Testing Hypotheses: Choosing a Test Statistic - Common Errors in Statistics - page 68

FIGURE 5.1 A 2 ¥ 3 Design with Three Observations per Cell.

FIGURE 5.2

A 2 ¥ 3 Design with Three Observations per Cell after p Œ P R .

The recent efforts of Salmaso [2003] and Pesarin [2001] have resulted
in a breakthrough that extends to higher-order designs. The key lies in the
concept of weak exchangeability with respect to a subset of the possible
permutations. The simplified discussion of weak exchangeability presented
here is abstracted from Good [2003].
Think of the set of observations { X ijk } in terms of a rectangular lattice
L with K colored, shaped balls at each vertex. All the balls in the same
column have the same color initially, a color which is distinct from the
color of the balls in any other column. All the balls in the same row have
the same shape initially, a shape which is distinct from the shape of the
balls in any other row. See Fig. 5.1.
Let P denote the set of rearrangements or permutations that preserve
the number of balls at each row and column of the lattice. P is a group. 7
Let P R denote the set of exchanges of balls among rows and within
columns which (a) preserve the number of balls at each row and column
of the lattice and (b) result in the numbers of each shape within each row
being the same in each column. P R is the basis of a subgroup of P . See
Fig. 5.2.
Let P C denote the set of exchanges of balls among columns and within
rows which (a) preserve the number of balls at each row and column of
the lattice and (b) result in the numbers of each color within each column
being the same in each row. P C is the basis of a subgroup of P . See Fig.
5.3.
Let P RC denote the set of exchanges of balls that preserve the number
of balls at each row and column of the lattice and which result in (a) an

7 See Hungerford [1974] or http://members.tripod.com/~dogschool/ for a thorough dis-
cussion of algebraic group properties.