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

Scale factor¼f
Allows forextra variationin Y:

varðYÞ¼fVðmÞ

IfYbinomial:f¼1 and
V(m)¼m(1m)

f>1 indicates overdispersion

f<1 indicates underdispersion

Equations

Allow extra

variation?

QL: “score-like” Yes
GLM: score No

Summary: ML vs. QL Estimation

Step

ML

Estimation

QL

Estimation

1 Formulate L –
2 For eachb,
obtain
@ln L
@b

–

3 Form score
equations:
@lnL
@b

¼ 0



Form “score-
like”
equations
using
var(Y)
¼fV(m)
4 Solve for ML
estimates

Solve for QL

estimates

The scale factorfallows forextra variation(dis-

persion) in the response beyond the assumed

mean variance relationship of a binomial

response, i.e., var(Y)¼m(1m). For the bino-

mial distribution, the scale factor equals 1. If the

scale factor is greater (or less) than 1, then there

is overdispersion or underdispersion compared

to a binomial response. The “score-like” equa-

tions are therefore designed to accommodate

extra variation in the response, in contrast to

the corresponding score equations from a GLM.

The process of ML and QL estimation can be

summarized in a series of steps. These steps

allow a comparison of the two approaches.

ML estimation involves four steps:

Step 1. Formulate the likelihood in terms of the

observed data and the unknown parameters

from the assumed underlying distribution of

the random data

Step 2. Obtain the partial derivatives of the log

likelihood with respect to the unknown

parameters

Step 3. Formulate score equations by setting

the partial derivatives of the log likelihood

to zero

Step 4. Solve the system of score equations to

obtain the maximum likelihood estimates.

For QL estimation, the first two steps are

bypassed by directly formulating and solving

a system of “score-like” equations. These

“score-like” equations are of a similar form as

are the score equations derived for GLM. With

GLM, the response follows a distribution from

the exponential family, whereas with the

“score-like” equations, the distribution of the

response is not so restricted. In fact, the distri-

bution of the response need not be known as

long as the variance of the response can be

expressed as a function of the mean.

Presentation: XII. Generalizing the “Score-like” Equations to Form GEE Models 523