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FIGURE 5.3
A 2 ¥ 3 Design with Three Observations per Cell p Œ P c .
exchange of balls between both rows and columns (or no exchange at all),
(b) the numbers of each color within each column being the same in each
row, and (c) the numbers of each shape within each row being the same in
each column. P RC is the basis of a subgroup of P .
The only element these three subgroups P RC , P R ,and P C have in
common is the rearrangement that leaves the observations with the same
row and column labels they had to begin with. As a result, tests based on
these three different subsets of permutations are independent of one
another.
For testing H 3 : g ij = 0 for all i and j , determine the distribution of the
values of S =S i £ i ¢£ I 1 S j £ j ¢£ I 2 ( X ij + X i ¢ j ¢ - X i ¢ j - X ij ¢ ) with respect to the
rearrangements in P RC . If the value of S for the observations as they were
originally labeled is not an extreme value of this permutation distribution,
then we can accept the hypothesis H 3 of no interactions and proceed to
test for main effects.
For testing H 1 : a i = 0 for all i , choose one of the following test statistics
as we did in the section on one-way analysis, F 12 =S i (S j S k x ijk ) 2 , F 11 =
S i |S j S k x ijk |, or R 1 =S j g [ i ]S j S k x ijk , where g [ i ] is a monotone function of i ,
and determine the distribution of its values with respect to the rearrange-
ments in P R .
For testing H 2 : b j = 0 for all j , choose one of the following test statistics
as we did in the section on one-way analysis, F 22 =S j (S i S k x ijk ) 2 , F 21 =
S j |S i S k x ijk |, or R 2 =S i g [ j ]S i S k x ijk , where g [ j ] is a monotone function of j ,
and determine the distribution of its values with respect to the rearrange-
ments in P C .
Tests for the parameters of three-way and higher-order experimental
designs can be obtained via the same approach; use a multidimensional
lattice and such additional multivalued properties of the balls as charm and
spin. Proofs may be seen at http://users.oco.net/drphilgood/resamp.htm.
Unbalanced Designs
Unbalanced designs with unequal numbers per cell may result from
unanticipated losses during the conduct of an experiment or survey (or
from an extremely poor initial design). There are two approaches to their
analysis:
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