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

Chapter 10: Assessing Discrimi-
natory Performance of a Binary
Logistic Model

We suggest that you review the material cov-

ered here by reading the detailed outline that

follows. Then do the practice exercises and

test.

In the next chapter (Chap. 10), we describe

methods for assessing the discriminatory per-

formance is of a binary logistic model using

misclassification tables and ROC curves.

VII. Appendix: Derivation
of the Subject-
Specific (SS) Deviance
Formula

DevSSðb^Þ¼ 2 ~

n

k¼ 1

P^ðXiÞln

^PðXkÞ

1 P^ðXkÞ

"!

þln 1 P^ðXkÞ



Proof. We first write the Deviance formula in a

convenient form as follows:

DevSSðb^Þ¼ 2 ln

L^C

L^MAX



¼ 2 lnL^Csince lnL^MAX 0

 2 ln;L^C ¼

definition

 2 ~

n

k¼ 1

h

Yk lnP^ðXkÞ:

þð 1 YkÞlnð 1 ^PðXkÞ

i

¼

algebra

 2 ~

n

k¼ 1



Yk ln

P^ðXkÞ

1 ^PðXkÞ

!

þlnð 1 P^ðXkÞ



:

We now write the log of the logistic likelihood

function in a convenient form and take its

derivative:

LðbÞ¼

definitionY

k

P^ðXkÞYkð 1 P^ðXkÞÞ^1 Yk

so

lnLðbÞ¼~

k

YklnP^ðXkÞþð 1 YkÞlnð 1 P^ðXkÞÞ:

Presentation: VII. Appendix: Derivation of the Subject-pecific (SS) 327