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Denominators for COVID-19 vaccination statistics

To allow vaccine uptake percentages to be calculated, data on the size of the population is needed for the denominator in the calculation. This paper explains the different denominators that are used in the COVID-19 vaccinations publications, including the strengths and weaknesses of each and implications for the quality of percentage uptake figures calculated using these denominators.


Denominators for demographic characteristics and geographic areas

The weekly and monthly COVID-19 vaccinations publications include estimates of population sizes by certain demographic characteristics and at different geographical levels. These estimates can be used as denominators to calculate approximate vaccine uptake percentages. Two different sources of denominators are used, and the publications provide the best available source for each breakdown as management information. Vaccine uptake rates should be calculated using the most appropriate denominator provided as described below, as the two sources will provide different results. Vaccine uptake percentages calculated using these denominators should be considered as estimates only, as there are known issues with both sources:


1. Office for National Statistics (ONS) 2020 mid-year population estimates

The ONS 2020 mid-year population estimates are the most recent Official Statistics on population size, and the best publicly available population estimates. Prior to 23rd September 2021 the ONS 2019 mid-year estimates were used as denominators, as they were the most recent Official Statistics on population size when the COVID-19 vaccination programme began, and were consistent with the estimates used in the UK COVID-19 vaccines deployment plan. ONS 2020 mid-year estimates are now used to calculate vaccine uptake percentages as they provide more up-to-date estimates of population size.

As the ONS population estimates are based on the 2011 Census, they are subject to a degree of uncertainty, and do not reflect changes to the population since 2020. The ONS 2020 mid-year population estimates are likely to be an underestimate.

The ONS estimates are used as denominators for national, regional, Integrated Care System (ICS) / Sustainability Transformation Partnership (STP) and Clinical Commissioning Group (CCG) geographies only, as they are less robust at smaller areas. They are also used as denominators for age and gender breakdowns, and the breakdown by Index of Multiple Deprivation (IMD) decile. 

Some of the uptake rates calculated using the mid-2020 ONS denominators are reported as 100%* in the publications. In these instances, the number of people who have been vaccinated exceeds the ONS population estimate for that group. This predominantly happens in the 75-79 age group, although there are also other instances of uptake rates exceeding 100%. The impact is largest in the 75-79 age group because of a large number of people born soon after the second world war who were in the 70-74 age group in mid-2020 but are now in the 75-79 age group. These people are counted in the number of people vaccinated for age 75-79, but in the denominator for age 70-74, resulting in an apparent vaccine uptake rate of more than 100%. Conversely, uptake rates for the 70-74 age group will appear lower as a result.


2. National Immunisation Management System denominators

National Immunisation Management System (NIMS) denominators are the numbers of individuals registered with the NHS who are currently alive in the resident population. Overall they likely overestimate the population and so underestimate vaccine uptake percentages, as death registration data is subject to a reporting lag (more information on this can be found on the NHS Digital website) and there are also concerns about people who are no longer resident in England still being counted in NIMS. Coverage can therefore be viewed as being ‘at least’ the figures calculated using the NIMS denominators.

Unlike the ONS denominators which are fixed, the NIMS denominators are updated in each weekly and monthly publication, to reflect known changes to the current resident population.

The NIMS denominators do not include those without an NHS number and so do not cover the whole population. However, this aligns with the number of people vaccinated reported in the weekly and monthly publications.

alloneword » Wed Feb 02, 2022 11:13 pm wrote: Certainly worth keeping an eye on, should they ever decide to publish an updated dataset.

..What happened to the pandemic? The ‘pandemic’ seems to squeeze into a nodule of activity during April/May of 2020 and that’s yer lot!

So where are the second wave and the third waves? Why are adults with respiratory conditions (which include COVID-19) not being shipped into the department at record-breaking levels? Why are things so flat after June 2020? How can this emergency department claim to have been so busy with COVID-19 admissions?..

..So how does all this translate in hospitalisation rates for adult respiratory admissions over time?..



What we have here are square waves that are always indicative of administrative, managerial, protocol or policy change i.e. anything and everything non-medical. A decision was clearly made in 2017/w39 to stop giving beds to adult respiratory admissions. Ward closure can do this, along with a staffing crisis or even a transport crisis. The problem was suddenly solved in 2019/w5 until a hiccup spanning 2019/w39 to 2019/w47, followed by an odd middle period lasting until 2020/w11. So here’s my barbed question: did the pandemic really strike after 2020/w11 or are we simply observing a NHS Trust solving patient management issues that then gives the impression of elevated bed use?

Whatever the reason for this jiggery-pokery we can plainly see that hospitalisation rates post-pandemic for COVID and non-COVID respiratory admissions are nothing special. Rather odd for a novel SARS coronavirus don’t you think?

stickdog99 » Thu Feb 10, 2022 8:21 am wrote:https://dailysceptic.org/2022/02/09/heart-problems-after-covid-are-much-worse-for-the-vaccinated-nature-study-shows-but-its-hidden-in-the-appendix/
Nature published a comprehensive study this week on cardiovascular risk including a total of over 11 million patients that has made a few headlines. The aim was to identify the cause of increased cardiac pathology. It should have been a very simple study comparing four groups:

Not infected and never vaccinated
Not infected and vaccinated
Infected but not vaccinated
Infected and vaccinated
It is hard to believe the authors did not look at these groups, but whatever was found when comparing them remains a mystery.

Instead, the following groups were compared:

Not infected and never vaccinated data from 2017
Not infected, including vaccinated and not vaccinated
Infected but not vaccinated
Infected with vaccinated people included but using modelled adjustments
When studies with huge datasets use modelling and fail to share data prior to their adjustments alarm bells should start ringing. Therefore, I took a deeper dive to see what else was questionable.

There were serious biases in the paper which need addressing but first let’s look at the critical question of myocarditis (heart inflammation).

Because of the known risk of myocarditis from vaccination it is worth looking particularly closely at the data presented on this. Oddly, for the issue of the day, the data on myocarditis was all hidden in the supplementary appendix to the paper.

The risk of myocarditis appears to be an autoimmune (the immune system attacking the heart after interaction with the spike protein) rather than direct damage by the virus/vaccine spike protein. Therefore, myocarditis could result from the virus or the vaccine. The key question that needs answering is whether vaccination protects or enhances the risk from the virus.

The authors report 370 per million risk of myocarditis after Covid infection in the unvaccinated. The contemporary control rate was 70 per million and the historic one was 40 per million. What was wrong with the contemporary controls?

They made it clear they removed those who had been vaccinated from the calculation in the Covid arm but they did not state they did this for the control arm. Did vaccination lead to a 30 per million increase in myocarditis in the control arm? Given the cohort appears to be old and we know myocarditis incidence is worse in the young a one in 30,000 incidence is significant.

What about those who were vaccinated and had Covid? Once vaccination (and modelling) were included, the rate rose to 500 per million. It is not entirely clear whether supplementary Table 22 excludes those who were not vaccinated, but given that it does not state the unvaccinated were excluded from this data it is fair to assume the 500 per million relates to the whole population.

Given the higher risk of myocarditis after vaccination one might wonder whether this study showed protection from infection due to vaccination, as this would lower risk from the virus. Hidden in the legends of the supplementary tables the authors reveal that 62% of the Covid patients had been vaccinated compared to 56% of the non-infected controls (not a great advert for vaccine effectiveness against infection).

Using the fact that 62% of the Covid cohort were vaccinated and that the unvaccinated had a rate of 370 per million, to get to an overall rate of 500 per million the vaccinated 62% must have had a rate of 580 per million (580×0.62 + 370×0.38 = 500). Therefore, in those with Covid and vaccination the rate (even after modelling) was 210 per million higher (58% higher) than the unvaccinated with Covid. (If supplementary Table 22 did exclude the unvaccinated the incidence of myocarditis after Covid would have been 35% higher in the vaccinated.) An extra 210 per million works out as an additional risk from vaccination of one in 5,000 among a relatively old population. How high was it for the younger men? This critical question was left unanswered.

The data comprised medical records for U.S. veterans who were 90% male, three quarters white and had a mean age of 63 years.

Two control groups were selected:

Patients who had used healthcare in 2017 and were still alive in March 2018.
Patients who had used healthcare in 2019 and were still alive in March 2020.
These groups were compared to patients who tested positive for Covid after March 2020, with each patient being matched to one patient from each control and measuring beginning from the same day as the positive test but two years earlier for the 2018 control.

There was a significant bias between these two control groups and those who tested positive.

The Covid patients (not just those who were sick with it – all those who tested positive) were more obese, saw doctors more often, had more cancer, kidney disease, lung disease, dementia etc.



There are two ways to deal with such biases. One is to match the 150,000 Covid patients with similarly sick patients from the over five millions controls. This reduces the size of the control group but when it is already so large this should not be a concern. Instead, the authors modelled the data until the groups seemed similar. Using an algorithm they claimed the same total number of people were present in the Covid cohort, but whereas 49,407 actually had diabetes in the raw data, 11,903 (24%) no longer had diabetes according to the weighted data. Similarly, 14% were ‘cured’ of lung disease, 14% of cancer and a full 35% of the dementia patients no longer had dementia.

There was no discussion in the paper about the reasons for this unhealthy bias among the Covid patients. All positive test results were included and anyone can catch SARS-CoV-2, so the factors that increase the risk of serious disease and hospitalisation should not have biased a dataset based only on infection. Instead the authors discuss the hypothetical issue of people in the non-infected control group having Covid but not getting tested such that the damage caused by Covid could be worse than the paper reports.

It has been well established that hospital transmission dominates as a source of spread and SAGE has reported that up to 40.5% of cases could be traced back to hospital spread and a majority of hospitalised patients in June 2020 were linked to hospital spread. In Scotland, in December 2020, 60% of the acutely ill with Covid acquired the infection in hospital. Patients accessing hospital are highly likely to be less healthy than the general population. Indeed, we know that the Covid patients in the study accessed hospital more frequently than the controls. If the bias was related to hospital acquired infection then the whole study is called into question, as people who attend hospitals are more likely to be sick.

The authors picked some control conditions to attempt to show they had not introduced a bias. Given the study was about cardiovascular diseases, including those that are an immediate threat to life and those that are very common, I would have picked conditions that might kill you within a year, like lung, pancreatic or oesophageal cancer and common conditions e.g. urinary tract infections, diabetes or prostate cancer.

The authors chose three rare malignancies, all with a one-year survival rate of over 80%, and pre-invasive melanoma – why not include invasive melanoma? They then included rare conditions and odd selection of: hypertrichosis (‘werewolf syndrome’ with excessive facial hair), sickle cell trait and perforated ear drums. When the choices are so niche it begs the question of what the results would have been if more obvious choices had been selected.

The group that tested positive for Covid did badly: 13% ended up in (or began in) hospital and 4% in ICU. The mean age was 63 years which may explain part of the high percentage of sick Covid patients, but it does, again, suggest this group may have been more vulnerable than the control.

They then compared the risk of various cardiac outcomes against the controls. However, they used the same control to compare non-hospitalised patients as patients who had received ICU care. Of course, people who have needed ICU care will be more likely to have cardiovascular complications. Indeed, many of the patients may still have been in the ICU when the measuring period began 30 days after the positive test. A fair study would have only compared the ICU outcomes with the sickest people within the control group, not the average of the whole control group.



The risk to the non-hospitalised Covid patients was low for almost all the cardiovascular risk factors.

The risk to the hospitalised was higher (but remember the controls had significant biases).

Those on ICU had a much higher risk. What is not clear is how much of this is because of the virus.



It is not a surprise for people who have had an ICU stay to be unwell for some time afterwards. The risk of ICU admission for Covid was higher than for influenza, but it is important to understand how much of the cardiovascular risk resulted from the virus and how much from the stay in intensive care per se. How do these Covid ICU patients compare to other ICU patients? The paper did not say.

Similarly the paper makes no attempt to unpick how many of the Covid patients tested positive only after being admitted to hospital. If, as in other studies, a significant proportion acquired Covid in hospital, then a higher risk of being diagnosed with other conditions would be highly likely.

Having failed to examine the above two questions – how much cardiovascular disease was a confounder of hospital transmission and how much is secondary to ICU harm – the overall risk of consequent cardiovascular problems included all the above cardiovascular conditions and thereby inflated the average for the Covid population as a whole.

Nature has published this paper which presents data in an obtuse way that should never have passed peer review. The results were presented as showing how dangerous the Covid virus was for cardiovascular complications without suitable controls to enable that conclusion to be drawn. The evidence on vaccination risks was hidden and not presented in a meaningful way for different age groups. Even then, they demonstrated a significant risk of myocarditis after vaccination, particularly after then encountering the virus but this key finding was hidden in the supplementary appendix. Why?

..During the spring outbreak of 2020 asymptomatic COVID death greatly exceeds symptomatic by a factor of around x3 at peak. During this time COVID was billed as an acute respiratory disease but it is clear that this was far from the case! It would have been more accurate to portray it as an asymptomatic respiratory disease (i.e. a disease of something else) but then again the behavioural insights team wouldn’t have been able to create all those highly emotive blue-filtered images of older people gasping for their last breath.

Come the second and third wave and it is a different matter. We now see equal numbers of symptomatic and asymptomatic adult COVID in-hospital deaths, with symptomatic death reaching an all time pandemic peak. This clearly wasn’t a good time for those with significant comorbidities running into the winter season, nor for teams keeping ICU beds turning over. Even so, what the ONS are claiming as causal COVID death – and the death certification process itself - must be brought into question: at very best COVID causal death figures are half what they claim.

Take a look at this study:

Besides their effects on specific (adaptive) immune memory, certain vaccines such as Bacillus Calmette-Guérin (BCG) and the measles, mumps, and rubella (MMR) vaccine also induce long term functional reprogramming of cells of the innate immune system. (Netea et al., 2020). This biological process is also termed trained immunity when it involves increased responsiveness, or innate immune tolerance when it is characterized by decreased cytokine production (Ifrim et al., 2014). Although these effects have been proven mainly for live attenuated vaccines, we sought to investigate whether the BNT162b2 vaccine might also induce effects on innate immune responses against different viral, bacterial and fungal stimuli. One of the trademarks of trained immunity is an elevated production of inflammatory cytokines following a secondary insult (Quintin et al., 2012). Surprisingly, the production of the monocyte-derived cytokines TNF-α, IL-1β and IL-1Ra tended to be lower after stimulation of PBMCs from vaccinated individuals with either the standard SARS-CoV-2 strain or heterologous Toll-like receptor ligands (Figures 1 and 2). TNF-α production (Figure 1B-1G) following stimulation with the TLR7/8 agonist R848 of peripheral blood mononuclear cells from volunteers was significantly decreased after the second vaccination (Figure 1C). The same trend was observed after stimulation with the TLR3 agonist poly I:C (Figure 1D), although the difference did not reach statistical significance. In contrast, the responsesto the fungal pathogen Candida albicans were higher after the first dose of the vaccine (Figure 1G). The impact of the vaccination on IL-1β production was more limited (Figure 2A-2F), though the response to C. albicans was significantly increased (Figure 2F). The production of the anti-inflammatory cytokine IL-1Ra (Figure 2G-2L) was reduced in response to bacterial lipopolysaccharide (LPS) and C. albicans after the second vaccination (Figure 2K, 2L), which is another argument for a shift towards stronger inflammatory responses to fungal stimuli after vaccination. IL-6 responses were similarly decreased, though less pronounced (data not shown). The induction of tolerance towards stimulation with TLR7/8 (R848) or TLR4 (LPS) ligands by BNT162b2 vaccination may indicate a more balanced inflammatory reaction during infection with SARS-CoV-2, and one could speculate whether such effect may be thus useful to regulate the potential over-inflammation in COVID-19, one of the main causes of death (Tang et al., 2020). On the other hand, inhibition of innate immune responses may diminish anti-viral responses. Type I interferons also play a central role in the pathogenesis and response against viral infections, including COVID-19 (Hadjadj et al., 2020). With this in mind, we also assessed the production of IFN-α by immune cells of the volunteers after vaccination. Although the concentrations of IFN-α were below the detection limit of the assay for most of the stimuli, we observed a significant reduction in the production if IFN-α secreted after stimulation with poly I:C and R848 after the administration of the second dose of the vaccine (Figure 1H, 1I). This may hamper the initial innate immune response against the virus, as defects in TLR7 have been shown to result in and increased susceptibility to COVID-19 in young males (Van Der Made et al., 2020). These results collectively demonstrate that the effects of the BNT162b2 vaccine go beyond the adaptive immune system and can also modulate innate immune responses.

Note how they also tacitly explain how these vaccines actually reduce deaths: They don’t prevent you from getting infected, rather, these vaccines prevent the immune over-reaction that leads to acute respiratory distress syndrome in a handful of people who are infected. Your immune system learns to become more tolerant of the virus. That’s how these vaccines have mainly prevented deaths.

They can’t end the pandemic and they prevent the development of herd immunity, by decreasing the innate immune response against this virus. Rather, the vaccines reduce deaths by raising the white flag. The halfwits taught your immune system to tolerate the spike protein, tolerating the spike protein reduced deaths and so the halfwits accidentally booked a success and began injecting the whole population with this junk.

Rather than developing a vaccine, the halfwits at Moderna and Pfizer had accidentally developed an immunotolerance inducing therapy that appeared to help reduce deaths (as long as you make sure to label people as unvaccinated for the first 21 days after the first injection). The halfwits also made sure to exclude high risk elderly from their initial trials, so the vaccines looked far more effective than they actually were. They had achieved the most dangerous kind of success: Accidental success.

It wasn’t immediately obvious that the vaccines increase infection risk by disabling innate immunity, because initially your body is flooded with effective neutralizing antibodies against this virus. It’s only once the antibodies begin to wane, that it becomes clear that the main methods your body normally uses to prevent or abort infection have been disabled.

<cont.>

In the bar chart below I have plotted out all 1,574 symptomatic COVID-19 deaths according to the number of total diagnoses made in the electronic patient record (maximum of 10 fields) and according to whether other major morbidities were also diagnosed. The chart starts off at a minimum of two diagnoses by definition (one for COVID-19 and the other for any acute respiratory condition), there being a total of 89 such cases (5.7%). For symptomatic COVID-19 deaths with three diagnoses recorded we observe around half (51.5%) would be considered co-causal deaths, this fraction increasing with mounting case complexity. Co-causal symptomatic COVID death reaches a peak of 91.7% for records possessing a total of 9 diagnoses, there being 12 such symptomatic COVID deaths of which 11 were attributed with at least one other major morbidity.

Out of the 1,574 symptomatic COVID-19 deaths some 939 (59.6%) exhibited at least one major morbidity. Whilst we cannot attribute these deaths to stroke, heart attack, liver failure etc we cannot equally attribute them to SARS-COV-2 though, of course, this is precisely what the ONS do when they count certified deaths allegedly due to COVID. What this analysis usefully shows us is that around 60% of what the ONS claim are genuinely certified 100% COVID causal deaths should be categorised as co-causal at the very least!

How to Eff-Up your Efficacy - by Rudolph Rigger


You’ve all seen the stats and the claims - didn’t Prof Joe Brandon recently say that if you’re Goo-free you’re something like 97 times more likely to die of covid than the Goo-Goo’ed and boosted? I probably need to be ‘fact-checked’ on this, but lots of people make similar claims for Goo efficacy.

I think he was referring to charts from Our World in Data that look like this



Here the data for Switzerland seems to indicate you’re 48 times more likely to die if you’re Goo-free. This is praise indeed, for the Goo. Yet when you look at worldwide covid mortality data we don’t seem to be seeing anything like this high claimed efficacy.

Here’s a comparison of China, Israel and the UK



Didn’t China do well? No covid deaths since April 2020, without mRNA vaccines. I couldn’t possibly comment on whether CCP figures are accurate here - I attribute their great success to having removed bat from the menu.

What’s fascinating, though, is jab-happy Pfisrael. Two years of covid and consequent higher percentage of disease-acquired immunity, 4 lots of Goo, and still their covid deaths peaked higher than ever before in 2022. Hard to reconcile that with the superly-duperly efficacious Goo isn’t it?

I’m sure the pro-Goo will have ‘explanations’ on hand, but something doesn’t sit right with me here. This is not what I would expect to see after a high percentage of your population have been Goo’ed with a highly efficacious Goo - and especially not after an increased percentage of people with disease-acquired immunity after two years’ worth of exposure.

But these are murky waters - and it’s hard to trust any of the data fully. The UK’s relatively high covid death toll owes much to the absurd way in which deaths were registered, for example. And we also see similar sleights of hand with definitions of Goo’ed and Goo-free. Notice in the first chart that partially Goo’ed people have been excluded. In my view, this is a fundamental category error and I want to explain why in a bit more detail.

Things are going to get moderately technical but the TL;DR here is that :

mis-categorization can affect things quite drastically.

I’m going to phrase things in terms of probabilities (which can often be difficult to understand and sometimes counter-intuitive), but the actual calculation involved doesn’t require too much understanding of probability.

The basic question we’re asking is of the form “how much better is X than Y ?”

So if we’re shopping for our groceries and an item has been reduced from $12 to $8 we’re trying to answer the question “how big of a reduction is that?”

One way of answering that question is to take the difference 12 - 8 and to express that as a fraction of the original price. We would calculate (12 - 8)/12 which is equal to 1/3. So the item has been reduced by 1/3 (or about 33%) from its original price.

We can also do the same for price increases. Suppose the original price was $8 and it increased to $12, how much of an increase is that? In this case we do something similar and calculate (12 - 8)/8 giving us a factor of 1/2 (which is a 50% price increase).

The method is really to look at the difference between ‘new’ and ‘old’ and divide that by ‘old’.

Essentially these are exactly the same kind of calculations we do when computing vaccine efficacy.

Answering this question for the covid vaccines is fraught with difficulty - especially now we have 3 shots, and even 4 in some places. The relative populations in each group (unvaxxed, 1st dose, 2nd dose, etc) are time-dependent and we must also factor in things like age (the initial doses were rolled out in an age-dependent way, for example) and also other potential biases like health and behaviour.


Simplified Model

I want to try to simplify things a bit - to create a simplified model, so to speak - in order to get some idea of how changing definitions (i.e. mis-categorization) affects things.

I’m just going to consider 1 shot of the vaccine - and think of things in a “steady state” situation where the rollout has ended, so everyone who is going to get jabbed has been jabbed. Time-dependence will be ignored and I’m going to use probabilities. These probabilities can be thought of as a kind of ‘average’ over the period in question.

A word of caution here. Doing this properly really does require that some attention is paid to the time-dependence. As time progresses any ‘blip’ caused by limbo will become less and less significant, for example. Things like waning vaccine efficacy and prevalence also need to be accounted for. But this simple probability-based model I’m using does give us some insight into the issues with mis-categorization.

So, I’m going to be thinking of things like the probability that you die (of covid) if you are in the unvaccinated group. I’m also going to have a vaccine “limbo” period where you’ve had a jab, but are not yet considered to be “vaccinated” according to health officials. Finally, there will be a vaccinated group who have survived limbo.

One useful way to think about probabilities is to consider what’s called a “probability tree”. So you start with a ‘node’ and this splits off into two possibilities (say, jabbed and unjabbed, for example) which give us two nodes - and if there’s another choice at this point we have more ‘branches’ from these two nodes.

For this single-shot model the probability tree looks like this:



There are 5 possible outcomes we can have. Pick any person from the original population and they’re going to end up in one of these 5 outcome groups.

The issue is that the ‘official’ version of events is that you are not considered to be “vaccinated” until you have survived limbo. So here’s the ‘official’ version of the same tree:



You can see here how one of the branches has been moved from node V to node U. The vaccine limbo deaths just get added to the “unvaccinated, died” branch.

The question here is what is the impact of these changes when calculating vaccine efficacy?

Official bodies seem to be interested only in those who have survived limbo. As a potential vaccinee, however, you’re interested in whether getting something injected into your arm reduces your risk, and by how much.


Efficacy

So, there are going to be 2 different efficacies you can compute. The one based on the first tree I will call the ‘correct’ efficacy and label it as ε(c). The one based on the 2nd tree I will call the ‘official’ efficacy and label it as ε(o).

What we want to know is “how much better is the ‘official’ compared to the ‘correct’?”

We can apply the same reasoning as we did for the grocery shopping to get a kind of “efficacy of efficacies” - or a meta-efficacy, which I shall label E.

We’re going to calculate the initial efficacies, the official and correct versions, in terms of probabilities. There are 5 probabilities associated with the 5 possible outcomes. Since vaccine efficacy is determined by looking at death rates we only need to look at 3 of those probabilities

• P(d, u) : the probability that you died and were unvaccinated
  
  P(d, L) : the probability that you were vaccinated and died in limbo
  
  P(d, v, s) : the probability that you died and were vaccinated and survived limbo

Here’s what the squiggles look like:



Translating everything into math shorthand (the squiggles) makes it all look waaaaay more complicated than it actually is - it’s mostly a visual thing, not a conceptual one. It really is just the same calculation as the grocery store example.

We can see the effect of the definitional switch in the ‘official’ efficacy. First of all you’ve taken something off the top in the fraction. So if you work out (10 + 5)/5 you get 3. If you remove the 10, the answer becomes 1. This really helps because now you’re making the efficacy bigger (you subtract a smaller number from 1). But you compound the error by sticking the thing you removed into the denominator (the bottom bit). So (10 + 5)/5 becomes 5/(10 + 5) which now equals 1/3. So you go from 3 to 1 and then to 1/3 - all the time making the number you have to subtract smaller. For our calculation this makes the ‘official’ efficacy bigger than the ‘correct’ efficacy - a double whammy.

But we want to know by how much it improves things. So, we apply the efficacy measure again to get E, the meta-efficacy.

If the meta-efficiency is 0.5 this means that our ‘official’ efficacy is a 50% improvement on our ‘correct’ efficacy.

Now we’re going to write everything in terms of the unvaccinated probability. We do this by introducing scaling parameters. So, we will write : probability of dying in limbo equals some factor times the probability of dying in the unvaccinated group. In shorthand this looks like P(d, L) = λP(d, u) so the multiplicative factor is λ. We do the same for the probability of death in the vaccinated group so that P(d, v, s) = µP(d, u).

The parameter λ here is telling us about the risk in limbo.

The parameter µ here is telling us about the risk after you’ve survived limbo.

When you write everything in terms of these mu and lambda parameters here’s what you get (after a bit of algebra)



Some things to note here

• the efficacies are less than or equal to 1, but can be negative
  
  ε(o) is greater than ε(c) (or equal when λ = 0)
  
  the meta-efficacy goes to infinity when ε(c) = 0

Let’s have a look at this last special case when ε(c) = 0


No overall efficacy : ε(c) = 0

For this to happen we have to have µ + λ = 1

What does this mean? It means that at the end of the specified period (time in limbo plus time in vaccinated group) your risk of dying from covid has not changed. You are in exactly the same position as if you hadn’t been vaccinated.

This might happen, for example, if we have λ = 0.9 and µ = 0.1

If this were to be the case the ‘official’ view would be that the vaccines are “effective”. After all, now you’re defined to be fully-vaccinated you have only one 10th the risk of dying, compared to the unvaccinated.

We might expect λ to be less than 1. Suppose we consider an 8 week period. If you get jabbed right at the start then you might have 3 weeks of limbo compared to a total of 8 weeks if you remain unjabbed. Even if the jab makes it more likely to die (per day) of covid during limbo, you’re only in limbo for a fraction of the time.

Let’s look at a special case here:
λ = 1 and µ = 0
Here the limbo period significantly increases your risk of dying of covid (it is equal to the risk if you remain vax-free for the entire period), but if you survive this limbo period the vax works brilliantly and you don’t die of covid. In this special case we would have

ε(o) = 1 but ε(c) = 0

In other words, we would have the ‘official’ efficacy being 100% and the ‘correct’ efficacy being zero. Quite a significant difference!
The General Case
What does this look like in general? If you work out the ‘official’ efficacy when ε(c) = 0 you get ε(o) = 2λ/(1 + λ). Here’s what this curve looks like (and we note that λ can only vary between 0 and 1 in this special case condition of ε(c) = 0):



So in all cases here (except for λ = 0) the ‘official’ efficacy would be positive with the ‘correct’ efficacy being precisely zero.

The take-away here is that it’s quite possible to have zero efficacy overall (as measured by the ‘correct’ efficacy) but with a reasonably high efficacy as measured by the ‘official’ efficacy.

Mis-categorization of vaccinated vs unvaccinated with regards to covid death can have quite profound consequences. This is especially important because it is known that being in vaccine limbo elevates your risk of covid death compared to the same time period in the unvaccinated.


ε(o) - ε(c)

With a little bit of algebra it’s possible to prove that

ε(o) – ε(c) ≥ λ

which means that the difference between the official and correct versions of efficacy is at least as big as λ.

As time progresses these two measures will converge because λ is a scaling factor comparing unvaxxed risks with limbo risks.


High Claimed Efficacy

We’re going to consider the case where we compare only the vaccinated (and survived limbo) with the unvaccinated. This is what we’re seeing in the OWD chart for Switzerland above. Let’s, therefore, suppose that µ = 0.1

This means that covid deaths in the survived-limbo-vaxxed group are (proportionally) only a tenth of those in the unvaxxed group.

Here’s the meta-efficiency here - a measure of how much better the ‘official’ version is over the ‘correct’ version as a function of λ and with µ = 0.1



Notice that even with λ being a relatively modest 0.3 you’re getting over a 50% improvement in your ‘official’ efficacy compared to the ‘correct’ efficacy. That’s with a low value of µ - so it looks like the vax is reducing death by 90% compared to the unvaxxed, when in reality looking at overall risk the efficacy is not this high.


Summary

I apologise for the technical jiggery-pokery here. It’s something I’ve been working away on over the last few days. I really wanted to get to a point where there was a simple mathematical proof that you really can eff-up your efficacy by mis-categorization. I’m not saying anything new here - lots of people have pointed this out. I just wanted to try out a different perspective on it all.

The main take-away is that we really, really need to include the risks in vaccine limbo in any calculation of overall efficacy, because not doing so can give a hugely misleading result.

The UK Health Security Agency (UKHSA) kindly provide us with the case-rates per 100k population by vaccination status. However, their Week 2 – 2022 – Vaccine Surveillance Report was the last time they published the case-rates per 100k among the double vaccinated, instead choosing to only publish the case-rates per 100k among the triple vaccinated since.

But thankfully we can work out what these case rates among the double vaccinated ourselves.

Method

To calculate the case-rate per 100k among the double vaccinated we first have to work out the overall population size of the double vaccinated, and to do this we need to look at the numbers in the Week 2 – 2022 Vaccine Surveillance Report from the UKHSA.

In the week 2 report they provide combined figures for the double and triple vaccinated. So we need to work out the total population size for each age group provided by using the provided case-rate figure for the double/triple vaccinated and the provided total number of cases for the double/triple vaccinated.

Then we just divide the number of cases by the case rate, and then multiply the answer by 100,000 to work out the total population size of the double and triple vaccinated in the Week 2 -2022 Vaccine Surveillance Report.

Next we need to calculate the population size of each age group in the triple vaccinated demographic in the Week 5 -2022 Vaccine Surveillance Report using the same method.

Then we just subtract the triple vaccinated under 18’s in week 5 from the double and triple vaccinated in week 2, to work out the total population size of the double vaccinated in week 5.

Finally, to work out the case rate per 100k among double vaccinated in the week 5 report all we have to do is divide the total population size by 100,000 , and then divide the number of cases among the double vaccinated under 18’s by the answer to the previous equation.

Vaccines and Related Biological Products Advisory Committee Meeting
September 17, 2021
FDA Briefing Document
Application for licensure of a booster dose for COMIRNATY (COVID-19 Vaccine, mRNA)