## Better weight management through science

## 22 August 2009

Warning: I am not an expert in this field. I have merely pieced this knowledge together by trying to read a bit about it. Please do mistrust everything here, and talk to scientists in the field if you can. (Do tell me of places where I am wrong).

## The problem of weight management

Most affluent people in the world have a problem with weight management. Obesity is associated with higher odds of encountering a wide variety of deteriorative illnesses. As an example, if you have diabetes in your family, two factors which can change your odds of transitioning to full blown diabetes are obesity and muscle mass. Genetics may load the gun, but what fires the trigger is partly in your hands.

Generally, people `slow down' as they grow older. In a quiet and insiduous way, this leads to a weight gain if old diet habits are held intact.

Perhaps more than half of affluent people are on a time-trend of increasing body weight. Most people who try to lose weight fail to make a significant dent on the problem.

In the West, a body mass index of 25 is considered the threshold of obesity. If you're at 24.9, you should not be satisfied: you're at the threshold of obesity. Other ethnic groups need different standards. In Singapore and in India, the authorities are pushing 23 as the threshold. There are significant health benefits of pushing further down below a target of 23.

## Conservation of energy

The energy you take in, minus the energy you use, is your energy imbalance, also termed `energy gap'. This gets converted to fat. The exchange rate is 7.8 kcal (also written as Cal or food calories) per gram of fat. In other words, when your body has piled up 7800 kcal of extra energy, this shows up as 1 kg of weight gain. Conversely, to lose 1 kg requires an energy deficit over a set of days that adds up to 7800 kcal.

## Measurement of the energy you take in

This is done by adding up the calorific value of everything you eat. E.g. the US FDA has a good database which can guide you on the calories per gram of the various things that you might eat.

As a thumb rule, the energy-intensive foods are fat, cereal and meat. You can go far with eating fresh vegetables (which average 0.2 kcal/g), fresh fruit (which averages 0.5 kcal/g), egg whites (which average 0.5 kcal/g) and milk with 2% fat (which is 0.5 kcal/g).

Everything else is troublesome. Surprisingly, lean meat (e.g. trout at 1.2 kcal/g) is less dense than cereals (which are 1.3 kcal/g for boiled rice and twice that for the typical commercial bread). Commercial food which does not report calories per gram is dangerous because their path to making it tasty involves laying on the butter or cream. Butter is 7.2 kcal/g. A tablespoon of butter is 102 kcal, which is the same as 566 grams of tomato.

It's a bit painful to log everything that you eat. I tried to do measurement and it was hard to track the weight and energy content of a lot of things that we eat every day. Matters are made worse by the fact that cooking (or the lack thereof) modifies the calorific value of food. The energy gained from eating cooked food is not the linear combination of the energy content of the raw materials. In short, it's hard to have more than a hazy idea of how much energy we're taking in.

## Measurement of the energy that you use

The key term here is `metabolic equivalent' or METs. The energy cost of sitting quietly works out to 1 kcal per kg per hour. (I always found it surprising, how there was a round number here). In other words, if a person weighing 100 kg sits tight for one hour, he burns 100 kcal. Sitting tight is termed a 1 MET activity.

Scientists have compiled tables which show the energy intensity of various activities [pdf]. From this table, we see that walking at 5 kph is 3.3 METs. Running at 8 kph is 8 METs.

As an example, a person weighing 70 kg who runs at 8 kph is burning 560 kcal per hour or 9.33 kcal per minute. Since the human body stores fat at 7.8 kcal/g, this corresponds to losing 1.2 grams per minute of a 8 kph run.

## Kcal per km for running, across a wide range of velocities

Can one roughly guess what the METs associated with running at a certain speed? Suppose you do 1 km at x1 kph vs. x2 kph where x2>x1. The time taken to do this will be 60/x1 vs. 60/x2 minutes. To hold the energy expended to run 1 km constant, the energy outgo per minute has to be x2/x1 times. In other words, if you switch from 5 kph to 10 kph which halves the minutes per km, and the METs double, then your energy per kilometre is unchanged.

Remarkably enough, the human machine is rather efficient, and so the cost of running at (say) 10 kph turns out to be much like 2x the cost of running at 5 kph. (This came as a bit of a surprise to me; I had expected efficiency to degrade as you go faster. I guess at these velocities, air resistance doesn't matter :-)).

So for all people, running works out to roughly

1 kcal per km per kg. That is, a 70 kg person who runs 1 km burns roughly 70 kcal at a wide range of velocities.For a person who is 60 kg, running this same 1 km burns roughly 60 kcal at a wide range of velocities. The exchange rate of 7.8 kcal per gram then implies that a person with weight 60 kg uses 7.7 grams of fuel (i.e. fat) to run a kilometre. So for a person with weight 60 kg, eating one snickers bar (of 57 grams or 271 kcal) is counteracted by running 4.5 km. I await the day when a television advertisement for a snickers bar features an attractive 60 kg woman who holds the familiar brown widget up to eye level and says:

``It powers me for a full 4.5 km run''.## Mpg of the human machine

The human machine is rather efficient at converting fat into locomotion. A litre of human fat is 918 grams. This yields 7160 kcal. A 70 kg person running at 8 kph is burning 560 kcal an hour, so a litre of human fat can power him for 12.8 hours or 102 km. In other words,

the human machine obtains an efficiency of 102 kpl or 242 mpg, when a person weighing 70 kg runs at 8 kph.## Losing weight by eating ice

Here is another amusing calculation. Suppose you eat a kilo of ice. Your body spends 116 kcal in heating this up to body temperature. This corresponds to a weight loss of 15 grams.

## Good data about energy outgo is essentially infeasible

It's a bit painful to log your energy outgo. I tried to do measurement and it was hard to track activities and their METs for a lot of things that we do every day. So most of us only have a hazy idea of how much energy we're using every day.

## Input and output are hard to measure but the energy gap is not

So in short, your body is a system where energy goes in every day, where energy is used every day, and it's hard to get a numerical fix on what goes in and what is used. However, while the input and the utilisation are hard to measure, it

isactually feasible to measure the energy gap.The logic is simple: the body stashes away the energy gap at an `exchange rate' of 7.8 kcal per gram. So if you watch your weight, the rate of change of weight can be converted into a measure of your energy gap.

The input and the usage of energy are both hard to measure, but the change in weight is the summary statistic through which you can infer the energy gap.

## Measurement of energy gap

So what we're after is a measure of your weight gain or loss, measured in the units of grams per day. E.g. if you know that you were losing 10 grams a day, then you know that your energy gap is 78 kcal a day. This is not bad, for if you're losing 10 grams a day, then this is weight loss of a kilo every 100 days and 3.65 kilos a year, which is fine for most people.

The trouble is: there is considerable random noise in the observed weight. E.g. you drink a small glass of water and you've just added 200 grams to your weight. So on a day to day basis, it's hard to interpret anything in your weight data.

There is a good solution: track a time-series of your weight, and fit a linear regression y = a + b t + e. Here the slope `b' measures your weight change per day, on average. Now this will be measured with imprecision. The 95% confidence interval of `b' will tell you how sure you are about what is going on.

This requires buying a weigh scale and writing down your weight (say) when you wake up each morning.

## A system for watching yourself

I have written an Sweave file which takes in data for your weight, and generates a pdf slideshow with some interesting results. An example data file is supplied for you to experiment with it: Sweave file and example data file.

How to use this: You need to have R, latex, beamer and Sweave installed. The R package `ggplot' is used for making graphs and it must also be installed. On Unix, you first fire up R and use the Sweave command to generate a .tex file:

`$ R`

> Sweave("sl_analysis.Rnw")

> q()

$ pdflatex sl_analysis.tex

The first step involves going into R, and feeding the file.Rnw to the Sweave() function. This does the statistical computations and writes a slideshow as sl_analysis.tex, which is fed to pdflatex to get a pdf slideshow.

Here is an example of the resulting pdf slideshow: example pdf slideshow.

I believe R, latex, beamer, ggplot are all able to run on Windows also, but I am unable to give you those instructions.

## What this software does

## Overall regression results

Runs a regression to give you an estimate of how many grams per day you are gaining/losing, with a t statistic:

## A graph with the data and OLS regression

## The distribution of the estimated energy gap (using bootstrap inference)

## A regression spline model to give you some time variation of the energy gap

We try to utilise the full data so as to have the full history, and also look at the latest six weeks separately so as to have a sense of what is going on right now.

## How to utilise this information

You don't know your energy input and you don't know your energy outgo. But by using this procedure, you can have a fair sense of your energy gap.

Fairly small energy gaps add up to dangerous weight gain. E.g. an apparently innocuous energy gap of adding one extra tablespoon of butter every day is an energy gap of 102 kcal/day which is weight gain of 4.77 kg per year.

If your weight is on track, then your target energy gap is zero. If you are overweight, then your target energy gap might be something like -78 kcal/day so as to lose 3.65 kg/year. In either event, you should setup this system, and keep watching so that in the latest 6 weeks, the 95% confidence interval for your estimated energy gap feels okay. If the energy gap is not satisfactory, then go back into designing your life and modify the input or the energy utilisation or both (in a utility maximising way). Keep tinkering with your lifestyle until this summary statistic shows the values that you desire.

As an example, suppose you are presently averaging 15 minutes a day of walking which is 3.3 METs and suppose your weight is 70 kg. This corresponds to burning 60 kcal/day. Now suppose you switch to running for 15 minutes a day (on average), which is an 8 METs activity. This gives you an additional outgo of 82 kcal per day. In similar fashion, you can think about what lifestyle changes you want to make, focusing on the implications (measured in kcal/day) of alternative

changesthat you might consider. The key point is: you don't need to know how much you are taking in and how much you are spending every day. But you can solve your problem by measuring yourenergy gap, and thinking about ways tochangethis energy gap (through changes to intake or outgo) that are the most comfortable (i.e. utility-maximising) for you.As we get older, we `slow down'. Holding diet habits constant, our energy consumption goes down. Using this monitoring mechanism helps you raise red flags when this starts happening. This monitoring mechanism will give you continuous feedback and nudge you towards modifying your diet and/or modifying your fitness regime so as to stay on track.

There is a grand debate about whether exercise or diet matter to weight management. On one hand, it seems that exercise is pointless because a 30 minute jog is 300 calories which is just 3 tablespoons of butter. The useful way to think about this question is to focus on the energy gap. For most people, an energy gap of -20 grams a day or -150 kcal a day is a good achievement. Out of this, a 300 kcal exercise ritual can be an important component. Of course, you have to be careful to not reward yourself for a 30 minute run by eating a rich dessert.

## Is the weight time-series a random walk?

In the model described above, the time-series of your weight reflects the cumulation of the energy gap each day. Your weight on date t+1 is your weight on date t with a random shock added in. In other words, your weight should be I(1). This implies many strange properties, which we don't see in the data:

- The uncertainty of your forecasted weight should go up sharply as we discuss forecasts deeper into the future.
- If a person has an energy gap of +78 kcal per day, then he is adding 10 grams per day, 3.65 kg per year, or 109.5 kg over 30 years. But we seldom see such dramatic weight changes take place.
- It seems very unlikely that the food you take in and the energy you utilise will be exactly equal. Hence, nobody should be at an equilibrium weight. The time-series of human weight should always be bouncing around as a random walk.
But this is not what we see in reality. What is going on?

Suppose you want to lose 3.65 kg and your energy gap is right now at 0. Then it seems that you have to setup an energy gap of -78 kcal per day, hang tight for a year, and you're done. At the end, you can go back to your old ways, so as to get back to to your old state of a zero energy gap. Is that correct? (No, it isn't).

How do we resolve these questions?

Recall that the energy consumption is weight multiplied by the specific energy intensity of the activity undertaken. E.g. when a 60 kg person sits still for an hour, he burns 60 kcal. When the same person gains weight and becomes 70 kg, that same hour of sitting still involves burning 70 kcal.

So when you have an energy gap of +10% of your daily energy utilisation, once you have added 10% to your weight, that energy gap is gone (assuming diet stands still). Conversely, suppose you start at 80 kg, and establish an energy gap of -10% of your daily energy utilisation. At first, you lose weight. Once you have shed 8 kg or 10% of your weight, your energy gap has gone to zero and your weight loss stalls.

So one elegant way to lose 10% of your weight is to execute a lifestyle change which yields an energy gap of -10% of your daily energy expenditure. Initially, this will give you a rapid pace of weight loss. Gradually, as you get lighter, your energy outgo will go down, and your rate of weight loss will decline. There will be an exponential decay to the new long-run equilibrium. This strategy is elegant because you execute exactly one lifestyle change (worth 10% of your daily energy expenditure) and reach a new long-run equilibrium over time.

At the new equilibrium, you cannot switch back to your old lifestyle.

Sadly, operationalising this elegant model requires a measure of your daily energy expenditure at the outset. I am unable to find a convincing way to measure that.

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