Page 95 - Contributed Paper Session (CPS) - Volume 2

CPS1442 Uzuke C.A. et al.
would conform to a mixed effect model with one observation per cell that is
without replications, Esinga et al, (2017). Here this assumption is not necessary.
As in the Friedmans two-way ANOVA by ranks, the data being analysed
may be presented in the form of a two-way r  table with the row say
c
representing one factor A with r levels say and the columns say representing
the other factor B with c levels Daniels, (1990). Factors A and B may both be
fixed, both random, or one of the factor fixed and the other random. Here two
null hypotheses are to be tested. One is that there are no differences between
the c treatment effects for the column factor B and the second is that there
are no differences between the effects of the r levels of the row factor A.
To develop the test statistics for the column factor B based on the median
test we first pull all the observation in the table together and determine the
common median M as usual Oyeka & Uzuke, (2011).. Now if the jth level of
factor B has median M for j = I, 2, …, c then the null hypothesis to be tested
j
for factor B is

Ho: M1 = M2 = … = Mj … = Mc = M
(1)
Now let x be the score on observation on a randomly selected subject at
ij
the ith level of factor A and jth level of factor B on a certain numeric criterion
variable X, measured in at least the ordinal scale for I = 1, 2 , …, r; j = 1, 2, …, c
Let
 , 1 if x  M
 ij
U ij = 0 , if x ij = M (2)


 − , 1 if x ij  M
For i=1, 2, …, r; j = 1, 2, …, c
Let
+ 0 −
U
 j = P ( );  j = P ( U ij = ); 0  j = P ( U ij = − ) 1 (3)
ij
Where
+ 0 −
 j +  j +  j = 1 (4)
And
r
U j =  U (5)
ij
= i 1
Finally, let
c
r
c
W =  U j =  U (6)
ij
= j 1 = j 1 = i 1

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