Monday, June 17, 2013

What is your optimal weight? Maybe it is the one that minimizes your waist-to-weight ratio


There is a significant amount of empirical evidence suggesting that, for a given individual and under normal circumstances, the optimal weight is the one that maximizes the ratio below, where: L = lean body mass, and T = total mass.

L / T

L is difficult and often costly to measure. T can be measured easily, as one’s total weight.

Through some simple algebraic manipulations, you can see below that the ratio above can be rewritten in terms of one’s body fat mass (F).

L / T = (T – F) / T = 1 – F / T

Therefore, in order to maximize L / T, one should maximize 1 – F / T. This essentially means that one should minimize the second term, or the ratio below, which is one’s body fat mass (F) divided by one’s weight (T).

F / T

So, you may say, all I have to do is to minimize my body fat percentage. The problem with this is that body fat percentage is very difficult to measure with precision, and, perhaps more importantly, body fat percentage is associated with lean body mass (and also weight) in a nonlinear way.

In English, it becomes increasingly difficult to retain lean body mass as one's body fat percentage goes down. Mathematically, body fat percentage (F / T) is a nonlinear function of T, where this function has the shape of a J curve.

This is what complicates matters, making the issue somewhat counterintuitive. Six-pack abs may look good, but many people would have to sacrifice too much lean body mass for their own good to get there. Genetics definitely plays a role here, as well as other factors such as age.

Keep in mind that this (i.e., F / T) is a ratio, not an absolute measure. Given this, and to facilitate measurement, we can replace F with a variable that is highly correlated with it, and that captures one or more important dimensions particularly well. This new variable would be a proxy for F. One the most widely used proxies in this type of context is waist circumference. We’ll refer to it as W.

W may well be a very good proxy, because it is a measure that is particularly sensitive to visceral body fat mass, an important dimension of body fat mass. W likely captures variations in visceral body fat mass at the levels where this type of body fat accumulation seems to cause health problems.

Therefore, the ratio that most of us would probably want to minimize is the following, where W is one’s waist circumference, and T is one’s weight.

W / T = waist / weight


Based on the experience of HCE () users, variations in this ratio are likely to be small and require 4-decimals or more to be captured. If you want to avoid having so many decimals, you can multiply the ratio by 1000. This will have no effect on the use of the ratio to find your optimal weight; it is analogous to multiplying a ratio by 100 to express it as a percentage.

Also based on the experience of HCE users, there are fluctuations that make the ratio look like it is changing direction when it is not actually doing that. Many of these fluctuations may be due to measurement error.

If you are obese, as you lose weight through dieting, the waist / weight ratio should go down, because you will be losing more body fat mass than lean body mass, in proportion to your total body mass.

It would arguably be wise to stop losing weight when the waist / weight ratio starts going up, because at that point you will be losing more lean body mass than body fat mass, in proportion to your total body mass.

One’s lowest waist / weight ratio at a given point in time should vary depending on a number of factors, including: diet, exercise, general lifestyle, and age. This lowest ratio will also be dependent on one’s height and genetic makeup.

Mathematically, this lowest ratio is the ratio at which d(W / T) / dT = 0 and d(d(W / T) / dT) / dT > 0. That is, the first derivative of W / T with respect to T equals zero, and the second derivative is greater than zero.

The lowest waist / weight ratio is unique to each individual, and can go up and down over time (e.g., resistance exercise will push it down). Here I am talking about one's lowest waist / weight ratio at a given point in time, not one's waist / weight ratio at a given point in time.

This optimal waist / weight ratio theory is one of the most compatible with evidence regarding the lowest mortality body mass index (, ). Nevertheless, it is another ratio that gets a lot of attention in the health-related literature. I am talking about the waist / hip ratio (). In this literature, waist circumference is often used alone, not as part of a ratio.

35 comments:

Martin Ankerl said...

Do you use Inch/lbs or cm/kg? 32.2/160 is quite different from 83/73 :-)

Ned Kock said...

Hi Martin. It makes no difference, at least not for the application discussed in the post.

dearieme said...

I don't believe in any ideal ratio that claims to be independent of age, and sex, and maybe even race, and some measure of intrinsic body shape.

None of this matters for me since I am undoubtedly too fat. But I can remember when I was very slim - I remained tall, broad shouldered, big chested and wide hipped. I don't believe I've ever seen a proposed magic number that takes account of all those.

Anonymous said...

A moving average can be used to smooth out the measurement errors and provide a better indication of increase or decrease. Just choose a weight percentage like around 8%-16% for the influence of the latest measurement, call that n. The new moving average is n times today's measurement plus (1-n) times the old moving average.

Ned Kock said...

Hi dearieme. Based on your experience, what do you think is the problem with this particular ratio?

Ned Kock said...

Hi Anon. Yes, thanks. Probably combining a SMA with the unadjusted ratio would give a smooth and “rocky” view of the variations, which combined could be useful for other purposes – e.g., assess how something like intermittent LC would affect one’s weight.

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dearieme said...

Hello, Ned. Let me take it in sections.

First: ... empirical evidence suggesting that ... the optimal weight is the one that maximizes the ratio L / T, where: L = lean body mass, and T = total mass.

The empirical evidence is (I guess) epidemiological evidence; that is, I take it that there are no controlled experiments that have shown this? I admit that I am intrinsically sceptical about epidemiological arguments, an instinct supported by many conversations with a friend who is a retired epidemiologist. I have also asked myself about controlled experiments that I have done on different scales (from the lab bench to large industrial plant): often only controlled experiments got to the fundamentals right whereas mere observation was misleading.

Further, I have to say that my own experience of complex systems (but presumably a good deal less complex than human biology) has been that often one wants to find the value of an independent variable (e.g. an aspect ratio) that maximises a dependent variable (e.g. a life span). It's not all that common to achieve that without a trade-off, so that that the minimisation or maximisation of an independent variable is often not the best procedure. Put another way, the objective function is often not monotonic in the independent variable.

Before I extend my remarks, I'd better admit that there's one of your remarks that I don't understand: "Here I am talking about one's lowest waist / weight ratio at a give point in time, not one's waist / weight ratio at a give point in time." Can you elucidate, please?

Ned Kock said...

I hope to be able to illustrate this relationship in an upcoming post dearieme. In W / T = F1(T) and W = F2(T), F1 is not monotonic, but F2 likely is. This is what makes this relationship counterintuitive. After a certain point, W / T starts to go up, even as W keeps going down.

Ned Kock said...

I should say: “…F2 likely is, for the ranges that most people experience”. Clearly W will stop shrinking after a certain value of T is reached, but that point on the T axis will be achieved after the lowest W / T is passed. Again, I hope to be able to find a case where the data is public to illustrate this – I only help HCE users understand their data, I neither keep nor share that data.

ProudDaddy said...

My first comment on this post was lost in cyberspace, and my second ended up in the Jeckell and Hyde post. Darn tablet! Anyway, check Figure 2B in "Normal Weight Obesity...", Romero-Corral, et al., European Heart Journal 2010, for an astounding statement about waist size and health.

john said...

I think that optimal body composition--low body fat with high muscle--is protective against many "stresses," which can be seen with some rodent models. There are suggestive human correlations regarding LBM and mortality too.

Also, I'd guess that near-maximum attainable muscle (with low body fat and without strong anabolics) is also best for strength-to-weight and power-to-weight ratios...basically traits of athleticism besides endurance.

dearieme,

That sort of controlled experiment would be almost impossible.

dearieme said...

Thanks, Ned, I'll keep reading. I value your work highly.

tap said...

Interesting post. I only have my waist and weight measurements going back to 2007 but the ratio seems spot on. My current ratio of 0.1882 inches/pound (~12% body fat) is the lowest of all. My highest was 0.2044 inches/pound (~22% body fat) a few years ago. Body fat was based on the US Navy Circumference Method.
I wish I had data going back another decade when I was 30 pounds heavier.

Ned Kock said...

Do you have a link for that ProudDaddy?

Ned Kock said...

Hi John. Variations in L / T ratios in different individuals are very interesting, and the topic overall is fascinating. One thing is for sure, a very high ratio is not necessarily adaptive. We have genetic adaptations to suppress muscle growth – the myostatin gene.

Ned Kock said...

Thanks for sharing your experience tap.

ProudDaddy said...

Full text available at

http://eurheartj.oxfordjournals.org/content/31/6/737.full

The pdf option shows Figure 2 better, I think.

LeonRover said...

That EurHeart Study provided by ProudDaddy is a mine of information - particularly Figure 2.

I confess to having a penchant for Wt/Ht as an anthropomorphic measure of fatness. I had to play with data to get something from this - to wit, that Wt/Ht of 46% is the non-body-builder's healthy waist.

My own other preference for a ratio would be FatMass/LeanMass - and ideally, LeanBMI & FatBMI.

One can but hope.

David Isaak said...

LeonRover--

What do you mean by "Wt/Ht"?

Waist/Height?

Weight/Height?

Waist circumference at Time (t) / Hip circumference at Time (t)?

LeonRover said...

David

The nomenclature here is varied.

I wished, above, only to refer to W(ais)t/H(eigh)t - and only same in my comment.

So, your 1st choice is what was meant.

Time dependent waist and hip diameters?

Mmmm - when this happens in a female, one asks - is she pregnant?

Patrick Snook said...

LeonRover,

Very interesting to see that study, particularly the data in figure 2, as you point out. Thanks.

The table for waist measurements indicates a dramatic reduction in risk for the major "cardiometabolic" diseases associated with waist circumference < 79.9cm (~31.5inch) for men.

It's not clear to me how "waist circumference" was measured.

Is there a clinical standard practice for measuring waist circumference, I wonder? Searching on line brings up a few directions, all seemingly error prone (eg. taken at or one inch above the navel? or the narrowest circumference? at the top of the hip bone?).

And how did you extract that 46% ratio for waist/height from the data presented?

Patrick


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LeonRover said...

Patrick,

BMI is calculated by using 1/(H^2)in the denominator. I deliberately chose H from data in the top tertile, as this was the smallest H, where H=sqrt((Fat+Lean)/BMI). This gave me H=1.74 m.

In addition, 46% of my height represents my waist when at 14% fat in my active twenties.

The study quotes "at the high point of the iliac crest for the waist".

My reason for estimating Waist/Height is to have a more useful measure for the tall or the small!!

Ned Kock said...

Interesting data on the article, which I would recommend taking in with caution. The waist circumferences provided look too low for the equivalent BFs. “True” 31-inch waists are not very common among men. The relatively short follow-up period, of 8.8 years, led to too few deaths being recorded – e.g., 470, from a total sample of 6171. These are among the problems that I’ve noticed right away.

LeonRover said...

Romero-Corral, lead on the EurHeart aslo was lead on "Accuracy of BMI to Diagnose Obesity" -

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2877506/?report=reader - .

Figure 3 therein shows the variation in Fat% for Men and Women @ BMI of 25.

Figure 2.A appears to show a curved relation between Fat% & BMI for Women - curious.

Patrick Snook said...

Thanks, LeonRover, for the clarifications.

Patrick

David Isaak said...

The Roman-Corral study was a very good idea.

Too bad that the study used electrical impedance to measure the body fat percentage. It's probably better than just guessing (maybe), but what it's really good at is measuring your level of hydration.

LeonRover, I have to say that your 46% number seems about right to me--at least for my own body. When I'm at 13% BF (measured hydrostatically), my waist is around 46% of my height.



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Causes Of Poor Sleep said...

This is what complicates matters, making the issue somewhat counterintuitive. Six-pack abs may look good, but many people would have to sacrifice too much lean body mass for their own good to get there. Genetics definitely plays a role here, as well as other factors such as age.

Kevin said...

My own n=1 suggests waist/height is a better indication than waist/weight.

Just over 2 yrs ago I was obese with a lot of body fat (waist 127cm, weight 122.9kg, height 185cm). November 1 2013 (waist 100cm, weight 85.1kg, height 185cm)

5 6-monthly figures Nov 2011-Nov2013 each *1000 (eliminating decimals):
W/T: 1054, 1148, 1197, 1202, 1175
W/H: 676, 638, 562, 562, 541

So, W/T looks like my body composition was deteriorating(!) while W/H gives a more realistic idea.

Kevin said...

My own n=1 suggests waist/height is a better indication than waist/weight.

Just over 2 yrs ago I was obese with a lot of body fat (waist 127cm, weight 122.9kg, height 185cm). November 1 2013 (waist 100cm, weight 85.1kg, height 185cm)

5 6-monthly figures Nov 2011-Nov2013 each *1000 (eliminating decimals):
W/T: 1054, 1148, 1197, 1202, 1175
W/H: 676, 638, 562, 562, 541

So, W/T looks like my body composition was deteriorating(!) while W/H gives a more realistic idea.

Kevin said...

Apologies for the accidental duplication of comment.

Also, I note that I have 3 different waist measurements with an approximate 12cm difference between my trouser size and my measured waist (no 'pinching').

Today:
trousers: 85 cm (snug) & 91cm (loose)
measured waist: 99cm (if I 'pinch', pulling the tape tight, I can easily 'save' 4-5cm).