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

Pred¼obs for each g

ðphat¼pÞ

+

d^g¼dgcases

EXAMPLE

Fully parameterized model

(w/o conts. Xs):

 Above results support its use as

gold standard for assessing GOF

 HL statistic always 0 when

perfect group prediction

Second illustration

(Evans County data):

Model EC3(no interaction):

logit PðXÞ¼aþbCATþg 1 AGE

þg 2 ECGþg 3 SMK

þg 4 CHLþg 5 HPT

Model EC4:

logit PðXÞ¼aþbCATþg 1 AGE

þg 2 ECGþg 3 SMK

þg 4 CHLþg 5 HPT

þd 1 CATCHL

þd 2 CATHPT

Yet, as the table of probabilities indicates,

Model EC2 perfectly predicts the observed

probabilities obtained for each covariate pat-

tern. Equivalently, then, this model perfectly

predicts the observed number of cases (dg)

corresponding to each covariate pattern.

When none of the predictors are continuous, as

with these data, these results support the use of

a fully parameterized model defined from all

covariate patterns as the gold standard model

for assessing GOF. In particular, the HL statistic

reflects this framework, since the value of HL

will always be zero whenever there is perfect

group, rather than subject-specific, prediction.

We now provide a second illustration of GOF

assessment using the Evans County data (see

Computer Appendix) with models that involve

continuous variables. In particular, we con-

sider two previously considered models (see

Chap. 7) shown on the left that involve the

predictors CAT, AGE, ECG, SMK, CHL, and

HPT. Here, AGE and CHL are continuous,

whereas CAT, ECG, SMK, and HPT are binary

variables.

EXAMPLE

AGE and CHL continuous

+

Few subjects with identical values for

both AGE and CHL

+

Gnð¼ 609 Þ

SASs Proc Logistic automatically
outputs “Number of Unique
Profiles” (G)

Here,G¼ 599

Since both models EC3 and EC4 contain con-

tinuous variables AGE and CHL, there are not

likely to be many of the 609 subjects with iden-

tically the same values for these two variables.

Consequently, the number of covariate pat-

terns (G) for each model should be close to

the sample size of 609.

In fact, SASs Logistic procedure automatically

outputs this number (identified in the output

as the “number of unique profiles”), which

turns out to be 599 (see output, below left),

i.e.,G¼599.

Presentation: V. Examples of the HL Statistic 323