Hi again, it's Margriet. In
this video I want to talk to you

about interactions. Now you
might have already come across

interactions between variables
when you learned about ANOVA.

Um an interaction basically
describes a situation where the

influence of a predictor on the
outcome variable depends on

another predictor. So
interactions are about

relationships between

predictors. And how two or more
predictors together influence

the outcome variable.

Um, as an example, let's imagine
modeling plant growth.

As a function of two of two
predictors, water use and

exposure to sun.

Now neither of these predictors alone will have a great effect

on on plant growth.

It's only if there is both water
and sun that a plant will grow.

So the influence of the
sun exposure predictor

critically depends on
the water use predictor

and the other way round.

So.

We can incorporate an
interaction by multiplying the

two predictors, and for that we
use this Asterix sign.

So here we have the equation of
a model with two predictors,

this is an example of multiple regression, right?

And so we have X1 and X2. I haven't bothered with the

error term here.

Now and to incorporate an
interaction, you multiply the

two predictors by each other. So X 1 * X 2 and this is what

this looks like.

So here we have.

The Intercept effect of.

The first predictor, the second
predictor. And here we have the

interaction term X 1 * X Two.

And regression will then
estimate the corresponding

slope for this new term, right? For this new, so the

better erm Beta 3 here.

And the numerical value of
this slope describes the

strength, the strength of this multiplicative multiplicative

multiplicative effect.

So when beta three is close to 0. The interaction is weak and

the further away it is from
zero, the stronger the

interaction effect.

Now in psychology interactions are sometimes called moderator

variables because they moderate
the effects of other predictors.

You can think of it this way
by multiplying two

predictors with each other, you effectively

interlock them and the
coefficient beta three

specifies how the two
predictors are interlocked.

Now in the video about multiple
regression, we looked at a study

by winter and colleagues er 2017 on iconicity.

So you might remember that
iconicity describes the degree

to which a word form resembles
its meaning, so.

Onomatopoeic words like bang and beep. They are

iconic because this sound
an what they describe.

Now, one outcome of the
analysis we looked at then

also had words with more
sensory content were on

average more iconic than
words with less sensory

content.

Now in the plot on the slide,
the relationship between

iconicity on the Y axis.

and.

Sensory experience ratings is
shown separately for nouns on

the left and verbs on the right.

Now the lines inside these
plots, they show the linear

model fits of a simple
regression model.

So as iconicity.

Iconicity is a functional
sensory experience rating for

only nouns on the left, or only
verbs on the right.

So basically we have two regression models here.

Now the regression models estimate the slopes to be .63 for verbs

and .12 for nouns.

So the fact that sensory
experience rating predicts

iconicity differently depending on the part of speech.

Some nouns versus verbs that
hints at an interaction.

But you cannot simply compare
the slopes of separate models.

You have to model the
interaction explicitly

within one model.

So.

Let's first model the iconicity data as a function of sensory

experience rating and part of
speech without of the

interaction term. So that's
what we do here, right?

Iconicity as a functional
sensory experience rating and

part of speech noun versus
verb. That as kind of two

separate predictors.

And so this is what we get.

Let's then spend a little bit of
time interpreting these

coefficients. So first, here we have to intercept.

And which is the prediction for
nouns with sensory rate

experience ratings of 0 and you know that nouns are the

Intercept. Or the reference
level because it says.

Part of speech POSverb here in

the output. Also, you also know it because N comes

before V in the alphabet, so it will have taken the noun

level. As the reference level.

And a positive slope here of .60 shows that verbs are more iconic

than nouns and that is in this
case in this model, regardless

of which value of sensory experience rating you consider

there. So this is visible in the
plot in that the lines for nouns

and verbs are parallel.

So then we have, so that's
the Intercept. But here we

have the sensory rate
sensory experience rating,

slope that's estimated to be .23. That indicates a

positive relationship
between iconicity and

sensory experience. And we
can indeed seem kind of

upward sloping line.

Um, if you remember what the
slopes of the individual.

Models for simple regression
models for nouns and verbs were

like then you might remember
that this .23 is in between the

slope for nouns only, which was .12 and the one for verbs

only, which was .63.

From the previous slides.

It's a bit closer to the slope of the of the noun model

because there are more nouns in the data set.

So now we have, so we've modelled an intercept, we've modelled

The

Predictor sensory experience
rating and suggest that this

positive so lot of slightly
positive and then we have here

this part of speech predictor and it basically says because

it's positive and verb. Noun is
the reference level verbs are.

More iconic than nouns because there is no interaction

terms. These lines are parallel.

And by not including the
interaction term, the model ends

up mischaracterizing the sensory
experience rating slope for both

lexical categories, right?

Remember this. Well, we have two

individual models. Did this this
regression slopes are kind of

appropriate for the different
categories, but we don't see

that in if we put them together
in a model because we haven't

included this interaction term.

OK, so let's now do that and
include an interaction term so.

We do. We specify that by
including the asterisks in

the model here.

And the most striking difference
is the fact that the two lines

are no longer parallel.

So this means that sensory
experience is estimated to have

different effects on iconicity
for nouns and verbs.

It also means that the degree
to which nouns and verbs differ

from each other in terms of
iconicity depends on what

sensory experience rating value
you're looking at.

And if you just look at the
difference between these lines,

so the solid black line for nouns
and the dashed line for verbs,

and you look at that here. Here
exactly at this at the Intercept

verbs are less iconic then
nouns, and if you then look

across the X axis, you see that
that changes. And here verbs

are more iconic, so for
different levels of sensory

experience rating. The
difference between verbs and

nouns changes.

It basically means you cannot
interpret each of the predictors

in isolation anymore. You have
to look at them together.

So let's look at the
coefficients here on the right,

and there is a coefficient for

each predictor. And there is an

additional coefficient.
That uhm.

That that that that is for the
interaction term. So it is.

colon here indicates that we're
talking about the interaction

term, so that is the coefficient
for the interaction term.

And we of course we have to
intercept here. Remember that is

the prediction for nouns with
zero sensory experience rating.

That is the intercept

so crucially, now the lines are
not parallel anymore. The

meaning of the Sensory
experience rating predictor and the

part of speech predictor
are have changed right? So

and the slopes. The
interpretation of the slopes

has changed, so for the.

Sensory experience rating slope
that is not a slope of sensory

experience, only for nouns.

And the slope that goes with

the position of speed, uh part
of speech effect is the noun

verb difference only for words
with zero sensory experience.

But how can we interpret
the coefficient of the

interaction term?

So in this context, it is
appropriate to think of this

coefficient as a.

Slope adjustment term.

So the coefficient here is
.51, which means that the

sensory experience rating slope
is steeper for verbs.

So adding the .51 interaction
term to the slope for the nouns.

.12 yields the slope for the
verbs, which is .63.

Or put differently, the verbs
get an additional boost with

respect to the sensory
experience rating effect.

OK, now interpretation of models
with interactions is often

easier when you center the

continuous predictor. So you
might remember from a previous

video that centering sets the
intercept to the mean.

So if you center the sensory
experience ratings, you put the

Intercept to the center of kind
of the mass of this data. Of all

these data points, right? So you
can see that here on the right.

So notice how this central

sensory experience rating value.

For this. For this central
sensory experience rating value,

verbs are actually more iconic

than nouns. So this is
arguably a better

characterization of the noun
verb difference than

evaluating this difference at
a sensory experience rating

of zero, as as is the case in
the uncentered model.

In particular because.

There isn't even any
data at zero because the

scale started at one.

So centering sensory experience
ratings has reversed the sign of

the  Coefficient.

For the part of speech
predictor.

So that is now, here.

.72

and it that is now more
meaning, that that is now a

more meaningful
interpretation.

Because it is the difference
between nouns and verbs for

words with average sensory
rating experience ratings.

So rather than
difference between

nouns and verbs with
some arbitrary rating of

0 that doesn't exist.

So you can see that here.

Then the way the lines are
fitted and the way it is

represented. I'm guess you
better characterization of what

is going on in the data.

So the motto can be: if in

doubt, center you continues
variable.

Now.

Important to know is that in R, there're
alternative ways of specifying

interactions in regression

models. And there are two
different ways of doing this in

the model formulas. So in this
is what we what we used in the

previous slide. And you might
wonder where the main effects

of in this case sensory
experience rating and part of

speech have gone.

This is basically a compressed

form. A compressed form of this.

So. Here, it spells out that
you have a main effect of

sensory experience rating and of
part of speech, and then throws

a third term that.

involves the interaction.

So they do exactly the same
thing, and the first one is just

kind of a quicker. It's a bit of
a shortcut, and so if you put in

this it will, R will
automatically include these

terms as well. You just don't
have to write them down.

As with some other kind of
shortcuts in R, in particular,

in the beginning it is. It's a
good idea to use this slightly

longer form. You know just to
make it clear to yourself what

what is in the model.

OK, so in summary, interactions
describe a situation where the

influence of a predictor on the
response depends on another

predictor, so it looks at the
relationships between

predictors and their combined
effect on the outcome variable.

And in the in the formula
that is characterized by

multiplying those two
predictors and and that

involves the asterisk, the asterisk.

In R, in the model formula
there is two options and. This

is kind of the short version and
this is the long version.

Remember that it is makes it
easier to interpret your model

if you have interactions. If you
center your continuous predictor.

And it's also important to
remember that if an interaction

in your model is significant,

you can't interpret the
predictors in isolation anymore.

You have to look at their
combined effect.

And you can think of the slope
for the interaction effect

as a slope adjustment term when
you move from one category to

the other category.

Thank you very much for
your attention.