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

No interaction:

RORE*vs.E**¼exp [(E*E**)b]

If (E*E**)¼1, then ROR

¼exp(b)

e.g.,E*¼1 vs.E**¼ 0

orE*¼2 vs.E**¼ 1

Note that if SSU*equals 4 and SSU**equals 2,

then the odds ratio expression becomes 2bplus

2 times the sum of thedjtimesWj, which is the

same expression as obtained when SSU*equals 3

and SSU**equals 1. This occurs because the

odds ratio depends on the difference between

E*andE**, which in this case is 2, regardless of

the specific values ofE*andE**.

As another illustration, supposeEis the inter-

val variable systolic blood pressure denoted

by SBP. Again, to obtain an odds ratio, we

must specify two values of E to compare.

For instance, if SBP*equals 160 and SBP**

equals 120, then the odds ratio expression

becomes ROR equals e to the quantity

(160120) timesb plus (160120) times

the sum of thedjtimesWj, which simplifies

to 40 timesbplus 40 times the sum of thedj

timesWj.

Or if SBP*equals 200 and SBP**equals 120,

then the odds ratio expression becomes ROR

equals e to the 80 timesbplus 80 times the sum

of thegjtimesWj.

Note that in the no interaction case, the odds

ratio formula for a general exposure variableE

reduces to e to the quantity (E*E**) timesb.

This is not equal to e to thebunless the differ-

ence (E*E**) equals 1, as, for example, ifE*

equals 1 andE**equals 0, orE*equals 2 andE**

equals 1.

Thus, ifEdenotes SBP, then the quantity e tob

gives the odds ratio for comparing any two

groups that differ by one unit of SBP. A one

unit difference in SBP is not typically of inter-

est, however. Rather, a typical choice of SBP

values to be compared represent clinically

meaningful categories of blood pressure, as

previously illustrated, for example, by SBP*

equals 160 and SBP**equals 120.

One possible strategy for choosing values of

SBP*and SBP** is to categorize the distri-

bution of SBP values in our data into clinically

meaningful categories, say, quintiles. Then,

using the mean or median SBP in each quin-

tile, we can compute odds ratios comparing all

possible pairs of mean or median SBP values.

EXAMPLE

E¼SBP¼systolic blood pressure

(interval)

(A) SBP*¼160 vs. SBP**¼ 120

ROR 160 ; 120 ¼exp

h

ðSBP*SBP**Þb

þðSBP*SBP**Þ~djWj

i

¼exp

h

ð 160  120 Þb

þð 160  120 Þ~djWj

i

¼exp



40 bþ 40 ~djWj

(B) SBP*¼200 vs. SBP**= 120

ROR 200 ; 120 ¼exp

h

ð 200  120 Þbþð 200  120 Þ~djWj

i

¼exp



80 bþ 80 ~djWj

EXAMPLE (continued)

(C) SSU*¼4 vs. SSU**¼ 2

ROR 4 ; 2 ¼expðÞ 4  2 bþðÞ 4  2 ~djWj

¼exp 2 bþ 2 ~djWj

   

Note. ROR depends on the difference

(E*E**), e.g., (31) = (42) = 2

EXAMPLE

E¼SBP

ROR¼exp(b))(SBP*SBP**)¼ 1

not interesting"

Choice of SBP:

Clinically meaningful categories,

e.g., SBP*¼160, SBP*¼ 120

Strategy: Use quintiles of SBP

Quintile # 12345

Mean or

median

120 140 160 180 200