Needle To Door Time (part 1)

In this series I use survival analysis techniques to investigate delays between vaccination and death for 5,039 cases over the period 2020/w53 – 2021/w36 using data from an undisclosed NHS Trust

Today, on this first day of the new year, I want to do something a bit different with that lump of NHS data I keep hacking at. We’ve had logistic regression models coming out of our ears for weeks on end and I don’t think there’s anything more I can squeeze out on this front other than endless model refinement with dwindling value.

The bottom line is that the vaccines appear to magically confer benefits for all manner of condition until you start to iron out the many sources of bias. After ironing all that stock and share-boosting hyper magic disappears and we are left with products that are ineffective in terms of reducing risk of acute respiratory conditions. Not only that but they appear to increase the risk of severe COVID leading to acute respiratory conditions. Incredible as this may seem in view of the “safe and effective” publicity campaign this is what rigorous statistical analysis of 18,000+ in-hospital deaths in a single NHS Trust actually tells us.

Every NHS Trust in the country could undertake the same analysis to confirm this, and even if they didn’t have the necessary in-house statistical expertise I am sure we could arrange for a few seasoned bods to lend a hand. For a year’s supply of baked cherry cheesecake I might even come out of retirement and turn the handle in a drab office somewhere with an asbestos ceiling¹. But of course none of this will ever take place because the NHS is not going to want to know the answer, and I’d wager that the top brass have been commanded by HMG not to go seeking any answers.

Not finding the answer was a game I once played as a government suit. If somebody accidentally found an answer then the trick was to keep it from spreading, and especially ‘upwards’, with the final ring of upper guardians being the departmental assistant secretaries who ensured ministers never got to hear anything that would compromise their “well, we just don’t know” ploy. The uninformed minister is a powerful weapon.

Something Tangy

So what answers had we better not go seeking today? Right now, and with all this early morning damp, I fancy something tangy like the delay between vaccination and death.

Verily, for once upon a time I shouldered the responsibility of being the foundational Trust target holder for something we used to call ‘door-to-needle time’ (jabbing folk with thrombolytic drugs as quickly as possible following a heart attack). Today I am going to consider needle-to-door time; the door in question being the mortuary door.

A quick crunch tells me that I’m sitting on 5,039 adult vaccinated deaths over the period 2020/w53 – 2021/w36 that divide into 2,235 deaths after the first dose (44.4%) and 2,804 deaths after the second dose (55.6%). If we pool these sub-samples we get what I am going to call ‘last dose survival’ (LDS).

Basic stats for LDS fetch-up with a minimum delay of 0 days (jabbed earlier the same day), maximum delay of 236 days, median delay of 64 days and mean delay of 68.9 days. The distribution is positively skewed and looks like this:

If we adopt the fortnight as the basis for our bins then we find 397/5,039 (7.9%) of deaths occurred within the first 14 days and 1,035/5,039 (20.5%) occurred within the first 28 days of whatever their last dose was. I am expecting keen readers to enquire as to differences between doses so have baked this slide:

It is quite clear from this that early death following the first dose is more of a feature than early death following the second dose and I shall be examining this in more detail a little later on². For the time being we may note the mean and median delay of 57.11/51 days for dose 1 and 78.30/79 days for dose 2. However, we have to be mindful that, when comparing delays by dose, we are not necessarily comparing equivalent cohorts over time. In addition those who took the first dose very early during rollout but refrained from a second dose will have attained a lengthy survival period compared to those who took a second dose and died shortly before the sample endpoint of 2021/w36. This inequality will be dealt with later, but for now I just want to give a basic flavour and I recommend readers pay attention to the shape of the distributions rather than read anything deep and meaningful into bias-laden means and medians.

28-Day Survival

What we need to do now is to tackle the inequality arising from different timing points and I am going to do this using classic survival analysis. Back when a health service suit in support of a cardiac surgery and cardiology unit it was 28-day survival that was the thing so 28-day survival is what I’m going to use as my primary clinical outcome.

With a sample end week of 2021/w36 I am going to have to exclude all deaths whose final vaccination date hadn’t occurred during or before 2021/w32 otherwise these folk wouldn’t have had a chance to return a 28-day survival outcome. I can thankfully report that no case fell into this category, leaving me with a full complement of 5,039 vaccinated adult deaths.

If I flag all those deaths occurring within 28 days of vaccination and crosstabulate these by dose we arrive at this rather interesting table:

Thus, we observe 597 ‘early’ deaths within the dose 1 sample of 2,235 (26.7%) but only 439 early deaths within the dose 2 sample of 2,804 (15.6%). It shouldn’t need a stats test to tell us that these are very different rates but I can report that a classic Fisher Exact Test flew off the end of the scale with p<0.001. In plain English the survival response to dosing was different, with dose 1 being 1.71 times more risky in terms of early death (all things being equal).

We can turn those early death rates into survival percentages by subtracting them from 100% to arrive at 73.3% survival and 84.4% survival for dose 1 and dose 2 respectively. I agree that it’s rather wacky to talk about ‘survival’ in a sample of deaths but there you go; it is what it is, and it is rather important, IMHO, to determine who died quickly after dosing and (hopefully) why.

Kaplan-Meier

With this in mind I can now reach for a spanner beloved by medics the world over called Kaplan-Meier. Wiki has a nice little entry about this and so I have nicked the juicy bits:

The Kaplan–Meier estimator also known as the product limit estimator, is a non-parametric statistic used to estimate the survival function from lifetime data. In medical research, it is often used to measure the fraction of patients living for a certain amount of time after treatment.

A plot of the Kaplan–Meier estimator is a series of declining horizontal steps which, with a large enough sample size, approaches the true survival function for that population. The value of the survival function between successive distinct sampled observations ("clicks") is assumed to be constant.

An important advantage of the Kaplan–Meier curve is that the method can take into account some types of censored data, particularly right-censoring, which occurs if a patient withdraws from a study, is lost to follow-up, or is alive without event occurrence at last follow-up. On the plot, small vertical tick-marks state individual patients whose survival times have been right-censored. When no truncation or censoring occurs, the Kaplan–Meier curve is the complement of the empirical distribution function.

With all those words and concepts proving in the warm bottom draw of our mind-ovens let me now furnish a fancy little slide:

Start by looking in the top left hand corner. At the beginning of our sample everybody is still alive (apart from two individuals who died within hours of their jab), so the cumulative survival function starts out at 1.00 (100%). Now look along to the 28-day point. The blue line for dose 1 is sitting down at 0.733 (73.3% survival) whilst the orange line is sitting up at 0.844 (84.4% survival), these being the 28-day survival rates we’ve already met. Those little steps down to the 28-day figures represent the progression of deaths, for which it is quite obvious that the curve for the first dose is steepest.

There will be all manner of reason for this difference, starting with initial dosing of the most vulnerable, aged and sick members of society. Seasonal factors will be in there as will patient management, diagnosis and treatment during an extraordinary period of service withdrawal. Then there’s the spectre of vaccine harm.

At this point some readers may ask what all those little sticks represent beyond 28 days. These are the censored cases, being deaths that continue to happen after our 28-day clinical endpoint. Very keen readers will ask if the difference between these two 28-day survival functions is statistically significant and I shall respond by saying there are three excellent ways of determining this, as derived by Mantel-Cox, Breslow and Tarone-Ware. The good news is that they all agree that the difference is stonking (p<0.001):

Scrub-Up

I think that’s enough to get the ball rolling, the juices flowing and the pan hot. As the Carpenters once sang, we’ve only just begun, and no doubt folk will ask if there are differences between the sexes and how age impacts as well as background health status, disease prevalence and all the rest. To handle of all this I’ll introduce readers to another new spanner in the next article in this series that goes by the name of Cox Regression, for which your homework this week is to digest this Wiki entry on proportional hazards models.

Kettle On!

Summary

• A sample of 5,039 adult in-hospital deaths were investigated using survival techniques to assess delays between vaccination and death over the period 2020/w53 - 2021/w36 for an unknown NHS Trust.

• Basic statistics for the survival period following the last dose yield a minimum delay of 0 days (jabbed earlier the same day), maximum delay of 236 days, median delay of 64 days and mean delay of 68.9 days. The distribution is positively skewed.

• Some 397/5,039 (7.9%) of deaths occurred within the first 14 days and 1,035/5,039 (20.5%) occurred within the first 28 days of whatever their last dose was.

• We observe 597 early (28-day) deaths within the cohort of 2,235 who died after their first dose (26.7%) but only 439 early deaths within the cohort of 2,804 who died after their second dose (15.6%). These differences are highly statistically significant (p<0.001; Fisher Exact Test).

• Kaplan-Meier estimation was undertaken, this confirming a significant difference in the survival functions for initial and secondary doses (p<0.001; Log rank; Breslow; Tarone-Ware). These differences will have arisen from a variety of inter-related factors.

Now for a spot of tin rattling…

Being self-employed these days means the time I can dedicate to data preparation, analysis and article writing is directly proportional to the income I derive from subscriptions. My substack channel has to pay its way, and remains viable only if there are sufficient subscribers. My grateful thanks to those who have supported this project thus far.

1. My second NHS office was condemned property owing to the presence of asbestos but that didn’t stop management filling the floor with staff. The thoracic surgeons opposite would joke about who’d get cancer first.

2. What may also possibly distort results are folk who decided against a second dose being lumped in with those who didn't have a chance for a second dose.