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

VIII. Numerical Example

EVANS COUNTY, GA
n¼ 609

Note that the example described earlier involv-

ing Model 3 is a special case of the formula for

theE, V, Wmodel, withX 3 equal toE, X 1 equal

to bothV 1 andW 1 , andX 2 equal to bothV 2 and

W 2. The linear functionlfor Model 3 is shown

here, both in its original form and in theE, V,

Wformat.

To obtain the confidence interval for the Model

3 example from the general formula, the fol-

lowing substitutions would be made in the gen-

eral variance formula:b¼b 3 ,p 2 ¼2,W 1 ¼X 1 ,

W 2 ¼X 2 ,d 1 ¼b 4 , andd 2 ¼b 5.

Before concluding this presentation, we illus-

trate the ML techniques described above by

way of a numerical example. We consider the

printout results provided below and on the fol-

lowing page. These results summarize the

computer output for two models based on fol-

low-up study data on a cohort of 609 white

males from Evans County, Georgia.

The outcome variable is coronary heart disease

status, denoted as CHD, which is 1 if a person

develops the disease and 0 if not. There are six

independent variables of primary interest. The

exposure variable is catecholamine level (CAT),

which is 1 if high and 0 if low. The other inde-

pendent variables are the control variables.

These are denoted as AGE, CHL, ECG, SMK,

and HPT.

The variable AGE is treated continuously. The

variable CHL, which denotes cholesterol level,

is also treated continuously. The other three

variables are (0, 1) variables. ECG denotes elec-

trocardiogram abnormality status, SMK denotes

smoking status, and HPT denotes hypertension

status.

EXAMPLE

E, V, Wmodel (Model 3):

X 3 ¼E,

X 1 ¼V 1 ¼W 1

X 2 ¼V 2 ¼W 2

^l¼^b 3 þ^b 4 X 1 þ^b 5 X 2

¼b^þ^d 1 W 1 þ^d 2 W 2

b¼b 3 ,

p 2 ¼2,W 1 ¼X 1 ,W 2 ¼X 2 ,

d 1 ¼b 4 , andd 2 ¼b 5

EXAMPLE

D¼CHD (0, 1)

E¼CAT

Cs¼AGE, CHL, ECG, SMK, HPT

(conts) (conts) (0, 1) (0, 1) (0, 1)

Model A Output:

2lnL^¼400.39

Variable Coefficient S.E. Chi sq P

Intercept 6.7747 1.1402 35.30 0.0000

CAT 0.5978 0.3520 2.88 0.0894

AGE 0.0322 0.0152 4.51 0.0337

Vs

8

>>>>

><

>>>>

>:

CHL 0.0088 0.0033 7.19 0.0073

ECG 0.3695 0.2936 1.58 0.2082

SMK 0.8348 0.3052 7.48 0.0062

HPT 0.4392 0.2908 2.28 0.1310

Unconditional ML estimation

n¼609, # parameters¼ 7

146 5. Statistical Inferences Using Maximum Likelihood Techniques