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

No interaction model

RORE*vs:E**

¼exp½ðE* 1 E** 1 Þb 1 þðE* 2 E** 2 Þb 2

þ...þðE*k 1 E**k 1 Þbk 1 Š

Thegeneral odds ratio formulafor comparing

two categories,E*vs.E**of a general nominal

exposure variable in ano interaction logistic

model, is given by the formula ROR equals

e to the quantity (E 1 *E 1 **) times b 1 plus

(E 2 *E 2 **) timesb 2 , and so on up to (Ek* 1 

Ek** 1 ) timesbk 1. When applied to a specific

situation, this formula will usually involve

more than onebiin the exponent.

For example, when comparing occupational

status category 3 with category 1, the odds

ratio formula is computed as e to the quantity

(OCC 1 *OCC 1 **) times b 1 plus (OCC 2 *

OCC 2 **) timesb 2 plus (OCC 3 *OCC 3 **) timesb 3.

When we plug in the values for OCC* and

OCC**, this expression equals e to the quantity

(01) times b 1 plus (00) times b 2 plus

(10) timesb 3 , which equals e to1timesb 1

plus 0 timesb 2 plus 1 timesb 3 , which reduces to

e to the quantity (b 1 )plusb 3.

We can obtain a single value for the estimate of

this odds ratio by fitting the model and repla-

cingb 1 andb 3 with their corresponding esti-

matesb^ 1 and^b 3. Thus,ROR for this example isd

given by e to the quantity (^b 1 ) plusb^ 3.

In contrast, if category 3 is compared to

category 2, thenE*takes on the values 0, 0,

and 1 as before, whereasE**is now defined

by OCC 1 **¼0, OCC 2 **¼1, and OCC 3 **¼0.

The odds ratio is then computed as e to the

(0 0) timesb 1 plus (01) timesb 2 plus

(1 – 0) timesb 3 , which equals e to the 0 times

b 1 plus1 timesb 2 plus 1 times b 3 , which

reduces to e to the quantity (b 2 ) plusb 3.

This odds ratio expression involves b 2 and

b 3 , whereas the previous odds ratio expression

that compared category 3 with category 1

involvedb 1 andb 3.

EXAMPLE (OCC)

ROR 3 vs: 1 ¼exp

h

ðOCC

0

1 *OCC

1

1 **Þb 1

þ



OCC

0

2 *OCC

0

2 **



b 2

þ



OCC

1

3 *OCC

0

3 **



b 3

i

¼exp½ð 0  1 Þb 1 þð 0  0 Þb 2 þð 1  0 Þb 3 Š

¼exp½ð 1 Þb 1 þð 0 Þb 2 þð 1 Þb 3 Š

¼expðb 1 þb 3 Þ

RORd ¼exp^b 1 þ^b 3



E*¼category 3 vs. E**¼category 2:

E*¼(OCC 1 *¼0, OCC 2 *¼0, OCC 3 *¼1)

E**¼(OCC** 1 ¼0,OCC 2 **¼1,OCC** 3 ¼0)

ROR 3 vs: 2 ¼exp½ð 0  0 Þb 1 þð 0  1 Þb 2

þð 1  0 Þb 3 Š

¼exp½ð 0 Þb 1 þð 1 Þb 2 þð 1 Þb 3 Š

¼expðb 2 þb 3 Þ

Note. ROR3 vs. 1¼exp (b 1 +b 3 )

84 3. Computing the Odds Ratio in Logistic Regression