Introductory Biostatistics

(a) ForX¼acid, we find that

bb^ 1 ¼ 0 : 0204

from which the odds ratio forðacid¼ 100 Þversusðacid¼ 50 Þwould be

OR¼exp½ð 100 
 50 Þð 0 : 0204 ފ

¼ 2 : 77

(b) ForX¼log 10 ðacidÞ, we find that

bb^

1 ¼^5 :^1683

from which the odds ratio forðacid¼ 100 Þversusðacid¼ 50 Þwould be

OR¼expf½log 10 ð 100 Þ
log 10 ð 50 ފð 5 : 1683 Þg

¼ 4 : 74

Note: IfX¼acid is used, an SAS program would include these instruc-
tions:

PROC LOGISTIC DESCENDING
DATA = CANCER;
MODEL NODES = ACID;

where CANCER is the name assigned to the data set, NODES is the variable
name for nodal involvement, and ACID is the variable name for our covariate,
the level of acid phosphatase in blood serum. The option DESCENDING is
needed because PROC LOGISTIC models PrðY¼ 0 Þinstead of PrðY¼ 1 Þ.

The results above are di¤erent for two di¤erent choices ofXand this seems
to cause an obvious problem of choosing an appropriate measurement scale. Of
course, we assume alinear modeland one choice of scale forXwould fit better
than the other. However, it is very di‰cult to compare di¤erent scales unless
there were replicated data at each level ofX; if such replications are available,
one can simply graph a scatter diagram of log(odds) versus theXvalue and
check for linearity of each choice of scale of measurement forX.

9.1.4 Tests of Association

Sections 9.1.2 and 9.1.3 deal with inferences concerning the primary regression
coe‰cientb 1 , including both point and interval estimation of this parameter
and the odds ratio. Another aspect of statistical inference concerns the test of

SIMPLE REGRESSION ANALYSIS 321