Criticism of the Body Mass Index

The body mass index (BMI) is a metric proposed by Adolphe Quetelet during the mid-nineteenth century to assess human body shape. It is defined as a person's body mass in kilograms divided by the square of their height in metres. 

Since the 1800s, the BMI has seen continued use by health professionals as a quick assessment of one's health. Today, the BMI is still used to judge if a person is obese (BMI > 30 kg/m²), overweight (25 < BMI < 30), normal weight (18.5 < BMI < 25), or underweight (BMI < 18.5 kg/m²). But is the BMI a reasonable metric? Does it make sense for one's body mass to be proportional to the square of one's height? People of above average height, particularly men, often find that their BMI seems high despite being lean and fit. Shorter people, particularly women, similarly may find that their BMI seems low even with noticeable excess weight in the midsection. The BMI standards are most applicable for people who are close to average human height, but the standards lose their usefulness for everyone else. Considering that the average man and woman are already naturally about 8 cm above and below the average human height, respectively, it would seem that BMI immediately tends toward classifying men as overweight and women as underweight.

The BMI assumes this is true. The evidence says it isn't.

The simplest approach to estimating a power relation between mass and height would be to assume that the density of the human body is independent of size and that our bodies exhibit isometry. In simple terms, isometry means that height, breadth, and thickness all change in equal proportion. If a person gets 10% taller, they also get 10% wider across the shoulders and 10% thicker from front to back. The assumption of density's independence of size means that the average density of a tall person is the same for a short person. If both assumptions are true, then we would expect mass to be proportional to the cube of height. 

What we'd expect if all humans were scale copies of each other.

A lesser known index, known as Rohrer's Index or the Ponderal Index, actually makes the assumptions I've just mentioned above. Rohrer's Index is defined as mass divided by height cubed. For reference, the equivalent standards for underweight, overweight, and obese using Rohrer's Index instead of BMI are < 11.0 kg/m³, 14.9 < RI < 17.8, and > 17.8 kg/m³, respectively.

While people generally get wider and thicker as they grow taller, humans don't exhibit true isometry. There is a tendency for taller people to be narrower relative to their height than their shorter counterparts. Babies, with their comparatively large heads and short legs, are also far from being miniature adults. 

Their big heads and tiny legs are adorable, but they also force us to abandon the isometry hypothesis.

If you look at actual data, you find that neither index is very good, though Rohrer's Index seems to work better, especially in pediatrics. We'd like to have a working power law relation between mass and height because it would make the whole mass-to-height type index applicable to more people and probably a more useful health metric as a result. To find a power law relation, we can simply plot mass as a function of height and use the power curve fit option in Excel (or see if log(m) ÷ log(h) is approximately constant)

We're looking for a value of 'p' that has better correlation with data.

From Vital and Health Statistics (Series 11, No. 252), which contains anthropometric data from American adults and children collected between 2007 and 2010, we find that p = 2.48 with R² = 0.98 for males and p = 2.50 with R² = 0.97 for females. According to the data used to create the CDC Growth Charts (published in 2000), p = 2.52 and 2.54 for males and females, respectively (R² = 0.99 and 0.98). Data from Britain's 2003 Health Survey suggests that p = 2.49 and 2.69 fit best for males and females, respectively (R² = 0.97 for both sexes). 

Mass vs. Height of Males (2000 CDC Growth Chart Data)

Even the data which appears in Quetelet's Treatise on Man and the Development of His Faculties indicates 2 < p < 3. Using the data tables he gives for height and weight at different ages, p = 2.35 and 2.40 for males and females, respectively (R² = 0.97 for both sexes). Quetelet presents a separate table showing average weight for a given height. Based on that table, p = 2.21 for males (R² = 0.98) and p = 2.27 for females (R² = 0.96). It appears that the exponent is usually about the same in males as in females, so if we simply take the average of all the values of p we get 2.45. Therefore, the mass index formula we should be using is: 

With this formula, the standards become:

• Underweight (MI < 14.6)
• Normal weight (14.6 < MI < 19.8)
• Overweight (19.8 < MI < 23.7)
• Obese (MI > 23.7)

With a correct power law, we eliminate the issue of classifying people as overweight or underweight simply because they are significantly taller or shorter than the average human. While the mass index derived from statistical analysis is an improvement over the BMI, it still doesn't overcome the other serious flaws. First, women naturally have a higher body fat percentage than men. Basically, female hormones cause women to grow breasts full of fatty tissue while male hormones cause men to grow larger muscles. The result is, on average, fat accounts for more of a woman's body weight than a man's by about 6 percentage points.

Second, the index doesn't distinguish between lean mass and body fat. Muscle tissue is about 17% denser than fatty tissue, so athletes and gym rats can be lean and fit but still have a total mass that suggests they are overweight according to the BMI standards.

Finally, lean mass accounts for most of a person's mass (except perhaps in a few extreme cases). Any mass index therefore should only correlate well with body fat percentage among the morbidly obese, but for the majority of people mass index and adiposity will correlate poorly.

Health professionals are aware of issues with using a mass index. Romero-Corral et al. (2008) published a study of over 13,000 Americans in the International Journal of Obesity  to assess the accuracy of BMI as a diagnostic tool. Their discussion of the usefulness of BMI evolves around the inability to distinguish between lean mass and fat. They found that BMI > 30 classified 21% of the men and 31% of the women as obese. However, 50% of the men and 62% of the women were actually obese (defined as having greater than 25% or 35% body fat for men or women, respectively).

The accuracy of diagnostic tests is often assessed by positive and negative predictive values. Positive predictive value (PPV) is the probability that a positive test result indicates a correct diagnosis. Negative predictive value (NPV) is the probabilty that a negative result is correct. In Romero-Corral et al, PPV indicates the probability that BMI > 30 correctly identifies a person as obese and NPV indicates the probability that it correctly identifies a person as not obese. They found that the 30 kg/m² benchmark for obesity has a PPV of 87% for men and 99% for women. The NPV was 60% for men and 54% for women. What this all amounts to is a few false positives and a lot of false negatives; 50% of Americans are misidentified by the BMI-defined threshold for obesity.

Obesity diagnoses of the American population using BMI > 30 kg/m².
For every true positive obesity diagnosis using this test, there are 1.31 false negatives.

Despite well-known flaws and such poor accuracy as a diagnostic tool for obesity, the BMI remains commonplace. There are plenty of online BMI calculators. BMI is often included as part of a fitness assessment (I recall that it was calculated during fitness testing in high school gym class). All I can say is that I hope it goes away and that, however BMI classifies you, there's a pretty good chance that it's wrong.

Three Basic Principles

There are three basic principles every civil engineer must understand.

F = ma

The simple explanation:

Force is equal to mass times acceleration. Newton said so.

The real explanation:

The equation is just a mathematical expression of Newton's Second Law. It applies to everything in civil engineering. Beams, columns, cranes, entire bridges, flowing water, you name it. All of the forces acting on a non-accelerating system must sum to zero. If a = 0 then F = 0. That's where we get the equations of statics, which are so important that every engineer (not just the CivEs) learns them in their 1st year engineering mechanics class. Because if you don't satisfy statics, the structure will do it for you:

This crane had to tip over in order to satisfy the equations of statics. 

You can't push a rope

 The simple explanation:

If you try, you get this:

Ropes have a habit of offering no resistance to compression.

 The real explanation:

Obviously this doesn't mean that you can't push a rope in the sense that a coil of rope can't be pushed up a hill. But a rope can't be utilized to resist forces in compression, because ropes are made of a bunch of very slender fibers all woven together. To resist tension, the only important geometric parameter is the total cross-sectional area. So when it comes to tension, it doesn't matter if you have one 100 mm² rod or twenty 5 mm² wires. Compression is different. Cross-sectional area is only important when the thing you're squashing is short compared to its thickness. Otherwise, you get buckling and the important geometric parameter becomes the second moment of area (civil engineers call it "moment of inertia"). Providing the same total cross-sectional area with many smaller parts gives you a much smaller moment of inertia. For example, the 100 mm² rod has a moment of inertia of about 796 mm⁴, but the twenty 5 mm² wires have a combined moment of inertia of only about 40 mm⁴. That's a 95% reduction in buckling capacity. Ropes are usually pretty slender to begin with, and then when you go and divide the cross-section up into dozens of fibers, you end up with negligible capacity to resist compression.

Water flows downhill

The simple explanation:

Because of gravity.

It's how rivers work. All of them.

The real explanation:

Again, this doesn't mean water can't be forced to go uphill. How else would everyone get hot showers and flushing toilets in high-rise buildings? But to do so, you have to apply a force large enough to beat gravity. Otherwise, water flows downhill. It's the reason rivers flow from tributaries in the mountains down to lakes or seas below. It's the reason sewage lines are downward sloping wherever possible. Pumping waste uphill costs money, and if there's a problem, the local people are quick to complain about it. It's also the reason flat roofs aren't supposed to actually be flat. They're supposed to be gently sloped towards drains so that the roof doesn't become a swimming pool. If your yard slopes toward your house, you're much more likely to have a wet basement after a heavy rain. Water's everywhere. If not properly controlled, it causes a significant amount of property damage. Putting "water flows downhill" into practice is the best way to manage water and prevent damage. 

This roof slopes to the corner instead of the drain. Stagnant water provides a mosquito breeding ground and can eventually leak through flaws in the roof as it ages.

About Architects, Engineers, and Contractors

An architect is said to be a person who knows very little about a great deal and keeps knowing less and less about more and more until he knows practically nothing about everything.

On the other hand, an engineer is a person who knows a great deal about very little and goes along knowing more and more about less and less until finally he knows practically everything about nothing.

A contractor starts out knowing practically everything about everything, but ends up knowing nothing about anything due to his association with architects and engineers. 

Yet they still somehow manage to create things like these when you put them together:

Kinemax building at Futuroscope, Poitiers, France.

Habitat 67, Montreal, Quebec, Canada.

Civil Justice Center, Manchester, United Kingdom.

Sydney Opera House, Sydney, Australia.

Walt Disney Concert Hall, Los Angeles, California, USA.

Guggenheim Museum, Bilbao, Spain.

Dancing Building, Prague, Czech Republic.

Stata Center, Cambridge, Massachusetts, USA.

Darcons Headquarters, Delicias City, Mexico.

Worldwide Beer Production and Consumption

Which countries produce the most beer? Which countries drink the most? Which nation is the drunkest? If you've ever wondered about these questions, this post is for you.

Beer is the world's third-most popular beverage, after water and tea.

I obtained most of my data from the Kirin Institute of Food and Lifestyle, which has been collecting and publishing beer production and consumption data since 1975. When I checked a Wikipedia article on beer consumption, Kirin was listed as the source, but the article on Wikipedia contains several errors, including one which moved Canada up 18 places in the worldwide per capita beer consumption rankings.

This ancient Egyptian model of a brewery is over 4,000 years old.
Chemical evidence reveals that beer existed at least 7,000 years ago in ancient Iran.

Let's look at beer production first. According to Kirin, the top ten beer producing countries in 2010 by total volume were:

1. China (44,252,936 m³)
2. USA (22,898,177 m³)
3. Brazil (12,769,662 m³)
4. Russia (10,240,000 m³)
5. Germany (9,568,300 m³)
6. Mexico (7,988,900 m³)
7. Japan (5,850,450 m³)
8. United Kingdom (4,499,700 m³)
9. Poland (3,600,000 m³)
10. Spain (3,337,500 m³)

Canada was 18th with 1,964,700 m³. One cubic metre is equal to 1,000 litres, which is about 2,933 bottles (341 mL bottles). In other words, Canada produced over 5.7 billion bottles worth of beer in 2010, about 8.6% of what the Americans produced and only about 4.4% of the amount of beer produced by the Chinese. I created a pie chart to visualize how much each of the top beer-producing countries contribute to the world's beer supply. As you can see, the top five beer-producing countries account for more than half of the world's beer.

Half of the top 25 beer-producing countries contribute only about 1% each to the world's total beer production.

Not surprisingly, it turns out that the countries producing the most beer tend to also be consuming the most beer. According to Kirin, the ten largest consumers of beer in 2010 were:

1. China (44,683,000 m³)
2. USA (24,138,000 m³)
3. Brazil (12,170,000 m³)
4. Russia (9,389,000 m³)
5. Germany (8,787,000 m³)
6. Mexico (6,419,000 m³)
7. Japan (5,813,000 m³)
8. United Kingdom (4,587,000 m³)
9. Spain (3,251,000 m³)
10. Poland (3,215,000 m³)

As you can see, all of the top ten beer-producing countries were also the top ten beer-consuming countries, the only difference being that Poland and Spain traded 9th and 10th places. Canada consumed 2,311,000 m³ of beer in 2010, which placed us at 14th in the world. Here's another pie chart, this time showing consumption.

More than half of the world's beer is being drunk in just five countries.

Overall, this chart is pretty similar to the previous one. Some interesting changes are that Netherlands and Belgium, the 14th and 20th biggest beer-producers, respectively, don't even crack the top 25 when it comes to beer consumption. Apparently the Dutch like to sell beer abroad much more than they like drinking the stuff. Similarly, Argentina, which didn't appear among the 25 biggest producers, is the 20th biggest consumer of beer worldwide. This gave me the idea of looking at relative national beer surplus or deficit, shown below:

Belgium produces more than twice the amount of beer that it consumes.

It looks like most countries we've looked at consume roughly the same amount of beer they produce. Seventeen of the 26 countries shown consume within 10% of what they produce. Netherlands and Belgium are interesting because they each consume only about half of the amount of beer that they produce, a much larger disparity than any of the other countries I have data for. The French had the largest relative beer deficit, consuming about 26% more beer than they produce. We Canadians also have an appreciable beer deficit, drinking 18% more beer than we produce. 

Now to answer the most important question of all: which country's beer consumption has their population most intoxicated? To answer this, we need to look at beer consumption per capita. China's total beer consumption might be about 19 times that of Canada's, but they've also got about 40 times more people to do the drinking. Based on the 2010 data from Kirin, the ten countries boasting the top beer-drinking peoples are:

1. Czech Republic (131.7 L/person/year)
2. Germany (106.8 L/person/year)
3. Austria (105.8 L/person/year)
4. Ireland (103.7 L/person/year)
5. Estonia (90.6 L/person/year)
6. Lithuania (85.7 L/person/year)
7. Poland (83.6 L/person/year)
8. Australia (83.4 L/person/year)
9. Venezuela (83.0 L/person/year)
10. Finland (82.7 L/person/year)

This chart shows the top 35 consumers of beer per capita:

Americans, and most Europeans, drink more beer than us.

You might be wondering what happened to China, the country drinking the biggest share of the world's beer. Because of their large population, they rank only 49th in terms of beer consumption per capita. Canada was only 23rd with 68.4 L/person/year (not 5th with 98.5 L/person/year as Wikipedia had first led me to believe). Czech Republic's 131.7 L/person/year is really quite impressive when you realize this works out to an average of a little more than a bottle of beer per day! In one year, the average Czech drinks 107 more Imperial pints of beer than the average Canadian. We can't even take pride in beating the Americans; they were 12th with 78.2 L/person/year. My guess is that Americans, having been brainwashed into believing that baseball is a noteworthy pastime and not wishing for others to see them as "un-American", have to drink a lot of beer in order to suffer through all the baseball games they watch. I mean, I'd drink more too if I thought I had to watch dozens of 3-hour games of baseball to prove my allegiance to my country.

Watching baseball's only slightly more entertaining than watching golf. 

Of course, not all beer has the same alcohol content. Americans like light beers with comparatively low alcohol content. Eastern Europeans generally tend to prefer stronger beers. So perhaps you're wondering if there are any stats on the actual quantity of alcohol consumed from beer-drinking? The answer is 'yes'. The World Health Organization's Global Health Observatory makes global data related to several health topics readily available to anyone. This includes the Global Information System on Alcohol and Health. They've been collecting their own per capita beer consumption data, but report it in terms of litres of pure alcohol per year, which is a better indicator of which countries are the most intoxicated from their beer-drinking. According to WHO, the ten countries most drunk on beer in 2010 were:

1. Czech Republic (6.79 litres of pure alcohol per person per year)
2. Austria (6.10 L/person/year)
3. Germany (6.01 L/person/year)
4. Lithuania (6.00 L/person/year)
5. Poland (5.90 L/person/year)
6. Ireland (5.73 L/person/year)
7. Serbia (5.01 L/person/year)
8. Spain (4.87 L/person/year)
9. Estonia (4.68 L/person/year)
10. Slovenia (4.59 L/person/year)

Here's yet another bar chart, this time plotted using the data from WHO.

On average, Czechs get more than 101 Calories/day from the alcohol in the beer they drink.

There are some apparent discrepancies between the Kirin and WHO data sets. For instance, Serbians aren't even in the top 35 beer-drinking peoples according to Kirin, but are 7th on WHO's list. Spain also shows up 8th on WHO's list but was only 22nd on Kirin's list. I doubt that the Spanish are drinking such strong beer that it would make up the difference. But for the most part, the two lists agree pretty well. The Czechs are still decisively ahead of everyone else, followed most closely by Austria and Germany. It appears that the American taste for light beer sends them down to 16th place with 4.28 L/person/year. Canadians on the other hand move up to 18th place with 4.20 L/person/year. 

To summarize:

• China produces and consumes nearly one quarter of the world's beer
• Canada isn't as big of a beer-drinking nation as some of us would like to believe
• Don't challenge a Czech to a beer-drinking contest