The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

138 CHAPTER 8 Contingency Tables

know if it is added to the nicotine patch group or the group counseling
group.
Table 8.9 is a 2 × 2 table that counts the number of each of the four
possible pairs for this matched experiment.
Under the null hypothesis that success does not depend on the
treatment, we would expect the discordant observations (0, 1) and (1,
0) to be approximately equal. So the expected total given the discordant
total R + Y would be ( R + Y )/2. McNemar ’ s test statistic is
T = ( R − [ R + Y ]/2)^2 /{( R + Y )/2} + ( Y − [ R + Y ]/ 2 )^2 /{( R + Y )/2}. This
is just like the chi - square statistic summing “ the observed minus
expected squared divided by expected. ” After some algebra we see for
the 2 × 2 table, this simplifi es to ( R − Y )^2 /[2( R + Y )], and so the test is
equivalent to testing for large values of W = ( R − Y )^2 /( R + Y ). For a
more detailed account, see Conover ( 1999 , p. 166). In this example,
we get (92 − 48)^2 /(92 + 48) = (44)^2 /140 = 0.1936/140 = 13.82. Now
under the null hypothesis, T is asymptotically chi - square with 1 degree
of freedom. T = W /2 = 6.91. Consulting the chi - square table, we see
that the p - value is slightly less than 0.01.
For more than two categories in each group the idea of concordant
and discordant pairs extends, and McNemar ’ s test can be applied to an
R × 2 table.

8.6 RELATIVE RISK AND ODDS RATIO

Relative risks and odds ratios are important in medical research and

are common in epidemiology studies as well. For a detailed discussion

of these concepts, see Lachin ( 2000 ), and, from the epidemiologists

Table 8.9

Outcomes for Pairs of Subjects Attempting to Stop Smoking

Counseling failure Counseling success Nicotine patch total

Nicotine patch

failure

N = 143 (0, 0) R = 48 (0, 1) N + R = 191

Nicotine patch

success

Y = 92 (1, 0) Z = 17 (1, 1) Y + Z = 109

Counseling total N + Y = 235 R + Z = 6 5 N + Y + R + Z = 300