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

As described in later sections, theEViVjand

EVi product terms can be eliminated using

appropriate statistical testing methods.

However, decisions about theViandViVjterms,

which are potential confounders, should not

involve statistical testing.

Thestrategydescribed by this flow diagram is

called hierarchical backwardbecause we are

working backward from our largest starting

model to a smaller final and we are treating

variables of different orders at different steps.

That is, there is a hierarchy of variable types,

with three-factor interaction terms considered

first, followed by two-factor interaction terms,

followed by two-factor, and then one-factor

confounding terms.

As we go through the hierarchical backward

elimination process, some terms are retained

and some terms are dropped at each stage. For

those terms that are retained at a given stage,

there is a rule for identifying lower order com-

ponents that must also be retained in any fur-

ther models.

This rule is called thehierarchy principle.An

analogous principle of the same name has

been described by Bishop, Fienberg, and Hol-

land (1975).

To illustrate the hierarchy principle, suppose

the initial model contains three-factor products

of the formEViVj. Suppose, further, that the

termEV 2 V 5 is found to be significant during

the stage that considers the elimination of

unimportantEViVjterms. Then, the hierarchy

principle requires that all lower order compo-

nents of theEV 2 V 5 term must be retained in all

further models considered in the analysis.

EViandEViVj(interactions):

use statistical testing

ViandViVj(confounders): donot

use statistical testing

Hierarchical

3 factors: EViVj

2 factors: EVi

2 factors: ViVj

1 factors: Vi

Backward

Large starting

model

Smaller final

model

IX. The Hierarchy
Principle for Retaining
Variables

Hierarchical Backward Elimination

Retain terms Drop terms

Hierarchy principle

(Bishop, Fienberg, and Holland,
1975)

EXAMPLE

Initial model:EViVjterms

Suppose:EV 2 V 5 significant

Hiearchy principle. all lower order

components ofEV 2 V 5 retained

i.e.,E,V 2 ,V 5 ,EV 2 ,EV 5 , andV 2 V 5

cannot be eliminated

Note.Initial model must containV 2 V 5

to be HWF

Presentation: IX. The Hierarchy Principle for Retaining Variables 185