IfX 3 makes no contribution, then
L^ 2 L^ 1)
L^ 1
L^ 2 ^1
) LR2 ln(1)¼ 2 0 ¼ 0Thus,X 3 nonsignificant)LR 0
0 LR1
""
N:S: S:
Similar to chi square (w^2 )
LR approximatew^2 ifnlarge
How large? No precise answer.
In contrast, consider the value of the test sta-
tistic if the additional variable makes no con-
tribution whatsoever to the risk of disease over
and above that contributed byX 1 andX 2. This
would mean that the maximized likelihood
valueL^ 2 is essentially equal to the maximized
likelihood valueL^ 1.Correspondingly, the ratioL^ 1 divided byL^ 2 is
approximately equal to 1. Therefore, the likeli-
hood ratio statistic is approximately equal
to2 times the natural log of 1, which is 0,
because the log of 1 is 0. Thus, the likelihood
ratio statistic for a highly nonsignificantX 3
variable is approximately 0. This, again, is
what one would expect from a chi-square
statistic.In summary, the likelihood ratio statistic,
regardless of which two models are being com-
pared, yields a value that lies between 0, when
there is extreme nonsignificance, and þ1,
when there is extreme significance. This is the
way a chi-square statistic works.Statisticians have shown that the likelihood
ratio statistic can be considered approximately
chi square, provided that the number of sub-
jects in the study is large. How large is large,
however, has never been precisely documen-
ted, so the applied researcher has to have as
large a study as possible and/or hope that the
number of study subjects is large enough.As another example of a likelihood ratio test,
we consider a comparison of Model 2 with
Model 3. Because Model 3 is larger than
Model 2, we now refer to Model 3 as the full
model and to Model 2 as the reduced model.EXAMPLE
Model 2: logitP 2 (X)¼aþb 1 X 1 þb 2 X 2
(reduced model) þb 3 X 3
Model 3: logitP 3 (X)¼aþb 1 X 1 þb 2 X 2
þb 3 X 3 þb 4 X 1 X 3
þb 5 X 2 X 3(full model)Presentation: IV. The Likelihood Ratio Test 137