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

Note that the no interaction model we have

been focusing on may, in fact, be inappropriate

when we compare it to other models of interest.

In particular, we now compare the no interac-

tion model to the model described by the sec-

ond set of printout results we have provided.

We will see that this second model, B, which

involves interaction terms, is a better model.

Consequently, the results and interpretations

made about the effect of the CAT variable from

the no interaction Model A may be misleading.

To compare the no interaction model with the

interaction model, we need to carry out alike-

lihood ratio test for the significance of the inter-

action terms. The null hypothesis here is that

the coefficientsd 1 andd 2 of the two interaction

terms are both equal to 0.

For this test, the full model is the interaction

Model B and the reduced model is the no inter-

action Model A. The likelihood ratio test statis-

tic is then computed by taking the difference

between log likelihood statistics for the two

models.

From the printout information given on pages

146–147, this difference is given by 400.39

minus 347.23, which equals 53.16. The degrees

of freedom for this test is 2 because there are

two parameters being set equal to 0. The chi-

square statistic of 53.16 is found to be signifi-

cant at the: 01 level. Thus, the likelihood ratio

test indicates that the interaction model is bet-

ter than the no interaction model.

We now consider what the odds ratio is for the

interaction model. As this model contains

product terms CC and CH, where CC is CAT

CHL and CH is CATHPT, the estimated odds

ratio for the effect of CAT must consider the

coefficients of these terms as well as the coeffi-

cient of CAT. The formula for this estimated

odds ratio is given by the exponential of the

quantity^b plusd 1 times CHL plus^d 2 times

HPT, where^b(12.6894) is the coefficient of

CAT,^d 1 (0.0692) is the coefficient of the inter-

action term CC, and^d 2 (2.3318) is the coeffi-

cient of the interaction term CH.

EXAMPLE (continued)

No interaction model

vs.

other models?

Model B vs. Model A

LR test for interaction:

H 0 : d 1 ¼d 2 ¼ 0

whereds are coefficients of interaction

terms CC and CH in model B

Full Model Reduced Model

Model B Model A

(interaction) (no interaction)

LR¼2lnL^model A(2lnL^model B)

¼400.39347.23

¼53.16

df¼ 2

significant at .01 level

dOR for interaction model (B):

dOR¼exp^bþ^d 1 CHLþ^d 2 HPT



^b¼12.6894 for CAT

^d 1 ¼0.0692 for CC

^d 2 ¼2.3318 for CH

150 5. Statistical Inferences Using Maximum Likelihood Techniques