How much can you polish a turd?

In yesterday’s post we had a look at the great big steaming turd that is the UKHSA’s latest release of the vaccine effectiveness data. All over Twitter and elsewhere heads were shaken, and keyboards mercilessly battered, to tell us that although it looked like a great big steaming turd, it was in fact the most beautiful and exquisite diamond we’d ever seen.

Garbled mentions of base rate fallacies were liberally thrown at us, and several mutterings of Simpsons gleefully thrust upon us along with biases and confounders galore.

But can you really polish a turd this big?

Contrary to popular belief, you actually can polish a turd. When my eldest daughter was 8 years old I got it into my head that it would be a wonderfully educational thing to take her to one of Manchester’s museums of modern art. It seemed like a good idea at the time. The modern art I’d seen up till then mostly consisted of random splodges of colour someone having a seizure had managed to get on to a canvas.

I knew something was desperately amiss when shortly after entering this cavern of delight I looked down to find she had disappeared. The little bugger had this knack of knowing when you’d taken your eye off her for a millisecond. Off she’d go exploring. After a few minutes’ frantic searching I spotted her inside a structure on the main floor. It was a little room, a bit like a shed, all alone, and I could see her inside it. I got closer and heard the strains of a Beethoven symphony coming from within. Ahhh, I thought, glad to see she’s getting a bit of culture.

She was standing, transfixed, watching a black and white video projected on one internal wall of a naked man with headphones dancing whilst listening to some disco music whilst we could only hear Beethoven. His quite impressive member was bobbing up and down as if conducting. Never having been confronted with quite this situation before I wasn’t sure what to do.

I did eventually manage to extricate her from Mr Jiggles and his flouncing truncheon, although I can’t remember how. Maybe the other exhibits wouldn’t be as awkward, I thought. She ran off to look at the pretty whitish glass pebbles scattered on the floor in what looked like a kind of river. What’s this Daddy? I looked at the title of the piece and the brief description. It was called “Semen II”. I spent the next few minutes trying to divert her attention from the sign whilst spouting some bullshit about it being a representation of a river in Winter.

By now I was wondering just what kind of fucking hellhole I’d wandered into, and worse, taken my daughter along with me. Pictures. There’s a room with some pictures in. Thank God. Off we went.

Here we were treated to pictures painted using elephant turd and propped up on little balls of - you guessed it - elephant turd. These scatological plinths were lacquered and polished. So I learned two things that day. You really can polish a turd and, for God’s sake, never, ever, take your 8 year old daughter to a museum of modern art.

Just in case you, like me, are still thinking this couldn’t possibly be real - here’s one of the turd pictures

Anyway, after the cultural heights of modern art, and having learned that turds can be polished, we’re now going to descend into the depths of covid cultism and the gibbering nonsense that is the milieu of the pro-Goo.

Let’s see some of the data from the UKHSA again represented in graphical form.

What stands out for me is the column for deaths. I’ve never been too much interested in cases because of all the absurd testing malarkey coupled with the asymptomatic drivel. We can see that we have about 70% vaxxed (I’m going to use 70% for convenience, but it doesn’t significantly alter any conclusion) with 90% of the deaths occurring in the vaxxed.

It’s just because so many are vaccinated

One popular argument is that this is all just an artefact of the high numbers of people vaccinated. If you had 100% vax rates then ALL of the deaths would be amongst the vaxxed, is a frequently heard statement. They’re suggesting that some kind of base rate fallacy is in play here.

They’re wrong on this. There is no base rate fallacy and this should, I hope, become clearer as we proceed.

A proof of something obvious

So the first thing we’re going to do is to formally prove something obvious. It’s always worthwhile to attempt this even with what appear to be the most obvious of statements.

We’re going to prove the following statement :

If a vaccine is (at all) effective then the percentage of deaths in the vaccinated cannot be higher than the percentage of people vaccinated

We’re going to need a bit of notation.

n(v) : number of people vaccinated
n(u) : number of people unvaccinated
d(v) : number of deaths of people vaccinated
d(u) : number of deaths of people unvaccinated

We’re also going to let the total number of people we have be N and the total number of deaths be D. Instead of percentages we’re going to work with fractions so that, for example, 67% is equivalent to 67/100 = 0.67

So the vaxxed fraction, which we’re going to call λ is just given by
λ = n(v)/N
so that the unvaxxed fraction will be 1 - λ

The fraction of vaxxed deaths is just
µ = d(v)/D
so that the fraction of unvaxxed deaths is just 1 - µ

If the vax is at all effective at preventing death (even just a very tiny bit) then it means that the covid death rate in the vaxxed must be lower than the covid death rate in the unvaxxed.

In maths terms, then, this looks like this

If we write things in terms of λ and µ we can rewrite this as

With a teeny bit of algebra we find that µ < λ

Which means that IF we have an effective vaccine - even the tiniest level of effectiveness - then the percentage of vaxxed deaths cannot be greater than the percentage of people vaxxed.

This is what we set out to prove.

This is the central fact that the pro-Goo folk are trying to “explain away” with all their talk of base rate fallacies and Simpsons and confounders and biases.

Is there a base rate fallacy?

The base rate fallacy argument is really trying to suggest that even though there are a larger number of people in the vaxxed category dying, when you look at the proportions, you find that, actually, a higher percentage of people in the vaxxed category are dying. You can stop right there when you express the problem in terms of percentages like this - because we already have those proportions. But just in case you’re still not convinced, here’s a pictorial representation of what’s going on.

We start off on the LHS with a vaxxed and unvaxxed population. The deaths are in green and 90% of them are in the vaxxed population. We squash the unvaccinated population down a bit in the middle - you can see that the green area is now a higher proportion of the unvaxxed area.

We continue squashing - but there’s a limit. We hit a brick wall where we can squash no more because we get to a 30% / 70% unvaxxed/vaxxed which is what we have. The 90% portion is still going to be a higher proportion of the vaxxed area than the 10% portion is of the unvaxxed area. We can’t arbitrarily go beyond this - because the 70/30 ratio is fixed. It doesn’t matter what size green box we choose - 90% of it is still going to be in the vaxxed portion.

So, no, there’s no problem with any base rate fallacy going on here.

The only way out of this is to suggest the actual numbers are incorrect. But the only way to improve things for the pro-Goo side is to suggest that more people have been vaccinated than the 70% figure (if it were fewer than 70% it would make things worse for Team Pro-Goo).

We could also have grossly miscategorized deaths and have fewer vaccinated deaths - but that would be odd and not, in my view, feasible in the numbers required to make things look all rosy in Goo World. But more on this later.

What about the efficacy?

We’re now going to prove another obvious statement

If the percentage of vaxxed deaths is higher than the percentage of vaxxed people (i.e. µ > λ) then the efficacy is negative

Using the standard definition of efficacy and after a bit of dicking about with algebra we find the following for the efficacy in terms of our λ and µ

The bit on the bottom there can’t be negative because µ is less than 1 (strictly less than or equal to one). With our figures of λ = 0.7 and µ = 0.9 it’s very clear that we have negative efficacy.

So how are we going to polish this turd of negative efficacy?

How much polish do we need?

The question now remains

how different do these figures have to be to return us to something that looks like the Goo is awesome?

OK - to answer this question we’re going to re-write our efficacy to give us expressions for µ and λ. After yet more pratting about with squiggles here’s what we get

What these expressions will do will allow us to calculate things like:

(a) given that we want an efficacy of 50% and have 70% of our population vaxxed what percentage of deaths would be in the vaxxed?

(b) given that we want an efficacy of 50% and we have 90% of the deaths in the vaxxed population what percentage of people need to be vaccinated?

• We want 75% efficacy with 70% vaxxed

In this case we need µ = 0.37 (approx). In other words, in order for the vaccine to have a 75% efficacy we would need to have only 37% of the deaths in the vaccinated. That’s a whopping 53 percentage points difference in our stats here.

• We want 20% efficacy with 70% vaxxed

We’re aiming low here. An efficacy of just 20% is pretty shitty for a vaccine. We would need µ = 0.65 (approx). So here we’d need the actual statistics (with 70% vaxxed) to be a sizeable 25 percentage points lower.

• We want 50% efficacy with 90% deaths in the vaxxed

Here we have less wiggle room. If we have 90% of deaths in the vaxxed then we’ve shown above that we need to have the vaxxed percentage higher than this. To achieve this we would need to have approx 95% of the population vaccinated if 90% of the deaths were occurring in the vaxxed population.

So, basically, to get the steaming turd that is the Goo all nice and shiny we’re going to need a whole lot of polish.

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In conclusion then, what we’ve shown here is that you need to manipulate these stats an awful lot to even begin to get into a territory that looks anyway half-decent for the Goo. You can polish a turd, but the UKHSA figures show that there’s just too much Pfiber in the diet and we don’t have a nice chunk to work with - just a steaming great sloppy mess all over the place.

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