Di¼diagonal matrix, with vari-
ance functionV(mij) on diagonal
Wi¼working covariance matrix
(nini)
Wi¼fD(^12)
iCiD
(^12)
i
GEE: form similar to score
equations
IfK¼# of subjects
ni¼# responses of subjecti
pþ 1 ¼# of parameters
(bh;h¼0, 1, 2,...,p)
GEEh+1 = [Wi]–1[Yi – mi] = 0
∂mi
i= 1 bh
K
Σ
partial
derivative
covariance
residuel
where
Wi¼fD
(^12)
iCiD
(^12)
i
Yieldspþ1 GEE equations of the
above form
Di is a diagonal matrix whosejth diagonal
(representing thejth observation of theith sub-
ject) is the variance functionV(mij). An example
with three observations for subjectiis shown
at left. As a diagonal matrix, all the off-diagonal
entries of the matrix are 0. SinceV(mij)isa
function of the mean, it is also a function of
the predictors and the regression coefficients.
Wiis anninivariance–covariance matrix for
the ith subjects’ responses, often referred to
asthe working covariance matrix. The variance–
covariance matrixWican be decomposed into
the scale factor (f), times the square root ofDi,
timesCi,timesthesquarerootofDi.
The generalized estimating equations are of a
similar form as the score equations presented
in the previous section. If there areKsubjects,
with each subject contributingniresponses, and
pþ1 beta parameters (b 0 ,b 1 ,b 2 ,...,bp), with
bhbeing the (hþ1)st element of the vector
of parameters, then the (hþ1)st estimating
equation (GEEhþ 1 ) is written as shown on
the left.
There are pþ1 estimating equations, one
equation for each of thepþ1 beta parameters.
The summation is over theKsubjects in the
study. For each estimating equation, theith
subject contributes a three-way product involv-
ing the partial derivative ofmiwith respect to a
regression parameter, times the inverse of the
subject’s variance–covariance matrix (Wi),
times the difference between the subject’s
responses and their mean (mi).
EXAMPLE
ni¼ 3
Di¼
Vðmi 1 Þ 00
0 Vðmi 2 Þ 0
00 Vðmi 3 Þ
2
(^64)
3
(^75)
Presentation: XII. Generalizing the “Score-like” Equations to Form GEE Models 525