an introduction to generalized linear models - GDM@FUDAN ...

The least squares estimates are the solutions ofthe equations
∂S1
∂αj
∂S1
∂βj
= −2
= −2
K�
(yjk − αj − βjxjk) =0,
k=1
K�
xjk(yjk − αj − βjxjk) =0. (2.10)
k=1
The equations to be solved in (2.8) and (2.10) are the same and so maximizing
l1 is equivalent to minimizing S1. For the remainder ofthis example we will
use the least squares approach.
The estimating equations (2.10) can be simplified to
K�
K�
yjk − Kαj − βj xjk = 0,
k=1
k=1
K�
K�
xjkyjk − Kαj
k=1
K�
xjk − βj x
k=1
k=1
2 jk = 0
for j = 1 or 2. These are called the normal equations. The solution is
�
K k
bj = xjkyjk − ( � �
k xjk)( k yjk)
K �
k x2 ,
jk − (�k
xjk)
2
aj = yj − bjxj,
where aj is the estimate of αj and bj is the estimate of βj, for j = 1 or 2. By
considering the second derivatives of(2.9) it can be verified that the solution
ofequations (2.10) does correspond to the minimum ofS1. The numerical
value for the minimum value for S1 for a particular data set can be obtained
by substituting the estimates for αj and βj and the data values for yjk and
xjk into (2.9).
To test H0 : β1 = β2 = β against the more general alternative hypothesis
H1, the estimation procedure described above for model (2.7) is repeated
but with the expression in (2.6) used for µjk. In this case there are three
parameters, α1,α2 and β, instead offour to be estimated. The least squares
expression to be minimized is
J� K�
S0 = (yjk − αj − βxjk) 2 . (2.11)
j=1 k=1
From (2.11) the least squares estimates are given by the solution ofthe
simultaneous equations
K�
∂S0
= −2 (yjk − αj − βxjk) =0,
∂αj
∂S0
∂β
© 2002 by Chapman & Hall/CRC
= −2
k=1
J�
j=1 k=1
K�
xjk(yjk − αj − βxjk) =0, (2.12)