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Month: May 2022

Preston’s Paradox

Preston’s Paradox

This article is an excerpt from my book Probably Overthinking It, to be published by the University of Chicago Press in early 2023. This book is intended for a general audience, so I explain some things that might be familiar to readers of this blog – and I leave out the Python code. After the book is published, I will post the Jupyter notebooks with all of the details!

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Suppose you are the ruler of a small country where the population is growing quickly. Your advisers warn you that unless growth slows down, the population will exceed the capacity of the farms and the peasants will starve.

The Royal Demographer informs you that the average family size is currently 3; that is, each woman in the kingdom bears three children, on average. He explains that the replacement level is close to 2, so if family sizes decrease by one, the population will stabilize at a sustainable size.

One of your advisors asks: “What if we make a new law that says every woman has to have fewer children than her mother?”

It sounds promising. As a benevolent despot, you are reluctant to restrict your subjects’ reproductive freedom, but it seems like such a policy could be effective at reducing family size with minimal imposition.

“Make it so,” you say.

Twenty-five years later, you summon the Royal Demographer to find out how things are going.

“Your Highness,” they say, “I have good news and bad news. The good news is that adherence to the new law has been perfect. Since it was put into effect, every woman in the kingdom has had fewer children than her mother.”

“That’s amazing,” you say. “What’s the bad news?”

“The bad news is that the average family size has increased from 3.0 to 3.3, so the population is growing faster than before, and we are running out of food.”

“How is that possible?” you ask. “If every woman has fewer children than her mother, family sizes have to get smaller, and population growth has to slow down.”

Actually, that’s not true.

In 1976, Samuel Preston, a demographer at the University of Washington, published a description of this apparent paradox: “A major intergenerational change at the individual level is required in order to maintain intergenerational stability at the aggregate level.”

To make sense of this, I’ll use the distribution of fertility in the United States (rather than your imaginary island). Every other year, the Census Bureau surveys a representative sample women in the United States and asks, among other things, how many children they have ever born. To measure completed family sizes, they select women aged 40-44 (of course, some women have children in their forties, so these estimates might be a little low).

I used the data from 1979 to estimate the distribution of family sizes at the time of Preston’s paper. Here’s what it looks like.

The average total fertility was close to 3. Starting from this distribution, what would happen if every woman had the same number of children as her mother? A woman with 1 child would have only one grandchild; a woman with 2 children would have 4 grandchildren; a woman with 3 children would have 9 grandchildren, and so on. In the next generation, there would be more big families and fewer small families.

Here’s what the distribution would look like in this “Same as mother” scenario.

The net effect is to shift the distribution to the right. If this continues, family sizes would increase quickly and population growth would explode.

So what happens in the “One child fewer” scenario? Here’s what the distribution looks like:

Bigger families are still overrepresented, but not as much as in the “Same as mother” scenario. The net effect is an increase is average fertility from 3 to 3.3.

As Preston explained, “under current [1976] patterns a woman would have to bear an average of almost two children fewer than … her mother merely to keep population fertility rates constant from generation to generation”. One child fewer was not enough!

As it turned out, the next generation in the U.S. had 2.3 fewer children than their mothers, on average, which caused a steep decline in average fertility:

Average fertility in the U.S. has been close to 2 since about 1990, although it might have increased in the last few years (keeping in mind that the women interviewed in 2018 had most of their children 10-20 years earlier).

Preston concludes, “Those who exhibit the most traditional behavior with respect to marriage and women’s roles will always be overrepresented as parents of the next generation, and a perpetual disaffiliation from their model by offspring is required in order to avert an increase in traditionalism for the population as a whole.”

So, if you have fewer children than your parents, don’t let anyone say you are selfish; you are doing your part to avert population explosion!

My thanks to Prof. Preston for comments on a previous version of this article.

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The Student-t model of Long-Tailed Distributions

The Student-t model of Long-Tailed Distributions

As I’ve mentioned, I’m working on a book called Probably Overthinking It, to be published in early 2023. It’s intended for a general audience, so I’m not trying to do research, but I might have found something novel while working on a chapter about power law distributions.

If you are familiar with the topic, you know that there are a bunch of distributions in natural and engineered systems that seem to follow a power law. But ever since people have claimed to find power law distributions, other people have said, “Not so fast”. There are two persistent problems with power law distributions:

  • They generally don’t fit the left part of the distribution, only the tail.
  • They don’t fit the whole tail; in the extreme, the data usually drop off faster than a real power law.

On the other hand, a lognormal distribution often fits the left part of these distributions, but it’s not always good model for the tail.

Well, what if there was another simple, well-known model that fits the entire distribution? I think there is, at least for some datasets: the Student-t distribution.

For example, here’s the distribution of city sizes in the U.S. from the 2020 Census. On the left is the survival function (complementary CDF), with city size on a log scale. On the right is the survival function again, with the y-axis also on a log scale.

The dashed line is the data; the shaded area is a 90% CI based on the data; and the gray line is the model, with parameters chosen to match the data. On the left, the Student-t model fits the data over the entire range; on the right, it also fits the tail over five orders of magnitude.

Here’s the distribution of earthquake magnitudes in California (the Richter scale is already logarithmic, so no need to transform).

Again, the model fits the survival function well over the entire range, and also fits the tail over almost six orders of magnitude.

As another example, here’s the distribution of magnitudes for solar flares (logarithms of integrated flux in J/m²):

Again, the model fits the data well in the body and the tail of the distribution. At the top of the left figure, we can see that there are not as many small-magnitude flares as the model expects, but that might be because we don’t detect all of them.

Finally, here are the relative changes in the daily closing price of the Dow Jones Industrial Average (thanks to Samuel H. Williamson), going all the way back to 1896.

The Student-t model is a remarkably good fit for the data.

Notably, the model is capable of matching tail curves with different shapes:

  • The tail of the city size distribution drops off with increasing curvature.
  • The tail of the earthquake distribution is initially curved and then straight.
  • The tail of the solar flare distribution is initially curved, then straight, then drops off again.
  • The tail of the changes in stock prices curves upward over a large part of the range.

You might think it would take a lot of parameters to track these different shapes, but the Student-t model has only three: location, scale, and tail thickness. It’s a bit of a pain to fit the model to data — I had to break out some optimization tools from SciPy. But at least I didn’t have to fit it by hand.

I’m not sure how much of this discussion will end up in the book, but if you would like to receive infrequent notifications about Probably Overthinking It (and possibly a discount), please sign up for this mailing list.

How Gaussian Is It?

How Gaussian Is It?

This article is an excerpt from the current draft of my book Probably Overthinking It, to be published by the University of Chicago Press in early 2023.

If you would like to receive infrequent notifications about the book (and possibly a discount), please sign up for this mailing list.

This book is intended for a general audience, so I explain some things that might be familiar to readers of this blog – and I leave out the Python code. After the book is published, I will post the Jupyter notebooks with all of the details!

How tall are you? How long are your arms? How far it is from the radiale landmark on your right elbow to the stylion landmark on your right wrist?

You might not know that last one, but the U.S. Army does. Or rather, they know the answer for the 6068 members of the armed forces they measured at the Natick Soldier Center (just a few miles from my house) as part of the Anthropometric Surveys of 2010-2011, abbreviated army-style as ANSUR-II.

In addition to the radiale-stylion length of each participant, the ANSUR dataset includes 93 other measurements “chosen as the most useful ones for meeting current and anticipated Army and [Marine Corps] needs.” The results were declassified in 2017 and are available to download from the Open Design Lab at Penn State.

Measurements like the ones in the ANSUR dataset tend to follow a Gaussian distribution. As an example, let’s look at the sitting height of the male participants, which is the “vertical distance between a sitting surface and the top of the head.” The following figure shows the distribution of these measurements as a dashed line and the Gaussian model as a shaded area.

The width of the shaded area shows the variability we would expect from a Gaussian distribution with this sample size. The distribution falls entirely within the shaded area, which indicates that the model is consistent with the data.

To quantify how well the model fits the data, I computed the maximum vertical distance between them; in this example, it is 0.26 percentile ranks, at the location indicated by the vertical dotted line. The deviation is barely visible.

Why should measurements like this follow a Gaussian distribution? The answer comes in three parts:

  • Physical characteristics like height depend on many factors, both genetic and environmental.
  • The contribution of these factors tends to be additive; that is, the measurement is the sum of many contributions.
  • In a randomly-chosen individual, the set of factors they have inherited or experienced is effectively random.

According to the Central Limit Theorem, the sum of a large number of random values follows a Gaussian distribution. Mathematically, the theorem is only true if the random values come from the same distribution and they are not correlated with each other.

Of course, genetic and environmental factors are more complicated than that. In reality, some contributions are bigger than others, so they don’t all come from the same distribution. And they are likely to be correlated with each other. And their effects are not purely additive; they can interact with each other in more complicated ways.

However, even when the requirements of the Central Limit Theorem are not met exactly, the combined effect of many factors will be approximately Gaussian as long as:

  • None of the contributions are much bigger than the others,
  • The correlations between them are not too strong,
  • The total effect is not too far from the sum of the parts.

Many natural systems satisfy these requirements, which is why so many distributions in the world are approximately Gaussian.

However, there are exceptions. In the ANSUR dataset, the measurement that is the worst match to the Gaussian model is the forearm length of the female participants, which is the distance I mentioned earlier between the radiale landmark on the right elbow and the stylion landmark on the right wrist.

The following figure shows the distribution of these measurements and a Gaussian model.

The maximum vertical distance between them is 4.2 percentile ranks, at the location indicated by the vertical dotted line; it looks like there are more measurements between 24 and 25 cm than we would expect in a Gaussian distribution.

There are two ways to think about this difference between the data and the model. One, which is widespread in the history of statistics and natural philosophy, is that the model represents some kind of ideal, and if the world fails to meet this ideal, the fault lies in the world, not the model.

In my opinion, this is nonsense. The world is complicated. Sometimes we can describe it with simple models, and it is often useful when we can. Sometimes, as in this case, a simple model fits the data surprisingly well. And when that happens, sometimes we find a reason the model works so well, which helps to explain why the world is as it is. But when the world deviates from the model, that’s a problem for the model, not a deficiency of the world.

Differences and Mixtures

I have cut the following section from the book. I still think it’s interesting, but it was in the way of more important things. Sometimes you have to kill your darlings.

So far I have analyzed measurements from male and female participants separately, and you might have wondered why. For some of these measurements, the distributions for men and women are similar, and if we combine them, the mixture is approximately Gaussian. But for some of them the distributions are substantially different; if we combine them, the result is not very Gaussian at all.

To show what that looks like, I computed the distance between the male and female distributions for each measurement and identified the distributions that are most similar and most different.

The measurement with the smallest difference between men and women is “buttock circumference”, which is “the horizontal circumference of the trunk at the level of the maximum protrusion of the right buttock”. The following figure shows the distribution of this measurement for men and women.

The two distributions are nearly identical, and both are well-modeled by a Gaussian distribution. As a result, if we combine measurements from men and women into a single distribution, the result is approximately Gaussian.

The measurement with the biggest difference between men and women is “neck circumference”, which is the circumference of the neck at the point of the thyroid cartilage. The following figure shows the distributions of this measurement for the male and female participants.

The difference is substantial. The average for women is 33 cm; for men it is 40 cm. The protrusion of the thyroid cartilage has been known since at least the 1600s as an “Adam’s apple”, named for the masculine archetype of the Genesis creation narrative. The origin of the term suggests that we are not the first to notice this difference.

There is some overlap between the distributions; that is, some women have thicker necks than some men. Nevertheless, if we choose a threshold between the two means, shown as a vertical line in the figure, we find fewer than 6% of women above the threshold, and fewer than 6% of men below it.

The following figure shows the distribution of neck size if we combine the male and female participants into a single sample.

The result is a distribution that deviates substantially from the Gaussian model. This example shows one of several reasons we find non-Gaussian distributions in nature: mixtures of populations with different means. That’s why Gaussian distributions are generally found within a species. If we combine measurements from different species, we should not expect Gaussian distributions.

Although I generally recommend CDFs as the best ways to visualize distributions, mixtures like this might be an exception. As an alternative, here is a KDE plot of the combined male and female measurements.

This view shows more clearly that the combined distribution is a mixture of distributions with different means; as a result, the mixture has two distinct peaks, known as modes.

In subsequent chapters we’ll see other distributions that deviate from the Gaussian model and develop models to explain where they come from.

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