Homework (formerly Ending Lockdown)

Published here by Don at Tony’s request.

I am looking at the Andrew Marr programme this morning, on the BBC1, where a guest is the statistician who wrote the article in the Guardian newspaper which went viral amongst politicians, and was used by Boris the Bad to explain that you cannot compare levels of infections and death rates across different countries. His name is Sir David Spiegelhalter and he is on  the Sage Committee, advising the U.K Government on Covid 19

Some of his learned observations are most illuminating, as follows:

• Estimated infection rate for UK min 3,500,000 … poor data and testing a problem.
• Death rate best indicator … but use stats for all deaths and compare with stats for same months across many years.

Using this basis he estimates U.K has about a further 27,000 unexplainable deaths so far.
He drew attention to the evidence across many countries that this illness is mainly a fatal illness for the over 75 age group … lockdown policy should perhaps now be focused on vulnerable groups .

Like many academics he feels his article was misrepresented. It is possible to compare mortality and infection rates across countries with caveats. On that basis we can compare the different policy responses.

Increasingly I am coming to the view that the Politicians handed over control on policy to medical and scientific advisors who had a very limited understanding of the enormous implications of their advice. Medical Advice is just that … Advice!! Policy Response  should go  far wider than that but factor it in.

I think future generations will wonder how creating mass unemployment and a global recession was an appropriate policy response.

Please Read the following articles. There will be an online exam on Thursday.

Sweden tames R
We know everything
ending_lockdown

Lies, Damned Lies and Statistics

I want to put down my thoughts on the 50% virus propagation rate now claimed by the Authorities.

First a teentsy weentsy bit of maths: Here’s how propagation works. A group of people have the virus and, on average, each passes it on to ρ more people. If ρ is less than one, we expect the virus to eventually run its course. (I’m assuming there are sufficient uninfected individuals to pass it on to or it will run its course even sooner). The average number of people affected by each initial carrier will eventually be ρ / (1 – ρ). (NB: this formula is only valid when ρ is strictly less that 1).

So, for instance, if ρ = 0.8, one initial carrier will affect 0.8 others, who will affect 0.64, who will affect 0.512, who will … and so on until ultimately a total of 4 people will be affected. And that’s all!  Finito!

So, if ρ = 0.5, as claimed, the total affected by one initial carrier will be one more!

0.5 + 0.25 + 0.125 + … = 1

Here’s a little diagram illustrating how the one person on the left eventually produces one further infected person on the right.

This means that of all the people currently infected, assuming they have not yet had the indelicacy to infect another, they will still only infect one more.

Do you believe the claim that the rate is only 50%? We can think of it an an iceberg where we don’t have all the information but we are making a prediction that in spite of what we don’t know, we don’t expect to have any more that twice as many cases in total as those already in existence. This is a very confident claim, suggesting that the pandemic is virtually over.

Baye’s Theorem

Medical testing is much in the news. I thought I’d do a little note to remove the mystique around the so-called false positives and false negatives which seem to be widely misunderstood.

A false positive is a test result which says you have the condition but actually don’t. It causes anguish but is arguably less dangerous than a false negative which says you don’t have the condition when actually you do.

So if you test for some condition and you get a positive result, just how worried should you be? Imagine the test is known to produce false positives 5% of the time and false negatives 10% of the time. The condition is know to affect 2% of the population.

When 100,000 people are tested,  the test will report as follows:

So the test proves very reliable for eliminating people without the condition but usually further more detailed testing is required to home in on those who have it.

Of course the test returns only positive or negative. It is the effectiveness of the test that provides a warning about how accurate the result is.

Simpson’s Paradox

With all the talk of anti-viral drug research in the news, I thought I’d amuse you with a little drug research paradox attributed to briton Edward Hugh Simpson, but noticed earlier by scotsman George Udny Yule.

Two drugs are being compared. In a trial in Hospital A, the results favour Drug 1 as follows:
In a further trial in Hospital B, the results the results also favour Drug 1 as follows:

Somebody has the idea of combining the two tables. Now it is Drug 2 that heads the effectiveness stakes!

As Mark Twain noted: There are lies, damned lies and statistics.