Hi, it is Margriet Groen here.

Now, each statistical test or
statistical model relies on

assumptions. So, all claims made
on the basis of that statistical

test, such as correlation, are
contingent on satisfying these

assumptions to a reasonable

degree. And that is the first
thing that we will be talking

about. The assumptions that
underlie correlation analysis.

Then there is also some issues
with correlation analysis that

you should be aware of and
we will be talking about those.

And about one possible way of
overcoming some of these

problems by using Spearman's rho

rather than Pearson's r.

And finally, we will be talking
about intercorrelation, where you

look at relationships between
multiple variables at once.

OK, the assumptions underlying
correlation analysis. So in

order to conduct a
correlation analysis, you need

to check that's the data meet
pre-specified assumptions. And if

any of these assumptions are
violated, then Pearson's r may

provide misleading information
regarding the relationship

between the variables of

interest. So you need to
check it.

The first assumption is to do
with the type of variables for

which you can use

Pearson's r.

Sorry, lost my mouse.
Here we are.

So you have to think about
what type the variables are.

that you're
interested in evaluating the

relationship between.

So both variables
need to be at

an interval level
or a ratio level.

If they are ordinal or
nominal variables, then you

cannot use Pearson's r.

So if you would, for instance, be
trying to correlate a continuous

variable like the mean
smelliness of cheese with the

categorical variable, the
type of cheese, then you could

not use Pearson's r for that
in a valid way.

Now when evaluating this
you don't need to do anything

specific other than think
about what the measurement levels

of your variables are.

The second assumption is that
there is a data point for each

participant, on both the

variables. So.

if we go back to the data from
last week, we looked at this

data table where we had
participants here in rows and we

had a number for each variable.

So, each participant had a score
on each of the variables.

If that is not the case, these

participants will usually
automatically be excluded from

the correlation
analysis, but if you have these

missing data, you might think
that you are calculating your

correlation coefficient on the
basis of a large sample, but

then discover that your
degrees of freedom is

lower so that it's actually only
including the people who have

kind of full data for

the variables in your correlation.

So something else that
you need to check in your

data before you go ahead.

The third assumption is that the
data should be normally

distributed. So, the data,
both variables should followed a

normal distribution or at least
approximate a normal

distribution. So you might
remember the normal

distribution from last term.
It is this kind of z,

sorry, this type of Bell
shaped curve.

And you can check that with a
with a histogram. So if you plot

variables of interest, if you
make histograms for those,

then you can do a visual
check to see whether that

histogram approaches the
normal distribution.

Now, a second way to visually
check that your data are

normally distributed is by using
what is called the 'normal

quantile-quantile plot' or the 'QQ-
plot' in short.

What that does is that it

puts things in quantiles

and checks whether, whatever you

plotting, comes from
the same distribution.

So if you use the normal
QQ plot, it has this 45 degree

angled line here, so this is a
45 degree angle.

And if the data of your variable
comes from a normal

distribution, then those data
points should fall quite close

to this line.

Now it also gives you a
confidence interval around

that, so as long as the data
points fall within.

this range of the two striped

lines then you would usually
say that you're OK, your data

are normal enough so to say.

Now the 4th assumption is that
the that correlation analysis

using Pearson's r assumes that
the relationship between the

two variables is linear. So
kind of follows

A pattern of a straight line.

It doesn't have to be an exact
exact straight line and not all

the data points have to fall
exactly on that line, but the

general pattern should be that

of a straight line. So if
you look at this example

here, we have age
on the X axis,

and language competence on
the Y axis and as age goes up,

so does language competence, so
it looks like a

linear relationship. A positive
relationship: when age increases,

language competence increases.

But there are other patterns in
other relationships in

development that do not quite
follow that pattern.

So just kind of looking at
linear relationships wouldn't

provide you a full picture.
There are other relationships

that are kind of grouped
under the term curvilinear

relationships. And you do see
quite a few of those in

in the language
development. So, for instance,

if we look at language
development and we have age on

the X axis,

but for some aspects of language
development, like vocabulary,

it will start quite, it will
grow very very fast but then

flattens out. So that's one
example of a curvilinear

relationship.

Yet in other domains of
language, like

to what extent children
understand compound sentences,

development kind of starts
slowly and then there is a

later growth spurt, so you get
kind of a curvilinear

relationship in the other
direction.

And in yet another kind
of type of measure to do with

with language development, you
get what might look like quite a

strange developmental pattern.
Here we have age again on the X

axis and errors in language use
on the Y axis. So here you can

see that at an early age
children might make actually

quite few mistakes, and then
they start making more mistakes

when they get older and then
it drops off again. So kind of

an inverted U shaped pattern of

development. You might wonder
why that is the case,

but this is really a pattern
that does that does occur and

there is there a perfectly
sensible explanation for

explanations for it.

So if you were to look at this
with if you were to evaluate

this particular these particular
data with the Pearson's r

correlation, you would find that
there is a correlation

coefficient of minus .18, and
that it is not significant. But

as you can see in the

scatterplot, we can see a clear
relationship between those

variables, it's just not linear.

So to check that you really need
to look at the scatterplot.

The 5th assumption to
do with correlation is

around the spread so different
spread around the line of

best fit. So that spread needs
to kind of be

evenly across the line
of best fit. So in this example

here on the left in orange, you
see data points that kind of

vary around this line of best
fit in a similar fashion on all

sides. So the spread here is
similar to the spread here and

similar to this right here.

Now this: random clouds around
the line of best fit, that is

what we need. We would say
this thread is homoscedastic.

So that's good.

Here is an example where
That is not the case. You

can see that the variance or the
spread on

the endpoint here and here is

much wider then in the
middle, so you have a kind

of bow tie shaped pattern.

And

then a different type of
heteroscedasticity is when

the spread on one side is
much wider and then kind of

tapers off along the line of
best fit. So these two are

examples of patterns that
are of the spread that that

is called heteroscedastic.

Again, you would look for
this in your scatterplot.

OK. So what to do if your data
are not normally distributed?

Now, instead of using Pearson's
r you could use what

is called a nonparametric
equivalent of that. So a

nonparametric correlation
coefficient is Spearman's rho.

There's another one, but today
we'll just talk about Spearman's

Rho, and that is based on the
ranking of the scores. So what

that does it takes all the
scores on variable X.

and ranks them from
lowest to highest.

And then it will check whether
that participant also scored

lowest on variable Y.

ETC etc. But that won't. I
mean, if that is the case,

then you would have kind of a
perfect correlation again, but

it doesn't have to be the
case. Of course it could be

that participant one scores,
Or here we have participant two

who actually doesn't score
second lowest on y

So Spearman's rho can be
used when the data is not

normally distributed, when
your when when your

variable is is ordinal rather

than on the interval or ratio
level, and so it's it gives

a solution to some of these
issues.

OK.

Then there are some other issues
with correlation that it is a

good idea to be aware of.

Um? So. A correlation
analysis may be distorted by

outliers with very extreme
scores. So very extreme is for

instance, more than three
standard deviations deviations

away from the mean.

So if you look at the two graphs
in on this slide, then here on

the right, if we

calculate Pearson's r for this
scenario we would find that

Pearson's r is .86.

Now that suggests quite a strong

positive correlation,
but if you look

at the scatterplot,
you could find this:

A set of data points here at the
bottom left and there's one data

point at the bottom right.

And it is actually kind of

trying to fit a line between
these data points

comes up with a positive
positive line of best fit.

Whereas once you look at at the
scatterplot, you think like

that can't be right. This
score is clearly completely

different from all the other
scores, and if you look at just

those scores, then a negative
correlation is much more likely

that this is a bit of an extreme
example, but it is really

important to look at your
scatter plots to see whether you

think that you know there might
be outliers in the data that

influence your correlation
coefficient in an unduly

fashion.

Now the second issue that you
that it is important to be aware

of, is that the sample that you
used may not represent the true

variation present in the two
variables in the population. So

there is an example of it here.
So if you think of kind of exam

exam mark, for instance, you
know you might say OK, let's

look at the A level exam marks
for a group of people.

And if you were to

look at people from from
various walks of life and you

would measure IQ and you would
look at A level results,

then you might find something
like this. So a relatively

strong positive correlation
where if IQ goes up A level

score goes up.

But if you would
only be looking at a sample

of University students, then
you might find a lot smaller

correlation because

by virtue of being at
University, it is quite likely

that people have a slightly
higher IQ, so you would only

be sampling from kind of
This end of the of of of this

variable, and you wouldn't
have any data points here.

So in this case, the correlation
is a lot lower, but that is

to do with the sample rather
than the true relationship of

these variables in the whole

population. So it is really
important to think about OK,

how much variation is there
in the score in the variables

that I'm looking at in my
correlation analysis. Is there

enough variation for them to
be able to correlate? And

where does my sample come
from? If there is a

restricted

amount of variance, or a
restriction of range.

But you need to think about OK,
why is there not? Why do the

people that I have sampled not
vary on this particular

variable to a larger degree?

Now, this might also play out
in in a different way, so if

one range on one variable is
unusually large.

It is sometimes better to to
look to look at that in a

different way. So let's look at
the variables. Here are math

confidence or confidence in math
and part one statistics mark.

The variables are
not that important, but the

pattern is important so you
can see kind of a cluster of

scores here. So people who
don't feel very

confident about their
maths also tend to

score a bit lower on
statistics. And then you have

a cluster of scores here.

And that of people who score a
bit higher on maths

also score higher on part I statistics.

So it's
basically two groups of

scores. Now again, if you
would run Pearson's

correlation then it would
would give you a positive

correlation coefficient.

But really, it might be more
informative and better to create

two different to to create two
different groups and

use a different statistical
tests or look within these

groups to see whether

these variables are related.

This happens quite a lot as well
if you look at atypically

developing children. So children
who might do quite

poorly on reading they,

you know, they might all sit
here at the bottom and you

might be looking to explain
their reading score with

something else, whereas the
typically developing children

might sit up there and the
relationship of these

variables might be different
in a group of typically

developing children and a
group of atypically

developing children.

So something else to be aware of

and again, the scatterplot will tell you.

Now intercorrelation.

So, so far we've looked at
correlation as something that

tells you what the relationship
is between two variables.

But if you want to kind of look
at how something like job

performance is influenced you might,

be interested in more than one
variable, right? You could say

OK, maybe your motivation will
contribute to your job

performance, but so will
your IQ and your social support.

You might also be interested in
how these different variables

are related to each other.

So this suggests that you
have really six different

correlation coefficients
that you're interested in.

And that is what we what we can
do, right? We can look, we can

calculate Pearson's or for each
of these relationships, and we

already briefly looked at an APA
correlation matrix

last last time. So here you see
one with more than two

variables. So again you have

the variables both at
the top and at the side. So

here they are just

just numbers. These are used for
convenience because otherwise

the table would get very

wide, but this one corresponds
to this one, so we have

performance, performance,
motivation, motivation, etc. So

this is how I would report that.

This bit you can add, so
this is the means and standard

deviations for each of these
variables. That is quite

helpful for the reader, but
it's not absolutely necessary.

Now there is something to keep
in mind.

You can calculate multiple
correlations between different

variables. That is not a
problem, but you have to double

check that you are not inflating
your type 1 error.

Last term,

Tom talked quite a bit about
sampling and probability and

type one errors and I believe
in week six, so if you can't

quite remember what a Type 1
error is, please

revisit that material.

So what do we need to take into
account here for when we do

intercorrelation? So if you're
conducting many correlations

at once, that increases the
chances of the type one error,

and therefore you have to
apply what is called the

Bonferroni adjustment.

Now what that does is that it
can adjust the significance

level of p. So usually you
would say a correlation is

significant if my P value is
bigger. Sorry if my P value is

smaller than .05 and that's
called the Alpha, right? .05 is

Alpha, so if the P value
that goes with your correlation

coefficient is smaller than
Alpha, then you say,

my correlation is significant.

Now, if you conduct many
correlations using the

same data set,

then you need to adjust that
Alpha level using the Bonferroni

adjustment. It's quite simple,
it just works like this.

So if you have three different
variables, you need to divide

that p = .05, so your Alpha, by
three. So you divide it by the

number of variables that are
included in your

intercorrelation analysis.
Now if you divide .05 by 3,

you get .016.

Therefore, for a correlation
coefficient to be significant in

this particular analysis

It needs to be smaller than .016

If there happen to be 6 variables
in your intercorrelation, then

he would have to divide your
Alpha level, that is p = .05

by 6

Any correlation in that
particular analysis would have

to be smaller than .05 / 6.

OK, in summary.

We reviewed the assumptions that
are important to check when you

do correlation analysis, so
variables need to be at interval

level at least and each participant
needs to have a data point for

each variable. Both variables
need to be normally distributed

and the relationship that you are looking

at needs to be linear.

And finally, the spread
needs to be homoscedastic.

Now then, we briefly looked at
two issues to do with

correlation analysis. So you
have to check your scatter plot

to see whether there are any
outliers that influence your

correlation unduly, and
whether there is any range

restrictions that is there is
limited variation in one or the

other variable that might
influence your conclusions.

Yeah, think about where the
sample comes from or why there

might be limited variation on
this particular variable.

And then we looked at doing
intercorrelation or multiple

correlations among variables in
from the same data set.

And a correlation matrix can
show you multiple

correlations between
variables at once. But be

aware of the multiple
comparisons problem and the

need to use the Bonferroni
adjustment.

Thank you, that is that
that is it for now.