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

C. Table of observed and expect counts (Step 4)

Deciles

of risk

1

2

3

10

Oc3

Oc2

Oc1

Onc3

Onc10

Onc2

Onc1

Ec3

Ec10

Ec2

Ec1

Enc3

Enc10

Enc2

Enc1

Oc10

Obs.

cases

Obs. non

cases

Exp. non

cases

Exp.

cases

D. HL Statistic formula (Step 5):

HL¼~

Q

q¼ 1

ðOcqEcqÞ^2

Ecq

þ~

Q

q¼ 1

ðOncqEncqÞ^2

Encq

V. Examples of the HL Statistic (pages 320–325)

A. Two examples, each using the Evans County

data (n¼609).

B. Example 1 uses two models involving three

binary predictors with data layout in

events–trials format (G¼8).

i. The models

Model EC1 (no interaction): logit PðXÞ¼

aþbCATþg 1 AGEGþg 2 ECG

Model EC2:logit PðXÞ¼aþbCAT

þg 1 AGEGþg 2 ECGþg 3 AGEGECG

þd 1 CATAGEþd 2 CATECG

þd 3 CATAGEECG

ii. Model EC2 is fully parameterized,

which, as expected, perfectly predicts the

observed number of cases (dg)

corresponding to each covariate pattern.

iii. The HL test statistic for Model EC2 is

zero.

C. Example 2 uses two models that involve

continuous variables.

i. The models:

Model EC3 (no interaction):

logit PðXÞ¼aþbCATþg 1 AGE

þg 2 ECGþg 3 SMKþg 4 CHLþg 5 HPT

Model EC4: logit PðXÞ¼aþbCAT

þg 1 AGEþg 2 ECGþg 3 SMKþg 4 CHL

þg 5 HPTþd 1 CATCHL

þd 2 CATHPT.

ii. The number of covariate patterns (G) for

each model is 599, which is quite close to

the sample size (n) of 609.

332 9. Assessing Goodness of Fit for Logistic Regression