[Auto-generated transcript. Edits may have been applied for clarity.]
Okay, so we've looked at what it means for a distribution to or from data to come from a bivariate normal distribution.

Um, but what happens if that is not the case?

That is what I want to want to look at next. And I want to do that by looking at um, some visualisations.

So on this slide you see four scatter plots.

And in the top left here we have a scatter plot as we have from data that come from a bivariate normal distribution.

Here we have our lovely ellipse shape with kind of a. The density of observations is higher in the middle than at the edges.

And you could also tell from this scatter plot that it's a positive correlation.

because when x goes up, y also goes up. Um, okay.

So that is kind of the, the scenario that, um, this is what it looks like when the data comes from a bivariate normal distribution.

Then we have three additional scatter plots, each showing kind of, um, violation of that of that assumption.

Um, so here we have a scatter plot over whether relationship is not linear.

In the bottom left, we have a relationship where we have this fan shaped pattern of, oh, so this is heteroscedasticity, so unequal spread.

Um, and then here we have a situation of skewed data.

Now I would like to look at each of these in turn to understand better what, um,

what what what what this means for a kind of linearity, homosexuality and normality as we are able to check them.

Okay, so let's first look at the bivariate normal scenario.

Um, so in this on the on this on the screen.

Now you can kind of see that same pattern of dots, grey dots kind of in the background.

And on top of that we've um fitted a line of, of best fit.

Um, we have um, we have taken a kind of slices through these, through these points where there is a yellow bar.

And if we look at all the observation in this yellow bar and we create, um, and look at the distribution of those, that is this blue shape.

So you're kind of looking at these observations that are in yellow here.

Orange kind of. And when if you were to create some sort of histogram of those this is not a histogram.

It's a density plot but it's the same principle. And then this is what the distribution looks like.

And we've done that for three slices across this sample of data points.

Okay. So bivariate normal normal distribution.

We have a nice elliptical shape. Um and we have these um that, that we have been looking at.

If we look at these different slices. Um so we can see that the means.

So this is this diamond here shows, uh, the mean for the observations of this slice, the bit in yellow.

You can see that these fall on a straight line, Right.

So linearity principle.

These look like a kind of, um. The linearity assumption has been met.

Now, if we look at, um, the city, the spread of Y is kind of similar for every slice, right?

So we can we can see that here in the distribution of the observations from this slice.

If we can, you can almost kind of tilt your head and then uh, well probably that way actually.

Um, and you can see the distribution in blue of the observations in yellow here.

So the spread of Y is similar in each, um, slice.

Right. So the spread is about this wide here and here and here.

And you can also see that's why is kind of bell shaped for each slice.

Sorry, this is a bell shape and this is a bell shape. And this is a bell shape.

So that is the thing that we that that is what these things look like when the data come from a bivariate normal distribution.

Okay. So what do things look like when the linearity assumption is violated.

This means that the slice means. So these diamonds leave the straight line a straight line is here right.

So if you fit a line of best fit through these data points that are in the background,

right, you fit a line of straight fit a line of best fit through those.

This is your straight line. The means are not on that line.

Um, if we were to look at the the spread and the shape of the distribution of the slices.

They actually look all right. You know, there's the spreads and the kind of going from the maximum to the minimum.

The minimum of the maximum over here and over here and over here, that that spread is similar across the three slices.

So that is okay in this scenario. And also these shapes look roughly bell shaped.

So again it. Normality is of of y at each value of x is not the problem.

The problem is that um the slice means leave the straight line.

So uh, if you were to use Pearson's r, it would kind of mis describe this relationship because it is not linear.

Okay. So then let's look at an example of unequal spread.

So again we have our observations the grey dots in the background.

Then we have taken slices through those kind of three points of, of x.

And you can see that the spread and so kind of the width of this blue distribution here at

this point for this for the most leftward slice is much narrower than the one in the middle.

And, and the one on the right, the one on the right, the spread is really wide.

So this is a scenario. We have a fan shaped pattern in the scatter plot.

And what that means is that the spread is narrower on one side and wider on the other side.

So this this this data are not homeostatic.

And that is a problem. Um. When we look at linearity, that Seems to be fine, right?

The means fall on the straight line. Um, but yeah, the city is is the problem.

Okay, so what does this look like with skewed data?

So we looked at reaction time data before. And um, this is an example of web kind of why.

Did you know if you have reaction time for instance, on this variable, on this axis, this is um, yeah,

this is what that would look like in, in a scatter plot if actually if both variables are highly skewed.

So linearity is fine. The sample the slice means sit on a straight line.

That's not the problem. The spread also looks similar across these three different slices.

Right. And it's going to similarly this spread is similarly white hair as it is hair as it is hair.

This is not an exact site, right? Because you can see that there's a longer tail here, here.

But that this this is close enough. So linearity and homosexuality are fine.

But as you can see, these are not normal distributions. Right?

They are heavily skewed. So in this case um not the normality assumption isn't met.

Okay. So here you have all four in on one slide. Uh, you know, just as a, as an illustration.

And really you can tell a lot right from a scatter plot, which is why we keep telling you it's really important to make a scatterplot of your data.

Okay. So why is any of this a problem? Um, I would like to kind of go back to, to that question, as Rob talked about in his earlier lecture.

Significance testing and hypothesis testing is about checking how surprised you should be to find a correlation of the site that you did.

Is it surprising enough to reject the null hypothesis? Um.

So, um, we know what, uh, what no relationship data look like, right?

And how R varies from sample to sample. That is the sampling distribution of R, and that is um, yeah.

That's that's the model that we use. So with that bivariate normality model and a true correlation of zero.

Mathematicians can work out exactly how R will vary across samples.

So if our data look very different from the model the p value may not be trustworthy.

Right. Um.

And what this looks like is this.

If we go back to our scenarios.

So we have our bivariate scenario here, the skewed data here and kind of the funnel shaped here in the funnel is actually in the other direction now.

But that's fine. So here we have one variable a histogram for let's say IQ.

And we have another variable height here. Um and these things are not related.

Okay. We know because this is how I've simulated the data. And also we know that this is actually true in real life.

IQ score and height are not related. It's not the case that taller people are more intelligent okay.

Or shorter people are more intelligent. Um, here we have the scatter plot.

So IQ score on one axis. Height on the other axis.

And as we can see it is a big cloud kind of in the middle which suggests there is no correlation.

Both variables are normally distributed. We can can kind of check that in our histogram.

And this is what the scatter plot looks like. So this is our bivariate normal situation of two unrelated variables.

So if we were to calculate a Pearson's r here we would get something that's close to zero and not significant.

Okay. So now let's look at that same situation.

So we are looking at two variables that are not related a null hypothesis right.

Um but in a situation where both variables are skewed.

So let's look at a variable reaction time variables from reaction time on some sort of task and hours of social media use per week.

Both are heavily skewed, as you can see in the histograms here.

And this is what this looks like in the scatter plot. We have this collection of data points all on one side for the funnel um shaped situation.

Um, this is what this is what it looks like, right? We have our histograms of hours of practice.

Let's a scenario where we have, uh, you know, people practice something, um, some sort of task for a number of hours per week.

Um, and then they get better at it so they have fewer timing errors.

Um, and this, this is what the scatter plot looks like. So, um, for people who are have practice, practice less.

There is a lot of variation in, in timing errors.

Whereas for people who have practised a lot, they get a lot more consistent.

So that's quite a realistic, um, situation of, of kind of people practising a particular task.

And if you practice more, you get better at it and your your timing gets more consistent.

So here we have a fan shaped situation. Okay.

So now we look at um our type one error rates.

So you might you may remember from last week we want we want to make sure that we're not wrong too often.

So we say okay, um, we're going to be wrong sometimes, you know, we're going to find a correlation that is just random just by chance.

But we are going to, um, as a field in psychology,

we have agreed that we don't want to be wrong too often, and we've basically set that level at five 5%.

Right. So that is um, when there is no relationship between two variables.

And the data comes from a bivariate normal, bivariate normal distribution,

the number of times that we are going to find significant correlation M will be in 5% of the samples.

So P will be smaller than 0.05 in 5 percent of the samples.

If you have a situation, um, where the data come from, a bivariate normal distribution and we as a field, we've decided that that that is acceptable.

Now. When we have data that do not meet that bivariate normal, um, Distribution do not come from a bivariate normal distribution.

So it violates one of these. Uh, if it violates that assumption, then it messes up a type one error rate.

It increases our one type one error rate to different extents.

So in the middle here we have kind of the. The number of times that we would find a significant correlation um, with our skewed data.

And um in the red areas here.

Um, it is it illustrates the situation that um is basically a type one error, right?

So we find, we conclude that there is a significant correlation when really there isn't.

So a false positive. And as you can see in the bivariate normal distribution that is controlled at 5%.

And if we if our data are skewed we can if you can see that here there is just the bars are a little higher than the red line.

But on the other side they're a little lower. So on average, even with highly skewed data, Pearson's R is pretty robust.

Um, to that too. So even though that that, that, um assumption has been violated, it is really quite robust.

And our error rate remains at about 5% on average.

That is very different for the funnel shaped scenario. So if our data are not bivariate, do not come from a bivariate normal distribution.

Instead we have a funnel shape like like this.

Um, then the type one error rate really goes up.

So instead of us being wrong 5% of the time.

If there is no safety, we are wrong about almost 12% of the time.

So you can see that here these grey bars are higher than the red line.

If I zoom in on these things, this is what it looks like. So this is our bivariate normal situation.

This is the skew the situation where our data are skewed.

And this is the scenario where the data are not homeostatic or they are heteroscedasticity.

Right. The funnel shaped pattern. And you can see in the bivariate normal distribution R error rate R type one error rate is nicely controlled.

And we are If there is no relationship between the two variables.

That's our null hypothesis. We we will only be wrong.

We will only wrongly conclude that there is in 5% of the samples.

So just by chance. And that is what as a field, we've agreed is acceptable.

If you use Pearson's r to calculate correlation coefficient between two highly skewed variables,

then it's not perfect because you can see that the the grey bars kind of do not fit into the red area on this side,

but they that average out to to what's happening on the other side.

So on average we are we're still controlling our type one error rate.

And we're not wrong too often even if the data are highly skewed.

What that means is that, um, Pearson's r is quite robust against violations of um, yeah, the kind of that not normality assumption.

But if the, um, data come from, uh, uh, come from a distribution where we have this, uh, heteroscedasticity.

So this funnel shaped pattern in the spread, um, that really messes up our type one error rate.

As you can see on both sides, we we these grey bars go well above the red line,

which means that instead of our error rate being 5%, our error rate is almost 12%.

So it is much more likely that we conclude that there is a significant correlation between two variables in that,

um, in that case, when really there isn't. So we make a type one error rate.

We have a false positive.