Introductory Biostatistics

leading to a 95% confidence interval of

0 : 0026 Gð 1 : 96 Þð 0 : 0007 Þ¼ð 0 : 0012 ; 0 : 0040 Þ

(b) For non-OC users, the estimated rate was

p 2 ¼

7

10 ; 000

¼ 0 : 0007

with its standard error

SEðp 2 Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð 0 : 0007 Þð 1 
 0 : 0007 Þ

10 ; 000

s

¼ 0 : 0003

leading to a 95% confidence interval of

0 : 0007 Gð 1 : 96 Þð 0 : 0003 Þ¼ð 0 : 0002 ; 0 : 0012 Þ

It can be seen that the two confidence intervals, one for OC users and one
for non-OC users, do not overlap, a strong indication that the two population
MI rates are probably not the same.

In many trials for interventions, or in studies to determine possible e¤ects of
a risk factor, the comparison of proportions is based on data from two inde-
pendent samples. However, the process of constructing two confidence intervals
separately, one from each sample, as mentioned briefly at the end of the last
few examples, is not e‰cient. The reason is that theoverall confidencelevel may
no longer be, say, 95% as intended because the process involvestwoseparate
inferences; possible errors may add up. The estimation of thedi¤erence of pro-
portionsshould be formed using the following formula (for a 95% confidence
interval):

ðp 1 
p 2 ÞGð 1 : 96 ÞSEðp 1 
p 2 Þ

where

SEðp 1 
p 2 Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 1 ð 1 
p 1 Þ

n 1

þ

p 2 ð 1 
p 2 Þ

n 2

s

164 ESTIMATION OF PARAMETERS