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

Suppose we are interested in the effect of

AGEGP, controlling for the effects of ESTRO-

GEN and SMOKING. The odds ratio for the

effect of AGEGP in the comparison of Adenos-

quamous (D¼1) to Adenocarcinoma (D¼0) is

equal to e to the^b 11 or exp(0.9871) equals 2.68.

The odds ratio for the effect of AGEGP in the

comparison of Other type (D¼2) to Adenocar-

cinoma (D¼0) is equal to e to the ^b 21 or

exp(0.2823) equals 1.33.

Our interpretation of the results for the three-

variable model differs from that of the one-

variable model. The effect of AGEGP on the

outcome is now estimated while controlling

for the effects of ESTROGEN and SMOKING.

If we compare the model with three predictor

variables with the model with only AGEGP

included, the effect of AGEGP in the reduced

model is weaker for the comparison of Adeno-

squamous to Adenocarcinoma (ORd¼ 2 : 18 vs.

2.68), but is stronger for the comparison of

Other to Adenocarcinoma (dOR¼ 1 : 53 vs. 1.33).

These results suggest that estrogen use and

smoking status act as confounders of the rela-

tionship between age group and the tumor

category outcome. The results of the single-

predictor model suggest a bias toward the

null value (i.e., 1) for the comparison of Adeno-

squamous to Adenocarcinoma, whereas the

results suggest a bias away from the null for

the comparison of Other to Adenocarcinoma.

These results illustrate that assessment of con-

founding can have added complexity in the

case of multilevel outcomes.

EXAMPLE (continued)

Adenosquamous vs. Adenocarcinoma:

dOR 1 ¼exp½^a^1 þ^b^11 ð^1 Þþ^b^12 ðX^2 Þþ^b^13 ðX^3 ފ

exp½^a 1 þ^b 11 ð 0 Þþ^b 12 ðX 2 Þþ^b 13 ðX 3 ފ

¼exp^b 11 ¼expð 0 : 9871 Þ¼ 2 : 68

Other vs. Adenocarcinoma:

dOR 2 ¼exp½^a^2 þb^21 ð^1 Þþ

b^ 22 ðX 2 Þþb^ 23 ðX 3 ފ

exp½^a 2 þb^ 21 ð 0 Þþb^ 22 ðX 2 Þþb^ 23 ðX 3 ފ

¼exp^b 21 ¼expð 0 : 2823 Þ¼ 1 : 33

Interpretation of ORs

Three-variable vs. one-variable model

Three-variable model:

)AGEGPjESTROGEN;SMOKING

One-variable model:

)AGEGPjno control variables

Odds ratios for effect of AGEGP:

Model

Comparison

AGEGP

ESTROGEN

SMOKING

AGEGP

1 vs. 0 2.68 2.18

2 vs. 0 1.33 1.53

Results suggest bias for single-

predictor model:

 Toward null for comparison of

category 1 vs. 0

 Away from null for comparison of

category 2 vs. 0.

446 12. Polytomous Logistic Regression