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

Some computer packages compute

dvar^l



General CI formula for E, V, W
model:

dOR¼e^l;

where

l¼bþ~

p 2

j¼ 1

djWj

exp ^lZ 1 a 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dvar^l

q 



;

where

dvarl^



¼vard^b



þ~

p 2

j¼ 1

W^2 jdvar^dj



þ 2 ~

p 2

j¼ 1

Wjdcovb^;^dj



þ 2 ~

j

~

k

WjWkcovd ^dj;^dk



Obtaindvars andcovs from printoutd
butmust specifyWs.

For example, ifX 1 denotes AGE andX 2 denotes

smoking status (SMK), then one specification

of these variables isX 1 ¼30,X 2 ¼1, and a

second specification isX 1 ¼40,X 2 ¼0. Differ-

ent specifications of these variables will yield

different confidence intervals. This should be no

surprise because a model containing interaction

terms implies that both the estimated odds ratios

and their corresponding confidence intervals

vary as the values of the effect modifiers vary.

A recommended practice is to use “typical” or

“representative” values ofX 1 andX 2 , such as

their mean values in the data, or the means of

subgroups, for example, quintiles, of the data

for each variable.

Some computer packages for logistic regres-

sion do compute the estimated variance of lin-

ear functions like ^las part of the program

options. See the Computer Appendix for details

on the use of the “contrast” option in SAS and

the “lincom” option in STATA.

For the interested reader, we provide here the

general formula for the estimated variance of

the linear function obtained from theE, V, W

model. Recall that the estimated odds ratio for

this model can be written as e to^l, wherelis the

linear function given by the sum ofbplus the

sum of terms of the formdjtimesWj.

The corresponding confidence interval for-

mula is obtained by exponentiating the confi-

dence interval for^l, where the variance ofl^is

given by the general formula shown here.

In applying this formula, the user obtains the

estimated variances and covariances from the

variance–covariance output. However, as in

the example above, the user must specify

values of interest for the effect modifiers

defined by theWs in the model.

EXAMPLE (continued)

e.g.,X 1 ¼AGE,X 2 ¼SMK:

Specification 1:X 1 ¼30,X 2 ¼ 1

versus

Specification 2:X 1 ¼40,X 2 ¼ 0

Different specifications yield different

confidence intervals

Recommendation. Use “typical” or

“representative” values ofX 1 andX 2

e.g.,X 1 andX 2 in quintiles

Presentation: VII. Interval Estimation: Interaction 145