Latest UK COVID-19 stats roundup

 Another Update 15 Dec: The curious case of the missing regional test data

Updated 14 December 2020

I wanted to see some data comparing people with COVID-19 symptoms to the 'case' numbers. The first link I found with relevant data was from Buckinghamshire, whose dashboard provides details of different types of triage numbers. As you can this provides yet more evidence of why counting 'cases' (without considering number of people actually ill with the virus and number tested) is really dumb.

 

(while I was impressed with the Bukinghamshire dashboard I was surprised that it doesn't have data for numbers tested, which almost certainly have been steadily increasing as elsewhere).

After tweeting about that somebody pointed me to a national NHS dashboard for COVID-19 triage. And it turns out that the national figures also show the stark difference between number of 'cases' (high and increasing) and number of people actually ill with symptoms (consistently low since the usual September blip).

So the 'casedemic' argument grows ever stronger.

I was also interested to see Tim Spector's analysis of London prevalence rates based on his ZOE css app for monitoring people with symptoms. This confirms how ludicrous today's Government decision was to move the whole of London to Tier 3 lockdown (caveat: of course I don't belief there should be any lockdowns of any sort anywhere for the reasons given below).

 

Now on to my normal regular updated plots:
Using only the data from https://coronavirus.data.gov.uk/ today (but taking account of numbers tested) here is an update of the key trend plots (UPDATE: the website https://inproportion2.talkigy.com/dashboard/ provides similar information to what I show below)

First we have the new 'cases' (i.e. those testing positive) as a proportion of number of tests (this now includes lateral flow device tests as well as PCR tests): 

 

So a very slight uptick in proportion of positive tests, but it is important to see this article for why there may be multiple explanations for such changes.

And here is the latest plot of  'cases' (i.e. positive tests) and number of tests (obviously different linear scales used for each as shown; moving 7-day averages with 2-day delay on cases):

Next, the proportion of 'cases' which lead to hospital admissions (but, as usual, note that any person admitted to hospital who has a positive PCR test within 14 days prior to admission or within 28 days after admission is classified as a COVID admission irrespective of their reason for being in hospital). And we know that many patients now classified as COVID admissions are people who actually 'contracted' COVID in hospital.

 

Next we have the current case fatality rate (proportion of people currently dying having tested positive any time up to 28 days prior to death; so, again as with hospital admissions, these are people with 'COVID19'). This has been steady at around 2% for some time.

And finally the cumulative 'death rate', i.e. the proportion of all people who have died having tested positive any time up to 28 days prior to death. This continues to drop. 

 

 

 

Note that none of the above plots are provided on the government website. The most relevant graph provided on the Government website is the number of  'COVID deaths', but (as with hospital admissions) we do not know how many of these are actual deaths because of COVID.

As usual all the caveats discussed previously apply.  This includes the fact that the government data actually tells us very little about what we really need to know; for example, none of the above tell us what the real death rate or infection rate is - as explained in the following two graphics:

And of course the fact that we are still waiting for the evidence needed to determine whether the 'benefits' of lockdowns  exceed the 'costs':

   

Finally here is a summary of the data/evidence we really need to determine if the 'benefits' of lockdowns exceed the 'costs':

Oh and here is an interesting comparison between the UK and Sweden which suggests lockdowns do not reduce deaths.

See also

• Different explanations for increasing number of proportion of positive test results
  
• Why we know so little about COVID19 from the data provided
  
• Impact of false positives in Covid testing
• Covid19 hospital admissions data: evidence of exponenial increase?
  
• Don't panic: limits to what we know about Covid-19 PC testing, inferred infection rates and alse positive rates
  
• A privacy-preserving Bayesian network model for personalised COVID19 risk assessment and contact tracing
  Covid-19: Infection rates are higher, fatality rates lower than widely reported 
• Coronavirus: country comparisons are pointless unless we account for these biases in testing 
• Why most studies into COVID19 risk factors may be producing flawed conclusions - and how to fix the problem
• Causal explanations, error rates, and human judgment biases missing from the COVID-19 narrative and statistics
• Covid-19 risk for the black and minority ethnic community: why reports are misleading and create unjustified fear and anxiety
• UK Covid19 death rates by religion: Jews by far the highest and atheists by far the lowest 'overall' - but what does it mean?
• All COVID articles on this blog 
  

No - there is nothing especially unusual about this lottery outcome

 

Every so often we get a story like that today (from South Africa this time) about unusual lottery draws, with the implication that it is too incredible to believe this happened without some cheating going on.

The first thing to note is that the article does give the correct probability of this particular sequence being drawn on any one occasion, namely one in 42,375,200. This is calculated as follows:

• The probability the first ball drawn is 5 is 1/50 because there are 50 number in the main draw.
• The probability the second ball drawn is 6 (given that the first ball drawn is 5) is 1/49 because there are now 49 number left.
• The probability the third ball drawn is 7 is 1/48
• The probability the fourth ball drawn is 8 is 1/47
• The probability the fifth ball drawn is 9 is 1/46
• The probability the bonus ball drawn is 10 is 1/20 because there are only 20 bonus balls to choose from.
• So, the probability of getting the exact sequence 5,6,7,8,9,10 in that order is:

However, the particular order of the sequence does not matter, in the sense that 5,6,7,8,9,10 would still be the reported sequence even if the first 5 balls were chosen in a different order such as:

• 6,9,5,8,7, 10
• 9,8,6,7,5, 10
• 7, 9,6,8,5, 10
• etc

In fact, there are 5! (which is equal to 5 x 4 x 3 x 3 x 1) different possible permutations in which these numbers can be drawn. That is 120.

So, the probability of drawing the numbers 5,6,7,8,9 and bonus ball 10 is exactly:

Now this seems like an incredibly unlikely event. But, crucially, every possible set of winning numbers has exactly the same probability. Nobody would suspect anything if, say the numbers drawn were 3, 8, 21, 29, 47 and bonus ball 10. But the probability of getting that particular set of number is also one in 42,375,200.

What would be very unlikely is if, say, only one million lottery tickets had been sold and as many as 20 of these had all been for a winning sequence like 8, 21, 29, 47, 10. 

However, it is well known that most lottery players do not choose their numbers randomly. For example, numbers under 31 are far more commonly chosen than numbers over 31 because many people use birthdays of family members. That is why, when all the numbers in a winning draw are less than 32, there are far more winners than otherwise. Similarly, it is known that many people chose consecutive number sequences like 5,6,7,8,9,10. In fact, in the UK it is known that the most commonly chosen set of numbers is 1,2,3,4,5,6  (an average of 6,000 people chose this sequence for each draw in the early years of the UK lottery). So the fact that 20 people chose the consecutive sequence 5,6,7,8,9,10 is not at all unusual. In other words there is nothing at all to suggest the need for an 'investigation' as stated in the article.

But there is another important probability issue that most people completely misunderstand. While the probability of getting the particular numbers 5,6,7,8,9,10 in any one specific lottery draw is only one in 42,375,200, there are in fact 16 different ways we could get a consecutive sequence in any one draw (assuming the bonus ball must be at most 20), namely:

1,2,3,4,5,6

2,3,4,5,6,7

3,4,5,6,7,8

….

16,17,18,19,20

Any one of these sequences would have raised exactly the same concern. So, in any given draw, the probability of getting a consecutive sequence is actually one in 2,648,450 (not one in 42,375,200). While that is still a very small probability, we should also take account of how often lottery draws happen.

Let us assume there are 2 draws each week. Then, over a 10 year period there are 1040 such draws. The probability of getting a consecutive sequence at least once in a 10 year period is approximately 1 in 2547. But there are several hundred national and state lotteries in the world. If there were 300 such lotteries, then in any 10 year period the probability of getting a consecutive sequence at least once somewhere in the world is over 11.7% that’s a 1 in 9 chance**. So it really is not such an incredible 'coincidence' after all.

As explained in our book similar reasoning shows that real examples like:

• A woman in the USA wining the jackpot twice in 5 years
• A Bulgarian lottery in which the same numbers were drawn in two consecutive draws

are actually not particular unusual events (indeed the first is almost certain to happen somewhere in the world over any 10-year period). 

It's also interesting that this story seems to have grabbed the interest of academic statisticians far more than that of the US election - where there seems to be genuine statistical evidence of fraud.

Hat tip to Scott McLachlan for the story.

  **to be specific it is one minus the probability of getting NO consecutive sequence anywhere which is 1 -  (2546/2547) to the power of 300

 See also: 

• Can you improve your chances of winning the lottery even if you can afford just one ticket?
  
• The damaged winning lottery ticket story