Example 6.1 A group of investigators wish to explore the relationship be-
tween the use of hair dyes and the development of breast cancer in women. A
sample ofn¼1000 female beauticians 40–49 years of age is identified and fol-
lowed for five years. After five years,x¼20 new cases of breast cancer have
occurred. It is known that breast cancer incidence over this time period for
average American women in this age group isp 0 ¼ 7 =1000. We wish to test the
hypothesis that using hair dyesincreasesthe risk of breast cancer (a one-sided
alternative). We have:
- A one-sided test with
HA:p>7
1000
- Using the conventional choice ofa¼ 0 :05 leads to the rejection region
z> 1 :65. - From the data,
p¼20
1000
¼ 0 : 02
leading to a ‘‘zscore’’ of:z¼0 : 02 0 : 007
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð 0 : 007 Þð 0 : 993 Þ= 1000p¼ 4 : 93
(i.e., the observed proportionpis 4.93 standard errors away from the
hypothesized value ofp 0 ¼ 0 :007).- Since the computedzscore falls into the rejection regionð 4 : 93 > 1 : 65 Þ,
the null hypothesis is rejected at the 0.05 level chosen. In fact, the di¤er-
ence is very highly significant (p< 0 :001).
6.2 ANALYSIS OF PAIR-MATCHED DATA
The method presented in this section applies to cases where each subject or
member of a group is observed twice for the presence or absence of a certain
characteristic (e.g., at admission to and discharge from a hospital), or matched
pairs are observed for the presence or absence of the same characteristic. A
popular application is an epidemiological design called apair-matched case–
control study. In case–control studies, cases of a specific disease are ascertained
as they arise from population-based registers or lists of hospital admissions,
210 COMPARISON OF POPULATION PROPORTIONS