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

EXAMPLE (continued)

Model 3: logit P(X)¼aþbEþgV

þdEV

Largest possible model containing

binaryEand binaryV

Note:E^2 ¼EandV^2 ¼V

Fully parameterized model:
Contains maximum#of
covariates defined from the
main-effect covariates,

No. of parameters (kþ 1 )¼G
covariate patterns, where
G¼#of covariate patterns

Covariatepatterns(i.e.,subgroups):
Distinct specifications ofX

EXAMPLE

Model 3: logit P(X) = α + βE + γV + δEV

4 covariate patterns

X 1 : E=1, V= 1

X 2 : E=0, V= 1

X 3 : E=1, V= 0

X 4 : E=0, V= 0

Model 1: logit P(X) = α + βE

2 covariate patterns

X 1 : E= 1

X 2 : E= 0

Model 2: logit P(X) = a + bE + gV

4 covariate patterns

(same as Model 3)

k + 1 = 4 = G

fully

parameterized

k + 1 = 2 = G

fully

parameterized

k + 1 = 3 ≠ G= 4

not fully

parameterized

Assessing GOF:

Saturatedðkþ 1 ¼nÞ

vs:

fully parameterizedðkþ 1 ¼GÞ?

Let us now focus on Model 3. This model is the

largest possible model that can be defined con-

taining the two “basic” variables, E and V.

SinceEandVare both (0,1) variables,E^2 ¼E

andV^2 ¼V, so we cannot add to the model any

higher order polynomials inE orVor any

product terms other thanEV.

Model 3 is an example ofa fully parameterized

model, which contains the maximum number

of covariates that can be defined from the

main-effect covariates in the model.

Equivalently, the number of parameters in such

a model must equal the number ofcovariate

patterns(G) that can be defined from the covari-

ates in the model.

In general, for a given model with covariates

X¼(X 1 ,...,Xk), the covariate patterns are

defined by the distinct values ofX.

For Model 3, which contains four parameters,

there are four distinct covariate patterns, i.e.,

subgroups, that can be defined from the covari-

ates in the model. These are shown at the left.

Model 1, which contains only binaryE, is also

fully parameterized, providing Eis the only

basic predictor of interest. No other variables

defined fromE, e.g.,E^2 ¼E, can be added to

model 1. Furthermore, Model 1 contains two

parameters, which correspond to the two

covariate patterns derived fromE.

However, Model 2 is not fully parameterized,

since it contains three parameters and four

covariate patterns.

Thus, we see that a fully parameterized model

has a nice property (i.e.,kþ 1 ¼G): it is the

largest possible model we can fit using the vari-

ables we want to allow into the model. Such a

model might alternatively be used to assess

GOF rather than using the saturated model as

the (gold standard) referent point.

308 9. Assessing Goodness of Fit for Logistic Regression