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

OR¼

PðDgjX 1 ¼ 1 Þ=PðD<gjX 1 ¼ 1 Þ

PðDgjX 1 ¼ 0 Þ=PðD<gjX 1 ¼ 0 Þ

¼

expagþb 1 ð 1 Þ



expagþb 1 ð 0 Þ

¼expðagþb^1 Þ

expðagÞ

¼eb^1

General case
(levelsX* 1 andX 1 ofX 1 )

OR¼

expðagþb 1 X 1 **Þ

expðagþb 1 X* 1 Þ

¼

expðagÞexpðb 1 X** 1 Þ

expðagÞexpðb 1 X* 1 Þ

¼exp

h

b 1 ðX** 1 X* 1 Þ

i

CIs:same method as SLR to com-
pute CIs

General case (levels X 1 * and X 1
ofX 1 )

95% CI:

exp

h

^b

1 ðX

**

1 X

*

1 Þ ^1 :^96 X

**

1 X

*

1



s^b 1

i

This is calculated, as shown on the left, as the

odds that the disease outcome is greater than

or equal togifX 1 equals 1, divided by the odds

that the disease outcome is greater than or

equal togifX 1 equals 0.

Substituting the expression for the odds in

terms of the regression parameters, the odds

ratio forX 1 ¼1vs.X 1 ¼0 in the comparison of

disease levelsgto levels<gisthenetotheb 1.

To compare any two levels of the exposure

variable,X 1 **andX* 1 , the odds ratio formula is

e to theb 1 times the quantityX** 1 minusX* 1.

Confidence interval estimation is also analo-

gous to standard logistic regression. The gen-

eral large-sample formula for a 95% confidence

interval, for any two levels of the independent

variable (X** 1 andX* 1 ), is shown on the left.

Returning to our tumor-grade example, the

results for the model examining tumor grade

and RACE are presented next. The results were

obtained from running PROC LOGISTIC in

SAS (see Appendix).

We first check the proportional odds assump-

tion with aScore test. The test statistic, with

one degree of freedom for the one odds ratio

parameter being tested, was clearly not signifi-

cant, with aP-value of 0.9779. We therefore fail

to reject the null hypothesis (i.e., that the

assumption holds) and can proceed to examine

the model output.

EXAMPLE

Black/White Cancer Survival Study

Test of proportional odds assumption:

H 0 : assumption holds

Score statistic:w^2 ¼0.0008, df¼1,

P¼0.9779.

Conclusion: fail to reject null

Presentation: III. Odds Ratios and Confidence Limits 473