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test based on the original values of 95 observations. This is not a price
one would want to pay in human or animal experiments.
HIGHER-ORDER EXPERIMENTAL DESIGNS
Similar caveats hold for the parametric ANOVA approach to the analysis of
two-factor experimental design with two additions:
1. The sample sizes must be the same in each cell; that is, the design
must be balanced.
2. A test for interaction must precede any test for main effects.
Imbalance in the design will result in the confounding of main effects
with interactions. Consider the following two-factor model for crop yield:
X
ijk
=++++
mab g e
i
j
ij
jjk
Now suppose that the observations in a two-factor experimental design are
normally distributed as in the following diagram taken from Cornfield and
Tukey (1956):
()()
()()
NN
NN
01
,
21
,
21
,
01
,
There are no main effects in this example—both row means and both
column means have the same expectations, but there is a clear interaction
represented by the two nonzero off-diagonal elements.
If the design is balanced, with equal numbers per cell, the lack of signif-
icant main effects and the presence of a significant interaction should and
will be confirmed by our analysis. But suppose that the design is not in
balance, that for every 10 observations in the first column, we have only
one observation in the second. Because of this imbalance, when we use
the
F
ratio or equivalent statistic to test for the main effect, we will
uncover a false “row” effect that is actually due to the interaction
between rows and columns. The main effect is
confounded
with the
interaction.
If a design is unbalanced as in the preceding example, we cannot test
for a “pure” main effect or a “pure” interaction. But we may be able to
test for the combination of a main effect with an interaction by using the
statistic that we would use to test for the main effect alone. This com-
bined effect will not be confounded with the main effects of other unre-
lated factors.
Whether or not the design is balanced, the presence of an interaction
may zero out a cofactor-specific main effect or make such an effect impos-
