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

Alternative (Events–Trials) Devi-
ance formula: for the logistic
model:
Gcovariate patterns
Xg¼ðXg 1 ;Xg 2 ;...;XgpÞ;g¼ 1 ; 2 ;...;G
d^g¼ngP^ðXgÞ¼expected cases
dg¼observed cases

DevETðb^Þ

¼ 2 lnL^c;ETð 2 lnL^max;ETÞ

¼ 2 ~

G

g¼ 1

dgln

dg

d^g

"!

þðngdgÞln

ngdg

ngd^g

!

;

where2lnL^c,ETand2lnL^max,ET
are defined using events–trials
(ET) format

First, we present an alternative formula for the

devianceinalogisticmodelthatconsiderstheco-

variate patterns defined by one’s current model.

We assume that this model contains G covariate

patternsXg¼(Xg 1 ,Xg 2 ,...,Xgp),withngsubjects

having pattern g. As defined earlier,d^g¼ngP^ðXgÞ

denotes the expected cases, where P^ðXgÞ is

the predicted risk forX¼Xg,anddgdenotes

the observed number of cases in subgroup g.

The alternative deviance formula is shown here

at the left. This formula corresponds to the data-

set listed in events–trials format, where there

areGdatalines,dgandngdenote the number

of events and number of trials, respectively, on

the gth dataline.

EXAMPLE

Model 3: logit P(X)¼aþbEþgV

þdEV

X: EV ng

Exp.

Cases

Obs.

Cases

X 1 :11 10 d^ 1 ¼ 6 d 1 ¼ 6

X 2 :01 10 d^ 2 ¼ 4 d 2 ¼ 4

X 3 :10 10 d^ 3 ¼ 3 d 3 ¼ 3

X 4 :00 10 d^ 4 ¼ 7 d 4 ¼ 7

DevETð^bÞfor Model 3:

¼ 26 ln

6

6



þ 4 ln

4

4



 24 ln

4

4



þ 6 ln

6

6



 23 ln^3

3



þ 7 ln^7

7



 27 ln

7

7



þ 3 ln

3

3



¼ 0

Deviance formula DevETðb^Þ uses
events–trials format

+

Units of analysis are groups

ðnot subjectsÞ

Recall that for the data set onn¼40 subjects

described above, Model 3 hasG¼4 covariate

patterns. For each pattern, the corresponding

values forng,d^g, anddgare shown at the left.

Substituting the values in the table into the

alternative (events–trials) deviance formula,

we find that the resulting deviance equals zero.

How can we explain this result, since we know

that the fully parameterized Model 3 is not

saturated in terms of perfectly predicting the

0 or 1 outcome for each of the 40 subjects in the

dataset? The answer is that since the ET devi-

ance formula corresponds to an events–trials

format, the units of analysis being considered

by the formula are the four groups of covariate

patterns rather than the 40 subjects.

314 9. Assessing Goodness of Fit for Logistic Regression