I’ve been trying to understand some of the work done by Prof Fenton and colleagues. They have done some brilliant work pointing out some of the difficulties in interpreting the official UK data (an example of which can be found here). One of the things they’ve shown is that merely adding in a delay in the reporting of the data can have some strange statistical effects. I can follow the steps but haven’t, as yet, really managed to develop what I would call a deep intuitive understanding of this.
However, it’s also clear that mis-classification of deaths is going to have pretty much the same effect as a reporting delay. If you offset deaths in the ‘vaccinated’ by building a delay so that it takes a couple of weeks to be classed as ‘vaccinated’ then this should give us a similar sort of effect. Prof Fenton says it’s identical (I haven’t gone through the proof of that yet) - but at least, even without any formal proof, we can intuitively see why these things might be equivalent.
In order to understand some of these statistical shenanigans I decided to do something a little bit different and have a look at how mis-classification might affect a calculation of a weekly ‘efficacy’.
So we’re going to imagine the following.
We start with a large number of people in a stadium - let’s think of something like a million people. We’re going to assume that there is a weekly death rate. We’re going to call this r, but we might think of a figure of 0.001 so that if you start with a million people then you’ll see 1,000 of them popping their clogs in the first week.
We’re now going to imagine that at the start of a week we’ll pick a certain number (say maybe 20,000) and move them to another stadium. Each week we’ll do this. We’ll call those in our original stadium the U group - which might stand for ‘unclean’ or maybe even ‘unvaccinated’.
Those moved to the new stadium we’ll call the V group - which might stand for ‘Verily ye are the Holy chosen’ - or maybe just ‘vaccinated’.
Notice we’re not actually going to do anything to anybody - we’re just going to shift them to a new location.
Now if we left it like this - where we increased those in the V group by 20,000 every week and compared death rates in the U and V groups - we’d just see the same death rate in each group (which would be 0.001 in our hypothetical scenario).
But what happens if we start playing silly buggers with our classification?
Let’s suppose we introduce yet another stadium - it’s a holding, or ‘limbo’, stadium where those chosen have to wait a week before being moved on to the promised land of stadium V.
And here’s the slightly ghoulish part. What are we going to do with all those dead bodies? Well, it’s a stadium, so we’re just going to stack up these unfortunate souls in the centre of the pitch.
Furthermore, we’re going to keep limbo clean - anyone who dies in the L group, the limbo group, we’re going to stack them up in the middle of stadium U.
Here’s what this looks like in pictures
So at the start of week 1 everyone is in stadium U and we move 20,000 to stadium L. Some people in stadium U pop their clogs and some people in stadium L pop their clogs. In terms of numbers in this first week we’d have 980 laid peacefully to rest in the middle of stadium U, and 20 in the middle of stadium L.
These 20 poor souls in stadium L, who thought they were the chosen ones, are moved to lie with the other poor souls in stadium U.
Here’s how things proceed over the next week
The survivors in L get moved to the promised land, the stadium of virtue, V. By the end of the week, sadly, some of those will have popped their clogs and are laid to rest in stadium V.
The survivors get moved out of stadium L and another 20,000 hopefuls are shuffled in. By the end of the week another 20 will have popped their clogs and will be awaiting removal to stadium U for the start of week 3.
In stadium U, things are looking increasingly bleak. Not only have they the 980 bodies from week 1 on the pitch, they now have another 20 from L - and at the end of the week 2 another 960 have been added (we had 980,000 at the start of the week, but 20,000 were moved to stadium limbo).
Hopefully you can see that every single week we’re going to get 20 poor souls from limbo added to the total in stadium U. But every week the relative importance of this 20 is going to increase.
The weekly death rate is going to be just the original r in stadium V - that will stay constant each week.
However, the weekly death rate in stadium U is going to increase because each week we’re adding 20, artificially, to the total. However as the number of people in stadium U decreases every week, fewer numbers are laid to rest in the centre - and so that constant 20 added each week is going to make up an increasing proportion, week by week.
Here’s what it looks like when written with the delightful squiggles of math
The total ‘death’ in any week in group U is augmented by the addition from stadium L - so two factors go into it, the deaths we would see anyway plus the added deaths from L. These added deaths act as a kind of ‘fudge factor’ - it really ought to be zero (we shouldn’t be moving corpses, which is equivalent to a misclassification).
So now we can work out the weekly ‘efficacy’ of our classification procedure. Recall that other than shifting people about we have done absolutely nothing to them. They haven’t been injected with any Goo, glorious or otherwise, for example.
Here’s what this looks like
In any sensible world the death rates in stadium U and V should be the same and so we should see zero ‘efficacy’. But we’ve played silly buggers by moving the corpses from L in to stadium U.
The 2nd equation for E above is just a reworking of the formula in terms of numbers.
The weekly ‘efficacy’ is a monotonically increasing function of k. This just means it keeps going up without any temporary wiggles. Granted, it’s not a very big increase for the kind of numbers we’re considering, but it’s there nevertheless. We start off with a zero efficacy and this just keeps on getting bigger and bigger as the weeks progress and more and more people get their much-vaunted chosen status.
Right at the end, when there’s only a few tens of thousands of people in U left to be chosen this weekly ‘efficacy’ is going to get closer to 1 as the last few weeks of choosing happen.
Nobody has been vaccinated, nobody has had any medical intervention, nobody has been hit over the head with the police baton of covid enforcement. All we’ve done is to bugger about with how we classify things - and we see that just by doing this we have a remarkable (and increasing) health benefit for the chosen!
Isn’t statistics a lot of fun?
Definitions really do matter - a lot.
Some appallingly inappropriate chants in the stadium of doom:
- Abide with me; fast falls efficacy
- You'll never die alone
- When the dead go marching in
Not being British myself I have little knowledge of most things UK, but wisdom comes from many places including a Former UK Prime Minister. “There are three types of lies -- lies, damn lies, and statistics.” - Benjamin Disraeli