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

EXAMPLE

e.g.,q¼3,n 3 ¼4,^PðXiÞ: 0.30, 0.35,

0.40, 0.45

Ec 3 ¼~

n 3

i¼ 1

^PðXi 3 Þ¼ 0 : 30 þ 0 : 35 þ 0 : 40

þ 0 : 45 ¼ 1 : 50

andEnc3¼n 3 Ec3¼ 4 1.50

¼2.50

Step 5:

HL¼~

Q

q¼ 1

ðOcqEcqÞ^2

Ecq

þ~

Q

q¼ 1

ðOncqEncqÞ^2

Encq

Q¼ 10 ) 20 values in summation

Step 6:

HL ~ approx c^2 Q–2 under H 0 : Good fit

i.e., not enough

evidence to

indicate lack of fit

Q = 10 ⇒ df = Q – 2 = 8

V. Examples of the HL
Statistic

EXAMPLE

Evans County Data (n¼609)

(see previous chapters and

Computer Appendix)

Model EC1(no interaction):

logit P(X)¼aþbCATþg 1 AGEG

þg 2 ECG

Model EC2(fully parameterized):

logit PðXÞ¼aþbCATþg 1 AGEG

þg 2 ECG

þg 1 AGEGECG

þd 1 CATAGE

þd 2 CATECG

þd 3 CATAGEECG

For example, if the third decile contains four

subjects with predicted risks of 0.30, 0.35, 0.40,

and 0.45, then the expected number of cases

(Ec3) would be their sum 0.30þ0.35þ0.40þ

0.45¼1.50 (regardless of whether or not a

subject is an observed case). The expected

noncases in the same decile (Enc3) would be

4 1.50¼2.50.

In step 5, the HL statistic is calculated using

the formula at the left. This formula involves

summingQvalues of the general form (Oq

Eq)^2 /Eqfor cases and anotherQvalues for non-

cases. WhenQ¼10, the HL statistic therefore

involves 20 values in the summation.

In step 6, the HL statistic is tested for signifi-

cance by comparing the computed HL value to

a percentage point ofw^2 withQ2 degrees of

freedom. WhenQ¼10, therefore, the HL sta-

tistic is approximatelyw^2 with 8 df.

We now illustrate the use of the HL statistic

with the Evans County data (n¼609). This

dataset has been considered in previous chap-

ters, and is described in detail in the Computer

Appendix. SASs Logistic procedure was used

for the computations.

In our first illustration, we fit the two models

shown at the left. The outcome variable is CHD

status (1¼case, 0¼noncase), and there are

three basic (i.e., main effect) binary predictors,

CAT (1¼high, 0¼low), AGEG (1¼age55,

0 ¼age>55), and ECG (1¼abnormal, 0¼

normal). Recall that the Evan County dataset

is described in the Computer Appendix.

320 9. Assessing Goodness of Fit for Logistic Regression