Earlier this year the Finnish National Institute for Health and Wellfare published a report about the increased risk of narcolepsy observed among children and adolescents vaccinated with PandemrixR. In short, the conclusion was that the swine flu vaccine seemed to have had an unexpected side effect; the risk of narcolepsy, a sleeping disorder disease, was larger for vaccinated children in the 4-19 year age group than for unvaccinated children in the same age group.
Steven Novella at Science-Based Medicine wrote a great piece about this last week, discussing how this should be interpreted. I’m not going to go into a discussion about the findings themselves, but I would like to discuss the following part of the press release:
Sceptical commenters on blogs and forums have questioned whether a 9-fold increase in risk really was observed. Here’s the reasoning:
The estimated risk within a group is (the number of observed cases of the disease)/(size of the group). That is,
Risk for vaccinated child: 52/(n*0.7) = 1/n * 52/0.7
Risk for unvaccinated child: 8/(n*0.3) = 1/n * 8/0.3
where n is the number of children in the 4-19 age group.
So that the relative risk, i.e. the risk increase for the vaccinated children, is
Hang on a minute. 2.79? If a 9-fold increase in risk was observed the relative risk should be 9! It seems that the Finnish epidemiologists made a mistake.
…or did they?
Not necessarily. When analyzing this data, we need to take time into account. The report itself is only available in Finnish, but using Google Translate I gathered that the unvaccinated group were studied from January 2009 to August 2010 whereas the vaccinated individuals were studied from the date of vaccination and eight months on. In other words, the unvaccinated group had a longer time span in which they could fall ill.
That means that in order to calculate the relative risk, we need to divide the number of cases by the number of months that the groups were studied, to get the risk per month. That eliminates the time factor. After doing this, the relative risk becomes
That’s higher, but still not 9. Well, to complicate things a bit it seems that an individual was considered to be a part of the unvaccinated group until the date of vaccination, making the calculations a bit more difficult. When that is taken into account, along with other difficulties that no doubt occur when you have the actual data at hand, the relative risk probably becomes 9.
The full report is not yet available, so I can’t say how close the above approach is to the one that was actually used in the analysis. Nevertheless, I hope that this post can help shed some light on the statistics behind the statement about a 9-fold increase.
A problem with this approach is that the number of months under which the unvaccinated group was studied might affect the results, just as in the shark attack example that I wrote about last week. Changing the time span for the unvaccinated group to January 2008 to August 2010, say, does however not change the conclusion in this case. The analysis seems to be pretty robust to the length of time under which the control group were studied.
WHO issued some comments regarding the Finnish study that are well worth reading.