(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)
In the context of a report from a drug study, Stephen Senn writes:
The bare facts they established are the following:
The International Headache Society recommends the outcome of being pain free two hours after taking a medicine. The outcome of being pain free or having only mild pain at two hours was reported by 59 in 100 people taking paracetamol 1000 mg, and in 49 out of 100 people taking placebo.
and the false conclusion they immediately asserted is the following
This means that only 10 in 100 or 10% of people benefited because of paracetamol 1000 mg.
To understand the fallacy, look at the accompanying graph. This shows the simplest possible model describing events over time that is consistent with the ‘facts’. The model in question is the exponential distribution and what is shown is the cumulative probability of response for individuals suffering from tension headache depending on whether they are treated with placebo or paracetamol. The dashed vertical line is at the arbitrary International Headache Society critical time point of 2 hours. This intersects the placebo curve at 0.49 and the paracetamol curve at 0.59, exactly the figures quoted in the Cochrane review.
The model that the diagram represents is simplistic and almost certainly false. It is what would apply if it were the case that all patients given placebo had the same probability over time of headache resolution and ditto for paracetamol and an exponential model applied. However, the point is that for all we know it is true. It would take careful measurement over time for repeated headaches of the same individuals to establish the element of personal response (Senn 2016).
The curve given for placebo is what we would expect to find for the simple exponential model if it were the case that mean time to response were 2.97 hours when a patient was given placebo. The curve for paracetamol has a mean of 2.24 hours. It is important to understand that this is perfectly compatible with this being the long term average response time (that is to say averaged over many many headaches) for every patient and this means that any patient at any time feeling the symptoms of headache could expect to shorten that headache by 2.97-2.24=0.73 hrs or just under 45 minutes.
Is this a benefit or not? I would say, ‘yes’. And that means that a perfectly logical way to describe the results is to say, ‘for all we know, any patient taking paracetamol for headache will benefit. The size of that benefit is an increase of the probability of resolution at 2 hours of 10 percent or a reduction of mean headache time of 3/4 of an hour’.
The latter, of course, depends on the exponential model being appropriate and it may be that some alternative can be found by careful analysis of the data. The point is, however, that the claim that only 10% will benefit by taking paracetamol is completely unjustified.
Unfortunately, the combination of arbitrary dichotomies (Senn 2003) and naïve analysis continues to fuel misunderstandings regarding personalised medicine.
Interesting. Similar issues arise in the interpretation of fitted regression models in social science.
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