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8.5 THE FACTORIAL EXPERIMENT 357
respectively. The subscript i runs from 1 to a and j runs from 1 to b. The total number
of observations is nab.
To show that Table 8.5.3 represents data from a completely randomized
design, we consider that each combination of factor levels is a treatment and that
we have n observations for each treatment. An alternative arrangement of the
data would be obtained by listing the observations of each treatment in a separate
column. Table 8.5.3 may also be used to display data from a two-factor
randomized block design if we consider the first observation in each cell as
belonging to block 1, the second observation in each cell as belonging to block
2, and so on to the nth observation in each cell, which may be considered as
belonging to block n.
Note the similarity of the data display for the factorial experiment as shown
in Table 8.5.3 to the randomized complete block data display of Table 8.3.1. The
factorial experiment, in order that the experimenter may test for interaction, requires
at least two observations per cell, whereas the randomized complete block design
requires only one observation per cell. We use two-way analysis of variance to analyze
the data from a factorial experiment of the type presented here.
2. Assumptions. We assume a fixed-effects model and a two-factor completely randomized
design. For a discussion of other designs, consult the references at the
end of this chapter.
The Model The fixed-effects model for the two-factor completely randomized design
may be written as
x ijk = m + a i + b j + 1ab2 ij +P ijk
i = 1, 2, Á , a; j = 1, 2, Á , b; k = 1, 2, Á , n
(8.5.1)
where x ijk is a typical observation, m is a constant, a represents an effect due to factor A, b
represents an effect due to factor B, 1ab2 represents an effect due to the interaction of factors
A and B, and represents the experimental error.
Assumptions of the Model
a. The observations in each of the ab cells constitute a random independent sample
of size n drawn from the population defined by the particular combination
of the levels of the two factors.
b. Each of the ab populations is normally distributed.
c. The populations all have the same variance.
3. Hypotheses. The following hypotheses may be tested:
a.
b.
c.
P ijk
H 0 : a i = 0
H A : not all a i = 0
H 0 : b j = 0
H A : not all b j = 0
H 0 : 1ab2 ij = 0
H A : not all 1ab2 ij = 0
i = 1, 2, Á , a
j = 1, 2, Á , b
i = 1, 2, Á , a; j = 1, 2, Á , b