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

To illustrate this point, we return to the first

example considered above, where the model is

given by logit P(X) equalsaplusbEplusg 1 V 1

plusg 2 V 2 plus the product termsd 1 EV 1 plus

d 2 EV 2 plusd 3 EV 1 V 2. This model is not hier-

archically well formulated because it is missing

the termV 1 V 2. The highest-order term in this

model is the three-factor product termEV 1 V 2.

Suppose that the exposure variable E in

this model is a dichotomous variable. Then,

because the model is not HWF, a test of

hypothesis for the significance of the highest-

order term,EV 1 V 2 , may give different results

depending on whetherEis coded as (0, 1) or

(1, 1) or any other coding scheme.

In particular, it is possible that a test forEV 1 V 2

may be highly significant ifEis coded as (0, 1),

but be nonsignificant ifEis coded as (1, 1).

Such a possibility should be avoided because

the coding of a variable is simply a way to

indicate categories of the variable and, there-

fore, should not have an effect on the results of

data analysis.

In contrast, suppose we consider the HWF

model obtained by adding theV 1 V 2 term to

the previous model. For this model, a test for

EV 1 V 2 will give exactly the same result whether

Eis coded using (0, 1), (1, 1), or any other

coding. In other words, such a test is indepen-

dent of the coding used.

We will shortly see that even if the model is

hierarchically well formulated, then tests about

lower order terms in the model may still depend

on the coding.

EXAMPLE

logit P



X



¼aþbEþg 1 V 1 þg 2 V 2

þd 1 EV 1 þd 2 EV 2 þd 3 EV 1 V 2

Not HWF model:

V 1 V 2 missing

EXAMPLE (continued)

Edichotomous:

Then ifnotHWF model,

testing forEV 1 V 2 may depend on

whetherEis coded as

E¼(0, 1), e.g., significant

or

E¼(1, 1), e.g., not significant

or

other coding

EXAMPLE

HWF model:

logit P



X



¼aþbEþg 1 V 1 þg 2 V 2 þg 3 V 1 V 2

þd 1 EV 1 þd 2 EV 2 þd 3 EV 1 V 2

Testing forEV 1 V 2 isindependent of

codingofE: (0, 1), (1, 1), or other.

HWF model. Tests forlowerorder
terms depend on coding

Presentation: VII. Hierarchically Well-Formulated Models 183