Bonus Eruptus Redux! A Few Thoughts on COVID-19 Testing

SARS-CoV-2 is the virus that causes COVID-19.  For what it is worth, SARS-CoV-2 stands for severe acute respiratory syndrome coronavirus 2.  This post is not about the virus, it is about the mathematical and probabilistic problems the virus presents to those who wish to test for it.  Unfortunately, the news is not good.

The first thing that any researcher would want to know is the accuracy rates of the tests being given to the public at large.   What is the rate of false positives?  How about false negatives?  No test is 100% accurate, and some do not come close to approaching that figure.  So anyone taking a COVID-19 test is going to have some reasonable questions to ask the people administering it.

If I test positive, what’s the probability I have the virus?

If I test negative, what’s the probability that I have the virus in spite of the test results?

Two legitimate questions, don’t you think?  To answer those questions, scientists need one more piece of information to plug into the equations.  They need a really good estimate of the percentage of people taking the test who legitimately are infected.  Of those three questions, at this point in time, we don’t have a good answer for any of them.

The May 13, 2020 edition of The New York Times includes an article by Todd Haugh and Suneal Bedi,  business professors at Indiana University.  While they are not scientists, they have a strong interest in the economic aspects of getting the economy back up and running.  Here are the headlines of their article:

In a population whose infection rate is 5 percent, a test that is 90 percent accurate will deliver a false positive nearly 70 percent of the time.

Yes, you read that correctly.  How good is a test if it gives us a false positive approximately 70% of the time?  Good question, isn’t it?

On 11/18/19, I wrote a short essay on the problems with mass medical testing.  Like the rest of us, I had no idea what was coming.  Replace the dreaded Bonus Eruptus with COVID-19, and you will get some insight into the problems we are all facing.  Mathematics and its close cousin Probability Theory are tyrannical in nature.  We can’t bend Mathematical Laws to our will no matter how much we would like to.  We will have to live with the inherent uncertainty of the tests being developed now and in the future.

Here is my original post.

Bonus Eruptus!

Let me begin by letting everyone know that I love The Simpsons.  The show is now in year 31, and I still look forward to each week’s episode.  I will admit that a few years in the middle of the run were pretty lean, but the show is experiencing a renaissance.  The Simpsons are back on solid footing.

Some of you may remember when Dr. Nick introduced us to Bonus Eruptus.  It was episode 21 of season 7.  The episode is entitled 22 Short Stories about Springfield, and that is exactly what transpires, 22 vignettes about the characters populating Homer’s hometown.  I think it is very clever and I have always wanted them to do more episodes like that one.  This particular episode, one of my favorites, first aired on 4/14/96.  Wow, the show has been around a long time, hasn’t it?  I will gladly take another 30 years.

During that stellar episode, Dr. Nick defined Bonus Eruptus as “a terrible condition where the skeleton tries to leap out of the mouth and escape the body.” Apparently, Grandpa Simpson had this condition, at least that was the diagnosis of the esteemed Dr. Nick Riviera.  I want to take a closer look at the mathematics behind this ostensibly severe condition.  Why?  I think that we might be able to learn a thing or two about probability theory and the inherent problems that come along with mass medical testing.

Please indulge me for a moment. Let’s all pretend that we live in Springfield USA and that Bonus Eruptus is a legitimate concern.  I know I wouldn’t want my skeleton to try to take its leave of me.

Imagine that Mayor Quimby, in a transparent attempt to get reelected, offers free, yet mandatory, testing to all the inhabitants of Springfield.  Since I have no idea how many people live there, let’s say that 10,000,000 people are living in the greater Springfield area.  I know that is more of a Capital City number but just play along, OK?  Of those, let’s say that 50,000 of them have the dreaded Bonus Eruptus.

Now let’s imagine that Bonus Eruptus is caused by a virus, one easily detectable by a simple test.  Like all tests, though, it is not perfect.  Some people who have the virus will test negative, and a certain percentage of the people who are negative will, in fact, test positive.  Imagine that the false-positive result rate is 2%.  Also, the poor people who have the virus will test positive only 95% of the time. So, the simple question is:  If someone actually tests positive, e.g., Bumblebee Man or Jeff Albertson (extra points if you know who that is), what is the probability that they actually have the terrible disease?  Think about that a while before you go on.  As you might already have guessed, the answer is not nearly as straightforward as you might think.  After all, why else would I be writing about it?

So, here we go.  Of the 50,000 people who have the virus, only 47,500 of them will actually test positive.

50,000 x .95 = 47,500

We know that 9,950,000 total people do not have it.

10,000,000 – 50,000 = 9,950,000

Of the people who do not have the virus, there will be 199,000 who will test positive anyway (the false positives).

9,950,000 x .02 = 199,000

So now, we can do some simple addition and see we come up with a total of 246,500 people who will test positive for Bonus Eruptus.

47,500 + 199,000 = 246,500

Of those, we know that only 47,500 will actually have it.  So if you test positive for the virus, there is only a 19.3% chance that you actually have Bonus Eruptus!. D’oh!

47,500 / 246,500 = 19.3%

Isn’t that interesting?  Without walking through the math, there is no way that a 19.3% chance could be seen as a reasonable possibility.

It is time for me to go.  I have to prepare for this week’s show.  I hear that Homer is going to do something stupid, and Marge is going to get upset.  I am about to burst with excitement.