Actual Type 1 Error Probabilities for Various Tests in the Homogeneity Case of the 2 × 2 Contingency Table

Variants of chi‐square for 2 × 2 contingency tables

“…Against these criteria a number of clear conclusions can be drawn with regard to the usefulness of the four variants of X 2 in analysing 2 x 2 contingency tables. Although the data presented here have been confined to the conventional criterion level of significance (a=0.05), similar conclusions can be reached with other a levels (see also Camilli & Hopkins, 1978, 1979Garside & Mack, 1976;Overall et al, 1987;Rhoades & Overall, 1982;Roscoe & Byars, 1971).…”

“…Some researchers have followed Yates (1934) in claiming that the use of Pearson's X 2 in the case of 2 x 2 contingency tables tends to generate too many Type I errors, especially with small samples (e.g. Garside & Mack 1976;Haber, 1980). Nevertheless, the correction for continuity appropriate to any particular statistic will depend upon the range of possible values that the statistic may take, and in the case of 2 x 2 contingency tables this depends in turn upon the underlying statistical model.…”

“…A number of researchers have used similar methods in order to compare some of the variants of X 2 and the Fisher Exact Test in the case of Model I1(Berkson, 1978;Conover, 1974;Garside & Mack, 1976; Liddell, 1976; Overall, 1980~;Overall, Rhoades & Starbuck, 1987;Overall & Starbuck, 1983; E. S Pearson, 1947;Rhoades & Overall, 1982;. Upton, 1982).…”

Exact Sample Sizes for Use with the Fisher-Irwin Test for 2 x 2 Tables

“…When considering sample size determination for 2 X 2 contingency tables, one should be aware of the problem of discrepancies between the nominal and actual levels of significance. Garside and Mack (1976) give some numerical results which indicate that for 0.1 < P < 0.5 (P P1 P2) and n reasonably large, only minor differences are found. For example, with n = 100 and a = 0.05, the actual significance levels ranged from 0.0310 to 0.0384 for 0.1 < P < 0.5.…”

A Nonrandomized Unconditional Test for Comparing Two Proportions in 2×2 Contigency Tables

“…A brief description is given in Section 3 of the algorithm used to generate the critical regions (the algorithm is applicable to any combination of sample sizes but for larger values, i.e., N, > 100; i = 1, 2, excessive amounts of computer time are required). The latter two papers have recently been reviewed by Garside and Mack [5]. An alternate approach for small sample sizes has been proposed by Boschoo [2].…”