The total length of the videos in this section is approximately 38 minutes, but you will also spend time answering short questions while completing this section.

You can also view all the videos in this section at the YouTube playlist linked here.

Ways to approximate the reference distribution, rather than calculating it exactly

Question 1: If you don't convert the data to ranks, can you exactly anticipate the mean and variance of the reference distribution in a randomization/permutation test?

Uniform reference distribution

Try sketching a histogram of the numbers from 1 to 10. It's flat, right? One example of a uniform distribution is the distribution of the number you obtain when you roll a die: each number from 1 to 6 has the same probability of occurring.

Question 2: If we don't convert to ranks, does the data necessarily follow any particular distribution?

Central Limit Theorem - first mention in this course!

In the video, I said that the Central Limit Theorem works regardless of the distribution of the original data. This is true. But, then, why is it helpful that converting to ranks gives us a uniform distribution?

As we will see, the CLT says that when we draw a large sample from a data set and record the sum (or mean) of the values in the sample, if we repeatedly draw different samples, the sums (or means) will look approximately normal. However, the meaning of the word "large" depends on the distribution of the original data: the weirder the distribution, the larger sample size we need in order for the CLT to be true. So, the advantage of converting the data to ranks is that we know the uniform distribution is not too weird (no outliers, symmetric), and the CLT will work even for a small sample size.

This will make much more sense very soon when you run a CLT simulation in R.

Question 3: The Central Limit Theorem says that which of the following will be normal, for large sample sizes? Check all that apply.

• The data

• The sum of the values in a sample from the data, if we drew many samples

• The mean of the values in a sample from the data, if we drew many samples

Example of the rank sum test

Below you can download the code file showed in the video, along with the data set I used, which comes from the New York Times. I am providing this code in case you are interested in looking at it at any point. HOWEVER, this lecture is not about R, and many lines in this code file are beyond what I expect you to be able to do in R at this point. The purpose of the following video is for you to see an example of the rank sum test.

Question 4: What do we conclude about covid deaths and low v. high risk states from the example in the previous video?

Facts about the rank sum

Question 5: Are rank sum tests appropriate for large sample sizes?

Handling ties

Question 6: When should you consider a rank sum test? Check all that apply.

• small sample size

• outliers

• censoring

• lots of ties

• when you make a histogram of the data in your sample, it is very far from normal

• big sample size, no outliers or censoring or ties, normal-looking histogram - is the rank sum test still fine?

That's it for this section.

During this tutorial you learned:

• More about the rank sum test, which is an example of a randomization/permutation test

• 3 ways to generate a reference distribution for a non-parametric test (exact, approximate exact, normal approximation)

• Why the distribution of ranks is uniform

• A little about the Central Limit Theorem

• How to perform a rank sum test in R using New York Times COVID-19 data

• The benefits of the rank sum test

• How to handle ties when performing a rank sum test

Terms and concepts:

rank sum test, randomization/permutation test, reference distribution, exact way, approximate exact, normal approximation, Central Limit Theorem

Functions in review:

rank(), order(), sum(), choose(), wilcox.test()