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

In contrast, the model given by logit P(X)

equalsaplusbEplusg 1 V 1 plusg 2 V 2 plus the

product termsd 1 EV 1 plusd 2 EV 2 is hierarchi-

cally well formulated because the lower-order

components of each variable in the model are

also in the model. For example, the compo-

nents ofEV 1 areEandV 1 , both of which are

contained in the model.

For illustrative purposes, let us consider one

other model given by logit P(X) equalsaplus

bE plus g 1 V^21 plus g 2 V 2 plus the product

termd 1 EV^21 .Isthismodelhierarchicallywell

formulated?

The answer here can be either yes or no

depending on how the investigator wishes

to treat the variableV 12 in the model. IfV 12 is

biologically meaningful in its own right with-

out considering its componentV 1 , then the

corresponding model is hierarchically well

formulated because the variableEV 12 can be

viewed as having only two components,

namely,EandV 12 , both of which are contained

in the model. Also, if the variableV^21 is consid-

ered meaningful by itself, it can be viewed as

having no lower order components. Conse-

quently, all lower order components of each

variable are contained in the model.

On the other hand, if the variableV 12 is not

considered meaningful separately from its fun-

damental componentV 1 , then the model is not

hierarchically well formulated. This is because,

as given, the model does not containV 1 , which

is a lower order component ofV 12 andEV^21 , and

also does not contain the variableEV 1 , which is

a lower order component ofEV 12.

Now that we have defined and illustrated an

HWF model, we discuss why such a model

structure is required. The reason is that if the

model is not HWF, then tests about variables in

the model – in particular, the highest-order

terms – may give varying results depending on

the coding of variables in the model. Such tests

should beindependent of the codingof the vari-

ables in the model, and they are if the model is

hierarchically well formulated.

EXAMPLE

HWF model:

logit P



X



¼aþbEþg 1 V 1 þg 2 V 2

þd 1 EV 1 þd 2 EV 2

Components ofEV 1 :

E,V 1 both in model

EXAMPLE

logit P



X



¼aþbEþg 1 V^21

þg 2 V 2 þd 1 EV^21

HWF model?

Yes,ifV^21 is biologically meaningful

components ofEV^21 EandV^21

components ofV^21 : none

No,ifV^21 is not meaningful separately

fromV 1 :

The model does not contain

 V 1 , component ofV 12

 EV 1 , component ofEV 12

Why require HWF model?

Answer:

HWF?

Tests for highest-order

variables?

No dependent on coding
Yes independent of coding

182 6. Modeling Strategy Guidelines