CI for coefficient
vs.
3 CI for odds ratio
As an example, if we focus on theb 3 parameter
in Model 2, the 100 times (1a)% confidence
interval formula is given by^b 3 plus or minus
the corresponding (1a/2)th percentage point
ofZtimes the estimated standard error of^b 3.In this formula, the values forb^ 3 and its stan-
dard error are found from the printout. TheZ
percentage point is obtained from tables of the
standard normal distribution. For example, if
we want a 95% confidence interval, thenais
0.05, 1a/2 is 10.025 or 0.975, andZ0.975is
equal to 1.96.Most epidemiologists are not interested in get-
ting a confidence interval for the coefficient of
a variable in a logistic model, but rather want a
confidence interval for an odds ratioinvolving
that parameter and possibly other parameters.When onlyone exposure variable, is being con-
sidered, such asX 3 in Model 2, and this vari-
able is a (0, 1) variable, then the odds ratio of
interest, which adjusts for the other variables
in the model, is e to that parameter, for exam-
ple e tob 3. In this case, the correspondingcon-
fidence interval for the odds ratio is obtained by
exponentiating the confidence limits obtained
for the parameter.Thus, if we consider Model 2, and ifX 3 denotes
a (0, 1) exposure variable of interest andX 1 and
X 2 are confounders, then a 95% confidence
interval for the adjusted odds ratio e tob 3 is
given by the exponential of the confidence
interval forb 3 , as shown here.This formula is correct, provided that the vari-
ableX 3 is a (0, 1) variable. If this variable is
coded differently, such as (1, 1), or if this
variable is an ordinal or interval variable, then
the confidence interval formula given here
must be modified to reflect the coding.EXAMPLE
logitP 2 (X)¼aþb 1 X 1 þb 2 X 2 þb 3 X 3
X 3 ¼(0, 1) variable
)OR¼eb^3
CI for OR: exp(CI forb 3 )Model 2:X 3 ¼(0, 1) exposure
X 1 andX 2 confounders
95% CI for OR:
exp ^b 3 1 : 96 sb^ 3Above formula assumesX 3 is coded as
(0, 1)EXAMPLE
Model 2: logitP 2 (X)¼aþb 1 X 1
þb 2 X 2 þb 3 X 3
100(1a)% CI forb 3 :^b 3 Z 1 a
2
s^b 3
b^ 3 ands^b
3 : from printout
ZfromN(0, 1) tables,
e:g:; 95 %)a¼ 0 : 05
) 1
a
2
¼ 1 0 : 025
¼ 0 : 975
Z 0 : 975 ¼ 1 : 96Presentation: VI. Interval Estimation: One Coefficient 141