Category Archives: Healthcare

Covid-19 deaths

I wrote last week about how the number of cases of coronavirus were following a textbook exponential growth pattern. I didn’t look at the number of deaths from coronavirus at the time, as there were too few cases in the UK for a meaningful analysis. Sadly, that is no longer true, so I’m going to take a look at that today.

However, first, let’s have a little update on the number of cases. There is a glimmer of good news here, in that the number of cases has been rising more slowly than we might have predicted based on the figures I looked at last week. Here is the growth in cases with the predicted line based on last week’s numbers.

As you can see, cases in the last week have consistently been lower than predicted based on the trend up to last weekend. However, I’m afraid this is only a tiny glimmer of good news. It’s not clear whether this represents a real slowing in the number of cases or merely reflects the fact that not everyone showing symptoms is being tested any more. It may just be that fewer cases are being detected.

So what of the number of deaths? I’m afraid this does not look good. This is also showing a classic exponential growth pattern so far:

The last couple of days’ figures are below the fitted line, so there is a tiny shred of evidence that the rate may be slowing down here too, but I don’t think we can read too much into just 2 days’ figures. Hopefully it will become clearer over the coming days.

One thing which is noteworthy is that the rate of increase of deaths is faster than the rate of increase of total cases. While the number of cases is doubling, on average, every 2.8 days, the number of deaths is doubling, on average, every 1.9 days. Since it’s unlikely that the death rate from the disease is increasing over time, this does suggest that the number of cases is being recorded less completely as time goes by.

So what happens if the number of deaths continues growing at the current rate? I’m afraid it doesn’t look pretty:

(note that I’ve plotted this on a log scale).

At that rate of increase, we would reach 10,000 deaths by 1 April and 100,000 deaths by 7 April.

I really hope that the current restrictions being put in place take effect quickly so that the rate of increase slows down soon. If not, then this virus really is going to have horrific effects on the UK population (and of course on other countries, but I’ve only looked at UK figures here).

In the meantime, please keep away from other people as much as you can and keep washing those hands.

Covid-19 and exponential growth

One thing about the Covid-19 outbreak that has been particularly noticeable to me as a medical statistician is that the number of confirmed cases reported in the UK has been following a classic exponential growth pattern. For those who are not familiar with what exponential growth is, I’ll start with a short explanation before I move on to what this means for how the epidemic is likely to develop in the UK. If you already understand what exponential growth is, then feel free to skip to the section “Implications for the UK Covid-19 epidemic”.

A quick introduction to exponential growth

If we think of something, such as the number of cases of Covid-19 infection, as growing at a constant rate, then we might think that we would have a similar number of new cases each day. That would be a linear growth pattern. Let’s assume that we have 50 new cases each day, then after 60 days we’ll have 3000 cases. A graph of that would look like this:

That’s not what we’re seeing with Covid-19 cases. Rather than following a linear growth pattern, we’re seeing an exponential growth pattern. With exponential growth, rather than adding a constant number of new cases each day, the number of cases increases by a constant percentage amount each day. Equivalently, the number of cases multiplies by a constant factor in a constant time interval.

Let’s say that the number of cases doubles every 3 days. On day zero we have just one case, on day 3 we have 2 cases, and day 6 we have 4 cases, on day 9 we have 8 cases, and so on. This makes sense for an infectious disease epidemic. If you imagine that each person who is infected can infect (for example) 2 new people, then you would get a pattern very similar to this. When only one person is infected, that’s just 2 new people who get infected, but if 100 people have the disease, then 200 people will get infected in the same time.

On the face of it, the example above sounds like it’s growing much less quickly than my first example where we have 50 new cases each day. But if you are doubling the number of cases each time, then you start to get to scarily large numbers quite quickly. If we carry on for 60 days, then although the number of cases isn’t increasing much at first, it eventually starts to increase at an alarming rate, and by the end of 60 days we have over a million cases. This is what it looks like if you plot the graph:

It’s actually quite hard to see what’s happening at the beginning of that curve, so to make it easier to see, let’s use the trick of plotting the number of cases on a logarithmic scale. What that means is that a constant interval on the vertical axis (generally known as the y axis) represents not a constant difference, but a constant ratio. Here, the ticks on the y axis represent an increase in cases by a factor of 10.

Note that when you plot exponential growth on a logarithmic scale, you get a straight line. That’s because we’re increasing the number of cases by a constant ratio in each unit time, and a constant ratio corresponds to a constant distance on the y axis.

Implications for the UK Covid-19 epidemic

OK, so that’s what exponential growth looks like. What can we see about the number of confirmed Covid-19 cases in the UK? Public Health England makes the data available for download here. The data have not yet been updated with today’s count of cases as I write this, so I added in today’s number (1372) based on a tweet by the Department of Health and Social Care.

If you plot the number of cases by date, it looks like this:

That’s pretty reminiscent of our exponential growth curve above, isn’t it?

It’s worth noting that the numbers I’ve shown are almost certainly an underestimate of the true number of cases. First, it seems likely that some people who are infected will have only very mild (or even no) symptoms, and will not bother to contact the health services to get tested. You might say that it doesn’t matter if the numbers don’t include people who aren’t actually ill, and to some extent it doesn’t, but remember that they may still be able to infect others. Also, there is a delay from infection to appearing in the statistics. So the official number of confirmed cases includes people only after they have caught the disease, gone through the incubation period, developed symptoms that were bothersome enough to seek medical help, got tested, and have the test results come back. This represents people who were infected probably at least a week ago. Given that the number of cases are growing so rapidly, the number of people actually infected today will be considerably higher than today’s statistics for confirmed cases.

Now, before I get into analysis, I need to decide where to start the analysis. I’m going to start from 29 February, as that was when the first case of community transmission was reported, so by then the disease was circulating within the UK community. Before then it had mainly been driven by people arriving in the UK from places abroad where they caught the disease, so the pattern was probably a bit different then.

If we start the graph at 29 February, it looks like this:

Now, what happens if we fit an exponential growth curve to it? It looks like this:

(Technical note for stats geeks: the way we actually do that is with a linear regression analysis of the logarithm of the number of cases on time, calculate the predicted values of the logarithm from that regression analysis, and then back-transform to get the number of cases.)

As you can see, it’s a pretty good fit to an exponential curve. In fact it’s really very good indeed. The R-squared value from the regression analysis is 0.99. R-squared is a measure of how well the data fit the modelled relationship on a scale of 0 to 1, so 0.99 is a damn near perfect fit.

We can also plot it on a logarithmic scale, when it should look like a straight line:

And indeed it does.

There are some interesting statistics we can calculate from the above analysis. The average rate of growth is about a 30% increase in the number of cases each day. That means that the number of cases doubles about every 2.6 days, and increases tenfold in about 8.6 days.

So what happens if the number of cases keeps growing at the same rate? Let’s extrapolate that line for another 6 weeks:

This looks pretty scary. If it continues at the same rate of exponential growth, we’ll get to 10,000 cases by 23 March (which is only just over a week away), to 100,000 cases by the end of March, to a million cases by 9 April, and to 10 million cases by 18 April. By 24 April the entire population of the UK (about 66 million) will be infected.

Now, obviously it’s not going to continue growing at the same rate for all that time. If nothing else, it will stop growing when it runs out of people to infect. And even if the entire population have not been infected, the rate of new infections will surely slow down once enough people have been infected, as it becomes increasingly unlikely that anyone with the disease who might be able to pass it on will encounter someone who hasn’t yet had it (I’m assuming here that people who have already had the disease will be immune to further infections, which seems likely, although we don’t yet know that for sure).

However, that effect won’t kick in until at least several million people have been infected, a situation which we will reach by the middle of April if other factors don’t cause the rate to slow down first.

Several million people being infected is a pretty scary prospect. Even if the fatality rate is “only” about 1%, then 1% of several million is several tens of thousands of deaths.

So will the rate slow down before we get to that stage?

I genuinely don’t know. I’m not an expert in infectious disease epidemiology. I can see that the data are following a textbook exponential growth pattern so far, but I don’t know how long it will continue.

Governments in many countries are introducing drastic measures to attempt to reduce the spread of the disease.

The UK government is not.

It is not clear to me why the UK government is taking a more relaxed approach. They say that they are being guided by the science, but since they have not published the details of their scientific modelling and reasoning, it is not possible for the rest of us to judge whether their interpretation of the science is more reasonable than that of many other European countries.

Maybe the rate of infection will start to slow down now that there is so much awareness of the disease and of precautions such as hand-washing, and that even in the absence of government advice, many large gatherings are being cancelled.

Or maybe it won’t. We will know more over the coming weeks.

One final thought. The government’s latest advice is for people with mild forms of the disease not to seek medical help. This means that the rate of increase of the disease may well appear to slow down as measured by the official statistics, as many people with mild disease will no longer be tested and so not be counted. It will be hard to know whether the rate of infection is really slowing down.

Solving the economics of personalised medicine

It’s a well known fact that many drugs for many diseases don’t work very well in in many patients. If we could identify in advance which patients will benefit from a drug and which won’t, then drugs could be prescribed in a much more targeted manner. That is actually a lot harder to do than it sounds, but it’s an active area of research, and I am confident that over the coming years and decades medical research will make much progress in that direction.

This is the world of personalised medicine.

Although giving people targeted drugs that are likely to be of substantial benefit to them has obvious advantages, there is one major disadvantage. Personalised medicine simply does not fit the economic model that has evolved for the pharmaceutical industry.

Developing new drugs is expensive. It’s really expensive. Coming up with a precise figure for the cost of developing a new drug is controversial, but some reasonable estimates run into billions of dollars.

The economic model of the pharmaceutical industry is based on the idea of a “blockbuster” drug. You develop a drug like Prozac, Losec, or Lipitor that can be used in millions of patients, and the huge costs of that development can be recouped by the  huge sales of the drug.

But what if you are developing drugs based on personalised medicine for narrowly defined populations?  Perhaps you have developed a drug for patients with a specific variant of a rare cancer, and it is fantastically effective in those patients, but there may be only a few hundred patients worldwide who could benefit. There is no way you’re going to be able to recoup the costs of a billion dollars or more of development by selling the drug to a few hundred patients, without charging sums of money that are crazily unaffordable to each patient.

Although the era of personalised medicine is still very much in its infancy, we have already seen this effect at work with drugs like Kadcyla, which works for only a specific subtype of breast cancer patients, but at £90,000 a pop has been deemed too expensive to fund in the NHS. What happens when even more targeted drugs are developed?

I was discussing this question yesterday evening over a nice bottle of Chilean viognier with Chris Winchester. I think between us we may have come up with a cunning plan.

Our idea is as follows. If a drug is being developed for a suitably narrow patient population that it could be reasonably considered a “personalised medicine”, different licensing rules would apply. You would no longer have to obtain such a convincing body of evidence of efficacy and safety before licensing. You would need some evidence, of course, but the bar would be set much lower. Perhaps some convincing laboratory studies followed by some small clinical trials that could be done much more cheaply than the typical phase III trials that enrol hundreds of patients and cost many millions to run.

At that stage, you would not get a traditional drug license that would allow you to market the drug in the normal way. The license would be provisional, with some conditions attached.

So far, this idea is not new. The EMA has already started a pilot project of “adaptive licensing“, which is designed very much in this spirit.

But here comes the cunning bit.

Under our plan, the drug would be licensed to be marketed as a mixture of the active drug and placebo. Some packs of the drug would contain the active drug, and some would contain placebo. Neither the prescriber nor the patient would know whether they have actually received the drug. Obviously patients would need to be told about this and would then have the choice to take part or not. But I don’t think this is worse than the current situation, where at that stage the drug would not be licensed at all, so patients would either have to find a clinical trial (where they may still get placebo) or not get the drug at all.

In effect, every patient who uses the drug during the period of conditional licensing would be taking part in a randomised, double-blind, placebo-controlled trial.  Prescribers would be required to collect data on patient outcomes, which, along with a code number on the medication pack, could then be fed back to the manufacturer and analysed. The manufacturer would know from the code number whether the patient received the drug or placebo.

Once sufficient numbers of patients had been treated, then the manufacturer could run the analysis and the provisional license could be converted to a full license if the results show good efficacy and safety, or revoked if they don’t.

This wouldn’t work in all cases. There will be times when other drugs are available but would not be compatible with the new drug. You could not then ethically put patients in a position where a drug is available but they get no drug at all. But in cases where no effective treatment is available, or the new drug can be used in addition to standard treatments, use of a placebo in this way is perfectly acceptable from an ethical point of view.

Obviously even when placebo treatment is a reasonable option, there would be logistical challenges with this approach (for example, making sure that the same patient gets the same drug when their first pack of medicine runs out). I don’t pretend it would be easy. But I believe it may be preferable to a system in which the pharmaceutical industry has to abandon working on personalised medicine because it has become unaffordable.

The amazing magic Saatchi Bill

Yesterday saw the dangerous and misguided Saatchi Bill (now reincarnated as the Access to Medical Treatments (Innovation) Bill) debated in the House of Commons.

The bill started out as an attempt by the Conservative peer Lord Saatchi to write a new law to encourage innovation in medical research. I have no doubt that the motivation for doing so was based entirely on good intentions, but sadly the attempt was badly misguided. Although many people explained to Lord Saatchi why he was wrong to tackle the problem in the way he did, it turns out that listening to experts is not Saatchi’s strong suit, and he blundered on with his flawed plan anyway.

If you want to know what is wrong with the bill I can do no better than direct you to the Stop the Saatchi Bill website, which explains the problems with the bill very clearly. But briefly, it sets out to solve a problem that does not exist, and causes harm at the same time. It attempts to promote innovation in medical research by removing the fear of litigation from doctors who innovate, despite the fact that fear of litigation is not what stops doctors innovating. But worse, it removes important legal protection for patients. Although the vast majority of doctors put their patients’ best interests firmly at the heart of everything they do, there will always be a small number of unscrupulous quacks who will be only too eager to hoodwink patients into paying for ineffective or dangerous treatments if they think there is money in it.

If the bill is passed, any patients harmed by unscrupulous quacks will find it harder to get redress through the legal system. That does not protect patients.

Although the bill as originally introduced by Saatchi failed to make sufficient progress through Parliament, it has now been resurrected in a new, though essentially similar, form as a private member’s bill in the House of Commons.

I’m afraid to say that the debate in the House of Commons did not show our lawmakers in a good light.

We were treated to several speeches by people who clearly either didn’t understand what the bill was about or were being dishonest. The two notable exceptions were Heidi Alexander, the Shadow Health Secretary, and Sarah Wollaston, chair of the Health Select Committee and a doctor herself in a previous career. Both Alexander and Wollaston clearly showed that they had taken the trouble to read the bill and other relevant information carefully, and based their contributions on facts rather than empty rhetoric.

I won’t go into detail on all the speeches, but if you want to read them you can do so in Hansard.

The one speech I want to focus on is by George Freeman, the Parliamentary Under-Secretary of State for Life Sciences. As he is a government minister, his speech gives us a clue about the government’s official thinking on the bill. Remember that it is a private member’s bill, so government support is crucial if it is to have a chance of becoming law. Sadly, Freeman seems to have swallowed the PR surrounding the bill and was in favour of it.

Although Freeman said many things, many of which showed either a poor understanding of the issues or blatant dishonesty, the one I particularly want to focus on is where he imbued the bill with magic powers.

He repeated the myths about fear of litigation holding back medical research. He was challenged in those claims by both Sarah Wollaston and Heidi Alexander.

When he reeled off a whole bunch of statistics about how much money medical litigation cost the NHS, Wollaston asked him how much of that was specifically related to complaints about innovative treatments. His reply was telling:

“Most of the cases are a result of other contexts— as my hon. Friend will know, obstetrics is a big part of that—rather than innovation. I am happy to write to her with the actual figure as I do not have it to hand.”

Surely that is the one statistic he should have had to hand if he’d wanted to appear even remotely prepared for his speech? What is the point of being able to quote all sorts of irrelevant statistics about the total cost of litigation in the NHS if he didn’t know the one statistic that actually mattered? Could it be that he knew it was so tiny it would completely undermine his case?

He then proceeded to talk about the fear of litigation, at which point Heidi Alexander asked him what evidence he had. He had to admit that he had none, and muttered something about “anecdotally”.

But anyway, despite having failed to make a convincing case that fear of litigation was holding back innovation, he was very clear that he thought the bill would remove that fear.

And now we come to the magic bit.

How exactly was that fear of litigation to be removed? Was it by changing the law on medical negligence to make it harder to sue “innovative” doctors? This is what Freeman said:

“As currently drafted the Bill provides no change to existing protections on medical negligence, and that is important. It sets out the power to create a database, and a mechanism to make clear to clinicians how they can demonstrate compliance with existing legal protection—the Bolam test has been referred to—and allow innovations to be recorded for the benefit of other clinicians and their patients. Importantly for the Government, that does not change existing protections on medical negligence, and it is crucial to understand that.”

So the bill makes no change whatsoever to the law on medical negligence, but removes the fear that doctors will be sued for negligence. If you can think of a way that that could work other than by magic, I’m all ears.

In the end, the bill passed its second reading by 32 votes to 19. Yes, that’s right: 599 well over 500* MPs didn’t think protection of vulnerable patients from unscrupulous quacks was worth turning up to vote about.

I find it very sad that such a misguided bill can make progress through Parliament on the basis of at best misunderstandings and at worst deliberate lies.

Although the bill has passed its second reading, it has not yet become law. It needs to go through its committee stage and then return to the House of Commons for its third reading first. It is to be hoped that common sense will prevail some time during that process, or patients harmed by unscrupulous quacks will find that the law does not protect them as much as it does now.

If you want to write to your MP to urge them to turn up and vote against this dreadful bill when it comes back for its third reading, now would be a good time.

* Many thanks to @_mattl on Twitter for pointing out the flaw in my original figure of 599: I hadn’t taken into account that the Speaker doesn’t vote, the Tellers aren’t counted in the totals, Sinn Fein MPs never turn up at all, and SNP MPs are unlikely to vote as this bill doesn’t apply to Scotland.

What my hip tells me about the Saatchi bill

I have a hospital appointment tomorrow, at which I shall have a non-evidence-based treatment.

This is something I find somewhat troubling. I’m a medical statistician: I should know about evidence for the efficacy of medical interventions. And yet even I find myself ignoring the lack of good evidence when it comes to my own health.

I have had pain in my hip for the last few months. It’s been diagnosed by one doctor as trochanteric bursitis and by another as gluteus medius tendinopathy. Either way, something in my hip is inflammed, and is taking longer than it should to settle down.

So tomorrow, I’m having a steroid injection. This seems to be the consensus among those treating me. My physiotherapist was very keen that I should have it. My GP thought it would be a good idea. The consultant sports physician I saw last week thought it was the obvious next step.

And yet there is no good evidence that steroid injections work. I found a couple of open label randomised trials which showed reasonably good short-term effects for steroid injections, albeit little evidence of benefit in the long term. Here’s one of them. The results look impressive on a cursory glance, but something that really sticks out at me is that the trials weren’t blinded. Pain is subjective, and I fear the results are entirely compatible with a placebo effect. Perhaps my literature searching skills are going the same way as my hip, but I really couldn’t find any double-blind trials.

So in other words, I have no confidence whatsoever that a steroid injection is effective for inflammation in the hip.

So why am I doing this? To be honest, I’m really not sure. I’m bored of the pain, and even more bored of not being able to go running, and I’m hoping something will help. I guess I like to think that the health professionals treating me know what they’re doing, though I really don’t see how they can know, given the lack of good evidence from double blind trials.

What this little episode has taught me is how powerful the desire is to have some sort of treatment when you’re ill. I have some pain in my hip, which is pretty insignificant in the grand scheme of things, and yet even I’m getting a treatment which I have no particular reason to think is effective. Just imagine how much more powerful that desire must be if you’re really ill, for example with cancer. I have no reason to doubt that the health professionals treating me are highly competent and well qualified professionals who have my best interests at heart. But it has made me think how easy it must be to follow advice from whichever doctor is treating you, even if that doctor might be less scrupulous.

This has made me even more sure than ever that the Saatchi bill is a really bad thing. If a medical statistician who thinks quite carefully about these things is prepared to undergo a non-evidence-based treatment for what is really quite a trivial condition, just think how much the average person with a serious disease is going to be at the mercy of anyone treating them. The last thing we want to do is give a free pass for quacks to push completely cranky treatments at anyone who will have them.

And that’s exactly what the Saatchi bill will facilitate.

Hospital special measures and regression to the mean

Forgive me for writing 2 posts in a row about regression to the mean. But it’s an important statistical concept, which also happens to be widely misunderstood. Sometimes with important consequences.

Last week, I blogged about a claim that student tuition fees had not put off disadvantaged applicants. The research was flawed, because it defined disadvantage on the basis of postcode areas, and not on the individual characteristics of applicants. This means that an increase in university applications from disadvantaged areas could have simply been due to regression to the mean (ie the most disadvantaged areas becoming less disadvantaged) rather than more disadvantaged individual students applying to university.

Today, we have a story in the news where exactly the same statistical phenomenon is occurring. The story is that putting hospitals into “special measures” has been effective in reducing their death rates, according to new research by Dr Foster.

The research shows no such thing, of course.

The full report, “Is [sic] special measures working?” is available here. I’m afraid the authors’ statistical expertise is no better than their grammar.

The research looked at 11 hospital trusts that had been put into special measures, and found that their mortality rates fell faster than hospitals on average. They thus concluded that special measures were effective in reducing mortality.

Wrong, wrong, wrong. The 11 hospital trusts had been put into special measures not at random, but precisely because they had higher than expected mortality. If you take 11 hospital trusts on the basis of a high mortality rate and then look at them again a couple of years later, you would expect the mortality rate to have fallen more than in other hospitals simply because of regression to the mean.

Maybe those 11 hospitals were particularly bad, but maybe they were just unlucky. Perhaps it’s a combination of both. But if they were unusually unlucky one year, you wouldn’t expect them to be as unlucky the next year. If you take the hospitals with the worst mortality, or indeed the most extreme examples of anything, you would expect it to improve just by chance.

This is a classic example of regression to the mean. The research provides no evidence whatsoever that special measures are doing anything. To do that, you would need to take poorly performing hospitals and allocate them at random either to have special measures or to be in a control group. Simply observing that the worst trusts got better after going into special measures tells you nothing about whether special measures were responsible for the improvement.