Introductory Biostatistics

Confidence intervals are derived from the normal approximation to the sam-
pling distribution of lnðORÞwith variance

Var½lnðORފF

1

a

þ

1

b

þ

1

c

þ

1

d

(ln islogarithm to base e, or thenatural logarithm.) Consequently, an approxi-
mate 95% confidence interval, on the log scale, for odds ratio is given by

ln

ad

bc

G 1 : 96

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

a

þ

1

b

þ

1

c

þ

1

d

r

A 95% confidence interval for the odds ratio under investigation is obtained by
exponentiating(the reverse log operation or antilog) the two endpoints:

ln

ad

bc

1 : 96

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

a

þ

1

b

þ

1

c

þ

1

d

r

and

ln

ad

bc

þ 1 : 96

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

a

þ

1

b

þ

1

c

þ

1

d

r

Example 4.11 The role of smoking in pancreatitis has been recognized for
many years; the data shown in Table 4.8 are from a case–control study carried
out in eastern Massachusetts and Rhode Island in 1975–1979 (see Example
1.14). We have

(a) For ex-smokers, compared to those who never smoked,

OR¼

ð 13 Þð 56 Þ

ð 80 Þð 2 Þ

¼ 4 : 55

and a 95% confidence interval for the population odds ratio on the log

scale is from

TABLE 4.8

Use of Cigarettes Cases Controls

Current smokers 38 81

Ex-smokers 13 80

Never 2 56

166 ESTIMATION OF PARAMETERS