L^similar toR^2
lnL^ 1 lnL^ 2 lnL^ 3
2lnL^ 3 2lnL^ 2 2lnL^ 1
2lnL^¼log likelihood statistic
used in likelihood ratio (LR) test
IV. The Likelihood Ratio
Test
2lnL 1 (2lnL 2 )¼LR
is approximate chi square
df¼difference in number of para-
meters (degrees of freedom)
Model 1: logitP 1 (X)¼aþb 1 X 1 þb 2 X 2
Model 2: logitP 2 (X)¼aþb 1 X 1
þb 2 X 2 þb 3 X 3
Note. special case¼subset
Model 1 special case of Model 2
Model 2 special case of Model 3
This relationship among theL^s is similar to the
property in classical multiple linear regression
analyses that the more parameters a model
has, the higher is theR-square statistic for the
model. In other words, the maximized likeli-
hood valueL^is similar toR-square, in that the
higher theL^, the better the fit.It follows from algebra that ifL^ 1 is less than or
equal toL^ 2 , which is less thanL^ 3 , then the same
inequality relationship holds for the natural
logarithms of theseL^s.However, if we multiply each log ofL^by2,
then the inequalities switch around so that
2lnL^ 3 is less than or equal to2lnL^ 2 ,
which is less than2lnL^ 1.The statistic2lnL^ 1 is called thelog likelihood
statisticfor Model 1, and similarly, the other
two statistics are the log likelihood statistics
for their respective models. These statistics
are important because they can be used to
test hypotheses about parameters in the
model using what is called alikelihood ratio
test, which we now describe.Statisticians have shown that the difference
between log likelihood statistics for two mod-
els, one of which is a special case of the other,
has an approximate chisquare distribution in
large samples. Such a test statistic is called a
likelihood ratioorLRstatistic. The degrees of
freedom (df) for this chi-square test are equal
to the difference between the number of para-
meters in the two models.Note that one model is considered a special
case of another if one model contains a subset
of the parameters in the other model. For
example, Model 1 above is a special case of
Model 2; also, Model 2 is a special case of
Model 3.134 5. Statistical Inferences Using Maximum Likelihood Techniques