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

Alternative form (w/o summation
signs):

Logit P(X) = a^ +^ b 1 E 1 +^ b 2 E 2 + ... +^ bqEq

+ g 1 V 1 + g 2 V 2 + ... + gp 1 Vp 1

Es

Vs

+ d 11 E 1 W 1 + d 12 E 1 W 2 + ... + d 1 ,p 2 E 1 Wp 2

+ d 21 E 2 W 1 + d 22 E 2 W 2 + ... + d 2 ,p 2 E 2 Wp 2

+ dq 1 EqW 1 + dq 2 EqW 2 + ... + dq,p 2 EqWp 2

+ ... EWs

EEs

iπi’

+ d∗∗ 12 E 1 E 2 + d 13 E 1 E 3 + ... + d∗ 1 q E 1 Eq

+ d∗∗ 23 E 2 E 3 + d 24 E 2 E 4 + ... + d 2 ∗q E 2 Eq

+ ... + dq–∗ 1 ,q Eq– 1 Eq

EXAMPLE

MRSA example:

Logit P(X) = a + b 1 E 1 + b 2 E 2

+ d 11 E 1 W 1 + d 12 E 1 W 2 + d 21 E 2 W 1

+ d∗E 1 E 2

q= 2

+ g 1 V 1 + g 2 V 2 p 1 = 2

4 EWs

1 EE

+ d 22 E 2 W 2

Next step in modeling strategy?

Step 2: Assess interaction

Questions regardingEWs andEEs?

 Consider separately or
simultaneously?

 If separately,EWsorEEs first?

Answer: It depends!
Several reasonable options.

This same model is alternatively written with-

out summation signs here and is divided into

four groups of predictor variables:

The first group lists theEvariables.

The second group lists theVvariables.

The third group lists theEWvariables, the first

line of which contains products ofE 1 with each

of theWjs, the second line contains products of

E 2 with each of theWjs, and so on, with the last

line of the group containing products ofEq

with each of theWjs.

Finally, the fourth group lists theEEvariables,

with the first line containing products ofE 1

with all otherEs, the second line containing

products of E 2 with all other Es exceptE 1 ,

and so on, with the last line containing the

single product termEq 1 Eq.

Returning to our initial model for the MRSA

data, there areq¼ 2 Evariables,

p 1 ¼ 2 Vvariables,

p 2 ¼ 2 Wvariables, which yields 4EWvariables,

and a singleEEvariable.

So, how do we proceed once we have identified

our initial model? Following our previous

strategy for one Evariable, we recommend

assessing interaction as the next step. But,

since there are two types of product terms,

EWs andEEs, should we consider these types

separately or simultaneously, and if separately,

do we first considerEWsorEEs?

The answer, not surprisingly, is it depends!

That is, there are several reasonable options.

Presentation: II. Modeling Strategy for Several Exposure Variables 247