Many quantities tend to be larger for larger countries and smaller for smaller countries: gross domestic product, CO2 emissions, number of sheep, number of international airports, number of buses. Simply asking if the USA has a higher GDP or more sheep or more buses than New Zealand or Ireland or Kiribati is not typically a sensible question, because of the difference in size.

There are two different kinds of interesting question you can ask about these quantities: are they higher per capita and are they higher than you would expect for a country of that size. These questions aren't the same, because some quantities tend to increase faster or slower than population. For example, the largest countries have lower GDP per capita than average, which means that a country with twice or ten times the population doesn't quite have twice or ten times the GDP. That's just another way of saying that large countries have lower GDP per capita: their GDP is lower than you would expect from a straight-line relationship with population.

Kratochvíl and Havlíček are arguing this trend necessarily implies the per-capita scaling is wrong and misleading. They want to draw a graph of GDP against population and find a mathematical curve showing how GDP typically differs between countries with large and small populations. When they do this, they find that China, whose GDP per capita is slightly under the world average, excels. It has more than five times the GDP they would predict from its population alone. Relationships of this kind are a long-standing area of research in biology and human health, and have been extended to studying populations of cities. For local examples, scientists have argued about whether the size of kiwi eggs is explained by the relationship to body size among its relatives and Dr Shaun Hendy is among those who have written about the importance of community size to scientific research and development.

If there is a good reason to think that the mathematical curve relating population to GDP or CO2 emissions or number of sheep reflects some stable underlying cause, it will be useful to estimate that curve, and to use it for international comparisons and perhaps to decide policy. For GDP at the national level, I'm not convinced. The graph of GDP against population would look very different across the decades, so the scaling rules would have to keep changing. There are also problems with subnational units such as states: the 50 US states or the 36 states and territories of India have per-capita GDP that averages out to the whole country value just by maths, but that would no longer be true for the non-linear scaling. A whole country could, for example, be above the population curve for GDP even if all its states were below the curve.

More importantly, even if the relationship is intrinsically not proportional, so large countries intrinsically must have lower GDP per capita on average for some fundamental reason, they still really do have lower GDP per capita. Lower GDP per capita means there is less money available per person, and that's an important fact about the world no matter why it arises. The same is true for CO2 emissions and Covid deaths: it matters how much CO2 the average Kiwi is responsible for, and it matters how many of your friends died of Covid.

Per capita comparisons are important because we care about typical people. Non-linear scaling rules can also be important for research and analysis, and policy-makers should know about them, but they don't make per-capita comparisons intrinsically misleading or wrong.

Are we a-head or not? Use and misuse of per capita measures

Any attempt to simplify and describe reality will have problems. British statistician George Box said, “All models are wrong, but some are useful”, and this very much applies to per capita measures and to the regression-based approach proposed by Kratochvíl & Havlíček.

Kratochvíl & Havlíček propose that organisations like the World Health Organisation and Wikipedia move away from reporting per capita measures and related rankings and instead use a statistical method that describes the relationship between the measure of interest (e.g., GDP) and the population size. This method allows us to identify overall trends and then comment on how a country over or under-performs expectations for its size.

The motivation for normalising a different approach is based on issues of misleading conclusions from traditional per capita measures, specifically, when it is not appropriate to assume the measure of interest changes proportionally with the population size.

Per capita measures will always retain usefulness as a ‘quick and dirty’ way to make comparisons and communicate interesting facts (e.g., the below 5:1 ratio of sheep to New Zealanders), but I support the article’s overall message that more sophisticated methods should be considered by agencies that make decisions using these statistics, some of which are literally ‘life-or-death’. Creators and users of these statistics should be aware that communicating how to interpret them to the public will be somewhat more difficult than familiar per capita measures.

-- What is a regression-based approach? 

Regression is a method that allows us to describe the relationship between a measure we’re interested in, and other measure(s) that might explain it. If you’ve ever seen a straight line drawn on a scatter plot, you’ve seen simple linear regression. The age of children might be useful for explaining their height, with older children tending to be taller, but with some being tall for their age, or short, etc. We could use a straight line to explain this relationship and then actually make claims that a child was tall relative to what we’d expect for their age.

The method in Kratochvíl & Havlíček (2024) uses a specific flavour of regression called log-log regression. This means we change the scale of both of the measures we’re interested in.

Log-log regression is a well-established method when your data has a lot of smaller values and a few big values. For example, data on income for adult New Zealanders is often ‘logged’ when used in statistical analyses because most people earn less than $100,000, but of the people earning more, some will be in the many millions of dollars. Changing the scale helps us “squash” down a long tail of data points and allows us to use and interpret standard statistical tools.

-- What problem is this method trying to solve?

Two key issues with the per capita approach that the proposed regression approach is trying to deal with:

1. Comparability of per capita measures often relies on unmet assumptions — the worst culprit usually being the assumption that the measure of interest is proportional to the population.

2. Per capita ratios can change a lot from year to year for small countries without the changes actually being meaningful, while even fairly large changes in large countries may not be particularly obvious. There is a lot of variability in per capita values for small countries, and less variation amongst bigger countries.

Fitting a regression line on the log-scale data lets us make comments about how much a country over or under-performs on the relevant metric for its size, and rankings based on this tend to better align with our expectations of what the ranking should be telling us and how it performs.

-- Positives of per capita measures over regression-based methods include:

1. that they are relatively easy to understand in comparison to the regression-based methods, and

2. you don’t need data for all countries to compare two countries directly, and

3. minimal statistical literacy is needed to calculate and interpret them.

-- What both per capita and regression-based approaches miss

Neither of these approaches consider a few key contextual aspects that any keen person should ask themselves about:

1. Distributions within populations
2. Uncertainty
3. Data quality, measurement differences and/or missing data

Neither of these approaches capture important features of underlying distributions in these populations. For example, New Zealand’s gross domestic product (GDP) was 48,781.03 USD per person in 2021. But consider life in a country with the same GDP per capita as us, but that is very unequal; all the benefits of economic prosperity go to a small elite and everyone else lives below the poverty line no matter what. Or consider a country with the same GDP per capita but with an extremely flat structure where a strong economy benefits everyone equally?