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

SLR: var(Y)¼m(1m)

GEE logistic regression

varðYÞ¼fmð 1 mÞ

fdoes not affect^b

faffectss^biff 6 ¼ 1

f>1: overdispersion

f<1: underdispersion

aandbestimated iteratively:

Estimatesupdated alternately

) convergence

To run GEE model, specify:

 g(m)¼link function
 V(m)¼mean variance
relationship

 Ci¼working correlation
structure

GLM – no specification of a corre-
lation structure

GEE logistic model:

logit PðD¼ 1 jXÞ¼b 0 þ~

p

h¼ 1

bhXh

acan affect estimation ofbands^b

but

b^iinterpretation same as SLR

For a standard logistic regression (SLR), the

variance of the response variable is assumed

to be m(1m), whereas for a GEE logistic

regression, the variance of the response vari-

able is modeled asfm(1m) wherefis the

scale factor. The scale factor doesnotaffect

the estimate of the regression parameters but

it does affect their standard errors (s^b) if the

scale factor is different from 1. If the scale

factor is greater than 1, there is an indication

of overdispersion and the standard errors of

the regression parameters are correspondingly

scaled (inflated).

For a GEE model, the correlation parameters

(a) are estimated by making use of updated

estimates of the regression parameters (b),

which are used to model the mean response.

The regression parameter estimates are, in

turn, updated using estimates of the correla-

tion parameters. The computational process is

iterative, by alternately updating the estimates

of the alphas and then the betas until conver-

gence is achieved.

The GEE model is formulated by specifying a

link function to model the mean response as a

function of covariates (as in a GLM), a variance

function which relates the mean and variance

of each response, and a correlation structure

that accounts for the correlation between

responses within each cluster. For the user,

the greatest difference of running a GEE

model as opposed to a GLM is the specification

of the correlation structure.

A GEE logistic regression is stated in a similar

manner as a SLR, as shown on the left. The

addition of the correlation parameters can

affect the estimation of the beta parameters

and their standard errors. However, the inter-

pretation of the regression coefficients is the

same as in SLR in terms of the way it reflects

the association between the predictor variables

and the outcome (i.e., the odds ratios).

Presentation: XII. Generalizing the “Score-like” Equations to Form GEE Models 527