one of four ways, ranging from the simple counting of letters in words to forming a
visual image of each word. Participants in a fifth Condition were not given any in-
structions about what to do with the items. A second dimension of the experiment
compared Younger and Older participants in terms of recall, thus forming a 2 3 5 fac-
torial design.
The dependent variable was the number of items recalled after three presentations of
the material. There was a significant Age effect (F(1,90) 5 29.94, p,.05, v^25 .087),
with younger participants recalling more items than older ones. There was also a signifi-
cant effect due to Condition (F(4,90) 5 47.19, p,.05, v^25 .554), and visual inspec-
tion of the means shows that there was greater recall for conditions in which there was a
greater degree of processing. Finally the Age by Condition interaction was significant
(F(4,90) 5 5.93, p,.05, v^25 .059), with a stronger effect of Condition for the
younger participants.
A contrast of lower levels of processing (Counting and Rhyming) with higher levels
of processing (Adjective and Imagery) produced a clearly statistically significant effect
in favor of higher levels of processing (t(90) 5 8.04, p,.05). This corresponds to an
effect size of 5 2.07, indicating that participants with higher levels of processing out-
perform those with lower levels of processing by over two standard deviations. This
effect is even greater if we look only at the younger participants, where 5 2.63.13.11 Unequal Sample Sizes
Although many (but certainly not all) experiments are designed with the intention of hav-
ing equal numbers of observations in each cell, the cruel hand of fate frequently intervenes
to upset even the most carefully laid plans. Participants fail to arrive for testing, animals
die, data are lost, apparatus fails, patients drop out of treatment, and so on. When such
problems arise, we are faced with several alternative solutions, with the choice depending
on the nature of the data and the reasons why data are missing.
When we have a plain one-way analysis of variance, the solution is simple and we have
already seen how to carry that out. When we have more complex designs, the solution is
not simple. With unequal sample sizes in factorial designs, the row, column, and interac-
tion effects are no longer independent. This lack of independence produces difficulties in
interpretation, and deciding on the best approach depends both on why the data are miss-
ing and how we conceive of our model.
There has been a great deal written about the treatment of unequal sample sizes, and
we won’t see any true resolution of this issue for a long time. (That is in part because
there is no single answer to the complex questions that arise.) However, there are some
approaches that seem more reasonable than others for the general case. Unfortunately,
the most reasonable and the most common approach is available only using standard
computer packages, and a discussion of that will have to wait until Chapter 15. I will,
however, describe a pencil-and-paper solution. This approach is commonly referred to as
an unweighted means solution or an equally weighted means solution because we
weight the cell means equally, regardless of the number of observations in those cells.
My primary purpose in discussing this approach is not to make you get out your pencil
and a calculator, but to help provide an understanding of what SPSS and SAS do if you
take the default options. Although I will not work out an example, such an example can
be found in Exercise 13.17. And, if you have difficulty with that, the solution can be
found online in the Student Manual (www.uvm.edu/~dhowell/methods7/StudentManual/
StudentManual.html).dNdN444 Chapter 13 Factorial Analysis of Variance
unweighted
means
equally weighted
means