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