## Introduction

This article is an extension of an article that I posted a few days ago. The idea that I want to discuss is the one below:

We have data from the time evolution of two parameters, named A and B. Some people believe that there is a dependence relationship between A and B so that A has a significant effect on B, but we know that A suffered a trend change and B didn’t, so other people say that this fact suggests that it is highly unlikely that A has a significant effect on B.

For the purpose of explaining the failures in the previous idea, I will assume a very simple model of obesity: **we gain 3 g of body weight for every 100 g of sugar** consumed. Please, don’t bother to criticize this model: I will only use it as a tool to explain the errors in the idea explained above. What we will see is that, under the premise that sugar is determining our body weight gain, the correlation between sugar intake and body weight may be low. That is the main conclusion and the specific model used for the explanations is irrelevant.

## DAILY and CUMULATIVE TOTAL parameters

As said above, our assumption is that we fatten 3 g for every 100 g of sugar consumed. If our intake of sugar were 50 g, we would fatten half that amount: 1.5 g.

Given a specific sugar intake, the **DAILY** increase in body weight will be directly related to that **DAILY** sugar intake.

If I’ve consumed a certain amount of sugar **PER YEAR**, the** PER YEAR** increase in body weight would also be directly related to the** PER YEAR** sugar intake. For example, if one year’s sugar intake were 33 kg, our body weight would increase by 1 kg that year. Had I consumed only one third of those 33 kg, my body weight would have increased that year one third of 1 kg.

Let’s say that over the years our **DAILY** sugar intake has changed according to the blue curve in the graph below. In this scenario, our **DAILY** body weight gain would change as indicated by the red curve (calculated according to the hypothesis of that we fatten 3g per each 100 g of sugar consumed).

**DAILY** sugar intake and **DAILY** body weight gain are directly related variables. Their correlation, *i.e.* the mathematically-computed resemblance between them, is maximum.

If we compute (in blue) the **CUMULATIVE TOTAL** sugar intake since 1980 (*i.e.* for each year we compute the total amount of sugar consumed since 1980 until that year), versus (in red) the **CUMULATIVE TOTAL** body weight increase since 1980 until that year, we get this:

Again, as we saw with **DAILY** body weight increase and **DAILY** sugar intake, there is a direct relationship between** CUMULATIVE TOTAL** sugar intake and **CUMULATIVE TOTAL** body weight gain. This is also to be expected: under the premise that body weight gain is directly proportional to the sugar intake, if the **CUMULATIVE TOTAL** sugar intake over the past X years gets bigger you are expected to gain more weight, and if it gets smaller you are expected to gain less weight.

*In order to clarify what comes next, let’s assume we are filling a bucket by pouring into it daily glasses of water. Every day we pour into the bucket the contents of one glass of water. Let’s assume that the volume of water in the glass has been progressively rising, day after day, until reaching a peak at 110 ml, and then, for the last 15 days we have gradually reduced the volume of water in the glass until reaching 95 ml, is the cumulative total water in the bucket expected to be reduced at the end of those 15 days? Does anyone think that if I reduce the volume of water poured daily, the volume of water in the bucket has to decrease? When we confirm it doesn’t decrease, do we conclude that it is highly unlikely that the volume of water in the glass has an important effect on the volume of water in the bucket?*

Let’s get to the point, but first remember that the correlation between **DAILY** sugar intake and **DAILY** body weight gain is maximum, and remember, too, that the correlation between **CUMULATIVE TOTAL** sugar intake and **CUMULATIVE TOTAL** body weight gain is also maximum. Now, what kind of relationship exists between the **DAILY** sugar intake at a given year and the **CUMULATIVE TOTAL** body weight gain at that same year? A direct relationship is not to be expected: if the **DAILY** sugar intake is reduced, the **CUMULATIVE TOTAL** body weight is not expected to be reduced, in any case that year the body weight will** go up by a smaller amount**, but it will continue to increase, and the effect of a smaller **DAILY** sugar intake will also be small in relative terms, since we have only changed the data for one year that has to be accumulated to the rest of the years in the **CUMULATIVE TOTAL**: we have been accumulating body weight for years and the sugar intake during the last one of those years is expected to have a small effect in the **CUMULATIVE TOTAL**.

From another point of view, the **CUMULATIVE TOTAL **sugar intake —including the contribution from last year, but with a weight that depends on how many years are being considered—, is the variable which determines the** CUMULATIVE TOTAL** body weight gain to that year. It is nonsense to expect a direct relationship between **DAILY** sugar intake and **CUMULATIVE TOTAL** body weight gain. And, in fact, that relationship is not a direct one. Assuming that a man weighed 80 kg in 1980, the graph below shows his **CUMULATIVE TOTAL** body weight (red curve) versus his **DAILY** sugar intake (blue curve):

We know that the red curve is completely determined by the blue curve, but they don’t have a good correlation. Or in other words, we confirm that a low correlation tell us nothing about the existence of a dependence relationship between two variables. If for the same situation, we had chosen the two previous graphs, we would have concluded that the relationship between sugar intake and body weight gain is **undeniable**.

If we recall the idea at the beginning of the article, A (blue curve) has changed, and the effect on B (red curve) is apparently small, but **we know for a fact** that A completely determines B.

## What are we seeing?

In the last graph I presented above, the red curve is completely determined by the blue curve and the correlation between them is low. How is that possible? Because these two variables, while they are completely related, **they don’t have a direct relationship**. One variable is the **CUMULATIVE TOTAL** body weight gain, the effect of several years of fattening, while the other variable is just the **DAILY** sugar intake for one of those years. The fact that one of them goes on rising, albeit more slowly, when the other one decreases DOES NOT suggest that a cause-effect relationship between them is unlikely. The **CUMULATIVE TOTAL** body weight gain is not supposed to go down when the **DAILY** sugar intake is reduced: a “negative consumption of sugar” would be needed to produce such an effect on body weight. And, in any case, it would be the **level**,* i.e.* the fact that sugar intake is negative, what would produce a decrease in the ** CUMULATIVE TOTAL **body weight gain, not the **change**, *i.e.* the fact that sugar intake decreases. Even if it were possible to consume negative amounts of sugar, neither there would be a direct relationship between both variables nor a high correlation would be expected.

anyone who defends that sugar intake is a main cause of obesity and diabetes is proposing that there is a direct relationship between those variables

That’s a fallacy. The hypothesis that sugar is fattening means that the **DAILY** sugar intake affects the **DAILY** body weight gain. Nobody says that the **DAILY** sugar intake is directly related to the **CUMULATIVE TOTAL** body weight gain to that date: **it is stupid to expect that when your DAILY sugar intake goes down your CUMULATIVE TOTAL body weight has to go down**. This is not mathematics: it is just common sense.

Does the data presented by Guyenet in his graph suggest, as he says, that “*sugar is highly unlikely to be the primary cause of obesity*“? No, it does NOT suggest that. His graph is absolutely consistent with a direct effect of the **DAILY** sugar intake on the **DAILY** body weight gain: it can easily be seen that when the **DAILY** sugar intake was decreased, the **DAILY** body weight gain also decreased.

## Summary

The main Guyenet’s mistake, or the base of his attempt of deception, is that he assumes that the **CUMULATIVE TOTAL** body weight gain and the **DAILY** sugar intake are directly related, and that hypothesis is both nonsense and inconsistent with the hypothesis that he is trying to refute. No one proposes that the **DAILY** sugar intake has a direct relationship with the **CUMULATIVE TOTAL** body weight gain: this specific relationship is expected to be non-linear! Had he compared **DAILY** body weight gain with **DAILY** sugar intake, he would have found a direct relationship (consistent with the hypothesis that sugar is fattening). Had he compared **CUMULATIVE TOTAL** body weight gain with **CUMULATIVE TOTAL** sugar intake, he would have found a direct relationship (consistent with the hypothesis that sugar is fattening).

We have seen in this article, with help from a simple model of obesity, that although** CUMULATIVE TOTAL** body weight gain and **DAILY** sugar intake are not well correlated, that doesn’t suggest that there isn’t a causal relationship between both variables.

On the other hand, Guyenet confuses a lower intake of sugar with a negative consumption of sugar. Logic says that if sugar is fattening, reducing its consumption doesn’t make us slim down.

## Endnotes

- When in this article I use the term “direct” what I’m saying is that when one variable goes up the other variable also goes up and that when one variable goes down the other variable also goes down. Note that a direct relationship is not necessarily one of proportionality.
- the data used by Guyenet as support for his hypothesis is epidemiological. This is relevant, since when part of the population chooses to decrease their sugar intake, they probably take additional measures related to improving their health, such as not eating grains/flour, not consuming processed products, cooking more at home, less frequently eating out, etc. And, in addition, those who make these decisions have not been chosen at random: they are the ones who have decided to take care of themselves, so we’re not just comparing sugar intake: we are comparing lifestyles. That is an important difference with respect to a randomized controlled trial (RCT), where participants can’t decide if they decrease their sugar intake (and perhaps its replacement by another product). In the case of a RCT that reduction is supposed to be the only difference between goups. Guyenet’s data is far from being that case.
- We are talking about total sugar intake, regardless of its format, regardless of when it is consumed, regardless of which products accompany it in the mouth. A lot of information is missed.
- We are talking about the average sugar intake of a population and the percentage of adults above a specific level of obesity. The interesting data would be to compare individualized
**DAILY**sugar intake and**DAILY**body weight change, and we would want to have this data for a large number of people. - Guyenet says the variation in the percentage of obese adult has been small —he even expected a decrease! — but, actually, the change has been bigger than expected from the small reduction in the
**DAILY**sugar intake (gradually decreasing the intake from 110 g/d to 95 g/d is an average reduction of 9% relative to the baseline: it is not an 18% decrease!). May be people who have decreased their sugar intake have also taken, at the same time, other measures to improve their health, and those additional measures could be contributing to the undeniable trend change perceived from year 2000 in the percentage of obese adults.

Further reading:

Si bien no tiene relación con el artículo, quiero manifestar la tristeza ante la muerte de un grande en la investigación sobre nutricion, me acabo de enterar que Staffan Lindeberg murió. triste noticia

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Cumulative total sugar intake (computed using the data in the graph below) added to Guyenet’s graph:

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Annualincrease in the percentage of obese adults (orange curve)Annualsugar intake (blue curve)Me gustaMe gusta

Annualsugar-sweetened beverages intakeAnnualincrease in the percentage of obese adultshttp://www.nejm.org/doi/pdf/10.1056/NEJMhpr0905723

http://ajcn.nutrition.org/content/98/1/180.full.pdf+html

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Congratulations Vicente!Your response to Guyenet’s hypothesis was outstanding. I won’t comment on Mr. G because this is a family blog.

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Thank you, Johnny 🙂

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