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

The odds that the tumor grade is in a category

greater than or equal to category 2 (i.e., poorly

differentiated) vs. in categories less than 2 (i.e.,

moderately or well differentiated) is e to the

quantitya 2 plus the sum ofb 1 X 1 plusb 2 X 2.

Similarly, the odds that the tumor grade is in a

category greater than or equal to category 1

(i.e., moderately or poorly differentiated) vs.

in categories less than 1 (i.e., well differen-

tiated) is e to the quantitya 1 plus the sum of

b 1 X 1 plusb 2 X 2. Although the alphas are differ-

ent, the betas are the same.

Before examining the model output, we first

check the proportional odds assumption with

a Score test. The test statistic has two degrees

of freedom because we have two fewer para-

meters in the ordinal model compared to the

corresponding polytomous model. The results

are not statistically significant, with aP-value

of 0.64. We therefore fail to reject the null

hypothesis that the assumption holds and can

proceed to examine the remainder of the model

results.

The output for the analysis is shown on the left.

There is only one beta estimate for each of the

two predictor variables in the model. Thus,

there are a total of four parameters in the

model, including the two intercepts.

EXAMPLE (continued)

odds =

different as same

bs

P(D ³ 2 X)

P(D < 2 X)

odds =P(D^ ³^1 X)

P(D < 1 X)

= exp(a 2 + b 1 X 1 + b 2 X 2 )

= exp(a 1 + b 1 X 1 + b 2 X 2 )

Test of proportional odds assumption

H 0 : assumption holds

Score statistic:w^2 ¼0.9051, 2 df,

P¼0.64

Conclusion: fail to reject null

Variable Estimate S.E. Symbol

Intercept 1 1.2744 0.2286 ^a 2
Intercept 2 0.5107 0.2147 ^a 1

RACE 0.4270 0.2720 ^b 1
ESTROGEN0.7763 0.2493 ^b 2

Presentation: IV. Extending the Ordinal Model 477