6.6 FISHER’S EXACT TEST
Even with a continuity correction, the goodness-of-fit test statistic such as
Pearson’sX^2 is not suitable when the sample is small. Generally, statisticians
suggest using them only if no expected frequency in the table is less than 5. For
studies with small samples, we introduce a method known asFisher’s exact test.
For tables in which use of the chi-square testX^2 is appropriate, the two tests
give very similar results.
Our purpose is to find the exact significance level associated with an
observed table. The central idea is to enumerate all possible outcomes consis-
tent with a given set of marginal totals and add up the probabilities of those
tables more extreme than the one observed. Conditional on the margins, a
2 2 table is a one-dimensional random variable having a known distribution,
so the exact test is relatively easy to implement. The probability of observing a
table with cellsa,b,c, andd(with totaln)is
Prða;b;c;dÞ¼
ðaþbÞ!ðcþdÞ!ðaþcÞ!ðbþdÞ!
n!a!b!c!d!The process for doing hand calculations would be as follows:
- Rearrange the rows and columns of the table observed so the the smaller
total is in the first row and the smaller column total is in the first column. - Start with the table having 0 in theð 1 ; 1 Þcell (top left cell). The other cells
in this table are determined automatically from the fixed row and column
margins. - Construct the next table by increasing theð 1 ; 1 Þcell from 0 to 1 and
decreasing all other cells accordingly. - Continue to increase theð 1 ; 1 Þcell by 1 until one of the other cells
becomes 0. At that point we have enumerated all possible tables. - Calculate and add up the probabilities of those tables with cellð 1 ; 1 Þ
having values from 0 to the observed frequency (left side for a one-sided
test); double the smaller side for a two-sided test.
In practice, the calculations are often tedious and should be left to a com-
puter program to implement.
Example 6.14 A study on deaths of men aged over 50 yields the data shown
in Table 6.17 (numbers in parentheses are expected frequencies). An applica-
tion of Fisher’s exact test yields a one-sidedpvalue of 0.375 or a two-sidedp
value of 0.688; we cannot say, on the basis of this limited amount of data, that
there is a significant association between salt intake and cause of death even
though the proportions of CVD deaths are di¤erent (71.4% versus 56.6%). For
implementing hand calculations, we would focus on the tables where cellð 1 ; 1 Þ
FISHER’S EXACT TEST 229