castillo.  Comparación de proporciones en tablas 2x2 (una extensión particular al caso RXS) , 1987 .

[46]  Michael Haber,et al.  A comparison of some conditional and unconditional exact tests for 2x2 contingency tables , 1987 .

[47]  L. Sachs.  Alternatives to the Chi‐Square Test of Homogeneity in 2×2 Tables and to Fisher's Exact Test , 1986 .

[48]  V. Rohatgi,et al.  Small Sample Comparison of Some Tests for Equality of Two Proportions , 1986 .

[49]  Michael Haber,et al.  A Modified Exact Test for 2 × 2 Contingency Tables , 1986 .

[50]  H. Riedwyl,et al.  Small Sample Properties of Asymptotic Tests for Two Binomial Proportions , 1984 .

[51]  G. Upton.  A Comparison of Alternative Tests for the 2 Times 2 Comparative Trial , 1982 .

[52]  H. Burstein.  Binomial 2×2 test for independent samples with independent proportions , 1981 .

[53]  John J. Gart,et al.  THE COMPARISON OF PROPORTIONS: A REVIEW OF SIGNIFICANCE TESTS, CONFIDENCE INTERVALS AND ADJUSTMENTS FOR STRATIFICATION' , 1971 .

[54]  J. Gart.  Alternative Analyses of Contingency Tables , 1966 .

[55]  R. Fisher,et al.  STATISTICAL METHODS AND SCIENTIFIC INDUCTION , 1955 .

[56]  E. S. Pearson,et al.  The choice of statistical tests illustrated on the interpretation of data classed in a 2 X 2 table. , 1947, Biometrika.

[57]  G. Barnard.  Significance tests for 2 X 2 tables. , 1947, Biometrika.

[58]  R. Fisher,et al.  The Logic of Inductive Inference , 1935 .