Page 98 - Contributed Paper Session (CPS) - Volume 2

CPS1442 Uzuke C.A. et al.
Now following a similar approach as above we note that the observed
frequencies of occurrence of 1s, 0s and -1s in the ith level of factor A are
respectively
−
−
+
O i1 = t ; O i2 = t ; O i3 = t i 0 = c −t i + −t (15)
i
i
i
The expected frequencies under the null hypothesis of equation 14 are
respectively
ct + ct − ct 0 c (rc − t + t − − )
E = ; E = ; E = = (16)
i1
rc i2 rc i3 rc rc
As above if the null hypothesis of equation 14 is true, then the test statistic
 t + − ct +  2  − − ct −  2  0 − ct 0  2



t
t



r
r
r
3

r


2
 =  (O ij − E ij ) 2 =   i rc  +   i rc  +   i rc 
+
−
0
= i 1 = j 1 E ij = i 1 ct = i 1 ct = i 1 ct
rc rc rc
which has approximately chi-square distribution with 2(r-1) degrees of
freedom
Now using equations 15 and 16 and the corresponding probabilities with the
above expression we have as before
)
)
+
r
r
r
2
 = c  (p i + − p + 2 + c  (p i − − p − 2 +c  ( ( p i + − p + ) (p i − − − p − )) 2 (17)
+
−
+
= i 1 p = i 1 p = i 1 1 − p − p
When equation 17 is further simplified and evaluated as before we obtain the
test statistics for the null hypothesis in equation 14 as
c  r  2  r  2  r  
−
+

+
−
2
 = p 1 − p −  p + + ) +  p 1 − p +  p − − ) +  2 + p −  ( − p + )( − p − )   (18)
( ) ( − p
 p
p
( ) ( − p
p
+ p p − ( − p + − − ) p   = i 1 i   = i i   = i 1 i i   
1
which under the null hypothesis has approximately a chi-square distribution
with 2(r-1) degrees of freedom for sufficiently large r and c
Note as before that if no observations are exactly equal to the common
median M then the test statistic of equation 17 now reduces to simply
n
c  (p i + − p + ) 2
 2 = = i 1
+
( − p
p 1 + )
That is
r
r
2
c  (p − ) p 2 c  p − rc p 2
i
i
 2 = i=1 = i=1 (19)
p q p q
Now with only r-1 degrees of freedom where p = p; p = p = p; q =1 − p
+
+
i

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