Category Archives: health

Careful with that thing – you could kill someone

It’s been a while, but guest author Mark Lauer is returning to the Mule. While in COVID-induced lockdown, the mind naturally turns to armchair epidemiology, but here Mark goes beyond mere amateur probability to add a sprinkling of ethics.

So, you’re in lockdown during a COVID-19 outbreak in your city. And you’re wondering, now that most of the elderly are vaccinated, if all the fuss is really justified. After all, only a tiny proportion of the city has caught COVID so far, and even if you get it, statistically speaking it is unlikely to harm you. The number of people dying is a small fraction of the population, especially now that effective vaccines are being rolled out. So just how dangerous would it be if you popped down to see that friend you’ve been missing?

It turns out, if you happen to have COVID, it could be rather dangerous indeed.  It may not be too risky for you, or your friend, but let’s do some simple mathematics to see what the consequences might be for others if you do pass on the virus.

In what follows, I’ll focus on the current outbreak here in Sydney, which began on June 16. It’s unusual at this stage of the global pandemic, since the population has lived largely unrestricted for over a year and perhaps some have become complacent about dealing with the virus, despite the carnage and sacrifices of freedom seen overseas. But the general gist applies anywhere that has significant case counts which aren’t falling dramatically.

Please note though, I am not an epidemiologist. There are many more qualified people, building far more sophisticated models. Listen to them and follow their advice.

One obvious factor to consider is how likely it is that you’re infected.  This will vary depending on the number of cases in the outbreak, how many cases are near you, and how often you go shopping or meet others.  But remember it takes several days for testing to reveal where cases are, during which time the outbreak can spread far across the city.  Also many people with COVID are asymptomatic, or at least asymptomatic for a period while they are infectious. None of the people who’ve passed on the virus so far have thought they had it at the time.  And it seems the Delta strain may take as little as a few seconds of contact to transmit.  But let’s set that question aside, and look at what happens if you do transmit it.

So suppose you unknowingly have the virus, and choose between two courses of action, one that passes it on to another person, and the other that avoids doing so.  From an ethical stand point, just how bad is it if you opt for the former?

To start, let’s consider the average risk of death for the person you infect. Case fatality rates for COVID-19 are in the range of 1-3% in most countries, but of course these will vary depending on many factors: the standard and capacity of health facilities, who in the population is getting the disease, how many of those are vaccinated, and the virulence of the prevalent strain.

In the Sydney outbreak we’ve had relatively few deaths. As at July 26, there have been 10 in this outbreak, whereas total case counts are now above 2000. However, that neglects the delay between cases being identified and consequent deaths. A study in the Journal of Public Health published in March finds the average lag is 8 days (even longer if a lower proportion of those infected are over 60 years old).

So a more comparable estimate of cases might be the number of locally acquired cases reported up to July 18, which is 1364. That yields a case fatality rate in this outbreak of 0.73%, which is indeed low by global standards of COVID. But while it might seem like a small number, that’s 7300 micromorts, which is equivalent to spending over 7 months as a British soldier serving in Afghanistan.

Now perhaps you and your friend are vaccinated, in which case the mortality risk to you is substantially lower. But while vaccination helps prevent your death, it is far less effective against transmitting the virus. And ethically speaking we need to consider what happens if your friend then passes the virus on further. The probability of this will vary according to the situation. If your friend is actually someone you’re  keeping locked in your cellar as a slave, then there’s no way for them to pass it on, and you can feel relieved of any moral qualms about deaths due to passing on the virus further (we can set aside other moral considerations in this scenario, since we’re talking about manslaughter here, so why worry about a minor case of enslavement).

Since normally we have little control over how others behave, even friends, let’s assume the friend is exactly like the average other Sydneysider in this outbreak. We can roughly guess the effective reproduction rate of the virus in the conditions of this outbreak by looking at case counts over time. Here is a chart of the number of new locally acquired cases by date during the outbreak so far.

Bar chart of new locally acquired cases in NSW 16.6-26.7.2021

Source: NSW Government

In the 24 days through to the imposition of city-wide stay-at-home restrictions on July 10, new cases grew exponentially to reach 103. For the purpose of this argument, I’ll assume a fixed cycle of infection lasting 3 days (this is not essential, since values below are still valid albeit with slightly different timeframes if the cycle is longer or shorter). A quick calculation yields a reproduction rate, r = 1.8.  That is, each infected person infects an average of 1.8 other people every three days.

At this level of transmission, 100 people will infect 180 people in three days, who will infect another 324 people after six days, and so on. If this continues for 15 days, the total number of resulting infections will be 4126, or 41.26 people per original infected person. If each of the 41 people infected via our friend has a 0.73% chance of dying from the virus, there is over 25% chance that at least one person will die. And that’s only counting infections in the next 15 days.  Giving the virus to one person is significantly worse than Russian roulette under these conditions.

Of course, as Sydneysiders are uncomfortably aware, the government here has been instituting successively more stringent restrictions across the city. And in the two weeks or so since July 11, the growth in case counts has happily slowed somewhat. Unfortunately, lockdown efforts so far appear to be insufficient to bring case counts down dramatically, with over 170 new cases reported on July 25.

But let’s be wildly optimistic and say that the reproduction rate is now down to 0.9. In that case, 100 people infect 90 people who infect 81 people, so that after 15 days the expected total number of resulting cases is 469. Your single transmission to your friend then leads to around 3.4% chance of at least one death as a result of infection in the next 15 days.

While that’s much better than before the citywide restrictions, it is nothing to shrug off. It’s similar to the chance of dying:

Most would agree that all these events have a “reasonable chance of killing someone”. And so too does passing on the virus under the current Sydney outbreak conditions.

So please, please be careful. Your choices can save lives.

Shark season

Summer in Australia comes with cicadas, sunburn and, in the media at least, sharks. So far, I have learned that aerial shark patrols are inefficient (or perhaps not) and that the Western Australian government plans to keep swimmers safe by shooting big sharks.

Sharks are compelling objects of fear, right up there with spiders and snakes in the package of special terrors for visitors to Australia. As good hosts, we are quick to reassure: sharks may be the stuff of nightmares and 70s horror movies, but attacks are rare.

But, exactly how rare is death by shark? Over a Boxing Day lunch, I heard an excellent ‘statistic’, designed to reassure a visiting American. Apparently, more people are killed each year in the US by falling vending machines than are killed by sharks around the world. I was skeptical, but had no data to hand. Later, with the help of Google, I discovered that this statistic is 10 years old and the source? Los Angeles life guards. The tale has, however, become taller over time. Originally, vending machine deaths in the US were compared to shark attack fatalities in the US, not the entire world.

While data on vending machine related deaths are hard to come by, subsequent attempts to validate the story concluded that it was plausible, on the basis that there were two vending machine deaths in 2005 in the US but no fatal shark attacks.

Fun though the vending machine line may be, it is not relevant to Australia and, if you are on the beach contemplating a quick dip, then the risk of a shark attack is certainly higher in the sea than death by vending machine. Local data is in order.

According to the Taronga Zoo Australian Shark Attack File (ASAF):

 In the last 50 years, there have been 50 recorded unprovoked fatalities due to shark attack, which averages one per year.

Fatalities have been higher than average over the last couple of years. The ASAF recorded two deaths in 2012 and, although validated figures for 2013 are yet to be published, six deaths have been reported over the last two years, suggesting that fatalities rose further to four this year.

To compare shark fatalities to other causes of mortality, a common scale is useful. My unit of choice is the micromort. A one-in-a-million chance of death corresponds to a micromort of 1.0, a one-in-ten-million chance of death to a micromort of 0.1. Taking the recent average death rate of three per year (more conservative than the longer run average of one), and a population of 23 million in Australia leads to a figure of 0.13 micromorts for the annual risk of death for a randomly chosen Australian.

The most recent data on causes of death published by the Australian Bureau of Statistics (ABS) are for 2009. That year, three people were killed by crocodiles. Sharks are not specifically identified, but any fatal shark attacks would be included among the three deaths due to ‘contact with marine animals’. The chart below illustrates the risk of death associated with a number of ‘external causes’. None of these come close to heart disease, cancer or car accidents. Death by shark ranks well below drowning, even drowning in the bath, as well as below a variety of different types of falls, whether from stairs, cliffs or ladders.

Shark barplot

Annual risk of death in Australia (2009 data)*

Of course, you and I are not randomly chosen Australians and our choices change the risks we face. I am far less likely to suffer death by vending machine if I steer clear of the infernal things and I am far less likely to be devoured by a shark if I stay out of the water.

So, care should be taken when interpreting the data in the chart. Drug addicts (or perhaps very serious Hendrix imitators) are far more likely to asphyxiate on their own vomit than summer beach-goers. The fairest point of comparison is drowning in natural waters. At almost 3.5 micromorts, drownings in the sea (or lakes and rivers) is more than 25 times more common than fatal shark attacks. And the risk of both can be reduced by swimming between the flags.

What does that leave us with for conversations with foreign visitors? If you are headed to the beach, the risk of shark attack would be higher than death by vending machine, but it is still very low. The drive there (at 34.3 micromorts) is almost certainly more dangerous.

I will be taking comfort from my own analysis as I am heading to Jervis Bay tomorrow and sharks were sighted there this weekend:

Bendigo Bank Aerial Patrol spotted up to 14 sharks between 50 and 100 metres from shore at various beaches in Jervis Bay. [The] crew estimated the sharks at between 2.5 and 3.5 metres in length at Nelsons, Blenheim, Greenfields, Chinaman’s Beach and Hyams Beaches.

The beaches are un-patrolled, so wish me luck…but I don’t think I’ll need it.

* The figure for ‘Shark attack’ is based on the estimate of three deaths per year rather than the ABS data.

Are you mad, sir?

Even if you haven’t heard of Jon Ronson, you have probably heard of one of his books. He wrote The Men Who Stare At Goats, which has been made into a film starring George Clooney. I have just finished reading a more recent, if lesser known book by Ronson: The Psychopath Test. It is an intriguing, anecdotal exploration of the nature of madness, with a particular focus on psychopathy.

The book is loosely centred on the psychopath test of the title, better known as the Hare Psychopath Checklist in honor of its creator, Canadian psychologist Robert Hare or, more simply, PCL-R (“Psychopath Checklist – Revised”). On his journey towards a better understanding of anti-social madness, Ronson attended a training course in the use of the PCL-R led by Hare himself. Armed with this qualification, Ronson found his new ability to expertly identify psychopaths out in the wild gave him an exciting sense of power. It is a sense of power that readers such as myself can readily share: it wasn’t long before I was attempting to spot corporate psychopaths in the upper echelons of my own place of work.

Here is how the test works: through a more rigorous interview process than I have had the opportunity to perform, your potential psychopath is scored on a scale of 0 to 2 on each of the categories below.

  1. Glibness/superficial charm
  2. Grandiose sense of self-worth
  3. Pathological lying
  4. Cunning/manipulative
  5. Lack of remorse or guilt
  6. Shallow affect
  7. Callousness, lack of empathy
  8. Failure to accept responsibility for own actions
  9. Need for stimulation/proneness to boredom
  10. Parasitic lifestyle
  11. Poor behavioural control
  12. Lack of realistic long-term goals
  13. Impulsivity
  14. Irresponsibility
  15. Juvenile delinquency
  16. Early behaviour problems
  17. Revocation of conditional release
  18. Promiscuous sexual behaviour
  19. Many short-term (marital) relationships
  20. Criminal versatility

Roughly speaking, a score of 30 or more suggests you have a psychopath on your hands. Reading Ronson’s book, I got the impression that there are currently few treatment options for a psychopath and those that veer towards criminality rather than high-flying success in the corporate world tend to be locked up for a very long time.

Over a sherry at a recently opened Spanish bar in Sydney, with the help of a colleague, I attempted an analysis of the executive at my firm I considered to be the best prospect for a high score on the PCL-R. Sadly, we only managed to chalk up 20 points. Apparently not a psychopath after all and, while that score is still reasonably high, I have to further concede that there may have been some overly-enthusiastic interpretations of the checklist involved in the assessment.

My own attempt at psychopath diagnosis brought me to sympathise further with Ronson, who found that the thrill of power was, after a while, replaced by doubt. Perhaps things are actually a bit more complicated after all. Even an interview with Al Dunlap, initially a slam-dunk candidate for the label of corporate psychopath, particularly given his extensive collection of statues of animals of prey (psychopaths apparently tend to see the world in terms of predators and prey), left Ronson uncertain of the appropriateness of the label of madness.

The lesson then is that I should use my new-found knowledge of PCL-R with care. As should you. Even so, I will not delete the list from this post as a precaution. After all, you could easily find it on Wikipedia; such is the power and the peril of the internet.

John Graunt and the Birth of Medical Statistics

Dr John Carmody of the Department of Physiology at the University of Sydney, recently appeared on the ABC Radio National program, Occams Razor, speaking about John Graunt, a man many years ahead of his time. For those of you preferring the written to the auditory format, he has kindly provided his talk as a guest post for the Mule.

We become blind to what is familiar.

So dependent is modern medicine on accurate measurement that patients and doctors alike accept the fact without surprise or question, perhaps believing that it is inevitable. Yet the importance of numbers of any sort in medicine, let alone precise ones, is a concept that is little over 350 years old. In physiology, the most basic of medical sciences, this dates only from 1628 when William Harvey published his great book on the circulation, a discovery which he formulated and proved through numerical argument.

Then in London, in 1662, 350 years ago this year, John Graunt published a booklet which we can now understand was the beginning of medical statistics, of epidemiology, of medical demography. In the manner of those times he gave it the formidable title of Natural and political observations, mentioned in a following Index, and made upon the Bills of Mortality, to which he added the supplementary description, “With reference to the Government, Religion, Trade, Growth, Air, Diseases and the several changes of the said City”. His work was, therefore, far wider than establishing a new medical discipline. He was arguing for the necessary interaction of medicine, good government and sensible policy—indeed, perhaps for the discipline of quantitative economics, as well. We can realize how original Graunt’s work was when we remember that the only previous English census was the compilation of the “Domesday Book” in 1086 and that the first official census was not taken until 1801.

Graunt’s genius was to recognize—as none of his contemporaries had done—the immense importance of what we would now call a “database” which had existed in London for about 60 years. These were the so-called “Bills of Mortality” which the administrative clerks of the Church of England parishes in London had been obliged to keep scrupulously since James I became king in 1603. In fact, when James granted a charter to the Company of Parish Clerks in 1611, he legally obliged the members to be far more diligent in their recording than before his accession to the throne. These Bills recorded the christenings and the burials, parish by parish, each week. As well, the burials were accompanied by what Graunt called the “diseases and casualties” which brought about those deaths. He drew on the records of about 97 parishes within the city walls and 16 outside them and in a typical year he would have to deal with 20,000-25,000 burials and supposed causes of death.

He was very concerned with the reliability of those diagnoses which were rarely professionally reported. As he wrote, “When anyone dies, then, either by tolling, or ringing of a Bell, or by bespeaking of a Grave of the Sexton, the same is known to the Searchers corresponding with the said Sexton. The Searchers hereupon (who are ancient matrons, sworn to their office), repair to the place where the dead Corps lies, and by view of the same, and by other enquiries, they examine by what disease or casualty the corps died. Hereupon, they make their report to the Parish-Clerk.” Graunt keenly recognized the flaws in such a system and acknowledged that “I have heard some candid physicians complain of the darkness, which they themselves were in hereupon”. He also saw the possibility of corruption, the temptation, as he put it, for “the old-women searchers after the mist of a cup of ale and the bribe of a two-groat fee” to report, say, “Consumption” instead of the more shaming “infection of the spermatick parts”. In fact, he was convinced that syphilis, or the “French pox” was substantially under-reported.

Nevertheless, he decided that the incidence of such problems probably had changed little over the period which he was examining, so errors of those kinds were likely to be fairly consistent. “The ignorance of the Searchers is no impediment to the keeping of sufficient and usefull Accompts”. However, he saw other potential flaws in his data. Whereas corpses had to be disposed of for obvious reasons of health and amenity, and therefore burials provided a pretty reliable index of deaths, christenings did not reliably count births. This was because Catholics and Puritans, in particular, were reluctant to have their offspring baptized into a faith which they opposed. Furthermore, from 1649, when Charles I was executed, until 1660, when his son was restored to the Throne, the government of England was dominated by the Puritans, so many people were more confident to flout Anglican authority. Graunt was therefore obliged to make some corrections to his figures. Then, in attempting to make comparisons of births, deaths and diseases between London and the country, he had to deal with population disparities and calculate per capita rates in the absence of any census information. Another source of error, which was especially nettlesome during outbreaks of plague, was under-reporting of that disease—either because the affected households simply threw bodies into the streets, or because the “Searchers” were unwilling to inspect the bodies closely for fear of contracting the disease themselves. This meant, as Graunt recognized, that plague deaths were under-reported and the counts attributed to other causes were inflated.

Not content with simply aggregating and analyzing his data, Graunt drew up a synoptic list of 106 points in what he called his “Index”, several of which were recommendations for social and health policy.

He asserted, for example, that it would be “better to maintain all Beggars at the publick charge, though earning nothing, then to let them beg about the streets; and that employing them without discretion, may do more harm, than good”. He also found that “not one in two thousand are murthered in London”—a statistical finding which could be considered the birth of serious criminology. Even more importantly, he found that “the Rickets is a new disease, both as to name, and thing”. That diagnosis, he realized, did not appear at all in the Bills until 1634 and even then there were only 14 cases in that year; but by 1658 there were 476 cases. He seriously considered the possibility that previously it had been misdiagnosed but used his data to disprove that hypothesis. This is a remarkable reflection of the approach of William Harvey who had also used numbers to falsify arguments against his concept of the circulation of the blood.

Three years later, at the end of 1665, Graunt published, London’s dreadful visitation, or, A collection of all the Bills of Mortality for this present year, in which he applied the same analytical techniques to the demographic consequences of the “Great Plague of London”. Even today it is amazing and chilling reading: week by week, parish by parish, it documents the relentless surge of that awful disease from its first real appearance in May when 28 cases were recorded. Thereafter, the fatalities increased horrifyingly: about 340 in June; 4400 in July; 13,000 in August; 32,300 in September; 13,300 in October; 4,100 in November and 1,060 in December—a recorded total for that year of 68,600 deaths. And remember: in his earlier book, Graunt had decided that plague was, in such circumstances, seriously under-reported.

Its effects can be put into perspective by this contrast. For example, in the week from 29 August to 5 September, the Bills of Mortality reported 6,988 deaths from plague out of 8,252 burials recorded in the London parishes for that week, and in those 7 days a mere 167 christenings were recorded. Altogether, there were 9,967 christenings in that year and 97,306 burials—an almost 10-fold difference compared with the more usual disparity of less than two-fold and, according to Graunt’s estimates, those burials represented more than 22% of the population of London.

This catastrophic effect on the population of the capital could hardly be replenished by the usual birthrate because even in the first part of 1665 the christenings had been only 57% of the number of burials. In his earlier book, though, Graunt had found that there was substantial nett loss of population from the country to London. The result was that by 1675 the population of the capital was back to pre-plague levels.

In 1663, between the publication of Graunt’s extraordinary books, he had been elected as a Fellow of the Royal Society of London, though this seems not to have been an entirely straightforward matter. By profession, this genius was a haberdasher, whereas, according to the first history of the Royal Society, its membership was comprised principally of “gentlemen, free, and unconfin’d”. That self-congratulatory but diplomatic history which Thomas Sprat published in 1667, only 6 years after King Charles II had joined the society, says of Graunt’s election, “it is the recommendation which the King himself was pleased to make” adding that “his Majesty gave this particular charge to His Society, that if they found any more such Tradesmen, they should be sure to admit them all, without any more ado”. Those last words suggest to me that the “Gentlemen” of the Society required a little Royal “persuasion” which, the King seemed to be hinting, he did not wish to exert a second time.

Graunt was moderately active in the affairs of the Royal Society for a few years, but in the late 1660s he fell onto hard financial times, principally, I think, on account of his conversion to Catholicism. Certainly, this required him to relinquish his military commission as a Major and doubtless had adverse effects on his professional activities. He was eventually bankrupted and died in 1674.

His fading fame was not the only thing which then disappeared. So did some important records of the Worshipful Company of Parish Clerks. In his History of London, William Maitland noted, in 1739, that he had access to the Bills of Mortality only from 1664, stating that the Company “were of the opinion that the same was lent to Graunt…..but by some accident never returned”. He was neither the first nor the last scholar to forget to return borrowed materials to their owners. Nevertheless, the world of medicine remains forever in his debt. Graunt taught doctors that, for all of the importance of their focus on each individual patient, they must also shift their attention to understand what is happening to the whole population and to do so with the aid of the best possible statistics. The world is also in debt to King James, not only for the Bible which he commissioned, but for his insistence that the Parish Clerks should keep those good statistics. It is an unusual example of a beneficial combination of science and religion.

Have wheelchair, will travel…probably

Spending couple of weeks down the south coast of New South Wales, spotting dolphins and echidnas, has slowed down my blogging. Fortunately, regular contributor James Glover has once more come to the rescue with a guest post. This time his topic is wheelchairs and air-travel.

Perhaps you’ve heard of a recent court case in which a wheelchair user, Sheila King, took Jetstar to court (and lost) on the basis of the Disabilities Discrimination Act? If you are a wheelchair user and you book a flight on one of our airline carriers then a fairly obvious thing won’t happen. Unlike say a bus you won’t be able to board the aircraft in your chair and be strapped in for the journey. What actually happens is that when making the booking you tick a box (or tell the booker on the phone) that you are in a wheelchair. If there are seats available for wheelies when you get to the airport you will give up your chair and be made to use a specially designed “wheelchair” (its a chair, it has wheels) that is designed to be fit the narrow corridor of most planes which I am sure you are aware of – their narrowness, for you, only apparent when the person ahead of you is blocking the aisle loading 3 pieces of carry on luggage into the overhead lockers while chatting to their new friends in the seat they are meant to occupy. We all suffer this situation. These “wheelchairs” are not designed to be used without help, they are more like children’s toy carts and cannot be operated by the user as the wheels are very small and low down. For a wheelchair user to be taken out of their wheelchair in a public place can be quite discombobulating. Many wheelchair users develop a personal relationship with their chair – it is after all a place you spend many of your waking hours.

Digression. The very first time I was in a wheelchair outside the confines of a hospital ward (it was a hospital wheelchair but is the exact same model I now own, like I said it is personal) I was being pushed by none other than the proprietor of this very website! Without going into the details let’s just say it was a pretty dramatic event and we both learned a valuable lesson in wheelchair use and the wheelchair repair workshop at the hospital was kept busy. But I digress.

So here is the thing. According to Google about 1% of the population uses wheelchairs. And a Jetstar plane has about 200 seats so they expect to get about 2 wheelchair users on average per flight. So what is the problem with only allowing this same number on each flight, as some airlines do? Well the problem is that statistically wheelchair users don’t travel in pairs and sometimes there will be less than 2 users and sometimes there will be more. Just as if you toss 10 coins sometimes there will be fewer than 5 heads (the average or expected number) and sometimes there will be more. Only on average will there be 5. In fact it is a simple problem to work out the probability of there being, say, n wheelchair users, given the average of 1% on a 200 seat plane. This is called the Binomial Distribution. If you have access to Excel then the function Binomdist(n,200,1%) will tell you this probability. Before I give you some numbers I admit that the overall population average may not be the same as the average flying on planes. It may be less than 1% due to wheelchair users being put off flying. But maybe on some routes it is higher: but I am guessing the annual “snowbird” migration of retired people from the northern United States to Florida at the start of Winter would track above the 1% rate.

So here are the Binomial probability figures.

Count Probability
0 13%
1 27%
2 27%
3 18%
4 9%
5 4%
6+ 2%

Binomial Probabilities (N=200, p=1%)

For example, assuming a 1% chance of any given passenger a 200 hundred seat plane being in a wheelchair, the probability that there will be exactly 4 wheelchair passengers wanting seats is 9%. To work out the probability of a passenger being denied a seat on their preferred flight, we will assume that we’re dealing with an airline where more than two wheelchair passenges book on a flight, then at least on passenger will have to change their travel plans. From the table above, the chance of the flight only having 0, 1 or 2 wheelchair passengers totals 68%, so there’s a 32% chance that there will be at least one wheelchair passenger who cannot fly. For any one wheelchair passenger, there is a (n-1)/(n+1) chance of being bumped if n other wheelchair passengers book on the flight. Weighting that by the probably that there are n passengers and adding it up for all n>1 gives a probability of 27%. As a frequent flyer in a wheelchair, you can expect to miss out on a seat quite regularly! [Note: these calculations have been updated: the editor’s “corrections” were undone. Ed.]

I am quite fortunate now that I no longer need to travel in my wheelchair. But as I still use a walking stick I wait for everyone else to get off the plane. You sit there, looking behind you to see if everyone else has left. But there are always these strange people who seem to sit there at the back of the plane and wait for 10 minutes or more, after everyone has disembarked, before even moving. You wonder why the airline staff don’t just hurry them off? I assume they aren’t disabled because they are sitting at the back of the plane. If airlines really had a problem with the extra time that getting wheelies off the plane then they could make this up by just moving these people along.

When I first read about this case my initial response was that being disabled and traveling is a bit of challenge anyway and you just get on with it. But the more I thought about it I wondered if the airlines just took it for granted that wheelchair users would change their plans to fit in with the rules. I am glad Sheila King took the issue up!

Micromorts

Everyone knows hang-gliding is risky. How could throwing yourself off a mountain not be? But then again, driving across town is risky too. In both cases, the risks are in fact very low and assessing and comparing small risks is tricky.

Ronald A. Howard, the pioneer of the field of decision analysis (not the Happy Days star turned director) put it this way:

A problem we continually face in describing risks is how to discuss small probabilities. It appears that many people consider probabilities less than 1 in 100 to be “too small to worry about.” Yet many of life’s serious risks, and medical risks in particular, often fall into this range.

R. A. Howard (1989)

Howard’s solution was to come up with a better scale than percentages to measure small risks. Shopping for coffee you would not ask for 0.00025 tons  (unless you were naturally irritating), you would ask for 250 grams. In the same way, talking about a 1/125,000 or 0.000008 risk of death associated with a hang-gliding flight is rather awkward. With that in mind. Howard coined the term “microprobability” (μp) to refer to an event with a chance of 1 in 1 million and a 1 in 1 million chance of death he calls a “micromort” (μmt). We can now describe the risk of hang-gliding as 8 micromorts and you would have to drive around 3,000km in a car before accumulating a risk of 8μmt, which helps compare these two remote risks.

Before going too far with micromorts, it is worth getting a sense of just how small the probabilities involved really are. Howard observes that the chance of flipping a coin 20 times and getting 20 heads in a row is around 1μp and the chance of being dealt a royal flush in poker is about 1.5μp. In a post about visualising risk I wrote about “risk characterisation theatres” or, for more remote risks, a “risk characterisation stadium”. The lonely little spot in this stadium of 10,000 seats represents a risk of 100μp.

One enthusiastic user of the micromort for comparing remote risks is Professor David Spiegelhalter, a British statistician who holds the professorship of the “Public Understanding of Risk” at the University of Cambridge. He recently gave a public lecture on quantifying uncertainty at the London School of Economics*. The chart below provides a micromort comparison adapted from some of the mortality statistics appearing in Spiegelhalter’s lecture. They are UK figures and some would certainly vary from country to country.

Risk Ranking

Based on these figures, a car trip across town comes in at a mere 0.003μmt (or perhaps 3 “nanomorts”) and so is much less risk, if less fun, than a hang-gliding flight.

It is worth noting that assessing the risk of different modes of travel can be controversial. It is important to be very clear whether comparisons are being made based on risk per annum, risk per unit distance or risk per trip. These different approaches will result in very different figures. For example, for most people plane trips are relatively infrequent (which will make annual risks look better), but the distances travelled are much greater (so the per unit distance risk will look much better than the per trip risk).

Here are two final statistics to round out the context for the micromort unit of measurement: the average risk of premature death (i.e. dying of non-natural causes) in a single day for someone living in a developed nation is about 1μmt and the risk for a British soldier serving in Afghanistan for one day is about 33μmt.

*Thanks to Stephen from the SURF group for bringing this lecture to my attention.

Fertility Declines Don’t Reverse with Development

In this follow-up guest post on The Stubborn Mule, Mark Lauer takes a closer look at the relationship between national development and fertility rates.

STOP PRESS: Switzerland’s population would be decimated in just two generations if it weren’t for advances in their development.

At least, that’s what the modelling in a recent Nature paper projects.  The paper, widely reported in The New York Times, The Washington Post and The Economist, amongst others, was the subject of my recent Stubborn Mule guest post.  In that post, I shared an animated chart and some statistical arguments that cast doubt on the paper’s conclusion.  In this post, I’ll take a firmer stance: the conclusion is plain wrong.  But to understand why, we’ll have to delve a little deeper into their model.  Still, I’ll try to keep things as non-technical as possible.

First, let’s recap the evidence presented in the paper.  It comprised three parts: a snapshot chart (republished in most of the reportage), a trajectory chart, and the results of an econometric model.  As argued in my earlier post, the snapshot is misleading for several reasons, not least the distorted scales.  And the trajectory chart suffers from a serious statistical bias, also explained in my earlier post.  I’ll reproduce here my chart showing the same information without the bias.

FertilityNullTrajectories

That leaves the econometric model.  From reading the paper, where details of the model are sketchy, I had wrongly inferred that the model suffered the same statistical bias as the trajectory chart.  I have since looked at the supplementary information for the paper, and at the SAS code used to run the model.  From these, it is clear that a fixed HDI threshold of 0.86 is used to define when a country’s fertility should begin to increase.  So there’s no statistical bias.  However, I discovered far more serious problems.

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Is There a Baby Bounce?

In this first ever guest post on The Stubborn Mule, Mark Lauer takes a careful look at the relationship between national development and fertility rates.

Recently The Economist and the Washington Post reported a research paper in Nature on the relationship between development and fertility across a large number of countries.  The main conclusion of the paper is that, once countries get beyond a certain level of development, their fertility rates cease to fall and begin to rise again dramatically.  In this post I’ll show an animated view of the data that casts serious doubt on this conclusion, and explain where I believe the researchers went wrong.

But first, let’s review the data.  The World Bank publishes the World Development Indicators Online, which includes time series by country of the Total Fertility Rate (TFR).  This statistic is an estimate of the number of children each woman would be expected to have if she bore them according to current national age-specific fertility rates throughout her lifetime.  In 2005, Australia’s TFR was 1.77, while Niger’s was 7.67 and the Ukraine’s only 1.2.

The Human Development Index (HDI) is defined by the UN as a measure of development, and combines life expectancy, literacy, school enrolments and GDP.  Using these statistics, again from the World Bank database, the paper’s authors construct annual time series of HDI by country from 1975 until 2005.  For example, in 2005, Australia’s HDI was 0.966, the highest amongst all 143 countries in the data set.  Ukraine’s HDI was 0.786, while poor old Niger’s was just 0.3.

A figure from the paper was reproduced by The Economist; it shows two snapshots of the relationship between HDI and TFR, one from 1975 and one from 2005.  Both show the well-known fact that as development increases, fertility generally falls.  However, the 2005 picture appears to show that countries with an HDI above a certain threshold become more fertile again as they develop further.  A fitted curve on the chart suggests that TFR rises from 1.5 to 2.0 as HDI goes from 0.92 to 0.98.

Of course, this is only a snapshot.  If there really is a consistent positive influence of advanced development on fertility, then we ought to see it in the trajectories through time for individual countries. So to explore this, I’ve used a Mathematica notebook to generate an animated bubble chart.  The full source code is on GitHub, including a PDF version for anyone without Mathematica but still curious.  After downloading the data directly from Nature’s website, the program plots one bubble per country, with area proportional to the country’s current population.

Unlike with the figure in The Economist, here it is difficult to see any turn upwards in fertility rates at high development levels.  In fact, the entire shape of the figure looks different.  This is because the figure in The Economist uses axes that over-emphasise changes in the lower right corner.  It uses a logarithmic scale for TFR and a reflected logarithmic scale for HDI (actually the negative of the logarithm of 1.0 minus the HDI).  These rather strange choices aren’t mentioned in the paper, so you’ll have to look closely at their tick labels to notice this.

To help focus on the critical region, I’ve also zoomed in on the bottom right hand corner in the following version of the bubble chart.

One interesting feature of these charts is that one large Asian country, namely Russia, and a collection of smaller European countries, dart leftwards during the period 1989 to 1997.  The smaller countries are all eastern European ones, like Romania, Bulgaria and the Ukraine (within Mathematica you can hover over the bubbles to find this out, and even pause, forward or rewind the animation).  In the former Soviet Union and its satellites, the transition from communism to capitalism brought a crushing combination of higher mortality and lower fertility.  In Russia, this continues today.  One side effect of this is to create a cluster of low fertility countries near the threshold HDI of 0.86 in the 2005 snapshot.  This enhances the impression in the snapshot that fertility switches direction beyond this development level.

But the paper’s conclusion isn’t just based on these snapshots.  The authors fit a sophisticated econometric model to the time series of all 37 countries that reached an HDI of 0.85, a model that is even supposed to account for time fixed-effects (changes in TFR due only to the passage of time).  They find that the threshold at which fertility reverses is 0.86, and that beyond this

an HDI increase of 0.05 results in an increase of the TFR by 0.204.

This means that countries which develop from an HDI of 0.92 to 0.98 should see an increase in TFR of about 0.25.  This is only about half as steep as the curve in their snapshot figure, but is still a significant rate of increase.

However, even this rate is rather surprising.  Amongst all 37 countries, only two exhibit such a steep rise in fertility relative to development between the year they first reach an HDI of 0.86 and 2005, and one of these only barely.  The latter country is the United States, which manages to raise TFR by 0.211 per 0.05 increase in HDI.  The other is the Czech Republic, which only reaches an HDI of 0.86 in 2001, and so only covers four years.  Here is a plot of the trajectories of all countries that reached an HDI of 0.86, beginning in the first year they did this.  Most of them actually show decreases in TFR.

FertilityTrajectories

So how do the authors of the paper manage a statistically significant result (at the 0.1% level) that is so widely different from the data?  The answer could well lie in their choice of the reference year, the year in which they consider each country to have passed the threshold.  Rather than using a fixed threshold as I’ve done above, they express TFR

relative to the lowest TFR that was observed while a country’s HDI was within the window of 0.85–0.9.  The reference year is the first year in which this lowest TFR is observed.

In other words, their definition of when a country reaches the threshold depends on its path of TFR values.  In particular, they choose the year when TFR is at its lowest.

Does this choice statistically bias the subsequent trajectories of TFR upwards?  I leave this question as a simple statistical exercise for the reader, but I will mention that the window of 0.85–0.9 is wider than it looks.  Amongst countries that reached an HDI of 0.9, the average time taken to pass through that window is almost 15 years, while the entire data set only covers 30 years.

Finally I’d like to thank Sean for offering this space for my meandering thoughts.  I hope you enjoy the charts.  And remember, don’t believe everything you see in The Economist.

UPDATE:

To show that the statistical bias identified above is substantial, I’ve programmed a quick simulation to measure it.  The simulation makes some assumptions about distributions, and estimates parameters from the original data.  As such it gives only a rough indication of the size of the bias – there are many alternative possibilities, which would lead to larger or smaller biases, especially within a more complex econometric estimation.

In the simulation, each of the advanced countries begins exactly where it was in the year that it first reached an HDI of 0.85.  Thereafter, a trajectory is randomly generated for each country, with zero mean for changes in fertility.  That is, in the simulation, fertility does not increase on average at all¹.  As in the paper, a threshold is found for each country based on the year with lowest TFR within the HDI window.  All shifts in TFR thereafter are used to measure the impact of HDI on TFR (which is actually non-existent).

Here is a sample of the trajectories so generated, along with the fitted response from the paper.

FertilitySimulationExample

The resulting simulations find, on average, that a 0.06 increase in HDI leads to an increase of about 0.075 in TFR, despite that fact that there is no connection whatsoever.  The range of results is quite broad, with an increase of 0.12 in TFR also being a likely outcome.  This is half of the value found in the paper; in other words, simulations of a simplified case where HDI does not influence TFR at all, can easily generate half of the paper’s result.

Of course, if the result is not due to statistical bias, then the authors can easily prove this.  They need only rerun their analysis using a fixed HDI threshold, rather than one that depends on the path of TFR.  Until they do, their conclusion will remain dubious.

¹ For the technically minded, the HDI follows a random walk with drift and volatility matching those of advanced countries, and the TFR follows an uncorrelated random walk with volatility matching the advanced countries, but with zero drift.  The full source code and results have been uploaded to the Github repository.

FURTHER UPDATE:

More details can be found in the follow-up post to this one, Fertility Declines Don’t Reverse with Development.

Swine Flu on Swivel

I have now uploaded the swine flu data to a Swivel data set. I will update this data set periodically and so the rankings in the chart below should stay reasonably up to date.
Cases per Million Population by Country
Data sources: Guardian Data Blog, CIA World Fact Book.

UPDATE: A number of people have told me that in a number of places, including Victoria and much of the US, testing for swine flu has ceased. This means that the “lab confirmed” swine flu count will become increasingly meaningless over time, so I have decided to stop updating this data.

Swine Flu League Table

The Guardian have been publishing swine flu data on their Data Blog. They are sourcing their data from by the World Health Organisation, the US Centers for Disease Control, country health agencies and press reports, which makes their data the most up to date I have found. One thing missing from their data is a sense of scale for each country. Of course populous countries like the US have had a high number of infections, but this also means that  hidden in the smaller numbers are some fairly significant infection rates in countries with smaller national populations.

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