# Causality based on conflicting data

1. Jun 6, 2009

### JakeA

There's a common heart defect called a PFO, Patent Foreman Ovale.

It's thought to cause stroke and migraines in some people. 25% of the general population has a PFO, and 40% of people who have a had a cryptogenic stroke, one where the cause is unknown, have a PFO. The link between PFO and strokes is that a PFO will open a flap between the chambers of your heart periodically allowing venous blood clots that would otherwise be filtered through the lungs to go to your brain.

My question is what percentage of people who have a PFO and a cryptogenic stroke should be assumed to have it caused by the PFO?

At a minimum you could conclude that it's at least 37.5%, which is 15/40, the increment between the general population of 25% and the crypto stoke population of 40%.

The problem is that PFOs don't increase the risk of stroke for the general population, or at least don't do it significantly. The data is somewhat at conflict, I think because there are other risk factors for people with PFOs and strokes, i.e. you have to have something else that is pathogenic besides a PFO to be at increased risk of stroke.

But back to my question, what would you do with the other 62.5% of the people with PFO and stroke? Would you, in absence of any valid data take 50% and assume another increment of roughly 30%?

Thanks.

2. Jun 6, 2009

### D H

Staff Emeritus
What you did is statistically invalid. What you can do is ask whether the presence of PFO increases your risk of getting a cryptogenic stroke. By Bayes' Law,

$$P(\text{cryptogenic stroke}|\text{PFO}) = \frac {P(\text{PFO}|\text{cryptogenic stroke})\,P(\text{cryptogenic stroke})} {P(\text{PFO})}$$

where
• $P(\text{cryptogenic stroke}|\text{PFO})$ is the probability of having a cryptogenic stroke given that one has PFO
• $P(\text{PFO}|\text{cryptogenic stroke})$ is the probability that someone who has had a cryptogenic stroke also has PFO
• $P(\text{cryptogenic stroke})$ is the probability of having a cryptogenic stroke by any cause
• $P(\text{PFO})$ is the probability of having PFO.

Using $P(\text{PFO}|\text{cryptogenic stroke})=0.4$ and $P(\text{PFO})=0.25$ yields

$$P(\text{cryptogenic stroke}|\text{PFO}) = 1.6 \times P(\text{cryptogenic stroke})$$

In other words, people with PFO have a 60% increased chance of having a cryptogenic stroke than the population at large.