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

Illustration

01

a 1 = log odds D ³ 1

a 2 = log odds D ³ 2

a 4 = log odds D ³ 4

a 3 = log odds D ³ 3

234

01234

01234

01234

As the picture on the left illustrates, with

five categories (D¼0, 1, 2, 3, 4), the log odds

ofD1 is greater than the log odds ofD2,

since forD1, the outcome can be in cate-

gories 1, 2, 3, or 4, whereas forD2, the

outcome can only be in categories 2, 3, or 4.

Thus, there is one more outcome category (cat-

egory 1) contained in the first inequality. Simi-

larly, the log odds ofD2 is greater than the

log odds ofD3, and the log odds ofD3is

greater than the log odds ofD4.

Returning to our example, the 95% confidence

interval for the OR for AGE is calculated as

shown on the left.

Hypothesis testing about parameter estimates

can be done using either the likelihood ratio

test or the Wald test. The null hypothesis is that

b 1 is equal to 0.

In the tumor grade example, theP-value for the

Wald test of the beta coefficient for RACE is

0.002, indicating that RACE is significantly

associated with tumor grade at the 0.05 level.

EXAMPLE (continued)

95% confidence interval for OR

95 %CI¼exp½ 0 : 7555    1 : 96 ð 0 : 2466 ފ

¼ð 1 : 31 ; 3 : 45 Þ

Hypothesis testing

Likelihood ratio test or Wald test

H 0 :b 1 ¼ 0

Wald test

Z¼

0 : 7555

0 : 2466

¼ 3 : 06 ; P¼ 0 : 002

Presentation: III. Odds Ratios and Confidence Limits 475