The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

8.4 Fisher’s Exact Test 135

2. Use the above formula for the hypergeometric distribution to cal-
   culate the probability of these tables under the null hypothesis.


3. Sum the probabilities for all such tables.

This gives a one - sided p - value. To get a p - value for the other side,
sum all the probabilities of tables where cell (1, 1) has values greater
than or equal to x , and probability lower than the observed probability.
The two - sided p - value is just the sum of the two one - sided p - values
minus the observed probability, because the sum would count the
observed probability twice.
We now illustrate an example of testing skills at detecting order.
Now suppose as hypothesized by Agresti ( 1990 , p. 61) that an experi-
ment was conducted to test the null hypothesis of random guesses
versus the alternative of skill in detecting order. Agresti ’ s well - known
text was revised (see Agresti [ 2002 ]). The patient is given a medicine
and water. She is told that four of the cups have the water poured fi rst,
and four had the medicine place in the cup fi rst, and then the water was
added. The cups are numbered 1 – 8. Because she knows that four are
water fi rst, and four are medicine fi rst, the table is constrained to have
both row margins and both column margins totaling four. The table
looks like Table 8.7.
Here, the conditioning is uncontroversial, because the experimenter
told the patient the row and column constraints. There are now only
fi ve possible tables: (1) perfect guessing x = 4, (2) two mistakes out of
eight guesses when x = 1, (3) four mistakes out of eight guesses when
x = 2, (4) six mistakes out of eight guesses when x = 3, and (5) eight
mistakes out of eight guesses when x = 4. Clearly, the more mistakes
that there are, the further we are from the alternative hypothesis. The

Table 8.7

Patient Taking Medicine Experiment: Possible

2 × 2 Tables

Placed in cup

fi rst

Patient guesses

medicine

Patient

guesses water

Row totals

Medicine x 4 − x 4

Water 4 − x x 4

Column totals 4 4 8