Hi, it's Margriet. So far we have spoken about relationships

between the numerical variables.

Today we will be looking at
associations between categorical variables.

So a categorical variable is
something like snack preference,

snack preference, as in some
people might prefer chocolate over crips.

And others prefer crips over chocolate.

Another example is sex, biological sex,
so male and female.

So the plan for today is as
follows. We will be looking

at what is called a
Pearson Chi-squared test.

What is it and how do you apply it?

We will be looking at how to
carry it out by hand and in

a separate video. Also show you
how to do that in R.

And then we will be looking
at how to interpret the results.

So, how to report them
following APA guidelines.

Now finally, there's also a few
issues that need to be aware of

when using Chi-Squared tests and we will be
talking about those.

So what is a Chi-Squared
test? A Chi-Squared test is a

non-parametric test of
inference. Statistical inference

for categorical data.

So it measures the Pearson's
Chi-Square, or test of

independence, sometimes also
called for a two-by-two design.

That means two categorical
variables with two levels each.

That measures the
relationship between two

nominal or categorical
variables.

Nominal and categorical
are just synonyms.

They mean the same thing.

Um? And you ask: are
observations in one variable

contingent upon another
categorical variable.

So it tests whether the
frequency counts can be

expected that you observe by
chance, or whether there is

some relationship between
the categorical variables.

So let's look at some examples.

So, you could use as Chi-Square
test of independence to assess whether

gender is associated with
preferred study subject at the University.

Or whether dog ownership, as in "has a dog"

and "does not have a dog", is
associated with residence,

countryside vs. city.

Or is smoking associated with

drinking? Now this last
example is not how much

somebody smokes or how
much alcohol they drink.

It is whether or not they
smoke and whether or not

they drink alcohol.
That is the relationship

between the sets with
the Chi-Squared test.

So the null hypothesis for a Chi-Squared
test is that there is no association

between the two variables.

So you calculate the Chi-Squared
statistic and you look up

You look up in the significance
table whether that number

is bigger than what
you need for a particular

p-value to see whether that null
hypothesis can be rejected.

In which case the alternative
hypothesis becomes relevant.

And that is, the two
variables are associated.

OK, how do you do this by hand?

The first thing to do is to
construct what's called

a contingency table.

A contingency table represents
the frequencies for both

categorical variables in, in this way.
So, here we have an example of researchers

interested in assessing relationship between
children's record of fighting in school.

and their preference for violent
or non violent TV programmes.

So here we have one
variable: fighting in school

It has two levels. Has
been involved in fighting in

school or has not been
involved in fighting in school.

And here we have our other
variable: TV programme preference

Prefers a violent TV programme

or prefers a non violent TV programme.

So this number here means that 40
children in this particular example

were involved in fighting in school
and also prefered a violent TV programme.

So similarly, 15 children
were not involved in

fighting in school, and preferred
a violent TV programme. Etc. etc.

Now, the next thing we need to calculate
Chi-Squared by hand is the formula.

So, that we've got here in the middle.

And what Chi-Squared does
is that it looks at the observed

frequencies and works out how
unusual they are giving the

expected frequencies. So, the observed
frequencies are just the numbers

that we have observed.

In this example, there are
40 children who fall in that

category and 30 who fall in
that category etc etc. So that

that's the data, our observed frequencies.

The expected frequencies, we
need to calculate, and they

are determined by.

the formula here at the bottom,
by the column and the row

totals. And also, the total of
other observations in the study.

So how does that work?

Here, we will be calculating our
expected frequencies in the

following way, so if you add
40 and 30, you get a column total of 70.

So for this particular
cell, the column total is 70

And here it is row total, so 40 plus 15 is
55. So the row total for that cell is 55.

Now we we calculate the expected
frequency by multiplying the

column total by row total and
dividing that number by the

total sample size.

And then we get a number and
that number is the

expected frequency for this
particular cell.

We do that for all the
different cells, right. So here

in blue in brackets, you see the
expected frequency we've calculated.

For each particular cell.

So given the sample size and.

The number of, the total
number of children who were

involved in a fight, and who were not
involved in a fight

and who preferred a violent
TV programme and who did not

prefer a violent TV program.

Given those
numbers, those are the

frequency counts that you would expect

And.

The Chi-Squared statistic expresses how big
the difference between the observed numbers

and the expected numbers
are and whether the

difference is big enough
for it to be unlikely to

happen by chance.

So that's what we do next.

So here we have our table from the
previous slide. I've just put these numbers

these numbers in a
slightly different way of how

it calculates the next bit.

So we've got the observed numbers
for all the four cells and we have our

expected frequencies
for all the four cells.

And then we can.

Do this bit were we have our
observed minus our expected frequency

and we square that
number, and then we get this.

And then you divide that by
the expected frequency count for that cell.

If you then add all these
numbers up, you get your

Chi-Squared statistic. So this
case I'm not sure what the number

is but is 26 point something.

To look up in a significance
table, whether or not that

number is big enough for it to
be statistically significant,

you also need to know what the
degrees of freedom are for your

your particular study. And degrees
of freedom are determined as follows.

for a Pearson's Chi-Squared test. You
need to know what the number of rows is.

minus 1, multplied by the
number of columns minus 1.

Now we have two rows.

violent and non violent TV programme
preference and we had two columns.

Been involved in fighting and
not been involved in fighting.

For example, it is
2 minus 1 multiplied by 2 minus 1.

which is 1 multiplied by 1, so
we have 1 degrees of freedom.

Just a quick reminder of what
the degrees of freedom are. So they

refer to the number of
independent pieces of

information and went into
calculating the estimate.

Or, expressed slighty
different, the degrees of freedom

are the number of values that
are free to vary in that particular

calculation. So what do you mean
by that? If we say, OK, let's

pick a set of numbers, numbers,
3 numbers that have a mean of 10.

Now, if the first person
chooses nine and second

person say OK the second
number is going to be

10 then the third number

is given by these
two numbers. Because for

these three numbers to have a mean of

10, has to be 11.

Here, it has to be 12, right. 8 10 and
12 divided by 3 is mean of 10.

Here, it has to be 15.

So. Just to illustrate that if
you know a certain number of

pieces of information for a
particular population, others

are going to be given because
otherwise, the numbers don't add up.

And that's the degrees
of freedom that you've got.

Now if you look at our
significance table, then

we can figure out whether or not our
Chi-Squared that we've calculated here,

it is 26.15 in our example,
is big enough given the

degrees of freedom we've
got for it to be significant.

Here you look in the
degrees of freedom column

So, we had 1 degrees of freedom

So, here's for the 1-tailed test
and here is for the 2-tailed test.

The two-tailed test is much more common,
so these arrows should really be down here.

OK, we have 1 degrees of freedom. So for
our results to be significant

at the 5% level, it needs to be
bigger than this number.

This number is 3.841

Our number is 26.15.
So, this is beyond that.

So, it is significant at the 5% level.

And in this column on the right here,
we have the numbers for the

significance at the 1% level.

So if our number 26.15 is bigger than
this number.

6.635, we can conclude that it
is significant at the 0.01 level.

Which is the case in this example.

OK. Now I mentioned that a Pearson's
Chi-Square is a non parametric test

So, you don't need to worry about normally
distributed variables or.

homogenous variables. You do have graphs and
assumptions there, but they are different.

The first assumption is that the
data cannot be related, so it is

one participant can only
contribute a score to one cell

in your contingency table.

The second assumption is
that Chi-Squared has to be conducted on

raw frequencies and raw counts. Not
on percentages or anything like that.

And Thirdly, you have to check
that your expected cell

frequencies that you've calculated are no
less than 1, and no more than

20% of the cells should be less than 5.

Now, if you find that that
actually that you don't meet

this assumption, there are
certain things you can do.

For instance, if you had a
study where you compare for

instance recycling behaviour
for meat-eaters, vegetarians and vegans.

Do they recycle and do
they not recycle, for instance.

And it turned out that
the expected frequencies

would be lower than 5 in
more than 20% of the cells.

Then, you could say OK, I will
combined these two categories to

compare meat-eaters to non meat-eaters.

Alternatively, you can report
what is called Fisher's exact test.

So just as an extra reminder, is in the
Chi-Squared output. We will see that later.

OK, now a few words about other
things that you would want

to report when you report
Chi-Square analyses.

Um? I said, Chi-Square
should not, should be

conducted on raw frequencies
and not on percentages.

That is true, but it is useful
to report the percentages

in addition to the raw frequencies.
Just to kind of give people an idea

of the percentage of children
that are in a particular cell.

The measure of effect size for a
Pearson's Chi-Square is called a Cramer's V

And in the other video,
we will not ask how to get

that from R.

And in as done previously, it's
useful to report to the variance

accounted for. So by one
variable. So you can get that

by squaring the effect size. That
will tell you how much variance in

1 variable accounted for, or can
be described by the other variable.

And finally, what is useful to
look at when you are interpreting

your data is what
are called the standardised

residuals. So they help to
determine which cells

contribute to a significant association.

Our Chi-Squared was significant but looking
that's kind of an overall test.

From that, we can't
tell which of these cells

contributes to that
significant association.

By looking at the standardised
residuals, you can say something

about that. So the standardised residuals
basically are the size of these

differences for each of the cells
and then standardised in certain

ways that you can compare them
even though there might be many

more observations in
one cell than the other.

So, they are z scores and they indicate how
many standard deviations above

or below the expected
count, a particular observed count is.

Indicating how much they differ.

So if a standardised residual is
bigger in plus or minus 1.96.

We say that it is
significant at the .05 level.

If the standard residual is bigger
than

plus or minus 2.58, we would
say it's significant at the 0.01 level.

And if it's bigger than plus

or minus 3.29, then we can conclude that

that particular standardised residual

is significant at the 0.001 level.

And we don't gonna calculate
this by hand. I'll show you

where to look in the R output
for the standardised residuals.

OK, so how do we report this?

This is an example how you could
write up our particular example and there

was a significant Association
between school fighting and TV

programme preference
(violent vs. non-violent).

And here you have the technical bit.
This is the sign for Chi, so Chi-Squared

Open brackets one, that's one degrees of
freedom comma, and then the total sample size.

Equals this, that's the
number that we've calculated.

And that's significant at the 0.001 level.

Cramer's V. We didn't
calculate that by hand but

I will tell you what it
is. That's our measure

of effect size. So
you report that as well.

And then you go onto interpret,
the kind of the different cells what this.

What this significant
relationship means. So

above expectation 73% or 40 out of 55
children who preferred violent TV programmes

had also fought in school.

This is then where you report
your standardised residuals.

We will see from the R output
that our standardised residuals is this

and that is significant at the 0.01 level.

And significantly less than
expected has not fought.

Say something about that particular cell.

Conversely, 70% of children who
preferred non-violent TV programmes

had not engaged in fighting. More than
expected. And that was more than expected.

And those who had fought
were below expectations.

It is below and above that you can
see that in the standardised residuals

because if it's below
expectations it has a minus sign.

Analysis showed 17% of
variance in school fighting

could be accounted for by
TV programme preference.

So, this is Cramer's V

uhm squared.

OK then finally a few
issues that are important

to be aware of.

I mentioned before, you need to
make sure that you conduct the

Pearson's Chi Square on

raw frequency counts and not on
percentages. So this is what happens if

you do that. So here you have
uhm Chi Square

calculated on the frequency counts. So
that's waht you should do. And there

you have your number. This
is what you get out of that.

On the right side here these
are percentages, so 35% of

male students preferred
science as a subject.

uhm so these two are related
obviously because these

are the percentages and
these are the raw counts

But if you do the Chi Square on the
percentages, then that this would

actually not be significant.

So it's important to
keep that in mind.

Then a few words about how to
handle larger contingency tables.

If you have categories, you have more, your
categorical variables have more than two

levels. Um?

You need to be careful with because
they can be really difficult to interpret

We can help on understand the associations
in a few different ways. We can

use the standardised residuals.

This as mentioned before.

Another way to do that, to interpret
the bigger contingency table is to use

what is called partitioning. Basically
you're carrying out multiple

two-by-two Chi Squares.

So if we have an example here of
TV program preferences,

soap opera, crime, drama, and other

in male and female students

Now that would be a 3 by 2
contingency table, which

might be difficult to interpret

So, instead we could do
multiple 2 by 2 Chi Square tests.

So, soap and crime, soap
and other, crime and other.

If you do that, you do
need to use a Bonferroni

correction for the
significance levels to kind of

take into account that you're doing
multiple tests on the same data.

The final alternative to deal
with big contingency tables

is to combine categories, so
those need to be informed by

the theory on and they
need to make logical sense.

So, we could combine soap
opara and other or combine

crime and another
depending on your hypothesis.

Which kind of would shape
it into a two-by-two design.

So in summary, we looked at
Pearson's Chi Square or

the test of independence
and that investigates

or tests whether there
is an association between

two categorical or nominal variables.

Uhm. The assumptions for
the Chi Square we talked about.

Then to understand the association,
we need to look at the Chi Square statistic

the p-value, Cramer's V,
and the standardised residuals

If the assumption for the
minimum frequencies is not met,

you can use, you can
report Fisher's exact test

And we need to make sure that we
always use raw frequencies

rather than percentages to
calculate the Chi Square because

Otherwise, we get into trouble.

And when you're dealing with larger
contingency tables, it's better

we can better understand
our results using standardised

residuals or using partitioning
or combining categories.

Thank you very much for
your attention.