Probability primer: reading a test result
Why a positive troponin means different things in different patients, from sensitivity and PPV to likelihood ratios and Bayes' theorem. Move the sliders to see it.
Cardiac troponin is one of the most frequently ordered tests in hospitals, including for patients without anginal symptoms . A result only helps if you know how to turn it into a probability for this patient. That takes a little probability.
Independent versus dependent events
Flip a coin and get heads. The chance of tails on the next flip is still one in two. The two flips are independent, so the probability of both is the product of the two probabilities.
An elevated troponin and an acute MI are dependent: knowing one changes the probability of the other. That change is exactly what a conditional probability measures.
Reading the notation
- P(A) is the probability of A. Example: the probability of an acute MI among everyone tested.
- P(A and B) is the probability that both are true. Example: having an elevated troponin and an acute MI.
- P(A | B) is the probability of A given that B is true. Read the bar as "given".
Two of these have names you already know:
| Question | Notation | Name |
|---|---|---|
| How likely is MI, given an elevated troponin? | P(MI | positive) | Positive predictive value (PPV) |
| How likely is no MI, given a normal troponin? | P(no MI | negative) | Negative predictive value (NPV) |
| How likely is an elevated troponin, given an MI? | P(positive | MI) | Sensitivity |
| How likely is a normal troponin, given no MI? | P(negative | no MI) | Specificity |
The one formula underneath
P(A | B) = P(A and B) ÷ P(B)
Once B is known to be true, you only care about the cases where B is true, and ask what fraction of those also have A. For independent events P(A and B) = P(A) × P(B), so the P(B) cancels and P(A | B) = P(A): learning B tells you nothing. For a troponin and an MI it does not cancel, and that is why the test is useful.
Sensitivity and PPV are different questions with the same two ingredients. Sensitivity starts from patients who have the disease. PPV starts from patients who tested positive. Confusing the two is the most common error in reading a test result, and the next tab lets you see why it matters.