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

EXAMPLE

Edited Output (Model EC4):

(Variables – CAT, AGE, ECG, SMK, CHL, HPT

CAT×CHL, and CAT× HPT)

7.9914 8 0.4343

347.2295 590 1.0000

799.0580 590 <.0001

• 2 Log L 347.230

No evidence

Model EC4

has lack of fit

1 61 2 0.94 59 60.06

2 61 1 1.96 60 59.04

3 61 2 2.68 59 58.32

4 61 5 3.37 56 57.63

5 61 4 4.07 57 56.93

6 61 2 4.78 59 56.22

7 61 4 5.77 57 55.23

8 61 12 7.66 49 53.34

9 61 10 11.05 51 49.95

10 60 29 28.71 31 31.29

Deviance statistic (347.2295)
very different from
Pearson statistic (799.0580)

Both Models EC3 and EC4
do not have lack of fit:

Use LR test to compare Models EC3
vs. EC4

H 0 :d 1 ¼d 2 ¼0 in Model EC4

LR¼ 2 lnL^EC 3 ð 2 lnL^EC 4 Þ

w^22 dfunderH 0

Model EC3 Model EC4
2 LogL 400.394 347.230

LR¼400.394347.230¼53.165
(P< 0.001):
Model EC4 preferred over Model
EC3

Edited output for the interaction model EC4 is

now shown here at the left. This model con-

tains CAT, AGE, ECG, SMK, CHL, and HPT,

and the product terms CATCHL and CAT

HPT in addition to the six main effects.

From this output, as with Model EC3, we see

that the predicted risks have been divided

into (Q¼10) deciles, with about 61 subjects

in each decile. Also, the observed and expected

cases are somewhat different within each dec-

ile, and the observed and expected noncases

are somewhat different.

The HL statistic of 7.9914 has Q 2 ¼ 8

degrees of freedom and is nonsignificant

(P¼0.4343). Thus, there is not sufficient evi-

dence from this test to conclude that interac-

tion Model EC4 has lack of fit (as concluded for

Model EC3).

As with Model EC3, both the Deviance

(347.2295) and Pearson (799.0580) statistics

for Model EC4 are very different from each

other as well as from the HL statistic (7.9914).

Although we can conclude from the HL test

that both Models EC3 and EC4 do not indicate

lack of fit, we can decide between these two

models by performing an LR test that com-

pares corresponding log likelihood statistics

for the two models.

The null hypothesis is that the coefficients of

the two product terms in Model EC4 are both

zero. As previously seen in Chap. 7, the test

statistic is approximately chi square with

2 degrees of freedom under the null hypothesis.

The resulting LR value is 53.164, which is

highly significant (P<0.0001).

Consequently, the interaction Model EC4 is

preferred to the no-interaction Model EC3.

Presentation: V. Examples of the HL Statistic 325