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

2lnL^ 2 ,2lnL^ 3
""
Computer prints these
separately

V. The Wald Test

Focus on 1 parameter

e.g.,H 0 : b 3 ¼ 0

There are two additional parameters in the full

model that are not part of the reduced model;

these areb 4 andb 5 , the coefficients of the prod-

uct variablesX 1 X 3 andX 2 X 3 , respectively. Thus,

the null hypothesis that compares Models

2 and 3 is stated asb 4 equalsb 5 equals 0. This

is similar to the null hypothesis for a multiple-

partial F test in classical multiple linear

regression analysis. The alternative hypothesis

here is thatb 4 and/orb 5 are not 0.

If the variableX 3 is the exposure variableEin

one’s study and the variablesX 1 andX 2 are

confounders, then the product termsX 1 X 3 and

X 2 X 3 are interaction terms for the interaction

ofEwithX 1 andX 2 , respectively. Thus, the null

hypothesis thatb 4 equalsb 5 equals 0, is equiva-

lent to testing no joint interaction ofX 1 andX 2

withE.

The likelihood ratio statistic for comparing

Models 2 and 3 is then given by2lnL^ 2 minus

2lnL^ 3 , which also can be written as2 times

the natural log of the ratio ofL^ 2 divided byL^ 3.

This statistic has an approximate chi-square

distribution in large samples. The degrees of

freedom here equals 2 because there are two

parameters being set equal to 0 under the null

hypothesis.

When using a standard computer package to

carry out this test, we must get the computer to

fit the full and reduced models separately. The

computer output for each model will include the

log likelihood statistics of the form2lnL^.The

user then simply finds the two log likelihood

statistics from the output for each model being

compared and subtracts one from the other to

get the likelihood ratio statistic of interest.

There is another way to carry out hypothesis

testing in logistic regression without using a

likelihood ratio test. This second method is

sometimes calledthe Wald test. This test is usu-

ally done when there is only one parameter

being tested, as, for example, when comparing

Models 1 and 2 above.

EXAMPLE (continued)

H 0 : b 4 ¼b 5 ¼ 0

(similar to multiple–partialFtest)

HA: b 4 and/orb 5 are not zero

X 3 ¼E

X 1 ,X 2 confounders

X 1 X 3 ,X 2 X 3 interaction terms

H 0 : b 4 ¼b 5 ¼ 0 ,H 0 :no

interaction withE

LR¼ 2 lnL^ 2  2 lnL^ 3



¼ 2 ln

L^ 2

L^ 3



which is approximatelyw^2 with 2 df

under

H 0 : b 4 ¼b 5 ¼ 0

138 5. Statistical Inferences Using Maximum Likelihood Techniques