5.X.5. Example Problem

Example: Suppose a disease affects 1% of the population. A diagnostic test has:

• True Positive Rate = 99%
• False Positive Rate = 5%

This chunk introduces a specific example to demonstrate the application of Bayes’ Theorem. We begin with a scenario where 1% of a population is known to have a particular disease. The accuracy of a test for this disease is given with two critical measures: the True Positive Rate, which indicates a 99% chance that the test will correctly identify someone with the disease as positive, and the False Positive Rate, which shows that 5% of people without the disease will still test positive. These rates play a major role in calculating the final probability that a person has the disease given a positive test result.

Think of it as trying to spot a rare bird in a large park. You know that only 1% of the birds in this park are the rare kind you're looking for. Your binoculars (the test) are mostly accurate, spotting the rare bird 99% of the time when it's there, but they also make a mistake sometimes, indicating that a common bird (5% of the time) is the rare bird. Now, you need to figure out how likely it is that you actually found the rare bird if your binoculars say you did.

Let:

• 𝐷: Has the disease
• 𝐷′: Does not have the disease
• 𝑇: Test is positive Given:
• 𝑃(𝐷) = 0.01, 𝑃(𝐷′)= 0.99
• 𝑃(𝑇|𝐷) = 0.99, 𝑃(𝑇|𝐷′)= 0.05

In this chunk, we define specific variables to simplify our calculations using Bayes’ Theorem. 𝐷 represents the event of having the disease, while 𝐷′ indicates not having the disease. The variable 𝑇 signifies a positive test result. We then assign numerical probabilities to these events: 1% (0.01) of the population has the disease, meaning 99% (0.99) do not. We also note the likelihood of the test returning positive when the disease is present and when it is not, helping us set up the problem for the theorem.

Imagine you're at a health screening event. Each person has a chance to either be sick (1 in a hundred) or healthy (99 in a hundred). The screening test has a high accuracy, catching 99% of those who are sick and wrongly flagging 5% of the healthy people as sick. Defining 'sick' and 'not sick' makes it much clearer how to interpret the test results.

Now, use Bayes’ Theorem: 𝑃(𝑇|𝐷) ⋅𝑃(𝐷) 𝑃(𝐷|𝑇) = 𝑃(𝑇|𝐷)⋅𝑃(𝐷)+𝑃(𝑇|𝐷′)⋅𝑃(𝐷′) 0.99⋅0.01 𝑃(𝐷|𝑇) = 0.99⋅0.01+ 0.05⋅0.99 0.0099 = ≈ 0.1667 0.0594

This part reveals the actual application of Bayes’ Theorem. We substitute the known probabilities into the formula. For the numerator, we multiply the probability of a positive test given the disease by the probability of having the disease itself. For the denominator, we add the probability of testing positive when having the disease times the probability of having the disease to the probability of testing positive when not having the disease times the probability of not having the disease. Simplifying the calculations yields approximately 0.1667. This means that even with a positive test result, the probability that a person actually has the disease is 16.67%.

Using our bird-watching analogy again, let's say you looked through your binoculars and saw a rare bird. Bayes’ Theorem helps you understand that despite the view, there's still a high chance that what you saw was just a common bird because the rarity of your target means even a few mistakes will lead to a misinterpretation.

Conclusion: There’s only a 16.67% chance the person actually has the disease even after testing positive.

This conclusion highlights an important takeaway: despite a highly accurate test with a strong true positive rate, the overall chance of having the disease given a positive result is surprisingly low at 16.67%. This counterintuitive outcome emphasizes how prior probabilities (1% prevalence of the disease) can dramatically influence the interpretation of results.

Returning to our bird analogy, imagine you’re so excited to find a rare bird that you overlook the larger picture of common birds in the area. Even with good equipment, the low numbers of your target species make it easy to be misled. This example serves as a reminder to always consider the base rates of events when assessing probabilities.

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