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

XI. Score Equations and
“Score-like” Equations

L¼likelihood function

ML solves estimating equations
calledscore equations.

S 1 ¼

@lnL

@b 0

¼ 0

S 2 ¼

@lnL

@b 1

¼ 0







Spþ 1 ¼

@lnL

@bp

¼ 0

9

>>>

>>

>>

>>

>>

>>>

>=

>>

>>

>>>

>>

>>

>>

>>

;

pþ 1 equations in

pþ 1 unknowns

ðbsÞ

In GLM, score equations involve
mi¼E(Yi) and var(Yi)

K¼# of subjects

pþ 1 ¼# of parameters
(bh,h¼0, 1, 2,...,p)

Yields pþ1 score equations
S 1 ,S 2 ,...,Spþ 1

(see formula on next page)

The estimation of parameters often involves

solving a system of equations called estimating

equations. GLM utilizes maximum likelihood

(ML) estimation methods. The likelihood is a

function of the unknown parameters and the

observed data. Once the likelihood is formu-

lated, the parameters are estimated by finding

the values of the parameters that maximize the

likelihood. A common approach for maximiz-

ing the likelihood uses calculus. The partial

derivatives of the log likelihood with respect to

each parameter are set to zero. If there are

pþ1 parameters, including the intercept,

then there arepþ1 partial derivatives and,

thus,pþ1 equations. These estimating equa-

tions are calledscore equations. The maximum

likelihood estimates are then found by solving

the system of score equations.

For GLM, the score equations have a special

form due to the fact that the responses follow

a distribution from the exponential family.

These score equations can be expressed in

terms of the means (mi) and the variances

[var(Yi)] of the responses, which are modeled

in terms of the unknown parameters (b 0 ,b 1 ,

b 2 ,...,bp), and the observed data.

If there areKsubjects, with each subject con-

tributing one response, andpþ1 beta para-

meters (b 0 ,b 1 ,b 2 ,...,bp), then there arepþ 1

score equations, one equation for each of the

pþ1 beta parameters, with bh being the

(hþ1)st element of the vector of parameters.

Presentation: XI. Score Equations and “Score-like” Equations 521