The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

136 CHAPTER 8 Contingency Tables

null hypothesis expects four mistakes. This test is naturally one sided,
since the patient claims skill, meaning she get more right than four out
of eight, and less than four simply says that she is no better or worse
at guessing than by chance. We do the experiment and get a result where
she guesses 1 “ water fi rst ” wrong and 1 “ medicine fi rst ” wrong. Her
table looks like Table 8.8.
There is one table more extreme than the observed, and that is the
perfect guess table with x = 4. The p - value for this experiment is the
sum of the probabilities that x = 3 and x = 4. So p = C (4, 3) C (4, 1)/ C (8,
4) = (4!/[3! 1!])(4!/[1! 3!])/(8!/[4! 4!]) = 4(4) 4!/8 7 6 5) = 16/70 = 8 /
35 = 0.229. Perfect guessing has probability 1/70 = 0.0142. So the p -
value for the experiment is 0.229 + 0.014 = 0.243. This is not statisti-
cally signifi cant. Only a perfect score would have been signifi cant for
a sample size of eight, with four of each mixture.
Fisher ’ s exact test is an example of nonparametric procedures that
go by various names: permutation tests, randomization tests or reran-
domization tests. For a modern treatment of these procedures, see Good
( 2000 ). In Fisher ’ s original book (Fisher 1935 ), the exact problem is
presented except that instead of a patient taking medicine, it is a lady
tasting tea. The experiment itself is covered in detail in Salsburg (2001),
a beautiful story of the history of the development of statistics through
the twentieth entury.

8.5 CORRELATED PROPORTIONS AND

MCNEMAR ’ S TEST

When considering the paired t - test, we recognized the advantage of
reducing variance through the use of correlated observations. Since we

Table 8.8

Patient Taking Medicine Experiment: Observed

Table

Placed in cup

fi rst

Patient guesses

medicine

Patient

guesses water

Row totals

Medicine 3 1 4

Water 1 3 4

Column totals 4 4 8