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

E*(group 1) vs.E**(group 2)

RORE*vs:E**¼exp



ðE*E**Þb

þðE*E**Þ~

p 2

j¼ 1

djWj



Same as

RORE¼avs:E¼b¼exp



ðabÞb

þðabÞ~

p 2

j¼ 1

djWj



To obtain an odds ratio for such a generally

definedE, we need to specify two values ofE

to be compared. We denote the two values of

interest asE*andE**. We need to specify two

values because an odds ratio requires thecom-

parison of two groups– in this case two levels

of the exposure variableE– even when the

exposure variable can take on more than two

values, as whenEis ordinal or interval.

The odds ratio formula forE*vs.E**, equals

e to the quantity (E*E**) times b plus

(E*E**) times the sum of thedjtimesWj.

This is essentially the same formula as previ-

ously given for dichotomous E, except that

here, several different odds ratios can be com-

puted as the choice ofE*andE**ranges over

the possible values ofE.

We illustrate this formula with several exam-

ples. First, suppose E gives social support

status as denoted by SSU, which is an index

ranging from 0 to 5, where 0 denotes a person

without any social support and 5 denotes

a person with the maximum social support

possible.

To obtain an odds ratio involvingsocial support

status (SSU), in the context of ourE, V, W

model, we need to specify two values ofE.

One such pair of values is SSU*equals 5 and

SSU**equals 0, which compares the odds for

persons who have the highest amount of social

support with the odds for persons who have

the lowest amount of social support. For this

choice, the odds ratio expression becomes e to

the quantity (5 – 0) timesbplus (5 – 0) times

the sum of thedjtimesWj, which simplifies to

eto5bplus 5 times the sum of thedjtimesWj.

Similarly, if SSU*equals 3 and SSU**equals 1,

then the odds ratio becomes e to the quantity

(3 – 1) timesbplus (3 – 1) times the sum of

thedjtimesWj, which simplifies to e to 2bplus

2 times the sum of thedjtimesWj.

EXAMPLE

E¼SSU¼social support status (0–5)

(A) SSU*¼5 vs. SSU**¼ 0

ROR 5 ; 0 ¼exp SSU*SSU**

   

bþSSU*SSU**

   

SdjWj

¼expðÞ 5  0 bþðÞ 5  0 SdjWj

¼exp 5 bþ 5 SdjWj

   

(B) SSU*¼3 vs. SSU**¼ 1

ROR 3 ; 1 ¼expðÞ 3  1 bþðÞ 3  1 ~djWj

¼exp 2 bþ 2 ~djWj

   

80 3. Computing the Odds Ratio in Logistic Regression