# Apple Watch Series 4 ECG and Bayes’ Theorem

## Sep 14, 2018 · 512 words · 3 minutes read

The Apple Watch Series 4 can perform a single-lead ECG and detect atrial fibrillation. According to the Apple Heart Study by Apple and Stanford, the software can correctly identify 98% of cases of atrial fibrillation (true positives) and 99% of cases of non-atrial fibrillation (true negatives). However, what is the probability of a person having atrial fibrillation when atrial fibrillation is identified by the Apple Watch Series 4?

To answer this let’s use Bayes’ Theorem.

\[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \]

- \(P(A)\) is the probability of having atrial fibrillation
- \(P(B)\) is the probability of a positive test
- \(P(B|A)\) is the conditional probability of having a positive test when you have atrial fibrillation

We know \(P(B|A)\) is 0.98 from the Apple Heart Study and according to the CDC the prevalence of atrial fibrillation is 2% among people younger than 65 and 9% 65 and older – so we can use 0.02 for \(P(A)\). We also know from the Apple Heart Study the probability of the Apple Watch not detecting atrial fibrillation if a person doesn’t have atrial fibrillation is \(P(\neg B|\neg A)\) = 0.99. However we still need \(P(B)\), the probability the Apple Watch Series 4 *reports* atrial fibrillation.

If we had 100,000 random people, we’d expect 98,000 would not have atrial fibrillation. 97,020 (99%) would be correctly identified as without atrial fibrillation and the other 980 (1%) would be predicted to have atrial fibrillation Among the 2,000 people with atrial fibrillation 1,960 (98%) would be correctly identified. So the total number of people predicted to have atrial fibrillation would be 2,940 and we can conclude P(B) is 0.0294.

Predicted | atrial fibrillation | no atrial fibrillation |
---|---|---|

atrial fibrillation | 1960 | 980 |

no atrial fibrillation | 40 | 97020 |

Now we can apply Bayes’ Theorem:

\[ 0.6667 = \frac{0.98 \cdot 0.02}{0.0294} \]

The conditional probability of having atrial fibrillation when the Apple Watch Series 4 detects atrial fibrillation is about 67%. Stated a little differently, this means the probability of the Apple Watch Series 4 detecting atrial fibrillation when the person doesn’t have it is 33%.

Even with 99% sensitivity (correctly identifiying atrial fibrillation when a person has it) and 98% specificity (correctly identifiying *no* atrial fibrillation when a person doesn’t have it), consumers should be cautious that the Apple Watch Series 4 may mistakenly detect atrial fibrillation when atrial fibrillation isn’t present.

*Update (2018-09-15)*

The Apple Heart Study findings report the software has a true negative rate of 99% – when the Apple Watch Series 4 detects non-atrial fibrillation it’s correct 99% of the time. So most people don’t have atrial fibrillation (98% of people under 65) and their Apple Watch Series 4 *won’t* detect atrial fibrillation. But what’s the probability a person has atrial fibrillation but their Apple Watch Series 4 *doesn’t* detect it?

Among our 100,000 randomly selected people, we’d expect the Apple Watch Series 4 to predict the absence of atrial fibrillation for 97,060 of them. Of these people 40 (0.04%) would actually have atrial fibrillation. So users should feel confident they *don’t* have atrial fibrillation if the Apple Watch Series 4 doesn’t detect it.