CHAPTER 12 Comparing Proportions and Analyzing Cross-Tabulations 169

8.81 down to 7.63 and increases the p value from 0.0030 to 0.0057, which is still

very significant — the chance of random fluctuations producing such an apparent

effect in your sample is only about 1 in 175 (because 1 0 0057

175

/ .

).

Even though the Yates correction to the Pearson chi-square test is only applicable

to the fourfold table (and not tables with more rows or columns), some statisti-

cians feel the Yates correction is too strict. Nevertheless, it has been automatically

built into statistical software like R, so if you run a Pearson chi-square using most

commercial software, it automatically uses the Yates correction when analyzing a

fourfold table (see Chapter 4 for a discussion of statistical software).

Focusing on the Fisher Exact Test

The Pearson chi-square test described earlier isn’t the only way to analyze cross-

tabulated data. Remember that one of the cons was that it is not an exact test?

Famous but controversial statistician R.  A. Fisher invented another test in the

1920s that gives the exact p value for tables that can handle very small cell counts

(even cell counts of zero!). Not surprisingly, this test is called the Fisher Exact test

(also sometimes referred to Fisher’s exact test, or just Fisher).

Understanding how the Fisher

Exact test works

Like with the chi-square, you don’t have to know the details of the Fisher Exact

test to use it. If you have a computer do the calculations for you (which we always

recommend), you technically don’t have to read this section. But we encourage

you to read this section anyway so you’ll have a better appreciation for the

strengths and limitations of this test.

This test is conceptually pretty simple. Instead of taking the product of the mar-

ginals and dividing it by the total for each cell as is done with the chi-square test

statistic, Fisher exact test looks at every possible table that has the same marginal

totals as your observed table. You calculate the exact probability (Pr) of getting

each individual table using a formula that, for a fourfold table (using the notation

for Figure 12-6), is

Pr

(

!)(

!)(

!)(

!)

(

!)(

!)(

!)(

!)( !

1

2

1

2

1,1

1,2

2,1

2,2

R

R

C

C

Ob

Ob

Ob

Ob

T )