π = 3.14

room temperature = 72° F

an adult male needs 2,500 calories a day

an adult weighs 70 kilograms

...

We live with approximations, but all to often we don't worry about when we require better precision. 3.14 for π works for simple tasks around the home - after all - it offers better than one percent accuracy, but 2,500 calories for an adult male is very crude and can cause havoc with dieting. The 70 kilogram rule of thumb for adult weight is equally crude, but many medications are dosed based on that approximation.

Some nutrition experts argue that a calorie isn't a calorie confusing many who listen and often themselves. Strictly defined a calorie is a calorie. Physics works and energy is conserved any time it converts from one form to another, but you have to pay careful attention to the accounting. More than a century ago it was recognized our bodies weren't completely efficient metabolizing food and careful measurement, by the standards of the day, were made. It turned out that metabolism could liberate about four nutritional calories of energy from a gram of a carbohydrate or protein and about nine calories from a gram of fat. This roughly accounted for the energy that was lost to us in the process, but roughly held across food types and people. The problem of knowing how much usable energy was in a slice of bread or a bowl of ice cream was reduced to tallying up how many grams of carbohydrates, fats and proteins were present .^{1}

piece of cake...

It turns out there are several serious problems with this model. The energy assignments for each main component are crude averages. Additionally individual people absorb and metabolize food with different efficiencies and metabolic efficiency changes for a single person occur throughout life.^{2} When you look into the model a bit, you begin to wonder if it has any utility.

The trigger for this post was an article a reader sent. From the *Times of London*:

*Being short has always had its drawbacks. There’s the problem of the top shelf in the supermarket, the unflattering “short man” stereotype and the perils of running for the bus in heels. But things just got worse for Lilliputians — more precisely, they just got fatter.*

*According to a professor of numerical analysis at the University of Oxford, the formula used to calculate body mass index, is flawed in such a way that it makes tall people too fat and small people too thin.*

*...*

Body Mass Index again...

Like the assignment of caloric values to food types it has been around for more than a century and is know to be problematic. Usually you see caveats like "athletes tend to have artificially high BMIs", but you rarely see a discussion of what went into the model. *That* turns out to be interesting.

If you were to make a simple model that would address how people deviate from what might be an average weight you might start by noticing that people come in different heights, body shapes and weights. For adults height doesn't change very much, but body shape and weight does. A good place to start might be to notice that if you just scale up an object, its volume increases by the cube of any length you might measure on the original object. If you have a cube that is a foot on each side and scale it up by a factor of two, you now have a larger cube that is two feet on a side. The new cube can contain eight of the originals so its volume has increased from one to eight cubic feet.

Now assume you have a tall cylinder. The same rule holds. Scale it up by a factor of two and its height doubles as does its diameter. Its new volume is eight times that of the original. You can pick any measure of length to compare scaled objects so long as their basic shape doesn't change.

People have very complicated shapes, but for an individual their height doesn't change so it makes sense to use it as the measurement that is cubed to calculate changes in volume.

Given her height, Colleen is a good example. She's about 23% taller than the average American woman. If her shape was the same as someone who was of average height you might expect her to weight about 86% more. She doesn't and the fly in the ointment is her body shape is different. This isn't terribly unexpected, but how do you account for different body shapes?

A quick and very simple approximation is to remember a person's height is fixed, but the shape can change. If you divide their weight by the cube of their height the number will be larger if they are heavier and smaller if they are lighter. It is a nice trick to quantify something about your change in shape without going and carefully measuring the volume.

This is the rough idea behind BMI (also called the Quetelet Index) ... divide a person's weight by a proxy for their volume. The formula is

BMI = mass/height^{2}

where mass is measured in kilograms and height in meters. Converting to English units the relation is

BMI = 703 * weight/height^{2}

with weight measured in pounds and height in inches...

But wait! Didn't I just say you need to cube the length to get volume?

I visited the professor's site and noted his new BMI formula assumed the height was not squared or cubed, but raised to the 2.5 power and adjusted the final result by multiplying by a constant to make the new and old BMIs match for an adult of average height. for mass in kilograms and height in meters....

new BMI = 1.3 * mass/height^{2.5}

He didn't have any measurements to go from and noted, while he was far from an expert, he felt this was a better estimate. It was not published as a formal result and he didn't claim it was a piece of serious analysis.

Plotting the new and old BMIs over a range of heights for a weight of 175 pounds you notice that the new BMI is lower than the old one for tall adults and higher for short adults. I picked the weight because if I weighed that I would be average under the new model and overweight with the old one - the difference is nearly a one point change.

The professor did note new BMIs would change for short and tall people. Somehow a reporter saw the piece and believed it was a serious replacement for BMI. There must have been no little or no fact checking as it led to the dramatic headline and a bad model became widely reported. Too many people in the chain had no idea what BMI was...

It turns out analyses based on real data from large distributions of people suggest a good approximation to use in the model for adults is the square of the height. For young children the exponent is between 2.5 and 2.8.

The BMI model is ugly - a person with a technical background looks at it, sees the square, and thinks "area" until they look a bit more deeply and understand what the model really is.^{3}

It almost seems too messy to use, but if you are comparing one population to a similar population it can be a useful tool. More accurate tools exist, but they are expensive and time consuming. It would be too impractical to use the better estimates, but gathering BMIs is trivial- just weigh and measure people.^{ } Armed with those numbers you can look at serious health issues like changes in obesity levels in a country.

The connection with BMI and health problems is well established at very high BMIs, but less so in the overweight range and that presents a major problem for specialists and family physicians. While the measurement has considerable utility, rigid predictions may not apply in your case. A clear case where understanding what the measurement is and the domain where it is useful.

The same is true with nutritional calories (or ANUs). Going by the textbook may result in dietary guidance that doesn't work for you, but if you establish a baseline and recognizing the thing you might call a calorie really isn't a proper unit of energy, you can create guidance for yourself that can be very robust.

There are two messages.

° Models and approximations have their limits! It is best to understand them in detail before using them for anything important. In both of these cases the models and approximations are too simplistic and are very domain dependent. ** **

**It is very easy to get cut by Occam's razor:-)**

° Be very skeptical of science reporting in the popular press. From climate change to BMI it is best to look at what the experts say.

__________

^{1} This is a simplification of the approximation. Alcohols and a few other components appear in the model. Here is an earlier post on the subject.

^{2 }When Colleen and I were sorting out how to communicate with this our working term was to reliable the nutritional calorie as an *approximate nutritional unit* or ANU. We never came up with a clever name, so if there are any suggestions.

If you have been heavy and lose weight, your metabolism compensates by becoming more efficient. It doesn't seem fair, but it is common that someone who was one heavy must eat less food than if they had never been overweight if they want to avoid weight gain. This feature of humans prevents keeps many dieters from long term success and help to fuel the diet industry.

^{3} The model is far from perfect and your composition also changes as you gain or lose weight. Athletes have a much higher percentage of muscle, which tends to be dense. Different ethnic groups have different exponents. The exponent changes a bit with height, and so on...

^{4} Other methods of measuring things like body fat percentages are expensive and time consuming at the required levels of accuracy

__________

Recipe corner

It is Winter in the Northern Hemisphere and many of us have had, or will soon get snow. Assuming it is under 20° F and your snow is new-fallen and very clean, why not snow ice cream? I've made this several times - you have to adjust the recipe to the temperature and moisture in the snow, but kids can have fun with it. It is usually a bit slushy, but fun to make. I offer two recipes - both depend on the type of snow you're having so you may want to try half batches until you get the hang of it.

**Fresh Snow Ice Cream**

**Ingredients**

° 6 to 10 cups of new fallen snow (depending on moisture content)

° 1 can (14 oz) of sweetened condensed milk chilled to near freezing

° 1 - 2 tbl maple syrup and/or 1 tsp vanilla extract

**Technique**

° Put the snow in a large bowl that has been left outside to chill below freezing

° and the condensed milk and flavorings, mix with a big spoon

° serve in chilled bowls and eat immediately!

**Rich Snow Ice Cream**

**Ingredients**

° new fallen snow (depending on moisture content)

° 2 pasteurized eggs

° 2 cups of heavy cream or a half and half mixture of cream and whole milk or just whole milk - I've done all three, so it is between you and your cardiologist.

° 1-1/2 cups white sugar

° 3 tbl vanilla extract

**Technique**

° Beat the eggs in a large bowl, mix in cream, sugar and vanilla (it helps of the milk and sugar are pre-chilled to nearly freezing)

° add enough fresh snow for a good consistency and mix with a really big spoon

° serve in chilled bowls and eat immediately!

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