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

Model A: X¼(X 1 , X 2 , X 3 ) fully
parameterized but

Model B:X¼(X 1 ,X 2 ,X 3 ,X 4 ,X 5 ,X 6 )
“better fit” than Model A

EXAMPLE

e.g.,

LR

test▸

Model 1 :logit PðXÞ

¼aþbE

vs:

Model 3 :logit PðXÞ

¼aþbEþgVþdEV

8

>>>

>>>

<

>>>

>>>

:

LR¼ 2 lnL^Model 1 ð 2 lnL^Model 3 Þ

¼ 55 : 051  51 : 355 ¼ 3 : 696

w^2 ð 2 dfÞunderH 0 :g¼d¼ 0

ðP> 0 : 10 Þn:s:

Fitted Model 3:

logitP^ðXÞ¼^aþb^Eþ^gVþ^dEV;

where ^a¼^0 :^8473 ;^b¼^1 :^6946 ;

^g¼ 1 : 2528 ;^d¼ 2 : 5055

Covariate

pattern

Obs.

risk

Pred.

risk

X 1 :E¼1,

V¼ 1

p^ 1 ¼ 0 : 6 P^ðX 1 Þ¼ 0 : 6

X 2 :E¼0,

V¼ 1

p^ 2 ¼ 0 : 4 P^ðX 2 Þ¼ 0 : 4

X 3 :E¼1,

V¼ 0

p^ 3 ¼ 0 : 3 P^ðX 3 Þ¼ 0 : 3

X 4 :E¼0,

V¼ 0

p^ 4 ¼ 0 : 7 P^ðX 4 Þ¼ 0 : 7

E=1, V=1: P(X 1 )

E=0, V=1: P(X 2 )

ˆ

ˆ

• E=1: some D=1,

E=0: some D=1,

Model 3:

No perfect fit

= 0.6 ≠ 0 or 1

= 0.4 ≠ 0 or 1

some D= 0

some D= 0

Note, however, even if a model A is fully para-

meterized, there may be a larger model B con-

taining covariates not originally considered in

model A that provides “better fit” than model A.

For instance, although Model 1 is the largest

model that can be defined when only binaryE

is considered, it is not the largest model that

can be defined when binaryVis also consid-

ered. Nevertheless, we can choose between

Model 1 and Model 3 by performing a standard

likelihood ratio (LR) test that compares the

two models; the (nonsignificant) LR results

(P>0.10) are shown at the left.

Focusing now on Model 3, we show the fitted

model at the left. Note that these same esti-

mated parameters would be obtained whether

we input the data by individual subjects (40

datalines) or by using an events–trials format

(4 datalines). However, using events–trials

format, we lose the ability to identify which

subjects become cases.

The predicted risks obtained from the fitted

model for each covariate pattern are also

shown at the left, together with their cor-

responding observed risks. Notice that these

predicted risks are equal to their corresponding

observed proportions computed from the

observed (stratified) data.

Nevertheless, none of these predicted risks are

either 0 or 1, so the fitted model does not per-

fectly predict each subject’s observed outcome,

which is either 0 or 1. This is not surprising,

since some exposed subjects develop the dis-

ease and some do not.

Presentation: II. Saturated vs. Fully Parameterized Models 309