3.5 Deviance function and over-dispersion
(a) Deviance function
The full model is the model with different parameters
case, the estimate
~
ij
is
yij
~
 ij 
mi
certain link function, for example,
has the parameter estimate
 ij . In this
. The reduced model with
rij 

exp xij  


1  exp xij  
,
ˆ ij . Then, the deviance function is
D~, ˆ   2l ~   2l ˆ   2 yij log ~ij  2 yij log ˆ ij 
n
k
i 1 j 1
n
k
i 1 j 1
 yij 

 2 y ij log 


i 1 j 1
 miˆ ij 
n k
 y ij 
 2 y ij log  
 ˆ 
i 1 j 1
 ij 
n
where
k
~  ~11  ~1k  ~n1  ~nk ,
ˆ  ˆ11  ˆ1k  ˆ n1  ˆ nk  ,
and
ˆ ij  miˆ ij .
Under some regularity conditions (similar to
chapter 2), the deviance function has an approximate
distribution.
1
2
(b) Over-dispersion
Over-dispersion for polytomous responses can occur in exactly the
same way as over-dispersion for binary responses. Under the
cluster-sampling model, the covariance matrix of the observed
response vector is the sum of the within-cluster covariance matrix
and the between-cluster covariance matrix. Provided that these two
matrices are proportional, we have
E Y   m , CovY    2  ,
where

is the usual multinomial covariance matrix. The main
problem now is to estimate
 2 . The sensible estimate is
~ 2 
where
X2
X2
nk  1  p ,
is Pearson’s statistic and p is the number of unknown
parameters.
2