[Auto-generated transcript. Edits may have been applied for clarity.]
In this lecture, I will talk about the assumptions that underlie correlation analysis, as we've looked at so far using Pearson's R.

And we'll also talk about some issues with correlation analysis that you need to be aware of.

And then we'll talk about how to overcome such problems.

And finally we will talk about inter correlation where we look at multiple correlations between multiple variables.

Okay. So what are assumptions. And more specifically what are the assumptions for Pearson's are.

Now as the statistical test works,

by comparing the result that you've found for your sample to what we would expect to see if only chance were operating,

and to describe that expectation, we have to build a model of how the data would behave if only chance were operating.

If you think back to the lectures in in in the last week.

Um, Rob talked about, uh, how that is kind of the key logic of hypothesis testing on significance testing.

Now, the assumptions are the conditions that that model relies on.

So if your data meets those assumptions, the p value is an accurate answer to how surprising is this result?

If they don't meet those conditions, then you can still calculate a p value.

You won't. You won't get an error message, but the output will be misleading.

So phrased another way, you could say that a test only um is only guaranteed to work as advertised in the conditions it was designed for.

And the further your data strays away from those conditions.

The less you can trust the have put, or the less you can be assured that you draw the correct conclusions from that output.

Now, it's also important to realise that assumptions aren't kind of a pass fail situation.

Um, there's things we need to check and and then we need to kind of get an accurate picture.

Um, because many tests tolerate modest violations. So we need to look at how serious a violation is rather than just whether one exists.

Okay. So what are the assumptions underlying Pearson's? Are some of them excuse me?

Some of the um, some of those assumptions, uh, under relate to how the data were collected and some of them relate to the shape of the data.

So we'll first look at the assumptions that relate to how the data were collected.

Um, they have to do with that. Both variables should be continuous.

So at an interval ratio level, and that observations need to be independent of one another.

Now let's look at each of those in turn.

A few weeks ago, in one of the earlier lectures, you learned about different types of variables, and we divided variables into categorical variables.

So at the nominal or ordinal level and continuous variable variables at the interval or ratio level.

So if we have categories um there can be at a nominal level where basically we have groups of things that have no inherent order.

So I colour is is an example or diagnosis type.

Um, so basically there are labels for different groups.

Then a different type of categorical variable Is at the ordinal level.

You still have categories of things, but you can order them in a meaningful way.

Um, but the gaps between them are not necessarily equal.

So an example of that would be a level highest level of education.

So one, you could have one group of people who have completed their GCSEs.

Another group of people who, as their highest level of education have completed their A-levels.

And a third group who have completed a at their highest level of education, a university degree.

Let's say bachelor's degree. Um, so you have groups of things and they are ordered.

Um, because you could say that GCSE is lower than A-levels and A-levels is lower than a university education.

Um, but the gaps between them are not necessarily equal.

You can't really say that a university completing a university, uh, education is kind of twice as high and then completing your GCSEs.

Another example is um, kind of groups of symptoms or kind of level of symptoms.

So if symptoms or something could be cough it could be mild moderate or severe.

So you could divide people into those categories. But again Sofia isn't necessarily twice as bad as mild.

Okay. So those are categorical variables. Um when we look at continuous variables we have they can exist at the interval level or at a ratio level.

At the interval level. A variable has ordered values with equal gaps between them.

But there is no true zero. So an example of that is calendar year.

Um, the gap between 1990 and 2000 is the is ten years, which is the same as between 2010 10 in 2020.

Also ten years. So the gaps are equal. But what we mean by there isn't a true zero is that the year zero doesn't mean that no time has passed.

It's basically an arbitrary starting point. Now a ratio, um, a variable at ratio level is continuous.

It is ordered, it has equal intervals and it has a true zero.

So examples of of that are reaction times in milliseconds or number of errors or height.

Um zero. Mean reaction time of zero milliseconds truly means zero milliseconds.

No errors. Zero errors means there are no errors. Um, and something can have a height of or of zero, I guess.

Anyway, um, so that is the ratio level.

Now Pearson's r and describes the relationship between two variables that are continuous so that are on an interval or ratio scale.

You cannot use it for variables on a nominal or ordered ordinal scale.

So you wouldn't try to correlate a continuous variable like the smell of cheese with a categorical variable like the type of cheese comparing,

let's say cheddar, Brie, and some blue cheese. Similarly,

you can't use Pearson's R to correlate a continuous variable like reading ability with an ordinal variable like the highest level of education.

So how do you check this? You need to look carefully at the measurements used and think about what types of variables they are.

Okay.

The second assumption that has to do with how the data were collected is independent, is that the Pearson's are assumes independence of observations.

So it each participant's data should be independent of every other participants data.

So the T test for R treats every data point as new independent information.

And if observations are linked in some way, you effectively have less information than your sample size suggests.

So the p value may be misleading. Um, so an example of of this way where observations are independent is if we ask 100 university students,

um, to complete, let's say, a stress questionnaire and a sleep diary once.

So each university student contributes a kind of one data point on each of those measures.

If we were to ask, um, those kind of 20 students, the same 20 students to complete both measures five separate times on five separate days,

let's say we would also have 100 data points, but they would not be independent because they are basically repeated measures from the same person.

So we could not use Pearson's R to look at the correlation between the stress questionnaire and the sleep diary.

Similarly, if the data come from related people such as siblings, that would be a violation of the assumption.

Now how do we check this? You need to kind of look carefully at how the data were collected.

So who were the participants and is there anything about them to suggest that they are related?

So this information you would usually find in the participants section of the method section of a published paper.

Okay. So the first two assumptions related to the way the data were collected.

That is the way the study was designed. The third assumption relates to the shape of the data itself.

So correlation analysis using Pearson's r assumes that the pairs of schools come from a bivariate normal distribution.

We'll talk about what that is. And in practice we check what is implies.

So we check that the relationship follows a straight line. This is referred to as linearity.

We check the spread that the spread of y is the same at every value of x.

That's referred to as homogeneity city. And we check that Y scores are roughly normal at every value of x that is referred to as normality.

So these three um aspects linearity and normality, um, are an implication of the data kind of being derived from a bivariate normal distribution.

Now we'll look at each of these things in turn. Let's first remind ourselves what a normal distribution is.

Now this is what it looks like. It's a bell shaped pattern that many measurements follow.

And on the slide you see an example of a height. Most of the values are clustered around the middle.

So this would be the mean or the average. Most of the observations are here.

We're looking at a histogram here. Right. So each bar tells us how many people fall into this bin.

Now the further away from the middle the rarer the values become.

And another kind of, um, characteristic of the normal distribution is that it is symmetrical.

So, um, you can see it's being ten centimetres taller than average is about as unusual as being ten centimetres, um, shorter.

So. Two numbers describe this bell curve.

There's the mean where the middle is of the bell. The bell of where?

The middle of the bell sits. And the standard deviation. So how spread out the bell is roughly the typical pattern from the mean.

Um, so if the mean is here, um, let's say it's 170cm.

These are simulated data. They're not real data. Um, and, um, we have a standard deviation of ten centimetres.

Then, uh, about two thirds of values would fall within one standard deviation of the mean.

That means between 160 and 180. So about two thirds of all observations fall into this dark blue middle bit.

About 95% of observations would fall in between two standard deviations from the mean.

So in kind of the lighter blue, the part and almost all observations,

99.7% of all observations would fall within three standard deviations of the mean.

Now, you might you might wonder where is distribution where the normal distribution comes from?

So why does this bell shape shape appear so often?

And for that we need to look at um a mathematical theorem.

So the central limit theorem is a result from probability theory.

And it has been proven that when many in small independent influences are added up or averaged,

the result is approximately normally distributed, distributed whatever the shape of the individual individual influences.

Um, so The central limit theorem explains why things shaped by many small influences look bell shaped, and in the.

In the 1870s, Francis Galton invented what is referred to as the.

Galton board to illustrate this. So if we look at the slide here, um, this is.

So this is a Galton board. It has lots of different pegs. A ball is dropped from the top here.

And it falls through these rows of pegs and it bounces left and right at random and then pile up, um, in bins at the bottom.

And the pile forms a bell shape. Uh, many small chance nudges add up to a normal distribution.

And let's see whether we can briefly look at a demonstration of this.

So, um, here we have our Galton board.

And so a ball falls from the top, gets bounced, bounced left and right at random, and then add a kind of ends up in one of these pins at the bottom.

We can do that for a one ball. Or we can do it for ten.

Um, if we do it for lots of different balls, we will see that, um, together they build a bell shaped, um, distribution in these bins at the bottom.

So, um, this here we have pals.

We can also look at this from kind of a web where this looks more like a histogram.

So, um, you know, that's that's.

So now we have one ball fall down. So we have one observation in this bin.

Now let's do many balls and see what happens.

And let's say that this measure at the bottom here, um, that we're measuring reading ability of many children in the UK.

So reading ability is shaped by many small influences, such as the number of books children have at home, uh, the school environment,

how often a child reads, but also genes and how well a child learns it has to do is also influenced by nutrition.

So you can think of each of these influences, um, as a kind of being represented, being represented by these pags.

So the children are the bowls here. We have reading ability. Right.

The children each each ball is, is a child. And these pegs are the different influences that affect, um, how well a child, um, learns to read.

So each of these influences nudges a child's ability up or down a little.

So picture each child as a ball falling through the Galton, a Galton board where each peg is one of those influences,

and add up lots of small nudges, and the totals pile up in a bounce shape.

So most children end up here in the middle and a few end up kind of at the extremes.

So we've we've seen lots of children kind of go through this now and this this is what this,

these blue lines is what kind of an ideal or, um, hypothetical normal distribution would look like.

So you can see that the more observations you have, um, the closer it gets to that ideal version of the normal distribution.

Now, a small note. Strictly speaking, the Golden boot illustrates a random process of independent factors.

So with reading, with the reading example ability and not all of those factors are actually independent.

And in real life the different factors can interact also in complicated way.

So I wanted to give that example to make it less abstract, but it is a little bit of an approximation.

Okay. Let's go back to the presentation. Okay.

So we reminded ourselves what the normal distribution looks like.

Um, next um, let's have a look at what we mean by bivariate a bivariate normal distribution.

Basically, we can think of that as a normal distribution for two variables together.

So for any value of x the y scores are normal centred on a straight line with the same spread.

Um, so uh, you can picture that basically as a, as a hill, as you can see on the slide here.

Um, so here we have x and here we have y.

And each of these. So, so the distribution of x if you look from this side this is what that looks like.

And as you can see that very much looks like a normal distribution.

On this x axis we have y if you look at it from this side, uh this is what the distribution distribution of Y looks like.

Also very much looks like a normal distribution together. They built kind of a 3D shape.

Um, that looks like a hill. The higher the hill, the more common that pair of scores is.

If you look at this from above. So we're kind of looking from here.

Now then this is what you see. So the hill is, is an oval.

Um, and it's, it kind of has a, yeah.

This ellipse shaped form. So where most of the observations are here and then there's fewer towards the, the outer edges.

Um. Now if we uh, instead of kind of looking at a simulation of it or picture of it like this,

um, and we look at a sample from people that comes from this oval cloud dots.

Then, then this is what it looks like, right? So if this each dot here, uh, is an observation, kind of an observation from one participant.

So this participant here is called kind of in the middle of the on the x measure and quite low on the y measure, etc..

So each dot here is from one participant. So this is a sample of people.

And this is what a bivariate normal distribution would look like in a scatterplot.

So each of the variables on its own is normally distributed.

The bivariate normal distribution refers to kind of the normal distribution of these two variables together.

So for any value of x the y scores are normal centred and on a straight line with the same spread.

We will get into what that means exactly in a minute. Now it's the correlation.

So this this looked at a positive correlation right. So when x goes up, the value of y also goes up.

So you can kind of see yeah the relationship going from the bottom left to the top right.

So this is a positive correlation. Um but it it the view is quite similar when you um look at a negative correlation.

Right. Um, so the correlation sets kind of the tilt of, of, of the oval and the narrowness,

the strength of the correlation, um, is, is you can see it in, in how thin this, this oval is.

Um, but but the principles are the same.

So from uh, if we, if we say okay, these are we could look at whether each variable is normally distributed by looking at a histogram.

Those things would look the same independent of whether it is a positive or negative correlation.

Okay, so now let's see how we check whether the pairs of schools that we are working with come from a bivariate normal distribution in practice.

The first thing that we look at is linearity, or whether the relationship follows a straight line as um, yeah, as a general pattern.

So Pearson's r assumes that the relationship between x and Y is linear, or that the points fall on a straight line,

or the pattern follows a straight line, so not all of the points need to fall exactly on the straight line.

Rather, the general trend should be described by a straight line. So here we have an example where the s h goes up.

Language competence also goes up. So that would be a linear relationship.

So it would be appropriate for the assumption of linearity is is met here.

Pearson's R cannot provide a full picture of curvilinear relationships.

So those are basically other types of relationship for M.

And we'll look at two examples from language development in a minute that are not linear.

They're curvy. There is a curve rather than a straight line.

So for instance um, vocabulary or the number of word words a child knows grows rapidly at first and then flattens flattens out leaving,

leading to kind of a curvilinear relationship like that.

Or in contrast, um, if we look at how well children understand compound sentences, we see slow development at first and then a later growth spurt.

So we have again a curvilinear relationship, but in the other direction,

you would not be able to use Pearson's R to test or that relationship, or at least it would give an unreliable measure of it.

Um, yet another curvilinear relationship is you can see on the slide here.

Uh, so if we have H on the x axis again, and errors in language use on the y axis, uh, we, we,

we observe something that is referred to as an inverted U-shape, kind of a U-shape on its head.

So at a young age, children make few errors, then they start making more errors.

And then when they get older, uh, the number of errors they make reduces again.

If you were to calculate Pearson's r for, this relationship.

For these two variables, you would get a value of 0.18 and it wouldn't be significant.

So if you just looked at that number, you would say, well, um, there's no relationship between age and errors in language use.

Uh, if you look at the scatterplot you can see that it clearly is a relationship.

Um, but it is not a linear one, which is why Pearson's r doesn't pick it up.

So how do you check for linearity. You have to have a look at the scatterplot.

Okay. So what do we mean by that.

Um it means that the spread of Y is similar at every level of x.

So the spread around the line of best fit. If we think of this, um, you can think of, of, of these things as a scatter plot in a way.

Um, and this has the line of best fit and the spread around it.

Uh, around this line of best fit is, uh, kind of similar all the way across.

So if we look at this orange example, this is a stick.

So we have this the variance of of scores on this end of the variable is similar.

The as it is at this value of the variable. And also at this extreme.

So this, this kind of broad band of observations that have a similar spread across the whole range.

Now in blue and green here we have two examples of heteroscedasticity um distributions.

So um here the spread on the extremes is much wider here on the left and here on the right than it is in the middle.

So that is a heteroscedasticity pattern. Um, in green we have another example of heteroscedasticity and where we kind of have a fan shaped pattern.

So the spread of scores on one side is much wider than it is on the other side.

Again how do we check this. We have to look at the scatter plot.

Um. So then the third kind of implication for of of the pairs of scores coming from a bivariate normal distribution is to look, um,

is to do with normality and a good approximation of whether the normality assumption

is met is to check whether both variables are normally distributed using histograms.

This is an approximation, but it is in practice what we do.

Um, so we've been making histograms over the past few weeks.

So we have a history. We have three histograms here right. The one here on the left shows you what's a kind of a normal distribution would look like.

So here we have our bell shaped pattern that is roughly um symmetrical.

And then in the middle here and on the right we have two histograms that look clearly not normally distributed.

So in the middle we have highly skewed observations.

So a lot of observations to the left and so of lower values.

And then kind of this long tail of observations towards the right of the axis.

So a lot of fast reaction times and then a long tail with some very slow reaction times.

So reaction time data are often skewed. So the histogram will often kind of have this shape.

So these this variable is not normally distributed.

Um here on the right we have an example of a variable um that is uh kind of has very heavy tail.

So it's very it's much thinner than a normal distribution. And it goes out further.

So here this is an example of, uh, timing errors.

Um, and as you can see, yeah, there's some people who have, uh, kind of timing errors in one direction and a few people out here.

So it has. Yeah, it's called a kind of a distribution that has heavy tails.

So we can look at the histogram for each variable that we are using in our correlation analysis

individually and see what it I normally distributed to get an idea of whether that is the case.

Um, we can also create a QQ plot, and that basically compares your data with what a perfectly normal distribution looks like.

So, um, and we can see these at the bottom of, of the slide.

Um, so each point in this in the, in the QQ plot.

So like these ones here is an observation. And if the points full follow kind of this orange diagonal line the data are close to normal.

So you can see that for this normal distribution example here the the points fall very close to the line will never be perfect because,

you know real observations hardly ever are perfect.

But so but so they might divert a little bit from that line.

But broadly speaking it's kind of like a snake coiling around that orange line.

Now here you can see, um, the QQ plot for the skewed data.

So if this QQ plot goes with this histogram and you can see that here at the extremes,

um, the data points very clearly divert away from the orange line.

So that is a very clear indication that is not normally distributed. And similarly.

Similarly um, we have for for this histogram with the heavy tails, you can see how at the extremes do data points go away from the orange line.

So we can look at the histogram of each individual variable.

And we can also create a QQ plot for each individual variable to to check whether the normality assumption has been met.

Okay I will leave it at that for now. We will continue in a little bit.