Author Archives: Mark Lauer

About Mark Lauer

Risk consultant, would-be novelist and stay-at-home father of twins, upon whom he has so far resisted conducting evil, yet fascinating, experiments

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.

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.


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.

Continue reading

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.


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.


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.


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.


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