Over replications the only variability comes from the last term (eijk), which explains
why can be used as the denominator for all three Ftests. That will be important as
we go on.A Crossed Experimental Design with Fixed Variables
The original example is what we will class as a crossed experimental designwith fixed
factors. In a crossed design each level of one independent variable (factor) is paired with
each level of any other independent variable. For example, both older and younger partici-
pants are tested under each of the five recall conditions. In addition, the levels of the fac-
tors are fixed because these are the levels that we actually want to study—they are not, for
example, a random sample of ages or of possible methods of processing information.
Simply as a frame of reference, the results of the analysis of this study are shown in
Table 13.6. We see that was used as the test term for each effect, that it was based
on 90 df, and that each effect is significant at p,.05.A Crossed Experimental Design with a Random Variable
Now we will move from the study we just analyzed to one in which one of the factors is ran-
dom but crossed with the other factor. I will take an example based on one used by Judd and
McClelland (1989). Suppose that we want to test whether subjects are quicker to identify
capital letters than they are lower case letters. We will refer to this variable as “Case.” Case
here is a fixed factor. We want to use several different letters, so we randomly sample five of
them (e.g., A, G, D, K, W) and present them as either upper or lower case. Here Letter is
crossed with Case (i.e., each letter appears in each case), so we have a crossed design, but
we have randomly sampled Letters, giving us a random factor. Each subject will see only
one letter and the dependent variable will be the response time to identify that letter.
In this example Case takes the place of Age in Eysenck’s study and Letter takes the
place of Condition. If you think about many replications of this experiment, you would ex-
pect to use the same levels of Case (there are only two cases after all), but you would prob-
ably think of taking a different random sample of Letters for each experiment. This means
that the Fvalues that we calculate will vary not only on the basis of sampling error, but also
as a result of the letters that we happened to sample. What this means is that any interac-
tion between Case and Letter will show up in the expected mean squares for the fixed ef-
fect (Case). This will affect the expected mean squares for the effect of Case, and we need
to take that into account when we form our Fratios. (Maxwell & Delaney, 2004, p. 475 do
an excellent job of illustrating this phenomenon.)
To see the effect of random factors we need to consider expected mean squares, which
we discussed only briefly in Section 11.4. Expected mean squares tell us what is beingMSerrorMSerror432 Chapter 13 Factorial Analysis of Variance
Table 13.6 Analysis of variance of Eysenck’s basic fixed variable design
Source df SS MS F
A(Age) 1 240.25 240.250 29.94*
C(Condition) 4 1514.94 378.735 47.19*
AC 4 190.30 47.575 5.93*
Error 90 722.30 8.026Total 99 2667.79*p,.05crossed
experimental
design
expected mean
squares