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

LR statistic (likeFstatistic) com-
pares two models:

Full model¼larger model

Reduced model¼smaller model

H 0 : parameters in full model equal
to zero

df ¼ number of parameters set
equal to zero

In general, the likelihood ratio statistic, like an

Fstatistic in classical multiple linear regres-

sion, requires the identification of two models

to be compared, one of which is a special case

of the other. The larger model is sometimes

called thefull modeland the smaller model is

sometimes called thereduced model; that is, the

reduced model is obtained by setting certain

parameters in the full model equal to zero.

The set of parameters in the full model that is

set equal to zero specify the null hypothesis

being tested. Correspondingly, the degrees of

freedom for the likelihood ratio test are equal

to the number of parameters in the larger

model that must be set equal to zero to obtain

the smaller model.

As an example of a likelihood ratio test, let us

now compare Model 1 with Model 2. Because

Model 2 is the larger model, we can refer to

Model 2 as the full model and to Model 1 as

the reduced model. The additional parameter

in the full model that is not part of the reduced

model isb 3 , the coefficient of the variableX 3.

Thus, the null hypothesis that compares Mod-

els 1 and 2 is stated asb 3 equal to 0. This is

similar to the null hypothesis for a partialFtest

in classical multiple linear regression analysis.

Now consider Model 2, and suppose that the

variableX 3 is a (0, 1) exposure variableEand

that the variablesX 1 andX 2 are confounders.

Then the odds ratio for the exposure–disease

relationship that adjusts for the confounders is

given by e tob 3.

Thus, in this case, testing the null hypothesis

thatb 3 equals 0 is equivalent to testing the null

hypothesis that the adjusted odds ratio for the

effect of exposure is equal to e to 0 or 1.

To test this null hypothesis, the corresponding

likelihood ratio statistic is given by the differ-

ence2lnL^ 1 minus2lnL^ 2.

EXAMPLE

Model 1 vs. Model 2

Model 2 (full model):

logitP 2 (X)¼aþb 1 X 1 þb 2 X 2 þb 3 X 3

Model 1 (reduced model):

logitP 1 (X)¼aþb 1 X 1 þb 2 X 2

H 0 : b 3 ¼0 (similar to partialF)

Model 2:

logitP 2 (X)¼aþb 1 X 1 þb 2 X 2 þb 3 X 3

SupposeX 3 ¼E(0, 1) andX 1 ,X 2 are

confounders.

Then OR¼eb^3

H 0 : b 3 ¼ 0 ,H 0 :OR¼e^0 ¼ 1

LR¼2lnL^ 1 (2lnL^ 2 )

Presentation: IV. The Likelihood Ratio Test 135