Category Archives: Health

Covid, Health, Maths, Politics

Boris Johnson’s massive maths mistake over Covid deaths is an embarrassment

This post is adapted from my Indy Voice article of the same title originally published on 02/03/23

Yesterday many of us woke up to read headlines about Matt Hancock having ignored scientific advice about protecting care homes in the early stages of the pandemic (though a spokesperson for the former health secretary has since said that the reports are “flat wrong”, and that the interpretation of the messages’ contents is categorically untrue).

The story arose from a cache of over 100,000 WhatsApp messages that had been leaked to the Daily Telegraph.

Buried in WhatsApp conversations between then-prime minister Boris Johnson, his scientific advisers and Dominic Cummings, is an exchange which is arguably even more worrying than this headline-grabbing story.

On 26 August 2020, Johnson asked the group:

“What is the mortality rate of Covid? I have just read somewhere that it has fallen to 0.04 per cent from 0.1 per cent.”

He goes on to calculate that with this “mortality rate” if everyone in the UK were to be infected this would lead to only 33,000 deaths. He suggests that since the UK had already suffered 41,000 deaths at that point, this might be why the death rate is coming down – because “Covid is starting to run out of potential victims”.

In fact, death rates were still coming down as a result of the earlier fall in the number of cases brought about by lockdown. Already though, by the time this conversation took place cases were rising again in the early stages of what would become a catastrophic second wave.

Based on his faulty maths, Johnson questioned “How can we possibly justify the continuing paralysis to control a disease that has a death rate of one in 2,000?”. He was suggesting that anti-Covid mitigations could be relaxed at perhaps the worst possible time. His whole argument was based on two fundamental misunderstandings.

His first mistake was a mathematical one. Johnson had seen the figure 0.04 in the Financial Times and interpreted it as a percentage. In fact it was a fraction – the number of people who were dying of Covid-19 divided by the number of people testing positive. This is known as the case fatality ratio (CFR).

At 0.04 (or 4 in 100), the CFR calculated by the Financial Times was 100 times larger than Johnson had suggested – it was actually four per cent, not 0.04 per cent as he believed.

The chief scientific adviser Patrick Vallance patiently explained this to Johnson: “It seems that the FT figure is 0.04 (ie four per cent, not 0.04 per cent)”. Johnson replied “Eh? So what is 0.04 if it is not a percentage?” at which point Dominic Cummings had to jump in and break it down into even simpler terms. Even then the messages show no acknowledgement from Johnson that he had understood.

The other mistake that Johnson made in his calculation was to confuse the case fatality ratio with the infection fatality ratio (IFR). The IFR is the number of people who die from Covid-19 as a proportion of those who get infected.

Though they may sound similar, there is a big difference between the CFR and the IFR. In the CFR we divide the number of Covid deaths by the number of people who test positive. However, in the IFR we divide by the number of infected people.

Early on in the pandemic, when testing was not readily available, the number of people who tested positive was much lower than the number of people who were actually infected with the disease. Because of this, the CFR overestimated the IFR.

By mixing up percentages and proportions, Johnson’s calculation actually underestimated what the figure should have been by a factor of 100. If he had had the CFR correct he would have come to a very different conclusion – that over 3 million people in the UK would die.

In reality, to do this calculation you need the IFR, not the CFR. With a 1 per cent IFR (closer to the true figure), the correct version of Johnson’s simplistic calculation would suggest that 660,000 people might have died in the UK if everyone became infected – 20 times more than Johnson’s mistaken numbers suggested.

It is almost unimaginable that the leader of the United Kingdom could allow his thinking to be informed by calculations which contained such rudimentary errors. Mistaking the CFR and the IFR would perhaps have been understandable in the early stages of the pandemic, but this conversation took place long after the first wave had subsided.

To have made this mistake over six months into the UK’s pandemic response is indicative of a leader who has failed to fully engage with even the most basic science required to make important decisions surrounding the pandemic. It perhaps explains Johnson’s reluctance to institute stronger mitigations in the autumn of 2020 as were called for by his own scientific advisors.

Even less forgivable is his mathematical mistake, which is indicative of his failure to engage in scientific thinking more generally. At a time when other country’s leaders were going on national television and defining important epidemiological concepts the masses, we endured a prime minister who was making basic mathematical errors the likes of which most 11-year-olds would not succumb to.

When it came to scientific literacy – such a crucial currency in the response to the pandemic – this incident suggests we suffered under the worst possible leader at the worst possible time.

Health, Maths, Politics

How do we know health screening programmes work?

This post is adapted from my Conversation article of the same title originally published on 30/07/23

The UK is set to roll out a national lung cancer screening programme for people aged 55 to 74 with a history of smoking. The idea is to catch lung cancer at an early stage when it is more treatable.

Quoting NHS England data, the health secretary, Steve Barclay, said that if lung cancer is caught at an early stage, “patients are nearly 20 times more likely to get at least another five years to spend with their families”.

Five-year survival rates are often quoted as key measures of cancer treatment success. Barclay’s figure is no doubt correct, but is it the right statistic to use to justify the screening programme?

Time-limited survival rates (typically given as five-, ten- and 20-year) can improve because cancers caught earlier are easier to treat, but also because patients identified at an earlier stage of the disease would live longer, with or without treatment, than those identified later. The latter is known as “lead-time bias”, and can mean that statistics like five-year survival rates paint a misleading picture of how effective a screening programme really is.

A graphic to illustrate the impact of lead-time bias on the perceived survival length of a disease detected with screening v symptoms.
Lead-time bias can appear to make a treatment more effective than it actually is, if the perceived post-diagnosis survival time increases while the course of disease progression is unaffected. Kit Yates

My new book, How to Expect the Unexpected, tackles issues exactly like this one, in which subtleties of statistics can give a misleading impression, causing us to make incorrect inferences and hence bad decisions. We need to be aware of such nuance so we can identify it when it confronts us, and so we can begin to reason our way beyond it.

To illustrate the effect of lead-time bias more concretely, consider a scenario in which we are interested in “diagnosing” people with grey hair. Without a screening programme, greyness may not be spotted until enough grey hairs have sprouted to be visible without close inspection. With careful regular “screening”, greyness may be diagnosed within a few days of the first grey hairs appearing.

People who obsessively check for grey hairs (“screen” for them) will, on average, find them earlier in their life. This means, on average, they will live longer “post-diagnosis” than people who find their greyness later in life. They will also tend to have higher five-year survival rates.

But treatments for grey hair do nothing to extend life expectancy, so it clearly isn’t early treatment that is extending the post-diagnosis life of the screened patients. Rather, it’s simply the fact their condition was diagnosed earlier.

To give another, more serious example, Huntington’s disease is a genetic condition that doesn’t manifest itself symptomatically until around the age of 45. People with Huntington’s might go on to live until they are 65, giving them a post-diagnosis life expectancy of about 20 years.

However, Huntington’s is diagnosable through a simple genetic test. If everyone was screened for genetic diseases at the age of 20, say, then those with Huntington’s might expect to live another 45 years. Despite their post-diagnosis life expectancy being longer, the early diagnosis has done nothing to alter their life expectancy.

Overdiagnosis

Screening can also lead to the phenomenon of overdiagnosis.

Although more cancers are detected through screening, many of these cancers are so small or slow-growing that they would never be a threat to a patient’s health – causing no problems if left undetected. Still, the C-word induces such mortal fear in most people that many will, often on medical advice, undergo painful treatment or invasive surgery unnecessarily.

The detection of these non-threatening cancers also serves to improve post-diagnosis survival rates when, in fact, not finding them would have made no difference to the patients’ lives.

So, what statistics should we be using to measure the effectiveness of a screening programme? How can we demonstrate that screening programmes combined with treatment are genuinely effective at prolonging lives?

The answer is to look at mortality rates (the proportion of people who die from the disease) in a randomised controlled trial. For example, the National Lung Screening Trial (NLST) found that in heavy smokers, screening with low-dose CT scans (and subsequent treatment) reduced deaths from lung cancer by 15% to 20%, compared with those not screened.

So, while screening for some diseases is effective, the reductions in deaths are typically small because the chances of a person dying from any particular disease are small. Even the roughly 15% reduction in the relative risk of dying from lung cancer seen in the heavy smoking patients in the NLST trial only accounts for a 0.3 percentage point reduction in the absolute risk (1.8% in the screened group, down from 2.1% in the control group).

For non-smokers, who are at lower risk of getting lung cancer, the drop in absolute risk may be even smaller, representing fewer lives saved. This explains why the UK lung cancer screening programme is targeting older people with a history of smoking – people who are at the highest risk of the disease – in order to achieve the greatest overall benefits. So, if you are or have ever been a smoker and are aged 55 to 74, please take advantage of the new screening programme – it could save your life.

But while there do seem to be some real advantages to lung cancer screening, describing the impact of screening using five-year survival rates, as the health secretary and his ministers have done, tends to exaggerate the benefits.

If we really want to understand the truth about what the future will hold for screened patients, then we need to be aware of potential sources of bias and remove them where we can.

Health, Maths

Why you’ve probably been using sunscreen all wrong your whole life

This post is adapted from my Indy Voices article of the same title originally published on 06/07/23

Are you a 50, a 30, a 15 or a “chance it in the name of a tan” oil lover? We all identify with at least one of these, but do you really know what SPF means – or how to use it?

Most of us don’t, actually. And as a mathematician with two redheads in their immediate family, I know better than most how vital it can be to protect yourself from the sun’s rays; particularly now, when the UK is experiencing some of the hottest weather this island has seen since records began.

The Met Office has issued a heat “health alert” for this weekend, with temperatures expected to reach at least 30C.

And we’ve just lived through the hottest June on record – warmer even than the June which kicked off the notoriously stifling summer of 1976. In fact, the top four hottest summers on record in the UK have occurred within the last 20 years, with the summer of ’76 only scraping in at number five. Two of the top four occurred within the last five years (2018 and 2022).

So, to state the obvious: with this climate crisis-induced trend looking likely to continue, it’s important that we know how to look after ourselves. But the less obvious issue is that it isn’t always easy to know which suncream you should use, or how much, particularly when it comes to things like SPF numbers. As I discovered when writing my new book, How to Expect the Unexpected, many people aren’t aware of what constitutes appropriate protection and how to use it.

Here’s what you might not realise when you’re slathering on suncream: of the two types of ultraviolet radiation that reach the earth’s surface – UVA and UVB – UVB plays the most significant role in causing sunburn and skin cancers. The higher the SPF (sun protection factor) you use, the more damaging UVB radiation is blocked, but the relationship between the number on the bottle is not directly proportional to the amount of radiation screened out.

Factor 50, for example, is not twice as effective at blocking UVB radiation as factor 25. Factor 30 does not block three times as much UVB radiation as factor 10.

In fact (still with me? good), when applied correctly, factor 10 blocks out… 90 per cent of all UVB radiation. Factor 30 blocks out just over 97 per cent – and factor 50 blocks out 98 per cent.

That means that the higher you go, the smaller the level of increased protection you are afforded. The increase in SPF from 10 to 30 gains you more than 7 per cent more protection. But the increase by the same numerical margin – from 30 to 50 – gains you less than 1 per cent extra sun screening effectiveness.

Factor 30 is usually the baseline recommended SPF by dermatologists. Lower than that and the degree of protection afforded starts to drop off quickly.

The way SPFs are often explained is by talking about the increase in exposure times different factors allow. If your skin would burn when subjected to 10 minutes of exposure without any protection, then the idea is that SPF 10 would extend that time by a factor of 10 to 100 minutes. SPF 50 would extend it to 500 minutes.

The underlying maths is that you can find the total UVB radiation exposure by multiplying the exposure time and the intensity of radiation experienced. When you apply SPF 50, the duration of time you can theoretically spend in the sun without getting burned increases by a factor of 50 (hence the reason it is called a sun protection factor).

If the total exposure is to be the same, to compensate for this increased time, the intensity of radiation must decrease by the same factor – 50. So, factor 50 lets only 1/50 (or 2 per cent) of the UVB radiation through, which is where the figure of 98 per cent screening effectiveness comes from for factor 50. Similarly, factor 10 lets only 1/10 of the radiation though, blocking 9/10 or 90 per cent.

If the maths goes over your head, then focus on this bit: Most dermatologists would recommend reapplying sunscreen every two hours, since protection can diminish over time as it breaks down, dries out or is rubbed off your skin.

By talking about the link between SPF and extended duration of exposure (the idea that factor 10 allows you to stay out 10 times as long) rather than screening efficiency, the traditional explanation of SPF can be baffling and misleading.

We forget that effectiveness diminishes over time as the sunscreen wears off. Talking only about SPF and not the proportion of UVB rays screened out can cause us to gain a false sense of security about how long we can stay out in the sun safely.

It’s also worth remembering that the SPF only refers to protection against UVB rays, which cause most skin cancers and sunburn. It does not grant protection against the deeper-penetrating UVA rays, which are largely responsible for premature skin ageing – but also cause some sorts of skin cancers and contribute to sunburn.

So: enjoy those outdoor parties and BBQs that we so rarely get the opportunity for in the UK. Take advantage of the opportunity to pursue the sorts of outdoor activities that our inclement winter weather so often deprives us of. But trust me on the sunscreen: wear at least factor 30, and reapply every two hours. You’ll thank me later.