MM1F28 Seminar

Week 8

Table of Content

Bivariate relationships between numerical variables

Pearson’s R

Parametric requirements of Pearson’s R

Spearman’s Rho

Correlation vs causation

Exercises

Bivariate relationships between numerical variables

A bivariate relationship in the context of correlations is the symmetrical relationship between two numerical variables (or ordinal, but more on that later). There are two types of correlations: positive and negative.

Pearson’s R

Without showing the maths, Pearson’s R is a standardised measurement of the symmetrical relationship between two numerical variables. Let’s look at an example.

My hypothesis is that there is a significant correlation between professor salaries and years of service (makes sense I think?). Here is my hypothesis written out:

H0: No correlation between salary and years of service

H1: Correlation between salary and years of service

Let’s ignore tha parametric assumptions for now. Here is the result of Pearson’s

There are actually just two statistics we are interested in, the ‘cor’ and the p-value. Here is how we would interpret the output:

There is a medium and significant correlation between salary and years of service [r = 0.33, p < 0.05]. That’s basically it.

As a side note, the correlation coefficient r is actually an effect size estimate. Here is a table for effect sizes:

Parametric requirements of Pearson’s R

Pearson’s assumes bivariate normality. In other words, both variables need to be normally distributed. If they are not, use Spearman’s Rho instead.

Spearman’s Rho

This is a non-parametric alternative to Pearson’s R. If we go back to the previous example, you will find that salary is on the threshold of normality (if you check the QQplot), but years of service is arguably non-normal. Therefore, let’s use Spearman’s instead:

You can see that the correlation coefficient is actually larger than before, and the p-value is even smaller. The increase in statistical power is due to the parametric violations. Anyway, we report it in the same way as before, with a slight change:

There is a medium and significant correlation between salary and years of service [rs = 0.42, p < 0.05].

Correlation vs Causation

Finally, it’s worth noting that correlation doesn’t always mean causation. There is a lot of information on this in textbooks and online, and I encourage to do some reading on the subject. Found this on twitter:

Exercises

Question 1

What is the parametric assumption for running Pearson’s R?

Bivariate normality

Question 2

What is a non-parametric alternative to Pearson’s R?

Spearman’s Rho

Question 3

How large of an effect size is an r of 0.45?

Medium

Question 4

Interpret the output shown. Make sure to name the variables and summarise the findings.

There is a medium and significant correlation between salary and years since PhD [r = 0.42, p < 0.05]

Question 5

What do you think the 95% CI in the output from the previous question is referring to?

Ad infinitum, if we took correlations from equal sized samples, we would expect 95% of all correlation coefficients to be between 0.33 and 0.50

Question 6

Give an example of a non-causative correlation

Just get creative with this one. Check online for funny examples.