Could Increases in Influenza Vaccination Rates Give Rise to Exponentially-Related Corresponding Increases in Mortality Rates From Covid-19?
Dr Grahame Blackwell
A Pan-European Study incorporating data from over 350 million subjects

Summary

A preliminary analysis based on a document published in the BMJ indicated that overall death rates from Covid-19 in different European countries could be exponentially linked to those countries' influenza vaccination rates (percentages) for over-65s. [This is consistent with a Jan 2020 study in the journal Vaccine linking influenza vaccinations with 36% increased susceptibility to coronavirus (pre-Covid-19).]

This follow-up in-depth analysis, involving over 350 million subjects, considers all European countries for which reliable vaccination data was available on the OECD website at time of preparation.
Data, links and instructions are provided to enable readers to check this analysis for themselves. The steps are as follows:
(1) Identify which data should and shouldn't be used, in accordance with statistical best practice; calculate the Correlation Coefficient, R;
(2) Test the null hypothesis ("no actual connection") by checking the probability that this value for R could occur by chance;
(3) This probability turns out to be less than 0.00001 - so we must reject that hypothesis in favour of the alternative hypothesis:
"Overall mortality rates from Covid-19 do increase exponentially with increasing percentage vaccination rates of over-65s".
At this point we can add the information given by R2, the Coefficient of Determination: the significance of this figure is given at the bottom of this page.

Conclusion: Rate of total deaths/million of population from Covid-19 rises exponentially with increasing rate of percentage influenza vaccinations for over-65s across Europe. This could be cause-and-effect, or it could be due to some as-yet unidentified third factor linking these two.

It's difficult to imagine what that third factor might be, to give an exponential connection.

============================= [See notes on Statistical Inferencing, in relation to this analysis, here.

BMJ publication: Dr Allan Cunningham has collected data from reliable sources for 20 European countries. He's listed this in a Rapid-Response document published in the BMJ. [See it here.]

Dr Cunningham suggests analysing these 20 data pairs to see whether there's any connection between percentage of over-65s receiving the Influenza vaccine in a country and the death rate per million in that country from Covid-19. That analysis can be seen here, with step-by-step instructions on how to do it yourself. To avoid any suggestion of selection bias (cherry-picking sets of figures to give a preferred result), this analysis is updated below, again with step-by-step instructions on how to do it yourself (no specialist expertise needed), using all the latest reliable data as at 1st August 2020.

Figures used here are taken from the OECD website (for over-65s vaccination percentages) and the Worldometer site for mortality rates per million population. First we need to be sure that we have reliable properly-accredited figures. With regard to to over-65 percentage vaccination figures, neither Austria nor Poland have a vaccination percentage documented for later than 2014; Turkey's latest figure is for 2016. It seems possible, then, that any of these three figures could introduce an element of unreliability, so they should be omitted.

We also need to identify, and discard, any outliers - data points that lie so far out of the pattern set by the other points that they are clearly being heavily influenced by other factors, and so can't be regarded as part of that pattern. Outliers can be identified to some degree by eye - looking at the scattergram of the graph points - or more accurately by calculating whether they lie outside statistically-defined boundaries and so are most unlikely to be valid elements of the set.

***

Vacc %
72
62.7
68.5
54.3
52.7
60.8
52.2
51
52
39.8
49.5
34.8
24.1
38.2
21.5
12.5
12.9
14.8
7.7
10.2
59.1
49.4
56.2

Deaths/m
679
359
357
608
581
170
568
464
106
182
59
110
62
47
36
5
57
29
17
47
849
29
20

Log(d/m)
2.83187
2.555094
2.552668
2.783904
2.764176
2.230449
2.754348
2.666518
2.025306
2.260071
1.770852
2.041393
1.792392
1.672098
1.556303
0.69897
1.755875
1.462398
1.230449
1.672098
2.928908
1.462398
1.30103

The columns of figures to the left give: percentage vaccinations of over-65s (for 2018, or in two cases 2017); deaths per million of total population; logs of those death rates (to base 10). Stats are given for the 26 European countries listed by the OEDC, in the order they're listed (apart from Greece, marked ***), less the three countries noted above as having possibly unreliable figures: Austria, Poland, Turkey - i.e. 23 sets of figures in all.

That previous study over 20 countries made it quite clear that any relationship between vaccination rates and death rates will be exponential, so relationship between vaccinations and logs of death rates will be linear - if that's not so, our results will make that very clear by not giving a significant correlation.

We can get a best straight-line fit for these 23 sets of figures here. Simply copy-and-paste the 'Vacc %' column of figure into the 'XValues' box and the 'log(d/m)' figures into the 'YValues' box, skip past the 'Estimate' box and press the 'Calculate' button. You'll get the Regression Equation:      y(hat) = 0.022X + 1.11802 .

You'll also see this graph (but without the blue circle), showing the 'line of best fit' through the 23 data points. That line has the equation: Y = 0.022X + 1.11802 .

Four points appear by eye to be outside of the general pattern: Greece, Slovakia, Belgium and Iceland (the four furthest-out points on this scattergram); this may be why three of them were omitted from the earlier analysis.

The point circled in blue is significantly further from the best-fit line than any of the other points. You can confirm this simply by holding a ruler up to your computer screen: the distance from that point to the line is half as much again as the distance from the next nearest point (the one down by '12' on the baseline).

This suggests that the circled point (for Greece) may be an outlier.
The next step is to check mathematically for any outliers.

To check for outliers we have to:
(a) calculate distances of all the points from the line; [A formula for that can be found here.]
(b) put those distances in order; [Excel, or any similar software, can do that for you - as well as the calculations for (a).]
(c) find the upper and lower quartiles (UQ & LQ), and calculate the interquartile range (IQR); [Info on these can be found here.]
(d) identify any points that lie more than 1.5 times the IQR outside the interval from LQ to UQ. [Simple subtraction.]

To save you the trouble of doing all that: Greece is an outlier, none of the other countries are.

We can now re-calculate that straight-line fit of the log values (excluding Greece), and the exponential fit of the death rates, against percentage vaccinations for those 22 data pairs. First simply copy-and-paste the above 'Vacc %' column of figure into the 'XValues' box and the 'log(d/m)' figures into the 'YValues' box here, as before, but omitting the *** figures for Greece. This gives us the straight-line graph below for death-rate log values against % vaccination figure and tells us that the equation for this graph is:           y(hat) = 0.02384X + 1.09086.           [Raise as power of 10 to give exponential equation below.]

You could also go to here, where you'll find another stats tool where you can copy-&-paste the 'Vacc %' and 'deaths/m' figures in (omitting Greece) to see an identical graph to the second one below - the exponential graph linking vaccination rates to mortality figures. It also confirms the equation for that curve:
Y = 101.09086*100.02384X ,             which can be written more tidily as:            Y =12.3271*1.0564X             [* is 'times'].

In addition it gives the value for the Correlation Coefficient, R (0.7975), and the value of R2 (0.636) - see below these graphs for the relevance of this.   [This confirms the value of R derived directly from the straight-line fit in the detailed version of this analysis.]

First, though, go here and plug in your R value (0.7975) and number of data points (22). Press 'Calculate' and it'll tell you the p-value for this result (i.e. the probability that it's just by chance): less than 0.00001. There's a 99.999% probability that this exponential graph is showing a very real connection.

Log Line                                                                                 Exponential Line  R2:   The Coefficient of Determination

R-Squared - which is literally the square of the Correlation Coefficient, R - is a measure of how much the variation in the dependent variable (in our case deaths/million) is related to variation in the independent variable (in our case % vaccination of over-65s). In this case R = 0.7975, giving R2 = 0.636; this in turn gives an Adjusted R2 = 0.618*. In other words, variation in death rate from Covid-19 across European countries is over 60% attributable to its 99.999%-certain exponential link with vaccination rates of over-65s in those countries. Under 40% of that variation in death rate is attributable to other factors.
[* Giving increased precision.]

Note the careful wording here: The stats do not tell us that higher vaccination rates are the cause of higher death rates; stats cannot tell us causes - they can only tell us about connections. But the whole point of identifying connections - and the strength of those connections - is to inform us about things that merit our serious attention. The very fact that this strong connection exists tells us that we need to investigate it: either it is causal or there is some third factor linking these two. To ignore this warning, or to assume "It can't be that, there must be another factor" without checking is both highly irresponsible and highly un-scientific. Without clear evidence to the contrary (such as an obvious provable third factor linking these two) the Precautionary Principle requires that we act on the basis that it's very likely that higher vaccination rates could be causing higher mortality rates - exponentially higher mortality rates.

If this is so, then that exponential factor would imply that those vaccinated are not just at risk themselves, they are also acting as 'propagators' to spread the disease.

The Bottom Line: There is very significant evidence that death rate from Covid-19 is exponentially linked to over-65s influenza vaccination rate, compared across Europe.