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

EXAMPLE

“group-saturated”:

# of parameters = # of groups

sample size = 4

Model 3: logit P(X)¼aþbEþgV

þdEV

vs.

Model 4 (perfectly predicts 0 or 1

outcome):

logit PðXÞ¼o 1 Z 1 þo 2 Z 2 þo 3 Z 3 þ

þo 40 Z 40

Subject-specific deviance formula:

DevSSðb^Þ¼ 2 ~

n

i¼ 1



Yiln

Yi

Y^i



þð 1 YiÞln

1 Yi

1 Y^i



Yi¼observed (0, 1) response for
subjecti
Y^i¼predicted probability for the
subjecti

DevSSðb^Þ 6 ¼DevETð^bÞunlessG¼n

DevSSðb^Þ> 0 sinceG<<n

for Model 3

Equivalent formula forDevSS(b^):
(from calculus and algebra)

DevSSðb^Þ

¼ 2 ~

n

i¼ 1

P^ðXiÞln

^PðXiÞ

1 P^ðXiÞ

"!

þlnð 1 P^ðXiÞÞ

Consequently, Model 3 is “group-saturated” in

that the number of parameters in this model is

equal to the total number of groups being con-

sidered. That is, since the units of analysis are

the four groups, the sample size corresponding

to the alternative deviance formula is 4, rather

than 40.

So, then, how can we compare Model 3 to the

saturated model we previously defined (Model

4) that perfectly predicts each subject’s 0 or 1

outcome?

This requires a second alternative formula for

the deviance, as shown at the left. Here,

the summation (i) covers all subjects, not all

groups, andYiandY^idenote the observed and

predicted response for theith subject rather

than the gth covariate pattern/group.

This alternative (subject-specific) formula is

not identical to the events–trials formula

given above unless G¼n. Moreover, since

G<<nfor Model 3, the SS deviance will be

nonzero for this model.

We now show how to compute the subject-

specific (SS) deviance formula for Model 3.

However, we first provide an equivalent SS

deviance formula that can be derived from

both algebra and some calculus (see this

chapter’s appendix for a proof).

Presentation: III. The Deviance Statistic 315