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

Confidence intervals:

exp ^lZ 1 a 2

ffiffiffiffiffiffiffiffiffiffiffiffiffi

dvarð^lÞ

q

where

^l¼^bþ~

p 2

j¼ 1

^djWj

Plugging the estimated coefficients into the

odds ratio formula yields the expression: e to

the quantity12.6894 plus 0.0692 times CHL

plus2.3318 times HPT.

To obtain a numerical value from this expres-

sion, it is necessary to specify a value for CHL

and a value for HPT. Different values for CHL

and HPT will, therefore, yield different odds

ratio values. This should be expected because

the model with interaction terms should give

different odds ratio estimates depending on the

values of the effect modifiers, which in this

case are CHL and HPT.

The table shown here illustrates different odds

ratio estimates that can result from specifying

different values of the effect modifiers. In this

table, the values of CHL used are 200, 220, and

240; the values of HPT are 0 and 1. The cells

within the table give the estimated odds ratios

computed from the above expression for the

odds ratio for different combinations of CHL

and HPT.

For example, when CHL equals 200 and HPT

equals 0, the estimated odds ratio is given by

3.16; when CHL equals 220 and HPT equals 1,

the estimated odds ratio is 1.22. Each of the

estimated odds ratios in this table describes the

association between CAT and CHD adjusted

for the five covariables AGE, CHL, ECG,

SMK, and HPT because each of the covariables

is contained in the model asVvariables.

To account for the variability associated with

each of the odds ratios presented in the above

tables, we can compute confidence intervals by

using the methods we have described. The gen-

eral confidence interval formula is given by e to

the quantityl^plus or minus a percentage point

of theZdistribution times the square root of

the estimated variance of^l, wherelis the linear

function shown here.

EXAMPLE (continued)

dOR¼exp½Šbþd 1 CHLþd 2 HPT

¼exp½ 12 : 6894 þ 0 : 0692 CHL

þðÞ 2 : 3318 HPTŠ

Must specify

CHL and HPT

""

Effect modifiers

AdjustedOR:d

HPT

01

200 3.16 0.31

CHL 220 12.61 1.22

240 50.33 4.89

CHL¼200, HPT¼ 0 )ORd¼3.16

CHL¼220, HPT¼ 1 )ORd¼1.22

dOR adjusts for AGE, CHL, ECG, SMK,

and HPT (Vvariables)

Presentation: VIII. Numerical Example 151