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

Likelihood ratio test

Assess significance ofX 1

2 bs tested at the same time

+

2 degrees of freedom

The 95% confidence interval for OR 2 is calcu-

lated as 0.82 to 2.87, as shown on the left. The

95% confidence interval for OR 1 is calculated

as 1.04 to 4.58.

As with a standard logistic regression, we can

use a likelihood ratio test to assess the signifi-

cance of the independent variable in our

model. We must keep in mind, however, that

rather than testing one beta coefficient for an

independent variable, we are now testing two

at the same time. There is a coefficient for each

comparison being made (i.e.,D¼2 vs.D¼ 0

andD¼1 vs.D¼0). This affects the number

of parameters tested and, therefore, the

degrees of freedom associated with the test.

In our example, we have a three-level outcome

variable and a single predictor variable, the

exposure. As the model indicates, we have two

intercepts and two beta coefficients.

If we are interested in testing for the signifi-

cance of the beta coefficient corresponding to

the exposure, we begin by fitting a full model

(with the exposure variable in it) and then com-

paring that to a reduced model containing only

the intercepts.

The null hypothesis is that the beta coefficients

corresponding to the exposure variable are

both equal to zero.

The likelihood ratio test is calculated as nega-

tive two times the log likelihood (lnL) from the

reduced model minus negative two times the

log likelihood from the full model. The result-

ing statistic is distributed approximately chi-

square, with degrees of freedom (df) equal to

the number of parameters set equal to zero

under the null hypothesis.

EXAMPLE

3 levels ofDand 1 predictor

+

2 as and 2 bs

Full model:

lnPðD¼gjX^1 Þ

PðD¼ 0 jX 1 Þ



¼agþbg 1 X 1 ;

g¼ 1 ; 2

Reduced model:

ln

PðD¼gÞ

PðD¼ 0 Þ



¼ag; g¼ 1 ; 2

H 0 :b 11 ¼b 21 ¼ 0

Likelihood ratio test statistic:

 2 lnLreducedð 2 lnLfullÞw^2

with df¼number of parameters set

to zero underH 0

EXAMPLE (continued)

95% CI for OR 2

¼exp½ 0 : 4256  1 : 96 ð 0 : 3215 ފ

¼ð 0 : 82 ; 2 : 87 Þ

95% CI for OR 1

¼exp½ 0 : 7809  1 : 96 ð 0 : 3775 ފ

¼ð 1 : 04 ; 4 : 58 Þ

442 12. Polytomous Logistic Regression