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

V. Likelihood Function for
Ordinal Model

odds¼

P

1 P

so solving forP,

P¼

odds

oddsþ 1

¼

1

1 þ odds^1



The estimated odds ratio for the effect of

RACE, controlling for the effect of ESTRO-

GEN, is e to the^b 1 , which equals e to the

0.4270 or 1.53.

The 95% confidence interval for the odds ratio

is e to the quantity^b 1 plus or minus 1.96 times

the estimated standard error of the beta coeffi-

cient for RACE. In our two-predictor example,

the standard error for RACE is 0.2720 and the

95% confidence interval is calculated as 0.90 to

2.61. The confidence interval contains one, the

null value.

If we perform the Wald test for the significance

of^b 1 , we find that it is not statistically signifi-

cant in this two-predictor model (P¼0.12).

The addition of ESTROGEN to the model has

resulted in a decrease in the estimated effect of

RACE on tumor grade, suggesting that failure

to control for ESTROGEN biases the effect of

RACE away from the null.

Next, we briefly discuss the development of the

likelihood function for the proportional odds

model. To formulate the likelihood, we need

the probability of the observed outcome for

each subject. An expression for these probabil-

ities in terms of the model parameters can

be obtained from the relationshipP¼odds/

(oddsþ1), or the equivalent expression

P¼1/[1þ(1/odds)].

EXAMPLE (continued)

Odds ratio

dOR¼exp^b 1 ¼expð 0 : 4270 Þ¼ 1 : 53

95% confidence interval

95 %CI¼exp½ 0 : 4270    1 : 96 ð 0 : 2720 ފ

¼ð 0 : 90 ; 2 : 61 Þ

Wald test

H 0 :b 1 ¼ 0

Z¼

0 : 4270

0 : 2720 ¼^1 :^57 ; P¼^0 :^12

Conclusion: fail to rejectH 0

478 13. Ordinal Logistic Regression