Basic Statistics

136 CATEGORICAL DATA: PROPORTIONS

In the example,
.10 - ‘00
.05

=2

From Table A.2, the area in the upper tail is 1 - .9772 = .0228. For a two-sided test,

we have

P = 2(.0228) = ,0456

With a = .05, the null hypothesis is rejected and it is decided that the two treatments
differ in their recovery rates.
If we wish to include a correction for continuity, then 1/2(1/n1 + 1/722) is sub-
tracted from a positive value of (pl - p2) or added to a negative value of (p1 - p2).
Here,pl -p2 = +.lo and 1/2(1/100+1/100) or .01 wouldbesubtractedfrom.10.
Although generally a two-sided hypothesis is used, it is also possible to test a
one-sided hypothesis. The procedure is analogous to that given in Section 8.2.2,
where tests for the differences in two means were discussed. If the null hypothesis
Ho : 7r1 I 7r2 is tested, the entire rejection region is in the upper tail of the z
distribution.
When the sample sizes are small, the normal approximation is not accurate. Ex-
planations of how the data should be tested and the needed tables may be found in
Dixon and Massey [1983], Fleiss et al. [2003], and van Belle et al. [2004].

10.7 SAMPLE SIZE FOR TESTING TWO PROPORTIONS

The test of whether two population proportions are equal is one of the more com-
monly used tests in biomedical applications. Hence, in planning a study finding the
approximate sample size needed for this test is often included in proposals. To deter-
mine the sample size, we will need to decide what levels of a and @ to use so that our
chances of making an error are reasonably small. Often, a is chosen to be .05 and ,O
is chosen to be .20, but other values should be taken if these values do not reflect the
seriousness of making a type I or type I1 error. For a = .05: z[l - a/2] = 1.96 is
used for a two-sided test (see Section 8.5). For ,/3 = .20, z[1 - p] = .842. We will
also have to estimate a numerical value for 7r1 and for 7r2. Suppose that we plan to
compare performing an operation using endoscopy versus the conventional method.
The proportion of complications using the conventional method is known from past
experience to be approximately 7r1 = .12. We wish to know whether the use of
endoscopy changes the complication rate (either raises it or lowers it), so we plan on
performing a two-sided test. We certainly want to detect a difference if 7r2 = .20
since that would imply that the new method is worse than the conventional method.
Also, by taking a value of 7r2 that is closer to one-half, we will obtain a conservative
estimate of the sample size since values of 7r2 closer to one-half result in a larger
variance. The next step is to calculate

7r1 + 7r2 - .12 + .20

• r=- - = .16
  2 2