4.4 Comparison of two or more Poisson means
The connection between log-linear models for frequencies and
multinomial response models for proportions stems from the fact that
the binomial and multinomial distributions can be derived from a set
of independent Poisson random variables conditionally on their total
being fixed.
Suppose that Y  Y1 , Y2 ,, YK  are independent Poisson random
 
variables with means 1 ,  2 , ,  k , log  j   0  1 x j , and that
we require to test the composite null hypothesis
H 0 : 1   2     k  exp  0  
H 0 : log 1   log  2     log  k    0 
H 0 : 1  0
where x j ’s are given constants. The alternative hypotheses under
consideration are
H a : 1  0 ,
or
H a : 1  0
or
H a : 1  0 .
Standard theory of significance testing leads to consideration of the
test statistic T Y  
k
x Y
j 1
j
j
conditionally on the observed value of
k
m   Y j , which is the sufficient statistic for  0 . For example,
j 1
as H a
: 1  0
, the test is
T  y   C0 m , reject H 0
T  y   C0 m , reject H 0 with probabilit y wm 
T  y   C0 m , do not reject H 0
1
given m 
k
Y
j 1
j
, where
C0 m 
wm
and
can be obtained by
solving
k


P T Y   C0 m  |  Y j  m, H 0 is true 
j 1


k



 wm P T Y   C0 m  |  Y j  m, H 0 is true   
j 1


and
wm
is some constant depending on m.
Note that under null hypothesis, we regard the data as having
multinomial distribution with index m and parameter vector
1
1 1
 , ,  ,  independent of  0 , i.e.,
k
k k
k



P
Y

y
,

,
Y

y
|
Y

m
,
H
is
true

1
1
k
k
j
0


j 1


k

m
m!
k
y
j
!
1
  k 
yj

j 1
j 1

y
j
!
j 1
m!
k
y
m!
k
 yj
 1  j 1
 
k
j
!
1
km
j 1
Note:
Let Y  Y1 , Y2 ,, YK  has the probability density function or
probability distribution function
r


f  y   f  y1 , y2 ,, yk   C  ,   exp T  y     j M j  y  h y 
j 1


where
  1 , 2 ,, r  and M  y   M1  y , M 2  y ,, M r  y  .
2
H 0 :    0 v.s. H a :    0 , an UMP (uniformly
Then, for testing
most powerful) unbiased level-test given
M Y   m
is
T  y   C0 m , reject H 0
T  y   C0 m , reject H 0 with probabilit y wm 
T  y   C0 m , do not reject H 0
where the constants
C0 m 
and
wm
depending on m can be
obtained by solving
PT Y   C0 m  | M Y   m, H 0 is true 
 wm PT Y   C0 m  | M Y   m, H 0 is true   
Therefore, for independent Poisson random variables Y1 , Y2 , , Yk ,
 
with means 1 ,  2 , ,  k , log  j   0  1 x j , the probability
distribution function is
k
f y  
j 1
j
yj
y j!
e
 j


 k

 k
1
y 
 exp     j  exp  log  j j  k
 j 1

 j 1
  y j!
j 1
 k

 k
 1
 exp     j  exp  y j log  j  k
 j 1

 j 1
  y j!
j 1
 k

 k
 1


 exp     j  exp  y j  0  1 x j  k
 j 1

 j 1
  y j!
j 1
k
k
 k


 1
 exp     j  exp  1  x j y j   0  y j  k
j 1
 j 1

 j 1
  y j!
 C 1 ,  0  exp 1T  y    0 M  y h y 
3
j 1
where
k
k
 k



C 1 ,  0   exp     j , T  y    x j y j , M  y    y j , h y  
j 1
j 1
 j 1

1
k
y !
j
j 1
Note:
Under
H 0 : 1  0 , the unconditional moments of T Y 
are
 k
  yj
k
k
k
 k

E T Y   E   x jY j    x j E Y j    x j exp  0    x j  j 1
k
j 1
j 1
j 1
 j 1









k
y
 
since E Y j   j  exp  0  and
j 1
j
is the estimate of
k
j.
Also,
 k

y


j 
k
k
 k
 k 2
j

1


VarT Y   Var  x jY j    x j Var Y j    x 2j exp  0    x 2j 
k 
j 1
j 1
 j 1
 j 1




Note that the unconditional moments of
T Y  depend on  0 . On
the other hand, under the null hypothesis, denote
Y 
k
Y j , y 
j 1
k
y
j 1
j
 m.
Then,
 k

E T Y  | Y  y   E   x j Y j | Y  y  
 j 1

k
y 
  xj  
 k 
j 1
 k
  yj
k
j 1
  x j 
k
j 1


4






 x E Y
k
j 1
j
j
| Y  y 


since Y j | Y  y ~ B y ,
The
unconditio
1
y
  E Y j | Y  y    .
k
k
estimate
of
the
VarT Y  | Y  y 
 k

 Var   x jY j | Y  y 
 j 1

  Var x jY j | Y  y   2 Covx jY j , xiYi | Y  y 
k
j 1
j i
1
  x 2j y 1 
k
j 1
1
1 1

  2  xi x j   y   
k
k k

j i
y  k 2 

  x j 1 
k  j 1 
1
1
  2 xi x j 
k
k
j i
k

y
k
 k 2 1 k 2



x

x

2
x
x
 j

i j 
 j
k
j i
 j 1
 j 1

2
 
y  k 2 1  k
 x j    x j  

k  j 1
k  j 1  





y  k 2

2

 x j  kx   denote x 
k  j 1
 


k
y
2
   x j  x 
k j 1
 k
  yj
k
2  j 1
  x j  x  
k
j 1


k
x
j 1
k






5
j






 CovY j , Yi | Y  y 


1 1


   y  , j  i

k k


nal variance of
T Y  is quite different from the exact conditional
variance. The conditional variance is unaffected by the addition of a
constant to each component of
x j ’s. ◆
The Poisson log-likelihood function for  0 ,  1  in this problem is
lY  0 , 1     y j log  j    j 
k
j 1


  y j  0  1 x j   exp  0  1 x j 
k
j 1
  0  y j  1  x j y j   exp  0  1 x j 
Denote  
k
k
k
j 1
j 1
j 1
 exp 
k
j 1
 1 x j     j .
k
0
j 1
Then, the log-likelihood for  , 1  becomes
lY  , 1    0  y j  1  x j y j   exp  0  1 x j 
k
k
k
j 1
j 1
j 1
k
  0 y   1  x j y j  
j 1
k
 y log      1  x j y j   y log     0 y 
j 1
  


 y log      1  x j y j  y log 


exp

j 1
0 
 
k
 k

 y log      1  x j y j  y log  exp 1 x j 
j 1
 j 1

 lY    lY |Y 1 
k
6
where
lY    y log    
is the Poisson log-likelihood for  based on m  y  Y ~ P  and
 k

lY |Y 1   1  x j y j  y log  exp 1 x j 
j 1
 j 1

k
is the multinomial log-likelihood for  1 based on conditional
distribution, Y1 , , Yk | Y  m ~ Mm,  1 , ,  k  , Mm,  1 , ,  k  is
a multinomial distribution with index m and parameters
j 
exp 1 x j 
 exp 1 x j 
k
.
j 1
Note:
The Poisson marginal likelihood based on Y mainly depends only
on 
while the multinomial conditional likelihood based on
Y1 ,,Yk | Y depends only on 1 . Provided that no information is
available concerning the value of  0 and consequently of  , then
all of the information concerning  1 is in the conditional likelihood
based on Y1 ,,Yk | Y .
Note:
The Fisher’s information for  , 1  is
7
1

0
 

k
2
 0
 j x j  x  



j 1


and these parameters are said to be orthogonal. Under suitable
limiting conditions, the estimate ˆ and ˆ1 must be approximately
independent.
8