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

Importance ofV^ð^uÞ:

inferences require accounting

for variability and covariability

(3) Variable listing
Variable ML Coefficient S. E.

Intercept ^a s^a
X 1 ^b 1 s^b 1



Xk ^bk s^bk

The variance–covariance matrix is important

because the information contained in it is

used in the computations required for hypoth-

esis testing and confidence interval estimation.

In addition to the maximized likelihood value

and the variance–covariance matrix, other

information is also provided as part of the out-

put. This information typically includes, as

shown here,a listing of each variable followed

by its ML estimate and standard error. This

information provides another way to carry

out hypothesis testing and interval estimation.

Moreover, this listing gives the primary infor-

mation used for calculating odds ratio esti-

mates and predicted risks. The latter can only

be done, however, if the study has a follow-up

design.

An example of ML computer output giving the

above information is provided here. This out-

put considers study data on a cohort of 609

white males in Evans County, Georgia, who

were followed for 9 years to determine coro-

nary heart disease (CHD) status. The output

considers a logistic model involving eight vari-

ables, which are denoted as CAT (catechol-

amine level), AGE, CHL (cholesterol level),

ECG (electrocardiogram abnormality status),

SMK (smoking status), HPT (hypertension sta-

tus), CC, and CH. The latter two variables are

product terms of the form CC¼CATCHL

and CH¼CATHPT.

The exposure variable of interest here is the

variable CAT, and the five covariables of inter-

est, that is, theCs are AGE, CHL, ECG, SMK,

and HPT. Using ourE, V, Wmodel framework

introduced in Chapter 2, we haveEequals CAT,

the five covariables equal to theVs, and twoW

variables, namely, CHL and HPT.

The output information includes2 times the

natural log of the maximized likelihood value,

which is 347.23, and a listing of each variable

followed by its ML estimate and standard

error. We will show the variance–covariance

matrix shortly.

EXAMPLE

Cohort study – Evans Country, GA

n¼609 white males

9-year follow-up

D¼CHD status

Output:2lnL^¼347.23

Variable

ML

Coefficient S. E.

Intercept 4.0497 1.2550

CAT 12.6894 3.1047

AGE 0.0350 0.0161

CHL 0.0055 0.0042

Vs ECG 0.3671 0.3278

SMK 0.7732 0.3273

8

>>>

>>>

>><

>>>

>>>

>>:

HPT 1.0466 0.3316

CC 0.0692 0.3316

CH 2.3318 0.7427

CC = CAT × CHL and CH = CAT × HPT

Ws

118 4. Maximum Likelihood Techniques: An Overview