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.