Statistical Methods for Psychology

use the average sample size, and that is actually quite close. Actually you will use the

harmonic mean of the sample sizes. The harmonic mean is defined as

where the subscript “h” stands for “harmonic” and krepresents the number of observations

whose mean we are calculating. You can now use the formulae shown in Table 13.2 by re-

placing nwith nhand the row and column means with the means of the cells in those rows

and columns. For the current example the row means would be 17 and 17, the column

means would be 14 and 20, and the grand mean would be the mean of the cell means. The

one difference is that the error term ( ) is not obtained by subtraction; instead, we

calculate for each cell of the design and then sum these terms to obtain the sum

of squares due to error.

I am not recommending that you solve your problem with unbalanced designs this way,

although the answer would be very close to the answer given by the solution that I will rec-

ommend in Chapter 15. I present this approach here because I think that it helps to clarify

what SPSS and SAS do when you have unequal sample sizes and select the default option

(Type III sum of squares). I think that it also makes it easier to understand how a column

effect can actually show up as a row effect even when the cell means within columns do

not differ by row.

13.12 Higher-Order Factorial Designs

All of the principles concerning a two-way factorial design apply equally well to a three-

way or higher-order design. With one additional piece of information, you should have no

difficulty running an analysis of variance on any factorial design imaginable, although the

arithmetic becomes increasingly more tedious as variables are added. We will take a sim-

ple three-way factorial as an example, since it is the easiest to use.

The only major way in which the three-way differs from the two-way is in the presence

of more than one interaction term. To see this, we must first look at the underlying struc-

tural model for a factorial design with three variables:

In this model we have not only main effects, symbolized by , , and , but also two

kinds of interaction terms. The two-variable or first-order interactionsare , , and

, which refer to the interaction of variables Aand B, Aand C, and Band C, respec-

tively. We also have a second-order interactionterm, , which refers to the joint

effect of all three variables. We have already examined the first-order interactions in dis-

cussing the two-way. The second-order interaction can be viewed in several ways. Proba-

bly the easiest way to view the ABC interaction is to think of the ABinteraction itself

interacting with variable C. Suppose that we had two levels of each variable and plotted

theABinteraction separately for each level of C. We might have the result shown in

Figure 13.3. Notice that for we have one ABinteraction, whereas for we have a dif-

ferent one. Thus, ABdepends on C, producing an ABCinteraction. This same kind of

reasoning could be invoked using the ACinteraction at different levels of B, or the BC

interaction at different levels of A. The result would be the same.

As I have said, the three-way factorial is merely an extension of the two-way, with a

slight twist. The twist comes about in obtaining the interaction sums of squares. In the

C 1 C 2

abgijk

bgjk

abijagik

ai bj gk

Xijkl=m1ai1bj1gk1abij1agik1bgjk1 abgijk 1 eijkl

SSwithin cell

SSerror

Xh=

k

1

X 1

1

1

X 2

1

1

X 3

1 Á 1

1

Xk

446 Chapter 13 Factorial Analysis of Variance

first-order
interactions

second-order
interaction