Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

General CI formula:

exp ^lZ 1 a 2

ffiffiffiffiffiffiffiffiffiffiffiffiffi

dvarð^lÞ

q

Example:l¼b 3 þb 4 X 1 þb 5 X 2

General expression forl:

RORX 1 ;X 0 ¼e~

k

l¼ 1

biðÞX 1 iX 0 i

OR¼elwhere

l¼~

k

i¼ 1

biðÞX 1 iX 0 i

We can alternatively write this estimated odds

ratio formula as e to the^l, wherelis the linear

functionb 3 plusb 4 timesX 1 plusb 5 timesX 2 ,

and ^lis the estimate of this linear function

using the ML estimates.

To obtain a 100 times (1a)% confidence

interval for the odds ratio e tol, we must use

the linear functionlthe same way that we used

the single parameter b 3 to get a confidence

interval forb 3. The corresponding confidence

interval is thus given by exponentiating the

confidence interval forl.

The formula is therefore the exponential of the

quantity^lplus or minus a percentage point of

theZdistribution times the square root of the

estimated variance ofl^. Note that the square

root of the estimated variance is the standard

error.

This confidence interval formula, though moti-

vated by our example using Model 3, is actually

the general formula for the confidence interval

for any odds ratio of interest from a logistic

model. In our example, the linear functionl

took a specific form, but, in general, the linear

function may take any form of interest.

A general expression for this linear function

makes use of the general odds ratio formula

described in our review. That is, the odds

ratio comparing two groups identified by the

vectorsX 1 andX 0 is given by the formula e to

the sum of terms of the formbitimes the dif-

ference betweenX 1 iandX 0 i, where the latter

denotes the values of theith variable in each

group. We can equivalently write this as e to

thel, wherelis the linear function given by the

sum of thebitimes the difference betweenX 1 i

and X 0 i. This latter formula is the general

expression forl.

EXAMPLE

i.e.,dOR¼e^l,

where

l¼b 3 þb 4 X 1 þb 5 X 2

100 (1a)% CI for el

similar to CI formula for eb^3

exp^lZ 1 a 2

ffiffiffiffiffiffiffiffiffiffiffiffiffi

dvarð^lÞ

q

similar to exp^b 3 Z 1 a 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dvarð^b 3 Þ

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dvarðÞ

q

¼standard error

Presentation: VII. Interval Estimation: Interaction 143