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

Abstract

Standard statistics textbooks recommend the use of chi-square with 2X2 contingency tables only when the expected frequency of each cell is at least five. The Yates correction for continuity is also recommended to increase the accuracy of the resulting probability statement. The empirical consequences of these recommendations were explored on a wide variety of 2 X 2 contingency tables. Pearson's chi-square test was found to be very robust with small expected cell frequencies. As expected from theoretical considerations, the Yates correction decreases the accuracy of probability statements when either or both marginals are not fixed. The chi-square test for 2 X 2 contingency tables is widely used in research. Virtually all textbooks in statistical methods recommend that the chi-square test (Pearson's chisquare) not be employed for 2 X 2 contingency tables if the expected frequency in any cell falls below five. With small cell frequencies, the probability derived from the chi-square test is thought to be an unsatisfactory approximation to the true probability-— since large-sample theory is used in the derivation of chi-square, the probability statement will be precise only when the sample size is large. Often-quoted rules of thumb state that the minimum expected cell frequency should be at least 5 (Cochran, 1954; Fisher, 19S8) or 10 (Hays, 1973) before the probabilities associated with the chi-square statistic can be considered accurate. In addition to the minimum expected frequency question, there is a related issue concerning the Yates correction for continuity. Almost all textbooks (e.g., Dixon & Massey, p. 242) recommend the Yates correction, claiming that it increases the accuracy of the probability statements in all 2x2 contingency tables.