The Problem
You can see what our problem is if we take a very simple 2 3 2 factorial where we know
what is happening. Suppose that we propose to test vigilance on a simple driving task when
participants are either sober or are under the influence of alcohol. The task involves using a
driving simulator and having to respond when cars suddenly come out of driveways and
when pedestrians suddenly step into the street. We would probably expect that sober driv-
ers would make many fewer errors on this task than drivers who had been plied with alco-
hol. We will have two investigators working together on this problem, one from Michigan
and one from Arizona, and each of them will run half of the participants in their own
facilities. We have absolutely no reason to believe that participants in Michigan are any dif-
ferent from participants in Arizona, nor do we have any reason to believe that there would
be an interaction between State and Alcohol condition. I constructed the data with those
expectations in mind.
Suppose that we obtained the quite extreme data shown in Table 13.13 with unequal
numbers of participants in the four cells. The dependent variable is the number of errors
each driver made in one half-hour session. From the cell means in this table you can see
that the data came out as expected. The Drinking participants made, on average, 6 more
errors than the participants in the Non-Drinking condition, and they did so whether they
came from Michigan or Arizona. Similarly, you can see that there are no differences
between Michigan and Arizona participants, whether you look at the Drinking or the Non-
Drinking column. So what’s wrong with this picture?
Well, if you look at the column means you see what you expect, but if you look at the
row means you find that the mean for Michigan is 18.3, whereas the mean for Arizona is
only 15.9. It looks as if we have a difference between States, even after we went to such
pains to make sure there wasn’t one here. What you are seeing is really a Drinking effect
disguised as a State effect. And that is allowed to happen only because you have unequal
numbers of participants in the cells. Michigan’s mean is relatively high because they have
more Drinking participants, and Arizona’s mean is relatively low because they have more
Non-Drinking participants. Now I suppose that if we had used actual people off the street,
and Michigan had more drunks, perhaps a higher mean for Michigan would make some
sort of sense. But that isn’t what we did, and we don’t usually want State effects contami-
nated by Drinking effects. So what do we do?
The most obvious thing to do would be to calculate row and column means ignoring
the differing cell sizes. We could simply average cell means, paying no attention to how
many participants are in each cell. If we did this, the means for both Michigan and Arizona
would be (14 1 20)/2 5 17, and there would be no difference due to States. You could then
substitute those means in standard formulae for a factorial analysis of variance, but what
are you going to use for the sample size? Your first thought might be that you would justSection 13.11 Unequal Sample Sizes 445Table 13.13 Illustration of the contaminating effects of unequal sample sizes
Non-Drinking Drinking Row Means
Michigan 13 15 16 12 18 20 22 19 21
23 17 18 22 20 1. 5 18.3
11514 12520
Arizona 13 15 18 14 10 24 25 17 16 18
12 16 17 15 10 14 2. 5 15.9
21514 22520
Col Means X.1 514 X.2 520