Proof of Theorem 1

SSW

SSB  SSW
(1  c) 2
4

 1nn
0
 ( m n )n
0n( m n ) 

0( m n )( m n ) 
Therefore, the meaning of  is clear. It compares the difference between the covariance matrix SSW with
the difference matrix SSB. When this difference is big, we say that there are significant changes in the
mean of the two populations.
Let us have some more notations:
(1  c) 2
1 
4
 1nn
0
 ( mn )n
0 n( mn ) 
12 

    11


0 ( mn )( mn ) 
 21  22 
Where

(1  c) 2
1


4
11  
2
   (1  c)

4
12
(1  c) 2 

4 

 ;
2
(1  c) 
1
4  nn
1 
 1

 

;  21  12 ' ;

 1
1  n( m  n )

 22

1

 


 
1  ( m  n )( m  n )
Remembering that 1  11  22   2111112 , and after some detailed calculations, we arrive at:
  [1  (m  2)   (n  1)(m  n  1)  2  n(m  n) 12 ](1   ) m2
And
 n(( m  n  1)   1)(1  c) 2

 n(m  n) 12  (1   ) m 2

4


 [1  (m  2)   (n  1)( m  n  1)  2 ](1   ) m 2 , when m  2n
1  
2
  n(( n  1)   1)(1  c)  n(m  n)  2  (1   ) m  2
1 
 
4


2
m2
  [1  (m  2)   (n  1)( m  n  1)  ](1   ) when m  2n
It is then concluded that Wilks lambda is given by
1  (m  2)   (n  1)( m  n  1)  2  n(m  n) 12
n(( m  n  1)   1)(1  c) 2
1  (m  2)   (n  1)( m  n  1)  2 
 n(m  n) 12
4
(3)
When m  2n , and
1  (m  2)   (n  1)( m  n  1)  2  n(m  n) 12
n(( n  1)   1)(1  c) 2
1  (m  2)   (n  1)( m  n  1)  2 
 n(m  n) 12
4
(4)
When m  2n .
Eq. (3) and (4) yield some interesting conclusions. First of all, let us check whether the significance
score is a decreasing function of m when m  n , as we have conjectured in [3]. To this end, we first need
to calculate the derivatives of Wilks lambda with respect to m, when m  n .
Denote B  1  (m  2)   (n  1)(m  n  1)  2  n(m  n) 12 , then for m  2n

B
n(( m  n  1)   1)(1  c) 2
B
4
And
' 
For m  2n
n 2 12 (1  c) 2 (1   )

n(( m  n  1)   1)(1  c) 2 
4 B 

4


2

B
n(( n  1)   1)(1  c) 2
B
4
And
' 
n[   (n  1)  2  n12 ][( n  1)   1](1  c) 2

n(( n  1)   1)(1  c) 2 
4 B 

4


Obviously,   1, 1   , for both cases we conclude that
increasing function with respect to m, when
m  2n .
' 0 ,
2
which indicates that  is a monotonically
Then the significance score S ( A) is monotonically
decreasing with respect to m.
Secondly we want to check whether the significance score is an increasing function of n when m is fixed.
Again, we calculate the derivative of Wilks lambda with respect to n, then for
'  4
n
m
2
(1  c) 2 (1   )( n1  1    n  m )( n1  1    n  m )
4
2

n(( m  n  1)   1)(1  c) 2 
B 

4


2
(1  c) (1   )(   1    (2n  m))( n1  1     (m  n))

n(( m  n  1)   1)(1  c) 2 
B



4


2
0
For
m
nm
2
'  4
(1  c) 2 (1  (m  1)  )( 1  1    (n  1)( 1   ))( n1  1  (n  1)  )

n(( m  n  1)   1)(1  c) 2 
B



4


2
0
For both cases, ' 0 , we see that Wilks lambda is a decreasing function with respect to n when m is
fixed. Then the significant score
S ( A)
is a monotonically increasing function with respect to n.
Thirdly, for other fixed parameters, we can prove that the significance score is also a decreasing function
with respect to  .
Fourthly, we want to find out how to detect the hot-spot in our setup. Here the hot-spot is {1,2,, n} . Let
A  {1,2, , i}
and   0 . When i  n we see
( A) 
1  (i  2)   (n  1)( 1)  2
i(( 1)   1)(1  c) 2
2
1  (i  2)   (i  1)( 1)  
4
When i  n , we then have an extra term
n(i  n) 12
in the expression of Λ, i.e.
1  (i  2)   (n  1)(i  n  1)  2  n(i  n) 12
( A) 
n((i  n  1)   1)(1  c) 2
1  (i  2)   (n  1)(i  n  1)  2 
 n(i  n) 12
4
We thus conclude that the derivative of
 ( A)
(5)
is not continuous when
in
(6)
and   0 . When   0 , Eq. (5)
turns out to be
When
in
( A) 
1
i (1  c) 2
1
4
( A) 
1
n(1  c) 2
1
4
and
When i  n . Again the derivative of
have proved Theorem 1.
 ( A)
is not continuous when i  n . Summarizing the results above, we