The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

8.4 Fisher’s Exact Test 133

close to other asymptotic and exact nonparametric tests. However,

when some cells are sparse (i.e., 0 – 4 counts in those cells), Fisher ’ s

exact test for 2 × 2 tables and its generalization to the R × C table is a

better choice. That is the topic of the next section.

8.4 FISHER ’ S EXACT TEST

In contingency tables, the counts in each cell may be totally random,
and hence the row and column totals are not restricted. However, there
are cases where the row totals and column totals (called marginal totals
or margins) are fi xed in advance. In such cases, it makes sense to con-
sider as the sample space all possible tables that yield the same totals
for each row and each column. The distribution of such tables under
the null hypothesis of independence is known to be a hypergeometric
distribution.
So one could ask under the null hypothesis is our observed table
likely to occur or not based on the known hypergeometric distribution.
This idea goes back to Fisher ( 1935 ) for 2 × 2 tables, and can easily
be generalized to any R × C table. This idea has been applied even
when the rows and column need not be the same as in the observed
table, with the argument being that it still makes sense to condition on
the given values for the row and column totals. In fact, the Fisher exact
test gives nearly the same results as the chi - square when the chi - square
is appropriate as an approximation, and the chi - square test does not

Table 8.5

Association Between Ethnicity and Breast Cancer Stage From a

Registry Sample *

Stage of breast cancer

Ethnicity In situ Local Distant Total

Caucasian 124 (232.38) 761 (663,91) 669 (657.81) 1554
African
American

36 (83.85) 224 (239.67) 301 (237.47) 561

Asian 221 (64.87) 104 (185.42) 109 (183.71) 434
Total 381 1089 1079 2549

* Note: Count with expected count in parentheses.