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

One way to test for this interaction is to carry

out a chunk test for the significance of both

product terms considered collectively. This

involves testing the null hypothesis that the

coefficients of these variables, namelyd 1 and

d 2 , are both equal to 0.

The test statistic for this chunk test is given by

the likelihood ratio (LR) statistic computed as

the difference between log likelihood statistics

for the full model containing both interaction

terms and a reduced model which excludes

both interaction terms. The log likelihood sta-

tistics are of the form2lnL^, whereL^is the

maximized likelihood for a given model.

This likelihood ratio statistic has a chi-square

distribution with two degrees of freedom. The

degrees of freedom are the number of para-

meters tested, namely 2.

When carrying out this test, the log likelihood

statistics for the full and reduced models turn

out to be 60.23 and 60.63, respectively.

The difference between these statistics is 0.40.

Using chi-square tables with two degrees of

freedom, the P-value is considerably larger

than 0.10, so we can conclude that there are

no significant interaction effects. We can,

therefore, drop the two interaction terms

from the model.

Note that an alternative approach to testing for

interaction is to use backward elimination on

the interaction terms in the initial model.

Using this latter approach, it turns out that

both interaction terms are eliminated. This

strengthens the conclusion of no interaction.

At this point, our model can be simplified to

the one shown here, which contains only main

effect terms. This model contains the exposure

variable SMK, 38Vvariables that incorporate

the 39 matching strata, and 2Vvariables that

consider the potential confounding effects of

SBP and ECG, respectively.

EXAMPLE (continued)

Chunk test:

H 0 :d 1 ¼d 2 ¼ 0 ;

where

d 1 ¼coefficient of SMKSBP

d 2 ¼coefficient of SMKECG

LR¼ 2 lnL^R



 2 lnL^F



R¼reduced model F¼full model

ðno interactionÞðinteractionÞ

Log likelihood statistics

 2 lnL^

LRw^22

Number of parameters tested¼ 2

 2 lnL^F¼ 60 : 23

 2 lnL^R¼ 60 : 63

LR¼60.6360.23¼0.40

P>0.10 (no significant interaction)

Therefore, drop SMKSBP and

SMKECG from model

Backward elimination: same

conclusion

logit PðXÞ¼aþbSMKþ~g 1 iV 1 i

þg 21 SBPþg 22 ECG

402 11. Analysis of Matched Data Using Logistic Regression