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

ii. Saturated model (n¼40 parameters)

logit PðXÞ¼o 1 Z 1 þo 2 Z 2 þo 3 Z 3

þþo 40 Z 40

Zi¼

1 if subjecti; i¼ 1 ; 2 ...; 40

0 otherwise

(

:

III. The Deviance Statistic (pages 312–317)

A. Formula: Devðb^Þ¼ 2 lnðL^c=L^maxÞ, where

b^¼ð^b 0 ;^b 1 ;^b 2 ;...;^bkÞ

L^c¼ML for current model

L^max¼ML for saturated model

B. Contrasts the likelihood of the current model

with the likelihood of the model that perfectly

predicts the observed outcomes

C. The closerL^candL^maxare to one another, the

better the fit (and the smaller the deviance

D. Common to test for GOF by comparing

deviance withw^2 np 1 value, butquestionably

legitimate

E. There are two alternative formulae for the

deviance:

i. DevETðb^Þ

¼ 2 ~

G

g¼ 1

dgln

dg

d^g



þðngdgÞln

ngdg

ngd^g

 

uses events–trials format, where

d^g¼ngP^ðXgÞ¼#of expected cases,

dg¼#of observed cases,

G¼#of covariate patterns

ii. DevSSðb^Þ

¼ 2 ~

n

i¼ 1

Yiln YY^i

i



þð 1 YiÞln^11 YY^i

i

h i

uses subject-specific format, where

Yi¼observed (0, 1) response for

subjecti

and Y^i¼predicted probability for

subjecti¼P^ðXiÞ

iii. DevSSðb^Þ 6 ¼DevETðb^ÞunlessG¼n

iv. Fully parameterized model:

DevETðb^Þ¼ 0 always but

DevSSðb^Þis never 0

330 9. Assessing Goodness of Fit for Logistic Regression