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

The likelihood ratio test for the significance of

EV 3 compares a “full” model containingE, the

fourVs,EV 1 ,EV 2 ,EV 3 , andEV 4 with a reduced

model that eliminatesEV 3 from the full model.

The LR statistic is given by the difference in the

log likelihood statistics for the full and reduced

models. This statistic has a chi-square distribu-

tion with one degree of freedom under the null

hypothesis that the coefficient of theEV 3 term

is 0 in our full model at this stage.

Suppose that when we carry out the LR test for

this example, we find that theEV 3 term is not

significant. Thus, at the end of the interaction

assessment stage, we are left with a model that

containsE, the fourVs,EV 1 ,EV 2 , andEV 4 .We

are now ready to assess confounding for this

example.

Our initial model contained four potential con-

founders, namely,V 1 throughV 4 , whereV 4 is

the product termV 1 timesV 2. Because of the

hierarchy principle, some of these terms are

not eligible to be dropped from the model,

namely, the lower order components of higher

order product terms remaining in the model.

In particular, becauseEV 1 V 2 has been found

significant, we must retain in all further mod-

els the lower order componentsV 1 ,V 2 , and

V 1 V 2 , which equalsV 4. This leavesV 3 as the

only remaining potential confounder that is

eligible to be dropped from the model as a

possible nonconfounder.

To evaluate whetherV 3 can be dropped from

the model as a nonconfounder, we consider

whether the odds ratio for the model that con-

trols for all four potential confounders, includ-

ing V 3 , plus previously retained interaction

terms, is meaningfully different from the odds

ratio that controls for previously retained vari-

ables but excludesV 3.

EXAMPLE (continued)

LR test forEV 3 : Comparefull model

containing

E;V|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 1 ;V 2 ;V 3 ;V 4

Vs

;EV|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 1 ;EV 2 ;EV 3 ;EV 4

EVs

withreduced modelcontaining

E;V 1 ;V 2 ;V 3 ;V 4 ;EV|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 1 ;EV 2 ;EV 4

withoutEV 3

LR¼ð 2 lnL^reducedÞð 2 lnL^fullÞ

isw^21 dfunderH 0 :dEV 3 ¼ 0 in full model

SupposeEV 3 notsignificant

+

model after interaction assessment:

E, V 1 , V 2 , V 3 , V 4 , EV 1 , EV 2 , EV 4

where V 4 = V 1 V 2 potential

confounders

Hierarchy principle:

identify Vs not eligible to be

dropped – lower order components

EV 1 V 2 significant

+Hierarchy principle

RetainV 1 ,V 2 , andV 4 ¼V 1 V 2

OnlyV 3 eligible to be dropped

dORV 1 ;V 2 ;V 3 ;V 46 ¼?dORV 1 ;V 2 ;V 4

"

excludesV 3

218 7. Modeling Strategy for Assessing Interaction and Confounding