Another conditional probability problem for medics

The importance of Bayes Theorem in Medical Screening is a classic textbook exemplar. There’s a great article at Plus Magazine which runs through a scenario whereby:

• A particular disease affects 1% of the population
• A screening test is such that it gives a positive result for 90% of people who have the disease (True Positives)
• This screening test also gives a positive result for 5% of people who do not have the disease (False positives)

In this scenario, if you imagine you tested positive, many people are surprised to find that their risk of having the disease is 0.154 and not 0.9. This is clearly demonstrated with natural frequencies. We can make up a fictitious population of 10,000 people. Given the probabilities set out above, 100 of these would have the disease and 9,900 would be disease free. Of those who are disease free, 495 would test positive. Of those who have the disease, 90% do test positive, but these 90 people are lost in the crowd of false positives amongst the much larger number of disease free people.

Anyway, this is an utterly standard teaching example. My problem doesn’t concern disease screening, but rather understanding side effects of pharmaceuticals.

I have a number annoying but eminently manageable health conditions. These include AERD (Aspirin Exacerbated Respiratory Disease, which used to be called SAMTER’s triad), Dupuytren’s Contracture, Plantar Fibroma on my left foot and Plantar Fibromatosis on my right foot, some digestive issues I currently think are to do with having my Gall Bladder removed in 1994 and a problem which involves brain fog and fainting a lot (at one stage it was diagnosed as POTs but I think Orthostatic hypotension is more likely). My problem is that I never feel I get much support, I just seem to be on the not-serious-but-irritating pile.

So today’s problem arose through my Asthma which is worse when it’s cold, when I exercise and when my polyps have grown. The trouble is the ENT people are only interested in polyps (not asthma) and the asthma people are not interested in polyps (not asthma). My real, major frustration this year is that I don’t think I’ve trained enough to run the Plymouth Half Marathon; I ran it last year but this year it’s been a cold winter and my asthma has meant most of my training runs have turned into walks.

So, the obvious answer is to increase the amount of preventer inhaler I take to deal with the cold/exercise/polyps. The trouble is, as I’ve been saying for a few years now, that just makes various muscles cramp very badly. Now if the ENT people aren’t interested in asthma, and the asthma people aren’t interested in polyps this is where it all gets a lot worse because the reason for the fainting could well be that I don’t absorb sodium and other electrolytes from my diet. Remember I mentioned I thought Gall Bladder removal might have caused some digestive issues. So when I talk about cramps, what I mean is that small muscles like fingers, toes, calves cramp up a lot of the time, and I have to stretch them out. When I talk about big cramps I mean my entire upper leg cramps, and my entire back cramps. They’re big muscles and essentially a decent cramp there knocks me over. And I don’t want to talk about intercostal cramps in the middle of asthma attack because that is utterly terrifying.

Photo by CNOrdic from Pexels

Well, it’s taken me a long time to get here, but this is where the side effect story starts. My wife was a little shocked by my exhibition when my back cramped and called 111. It’s a nice thing to do, but ultimately pointless. The call handler was very helpful, and did note that inhalers were associated with cramps. It didn’t help that I got my inhalers muddled (in my defence I was in a state of shock). We got sent to an out of hours GP who was sceptical that cramps were a problem to anyone and sceptical that it could be the inhalers because they were listed as a “rare” side effect. It was only after my wife pointed out the call handler had made some other comments that she acknowledged inhalers could lower some metabolites and suggested some blood tests.

There’s a lot I could say here, probably more important things. But I got hung up on the conditional probability issue here. A side effect of cramping may indeed be rare from asthma inhalers. But that is the probability of getting cramp given you have prescribed an inhaler to an asthmatic. We had the opposite situation, I had cramps, and wanted to know the probability it was the inhalers that were contribution. [ P(Cramp|Prescription) \neq P(Prescription is a problem | Cramp) ].

This feels like a fundamentally basic error to make. There are 100 asthmatics walking around my town merrily using the inhalers. How many them are going to call a doctor to tell them they don’t have cramps? The only one who will call is the one who gets cramps. In other words, 1 in 100. As it turns out, I phoned Asthma UK for some advice and was told the side effect of cramp is not rare rather uncommon when using inhalers with Fomoterol (and these words rare and uncommon have a specific meaning) and they suggested I ask for a different combination inhaler. So, we shall see, in the meantime, I wonder if there’s a market for teaching medics how to use conditional probability correctly.