Afsnitsforfatter: Danielle J. Navarro and David R. Foxcroft
Factorial ANOVA 2: balanced designs, interactions allowed
figur 148 Qualitatively different interactions for a 2 × 2 ANOVA
The four patterns of data shown in figur 147 are all quite
realistic. There are a great many data sets that produce exactly those
patterns. However, they are not the whole story and the ANOVA model that we
have been talking about up to this point is not sufficient to fully account for
a table of group means. Why not? Well, so far we have the ability to talk about
the idea that drugs can influence mood, and therapy can influence mood, but no
way of talking about the possibility of an interaction between the two. An
interaction between A and B is said to occur whenever the effect of Factor A is
different, depending on which level of Factor B we’re talking about. Several
examples of an interaction effect with the context of a 2 × 2 ANOVA are shown
in figur 148. To give a more concrete example, suppose that
the operation of Anxifree and Joyzepam is governed by quite different
physiological mechanisms. One consequence of this is that while Joyzepam has
more or less the same effect on mood regardless of whether one is in therapy,
Anxifree is actually much more effective when administered in conjunction with
CBT. The ANOVA that we developed in the previous section does not capture this
idea. To get some idea of whether an interaction is actually happening here, it
helps to plot the various group means. In jamovi this is done via the ANOVA
Estimated Marginal Means option - just move drug and therapy
across into the Marginal Means box under Term 1. This should look
something like figur 149. Our main concern relates to the fact
that the two lines aren’t parallel. The effect of CBT (difference between
solid line and dotted line) when the drug is joyzepam (right side)
appears to be near zero, even smaller than the effect of CBT when a
placebo is used (left side). However, when anxifree is administered,
the effect of CBT is larger than that of no.therapy (middle). Is this
effect real, or is this just random variation due to chance? Our original ANOVA
cannot answer this question, because we make no allowances for the idea that
interactions even exist! In this section, we’ll fix this problem.
figur 149 jamovi screen shot showing how to generate a descriptive interaction plot
within the ANOVA (using Estimated Marginal Means) for the
clinicaltrial data set
What exactly is an interaction effect?
The key idea that we’re going to introduce in this section is that of an
interaction effect. In the ANOVA model we have looked at so far there are only
two factors involved in our model (i.e., drug and therapy). But when
we add an interaction we add a new component to the model: the combination of
drug and therapy. Intuitively, the idea behind an interaction effect is
fairly simple. It just means that the effect of Factor A is different,
depending on which level of Factor B we’re talking about. But what does that
actually mean in terms of our data? The plot in figur 148
depicts several different patterns that, although quite different to each
other, would all count as an interaction effect. So it’s not entirely
straightforward to translate this qualitative idea into something mathematical
that a statistician can work with.
As a consequence, the way that the idea of an interaction effect is formalised in terms of null and alternative hypotheses is slightly difficult, and I’m guessing that a lot of readers of this book probably won’t be all that interested. Even so, I’ll try to give the basic idea here.
To start with, we need to be a little more explicit about our main
effects. Consider the main effect of Factor A (drug in our running
example). We originally formulated this in terms of the null hypothesis
that the two marginal means µr. are all equal to each
other. Obviously, if all of these are equal to each other, then they
must also be equal to the grand mean µ.. as well, right? So
what we can do is define the effect of Factor A at level r to
be equal to the difference between the marginal mean µr.
and the grand mean µ... Let’s denote this effect by
αr, and note that
Now, by definition all of the αr values must sum to zero, for the same reason that the average of the marginal means µr. must be the grand mean µ... We can similarly define the effect of Factor B at level i to be the difference between the column marginal mean µ.c and the grand mean µ..
and once again, these βc values must sum to zero. The reason that statisticians sometimes like to talk about the main effects in terms of these αr and βc values is that it allows them to be precise about what it means to say that there is no interaction effect. If there is no interaction at all, then these αr and βc values will perfectly describe the group means µrc. Specifically, it means that
That is, there’s nothing special about the group means that you couldn’t predict perfectly by knowing all the marginal means. And that’s our null hypothesis, right there. The alternative hypothesis is that
for at least one group rc in our table. However, statisticians often like to write this slightly differently. They’ll usually define the specific interaction associated with group rc to be some number, awkwardly referred to as αβrc, and then they will say that the alternative hypothesis is that
where αβrc is non-zero for at least one group. This notation is kind of ugly to look at, but it is handy as we’ll see in the next section when discussing how to calculate the sum of squares.
Calculating sums of squares for the interaction
How should we calculate the sum of squares for the interaction terms, SSA:B? Well, first off, it helps to notice how the previous section defined the interaction effect in terms of the extent to which the actual group means differ from what you’d expect by just looking at the marginal means. Of course, all of those formulas refer to population parameters rather than sample statistics, so we don’t actually know what they are. However, we can estimate them by using sample means in place of population means. So for Factor A, a good way to estimate the main effect at level r is as the difference between the sample marginal mean Ȳrc and the sample grand mean Ȳ... That is, we would use this as our estimate of the effect
Similarly, our estimate of the main effect of Factor B at level c can be defined as follows
Now, if you go back to the formulas that I used to describe the SS values for the two main effects, you’ll notice that these effect terms are exactly the quantities that we were squaring and summing! So, what’s the analog of this for interaction terms? The answer to this can be found by first rearranging the formula for the group means µrc under the alternative hypothesis, so that we get this
So, once again if we substitute our sample statistics in place of the population means, we get the following as our estimate of the interaction effect for group rc, which is
Now all we have to do is sum all of these estimates across all R levels of Factor A and all C levels of Factor B, and we obtain the following formula for the sum of squares associated with the interaction as a whole
where we multiply by N because there are N observations in each of the groups, and we want our SS values to reflect the variation among observations accounted for by the interaction, not the variation among groups.
Now that we have a formula for calculating SSA:B, it’s important to recognise that the interaction term is part of the model (of course), so the total sum of squares associated with the model, SSM, is now equal to the sum of the three relevant SS values, SSA + SSB + SSA:B. The residual sum of squares SSR is still defined as the leftover variation, namely SST - SSM, but now that we have the interaction term this becomes
As a consequence, the residual sum of squares SSR will be smaller than in our original ANOVA that didn’t include interactions.
Degrees of freedom for the interaction
Calculating the degrees of freedom for the interaction is, once again, slightly trickier than the corresponding calculation for the main effects. To start with, let’s think about the ANOVA model as a whole. Once we include interaction effects in the model we’re allowing every single group to have a unique mean, µrc. For an R × C factorial ANOVA, this means that there are R × C quantities of interest in the model and only the one constraint: all of the group means need to average out to the grand mean. So the model as a whole needs to have (R × C) - 1 degrees of freedom. But the main effect of Factor A has R - 1 degrees of freedom, and the main effect of Factor B has C - 1 degrees of freedom. This means that the degrees of freedom associated with the interaction is
which is just the product of the degrees of freedom associated with the row factor and the column factor.
What about the residual degrees of freedom? Because we’ve added interaction terms which absorb some degrees of freedom, there are fewer residual degrees of freedom left over. Specifically, note that if the model with interaction has a total of (R × C) - 1, and there are N observations in your data set that are constrained to satisfy 1 grand mean, your residual degrees of freedom now become N -(R × C) - 1 + 1, or just N - (R × C).
Running the ANOVA in jamovi
Adding interaction terms to the ANOVA model in jamovi is
straightforward. In fact it is more than straightforward because it is
the default option for ANOVA. This means that when you specify an ANOVA
with two factors, e.g. drug and therapy then the interaction
component - drug × therapy (shown as drug * therapy) - is
added automatically to the model.[1] When we run the ANOVA with the
interaction term included, then we get the results shown in
figur 150.
figur 150 Results for the full factorial model, including the interaction component
drug × therapy
As it turns out, while we do have a significant main effect of drug:
F(2,12) = 31.7, p < 0.001, and therapy: F(1,12) = 8.6, p =
0.013), there is no significant interaction between the two: F(2,12) = 2.5,
p = 0.125).
Interpreting the results
There’s a couple of very important things to consider when interpreting the
results of factorial ANOVA. First, there’s the same issue that we had with
one-way ANOVA, which is that if you obtain a significant main effect of (say)
drug, it doesn’t tell you anything about how the levels of drug are
different to one another. To find that out, you need to run additional
analyses. We’ll talk about some analyses that you can run in sections
Different ways to specify contrasts and Post-hoc tests. The same is true for
interaction effects. Knowing that there’s a significant interaction doesn’t
tell you anything about what kind of interaction exists. Again, you’ll need
to run additional analyses.
Secondly, there’s a very peculiar interpretation issue that arises when you obtain a significant interaction effect but no corresponding main effect. This happens sometimes. For instance, in the crossover interaction shown in figur 148 (top-left panel), this is exactly what you’d find. In this case, neither of the main effects would be significant, but the interaction effect would be. This is a difficult situation to interpret, and people often get a bit confused about it. The general advice that statisticians like to give in this situation is that you shouldn’t pay much attention to the main effects when an interaction is present. The reason they say this is that, although the tests of the main effects are perfectly valid from a mathematical point of view, when there is a significant interaction effect the main effects rarely test interesting hypotheses. Recall from section What hypotheses are we testing? that the null hypothesis for a main effect is that the marginal means are equal to each other, and that a marginal mean is formed by averaging across several different groups. But if you have a significant interaction effect then you know that the groups that comprise the marginal mean aren’t homogeneous, so it’s not really obvious why you would even care about those marginal means.
Here’s what I mean. Again, let’s stick with a clinical example. Suppose that
we had a 2 × 2 design comparing two different treatments for phobias (e.g.,
systematic desensitisation vs. flooding), and two different anxiety reducing
drugs (e.g., Anxifree vs Joyzepam). Now, suppose what we found was that
Anxifree had no effect when desensitisation was the treatment, and Joyzepam had
no effect when flooding was the treatment. But both were pretty effective for
the other treatment. This is a classic crossover interaction, and what we’d
find when running the ANOVA is that there is no main effect of drug, but
a significant interaction. Now, what does it actually mean to say that
there’s no main effect? Well, it means that if we average over the two
different psychological treatments, then the average effect of Anxifree and
Joyzepam is the same. But why would anyone care about that? When treating
someone for phobias it is never the case that a person can be treated using an
“average” of flooding and desensitisation. That doesn’t make a lot of sense.
You either get one or the other. For one treatment one drug is effective, and
for the other treatment the other drug is effective. The interaction is the
important thing and the main effect is kind of irrelevant.
This sort of thing happens a lot. The main effect are tests of marginal means, and when an interaction is present we often find ourselves not being terribly interested in marginal means because they imply averaging over things that the interaction tells us shouldn’t be averaged! Of course, it’s not always the case that a main effect is meaningless when an interaction is present. Often you can get a big main effect and a very small interaction, in which case you can still say things like “drug A is generally more effective than drug B” (because there was a big effect of drug), but you’d need to modify it a bit by adding that “the difference in effectiveness was different for different psychological treatments”. In any case, the main point here is that whenever you get a significant interaction you should stop and think about what the main effect actually means in this context. Don’t automatically assume that the main effect is interesting.