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

Model 3.Fully parameterized model

+

#of expected cases

¼#of observed cases for

each covariate pattern

Covariate patternXg:ngsubjects
Xg¼values ofXin group g

dg¼observed cases in group g
(binomial)
^PðXgÞ: predicted risk in group g

d^g¼ng^PðXgÞ: expected cases in
group g

EXAMPLE

X: EV Exp. Cases Obs. Cases

X 1 :11 d^ 1 ¼ 10 ð 0 : 6 Þ¼ 6 d 1 ¼ 6

X 2 :01 d^ 2 ¼ 10 ð 0 : 4 Þ¼ 4 d 2 ¼ 4

X 3 :10 d^ 3 ¼ 10 ð 0 : 3 Þ¼ 3 d 3 ¼ 3

X 4 :00 d^ 4 ¼ 10 ð 0 : 7 Þ¼ 7 d 4 ¼ 7

Model 3.

Perfect

Prediction?

YiY^i 6 ¼ 0 for

alli(not

Individuals: No

saturated)

dgd^g¼ 0 for

all g (fully

Groups:

(Patterns)

Yes

parameterized)

Model 3 is “group-saturated”:

perfectly group outcomes

Two GOF approaches:
Compare fitted model to:

1. Saturated model: Provides
   perfectindividual prediction


2. Fully parameterized model:
   Provides perfectgroup
   prediction(based on
   covariate patterns)

However, we will show below that because

Model 3 is a fully parameterized model, it per-

fectly predicts the number of cases actually

observed for each covariate pattern. That is,

the expected number of cases for each pattern,

based on the fitted model, equals the observed

number of cases for each pattern.

More specifically, suppose ng subjects have

covariate pattern Xg, and dg denotes the

observed number of cases in group g. Then,

since dg has the binomial distribution, the

expected number of cases in group g is

d^g¼ng^PðXgÞ, where^PðXgÞis the predicted risk

for any subject in that group.

Thus, for Model 3, we expectd^ 1 ¼ 10 ð 0 : 6 Þ¼ 6

cases among subjects withE¼1 andV¼1,

d^ 2 ¼ 10 ð 0 : 4 Þ¼ 4 cases among subjects with

E¼0 and V¼1, and so on for the other

two covariate patterns. The corresponding

observed number of subjects are also shown at

the left. Notice that the corresponding observed

and expected cases are equal for Model 3.

So, even though Model 3 does not provide “per-

fect prediction” in terms of individual out-

comes, it does provide “perfect prediction” in

terms ofgroupoutcomes.

In other words, althoughYiY^i 6 ¼ 0 for all sub-

jects,dgd^g¼ 0 for all covariate patterns.

Another way of saying this is that Model 3 is

“group-saturated” in the sense that Model 3

perfectly predicts the group outcomes corres-

ponding to the distinct covariate patterns.

Thus, we see that an alternative gold standard

model for assessing GOF is a fully parameter-

ized (group-saturated) model containing the

covariates of interest rather than a (subject-

specific) saturated model that can rarely if

ever be achieved using these covariates.

310 9. Assessing Goodness of Fit for Logistic Regression