Hi, it's Margriet

In a previous lecture we looked at the interaction

between a continuous predictor
and categorical predictor.

Now in this video I want to
look at the interaction

between two continuous
predictors.

I'll just change my pointer.

There we go.

Now from last time you might
remember this example where

we imagined modeling plant
growth as a function of the

two predictors. Water use and
sunlight.

Neither of these predictors by
themselves will have a great

effect on plant growth only if
there is both water and

sunlight, the plant will grow.

So that is the influence of the
sun exposure predictor.

Critically depends on the water
use predictor and vice versa.

Now here is the equation of a
model with two variables X1 and

X2 without the error term.

And to incorporate the
interaction you might

remember that we use, we
multiply the two predictors.

So we multiply X1 first predicted by X2,

and regression will then estimate the corresponding

slope for this new predictor.
So that is beta three. That is,

the slope of the interaction
effect.

Now that slope describes the strength of of this kind of

multiplicative effect. So when
beta three is close to 0,

the interaction. Is weak, but the further away from zero it is,

either positive or negative.

The stronger the
interaction effect.

OK, today another example
involving iconicity. But

this time will use the data from a paper by Sidhu and 

Paxman published in 2018.

So.

Um, these researchers, they
discovered they looked at the

effect of sensory experience or
iconicity. But in addition, they

considered the role of what's
psycholinguists called the

semantic neighborhood density.

Now that term describes the idea
that there is that for certain

regions in your mental lexicon
where there are lots of words,

so it's quite crowded or dense and all those words are connected.

to each other by virtue of having similar meanings.

And other areas of your
mental lexicon will have

fewer words with similar
meanings, and are therefore

sparser. So have a lower
semantic neighborhood

density.

Now it has been proposed that.

Iconicity may lead to confusion because with many iconic forms,

similar meanings will also sound

similar. So therefore it's been proposed that 

Sidhu and Paxman reasoned that language should be biased

against iconicity, specifically
in semantically dense

neighborhoods. So where there is
lots of opportunity for

confusion. In sparse
neighborhood, iconicity is

not so dangerous as there is
less opportunity for

confusion between concepts.

OK, so now let's load the data from Sidhu and Pexman into R

And this is what the table looks like, so there are four columns

four variables here we have the word variable and that is just

the different items that are in the data set. We have the SER

variable that you might

recognize from. Previous
examples so that contains the

sensory experience ratings. Then
we have the iconicity ratings

here and the new one is this ARC variable. ARC stands for

average radius, radius of Co
occurrence and is the measure

for semantic neighborhood

density. So small values of ARC mean that this semantic

neighborhood is sparse and large
values mean that it's semantic

neighborhood is dense.

Now the hypothesis was that
given that iconicity can lead to

confusion, words with large ARC
values open dense neighborhoods

should be less iconic.

OK, so let's fit the
model with sensory

experience and
neighborhood density as

predictors and well, we're including their

interaction term, and you
might recognize this.

Formula notation here. As kind
of the shorthand, we could also

see say, SER + ARC plus SER, colon, ARC. So this

is just shorthand for that now
below here you see the

coefficients for for this model.

So there is the intercept and
that is the predicted iconicity

rating of a word with zero sonsory experience and zero

neighborhood density. And we
have the slope for SER. We have

the slope for ARC an etc etc.

Um and we have to the slope for
the interaction effect.

So this model is hard to
interpret for two reasons.

First of all, the predictors
aren't centered, so that means

that SCR and or sensory
experience and semantic

neighborhood density are
reported for some arbitrary 0.

Um, and in this case it really, the zero doesn't make much

sense. Words without any sensory
experience rating don't really

exist, and neither do words that
do not have any neighbors. So

words in and segmented
neighborhood density of zero

also don't exist, so they don't.
That zero is arbitrary and

doesn't make any sense.

And Secondly, so the second
reason why this model is hard to

interpret is that the slopes so
that the magnitude of the SL of

the sensory experience and
semantic neighborhood slopes are

difficult to compare because
each variable has a different

metric. So the one unit
change is a different step

size for each variable.

Now we can do something about
that. We can standardize our

continuous predictors and that
is what you see on the slide

here. So here at the top you can see the codes that does that

and here on the right you see
the two variables that now have

now been added, and this underscore Z just stands for

kind of standardized units. So
here we have the standardized

version of the sensory
experience variable. And here we

have the standardized version of

the. Censoring semantic neighborhoods

variable. If you can't quite remember what

centering and standardizing are about,

then please revisit an
earlier video about that.

Um, OK, so now that we've standardized our predictors

we can refit the model. So
here you see.

The code that does that
again, so now we're using

these standardized
predictors. Here, again

using the shorthand
notation.

Now let's have a look at these
coefficients down here.

So in this model they sense
semantic neighborhood density

and the sensory experience
coefficients are shown for each

others respective averages.
Because the meaning of 0 has

changed for both variables
because we sent it to date the

predictors. So after Centering
Zero, Sensory experience is now

the mean sensory experience

rating. And O for cementing,
neighborhood is now the mean

semantic neighborhood density.

Now let's first look at
the slopes for sensory

experience and.

Semantic neighbor density.

So the slope for sensory
experience is 0.07, that is

positive. And that means
that words in.

That more perceptual
words are more iconic.

Which kind of confirms
earlier findings.

Now the ARC underscore Z

Slope so a slope for semantic neighbor density is negative

If it's minus -0.32 and, that means that words in

denser neighborhoods are
predicted to be less iconic.

Now that, fits with the prediction that the authors had.

So moving on to the interactions
are at the bottom there an.

He the slope is negative
again, so it's minus 0.06, seven

or six 8.

and

That you can read that in the
following way, so when both.

sensory experience and semantic neighbor density

increase words are actually
predicted to be less iconic.

Because it's the slope is
negative.

So in a way that the two effects
cancel each other out.

Now it it helps to think about
one variable at a time when

you're interpreting an
interaction. So considering what

the interaction means for that

variable. So for example, if we
take a sensory experience slope.

On its own, it's positive,
but your interactions

negative, so the effect of
sensory experience is

diminished. For high levels
of semantic neighborhood

density.

Or put differently, in dense
semantic neighborhoods,

there is less of a sensory
experience effect.

Now. And the interpretation of
semantic of interactions can be

facilitated by visualizing.

The effects and Sidhu and Pexman opted to visualize at the

interaction via partial plot,
where the relationship between

semantic experience rating and
iconicity is shown for

representative levels of the
semantic neighborhood density.

So we have sensory experience on
the X axis.

We have iconic iconicity ratings
on the Y axis and we have 3

lines for three kind of
representative values of the

semantic net neighborhood

density measure. So here we have
the average semantic neighbor

density. Here we have sparse
densities as neighborhood

density, and here we have dense
semantic neighborhood density.

So for for high levels of
semantic neighborhood density.

This line is pretty flat, so
the relationship between

iconicity and sensory
experience rating is very

weak.

and, for kind of intermediate levels of semantic

neighborhood density, there is
a weak positive relationship.

And for.

The.

Sparse semantic neighborhood
densities. There is actually

strongly positive relationship
between sensory experience

ratings and iconicity.

Now.

This in this plot the
authors have just chosen

three what they considered.

Representative values of this
continuous variable semantic

neighborhood density.

In a perspective plot that you
can see here on the right that

it takes into account all the
values of all the all the three

variables so they take a little
bit of getting used to when

interpreting. So let me talk you

through it. We basically have
3 axis, so it's a 3D plot and

we have iconicity.

On the Y axis.

We have a.

Sensory experience rating on
this axis and then we have.

Semantic neighborhood
density on this axis.

So if we just focus on the
relationship between sensory

experience rating and iconicity, we look at that low value

of er very high values of.

Semantic neighbor density. So if
we look at this end of the plot,

this bottom edge, then you can
see that this line is pretty

flat, so not much going on.

On the other hand, if you move
up to kind of a very sparse

semantic neighborhood density,

then. This line is, it is.

Moderately positive or even
strongly positive so.

The relationship between.

Semantic, sorry, the
relationship between sensory

experience rating and iconicity
varies depending on where along

the semantic neighborhood

density. Axis you look.

And in a plot like this.

That is, is visualised quite nicely.

Now, interpreting an interaction
takes a little bit of time, so

it is. It is really worth it to
consider what the individual

coefficients mean and trying to use different plots to show these

show these effects.

OK, in summary.

We looked at a different type of
interaction between two

continuous variables. And
it's really important to

remember to center and
standardize your continuous

predictors to facilitate the
interpretation of the model.

Then I'll be. I will
reiterate that it is

important to spend some time
interpreting the

coefficients.

And Please remember that if the
interaction is significant, you

can't really interpret the
predictors in isolation anymore.

And finally, excuse me finally,
the slope for the interaction

can be read as if both

predictors increase. Then
something happens to the

outcome variable.

OK, that's it for now. Thank you
very much for your attention.