Basic Statistics

CHI-SQUARE TESTS FOR FREQUENCY TABLES: TWO-BY-TWO TABLES 157

square test is that for Fisher’s exact test the row and column totals are considered
fixed. Fisher’s exact test is available in SAS, SPSS, and Stata. The results are often
given for both a one- and a two-sided test. The two-sided test should be chosen if
one wishes to compare the results from the exact test with those from the chi-square
test. The programs will report the P values for the test. Fisher’s exact test tends
to be somewhat conservative compared to the chi-square test (see Wicken [1989]).
Fisher’s test should not be used with matched samples. For McNemar’s test, it is
recommended that the sum of b + c > 30 (see van Belle et al. [2004]).

11.3.6 The Continuity Correction: Two-by-Two Tables

As mentioned in Section 1 1.3.1, using Table A.4 results in approximate P values. For
frequency tables with 1 d.f. (two rows and two columns), the approximation is poorer
than for larger tables. To adjust for this approximation, a correction factor called the
Yates correction for continuity has been proposed. To use the correction factor, we
subtract .5 from each of the positive differences between the observed and expected
frequencies before the difference is squared. For example, if we examine Table 1 1.7,
the differences between the observed and expected frequencies for all four cells are
either +5.37 or -5.37. The positive differences are all +5.37. The corrected xz is
given by

= 3.71

2 (5.37 - .5)2 (5.37 - .5)2 (5.37 - .5)2 (5.37 - .5)2

xc = 89.63 + 85.37 + 15.37 + 14.63

or

all cells (lobserved - expected1 - .5)2

x:= c expected

The vertical line (1) is used to denote positive or absolute values. Note that for each
observed frequency minus the expected frequency, the value has been made smaller
by .5, so the computed chi-square is smaller when the correction factor is used. In
this case, the original chi-square was 4.50 and the corrected chi-square is 3.71. With
large sample sizes the correction factor has less effect than with small samples. In
our sample, the uncorrected chi-square had a P value of .034 and the corrected one
has a P value of .054. Thus with Q = .05, we would reject the null hypothesis for
the uncorrected chi-square and not reject it if we used the Yates correction.
There is disagreement on the use of the continuity correction factor. Some authors
advocate its use and others do not. The use of the continuity correction gives P
values that better approximate the P values obtained from Fisher’s exact test. Using
the continuity correction factor does not make the P values obtained closer to those
from the tabled chi-square distribution. The result is a conservative test where the
percentiles of the corrected chi-square tend to be smaller than the tabled values. In the
example just given, we suggest that both corrected and uncorrected values be given
to help the user interpret the results.