Learning Statistics with R - A tutorial for psychology students and other beginners, 2018a

16. Factorial ANOVA
Over the course of the last few chapters you can probably detect a general trend. We started out looking
at tools that you can use to compare two groups to one another, most notably the t-test (Chapter 13).
Then, we introduced analysis of variance (ANOVA) as a method for comparing more than two groups
(Chapter 14). The chapter on regression (Chapter 15) covered a somewhat different topic, but in doing
so it introduced a powerful new idea: building statistical models that have multiple predictor variables
used to explain a single outcome variable. For instance, a regression model could be used to predict the
number of errors a student makes in a reading comprehension test based on the number of hours they
studied for the test, and their score on a standardised IQ test. The goal in this chapter is to import
this idea into the ANOVA framework. For instance, suppose we were interested in using the reading
comprehension test to measure student achievements in three different schools, and we suspect that girls
and boys are developing at different rates (and so would be expected to have different performance on
average). Each student is classified in two different ways: on the basis of their gender, and on the basis of
their school. What we’d like to do is analyse the reading comprehension scores in terms of both of these
grouping variables. The tool for doing so is generically referred to as factorial ANOVA. However, since
we have two grouping variables, we sometimes refer to the analysis as a two-way ANOVA, in contrast to
the one-way ANOVAs that we ran in Chapter 14.
16.1
Factorial ANOVA 1: balanced designs, no interactions
When we discussed analysis of variance in Chapter 14, we assumed a fairly simple experimental design:
each person falls into one of several groups, and we want to know whether these groups have different
means on some outcome variable. In this section, I’ll discuss a broader class of experimental designs,
known as factorial designs, in we have more than one grouping variable. I gave one example of how
this kind of design might arise above. Another example appears in Chapter 14, in which we were looking
at the effect of different drugs on the mood.gain experienced by each person. In that chapter we did find a
significant effect of drug, but at the end of the chapter we also ran an analysis to see if there was an effect
of therapy. We didn’t find one, but there’s something a bit worrying about trying to run two separate
analyses trying to predict the same outcome. Maybe there actually is an effect of therapy on mood gain,
but we couldn’t find it because it was being “hidden” by the effect of drug? In other words, we’re going
to want to run a single analysis that includes both drug and therapy as predictors. For this analysis each
person is cross-classified by the drug they were given (a factor with 3 levels) and what therapy they
received (a factor with 2 levels). We refer to this as a 3 ˆ 2 factorial design. If we cross-tabulate drug by
therapy, usingthextabs() function (see Section 7.1), we get the following table:
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