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

Taking derivatives, we obtain

@lnLðbÞ

@bj

¼~

k

Yk

P^ðXkÞ

1 Yk

1 P^ðXkÞ

()

^PðXkÞ

ð 1 P^ðXkÞÞXjk;

which can be rewritten as

@lnLðbÞ

@bj

¼~

k

Ykð 1 P^ðXkÞÞð 1 YkÞ^PðXkÞ

no

Xjk

and further simplied to

@lnLðbÞ

@bj

¼~

k

ðYkP^ðXkÞÞXjk:

We can then write

~

j

bj

@lnLðbÞ

@bj

¼~

k

Yk^PðXkÞ



~

j

bjXjk

¼~

k

Yk^PðXkÞ



ln

^PðXkÞ

1 P^ðXkÞ

!

¼~

k

Yk^PðXkÞ



logitP^ðXkÞ:

Since@lnLð

b^Þ

@bj ¼^0 for the ML estimate

b^,we

can write~

k

ðYk^PðXkÞÞlogitP^ðXkÞ¼ 0.

It then follows that ~

k

Yklogitð^PðXkÞÞ¼

~

k

P^ðXkÞlogitð^PðXkÞkÞ.

We then replace ~

k

YklogitðP^ðXkÞÞ by

~

k

^PðXkÞlogitðP^ðXkÞÞ in the above simplified

formula for the deviance to obtain

DevSSðb^Þ¼ 2 ~

n

k¼ 1

^

PðXkÞlogitðP^ðXkÞÞ

þlnð 1 P^ðXkÞÞ

:

328 9. Assessing Goodness of Fit for Logistic Regression