Medical Tests, Bayes' Theorem, and How a First Guess Can Lead to an Absurdity

Name: Medical Tests, Bayes' Theorem, and How a First Guess Can Lead to an Absurdity
Uploaded: 2019-04-04T06:38:40.000Z
Duration: 36 min 3 s

Introduction

The video introduces Bayes theorem and presents a brain teaser that demonstrates how a first guess can lead to an absurdity.

Understanding Bayes Theorem

Bayes theorem is introduced as the main topic of the video.

Viewers are encouraged to check out other videos on Bayes theorem if it is new to them.

Problem Presentation

A problem is presented that most people would initially get wrong, and viewers are shown how to get the right answer using Bayes theorem.

Disease Testing Problem

A scenario is presented where a person goes to their doctor for testing for a disease.

The disease occurs randomly in 1 in 10,000 people, and there is no reason to think that the person being tested is more or less likely than average to have the disease.

The medical test has a false positive rate of 0.1% and a false negative rate of 0.1%.

Calculating Probabilities

Viewers are shown how to calculate probabilities related to the disease testing problem.

Probability Calculation Process

Viewers are asked what the probability is that someone has the disease given that they tested positive.

Two possibilities for testing positive are discussed: having the disease and getting a correct test result, or not having the disease but getting an incorrect test result.

Probabilities for each possibility are calculated using provided information about rates of false positives and negatives.

Applying Bayes Theorem

Viewers learn how Bayes theorem can be applied to solve problems like the one presented earlier.

Using Bayes Theorem for Probability Calculation

Bayes theorem is introduced as a way of calculating conditional probabilities.

An equation representing Bayes theorem is presented.

The equation is used to calculate the probability of having the disease given a positive test result.

Conclusion

The video concludes by summarizing the problem and solution, and emphasizing the importance of understanding Bayes theorem.

Key Takeaways

Viewers are reminded that even though a test result may be positive, it does not necessarily mean that someone has the disease.

The importance of understanding Bayes theorem for solving problems like this one is emphasized.

Bayes Theorem and Medical Testing

This section explains how to use Bayes' theorem to calculate the probability of having a disease given a positive test result.

Calculating Probability with Bayes' Theorem

The probability that a person has the disease is equal to the rate of the disease in the general population, which is one in 10,000 or 0.0001.

The probability that a person tests positive given that they have the disease is 0.999 (99.9%).

The probability that a person tests positive including both people who have the disease and people who don't is 0.010998 (1.0998%).

Using these numbers, we can apply Bayes' theorem to calculate that the probability of having the disease given a positive test result is approximately 9.1%.

Understanding Test Accuracy

This section explains why a high accuracy rate for medical testing does not necessarily mean that receiving a positive test result indicates having the disease.

Rare Diseases and Incorrect Test Results

Because this particular disease is rare, before taking into account any test results, there is already a higher chance of getting an incorrect test result than actually having the disease.

Therefore, many positive test results may come from people who do not have the disease but received an incorrect result.

However, even though there are many false positives, it's important to note that this test still gives correct results for 99.9% of people who have or do not have the disease.

Hypothetical Scenarios

If there were no people with the disease in a population, then 100% of incorrect test results would be false positives.

If only 1 in 10,000 people had the disease and a fake test always gave negative results, it would still be correct 99.99% of the time.

Misconceptions about Test Results

Despite its high accuracy rate, receiving a positive test result does not necessarily mean that you have the disease.

Believing that a positive test result means having the disease leads to an absurdity when applied logically.

Probability and False Positive Rates

In this section, the speaker discusses an example of a test for a disease with nonzero false positive and false negative rates. The speaker calculates the probability that you have the disease given a positive test result using Bayes theorem.

Test Accuracy vs Probability

A coin flip has no diagnostic utility whatsoever.

The accuracy of the test is not the same as the probability that you have received a correct result.

Most people's initial guess leads to an absurd result if used in the case where the tests accuracy is poor.

Example Calculation

Let's take the rate of the disease in the population to be 1 in 10,000.

Your doctor offers you a test that has a false positive rate of 50% and a false negative rate of 50%.

The test will cost $120.

You decide to take the test, so your doctor flips a coin and it comes up heads.

The doctor says it came up heads, and you've tested positive.

Do you still believe that your probability of having the disease is 50%?

No, your probability of having the disease must still be 1 in 10,000.

Conclusion

The results we've seen here might still be somewhat surprising. If you'd like to see a similar example presented from a somewhat different perspective, check out Bayesian Law to Get Rich.