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

Sh+1 = ∂mi[var(Yi)]–1[Yi – mi] = 0

bh

i= 1

K

Σ

partial
derivative

variance residual

Solution: iterative (by computer)

GLM score equations:

 Completely specified byE(Yi)
and var(Yi)

 Basis of QL estimation

QL estimation:

 “Score-like” equations

 No likelihood

var(Yi) = fV(mi)

scale

factor

function of m

 gðmÞ¼b 0 þ~

p

h¼ 1

bhXh

 solution yields QL estimates

Logistic regression:Y¼(0, 1)

m¼PðY¼ 1 jXÞ

VðmÞ¼PðY¼ 1 jXÞ½ 1 PðY¼ 1 jXފ

¼mð 1 mÞ

The (hþ1)st score equation (Shþ 1 ) is written

as shown on the left. For each score equation,

theith subject contributes a three-way product

involving the partial derivative of mi with

respect to a regression parameter, times the

inverse of the variance of the response, times

the difference between the response and its

mean (mi).

The process of obtaining a solution to these

equations is accomplished with the use of a

computer and typically is iterative.

A key property for GLM score equations is that

they are completely specified by the mean and

the variance of the random response. The

entire distribution of the response is not really

needed. This key property forms the basis of

quasi-likelihood (QL) estimation.

Quasi-likelihood estimating equations follow

the same form as score equations. For this

reason, QL estimating equations are often

called “score-like” equations. However, they

are not score equations because the likelihood

is not formulated. Instead, a relationship

between the variance and mean is specified.

The variance of the response, var(Yi), is set

equal to ascale factor(f) times a function of

the mean response,V(mi). “Score-like” equa-

tions can be used in a similar manner as score

equations in GLM. If the mean is modeled

using a link functiong(m), QL estimates can

be obtained by solving the system of “score-

like” equations.

For logistic regression, in which the outcome is

coded 0 or 1, the mean response is the proba-

bility of obtaining the event, P(Y¼1|X). The

variance of the response equals P(Y¼1|X)

times 1 minus P(Y¼1|X). So the relationship

between the variance and mean can be

expressed as var(Y)¼fV(m) whereV(m) equals

mtimes (1m).

522 14. Logistic Regression for Correlated Data: GEE