The Meta-Diet: Was Pollan Right?

Michael Pollan wrote a very popular book which made the claim that there are only three imperatives about diets that we can know with any certainty:

1. Eat Food: That is, avoid processed foods.
2. Not too much: limit calories.
3. Mostly plants. This one's pretty self explanatory.

Not that I don't trust Pollan and all, but I would've like to see a survey of nutritionists or something which would indicate that people who have formal training in the subject agree with him. Fortunately for us, US News and World Report surveyed a bunch of nutritionists and had them rank various diets. From this ranking, I tried to divine how nutritionists feel about his three commandments. The results:

Diet	Overall Rank	No processed	Limit calories	Limit meat
DASH	1	Y	Y	Y
Mediterranean	2	Y	Y	Y
TLC	2	Y	Y	Y
Weight Watchers	2	Y	Y	Y
Mayo Clinic	5	Y	Y	Y
Volumetrics	5	Y	Y
Jenny Craig	7		Y	Y
Ornish	8	Y		Y
Vegetarian	9		Y	Y
Slim Fast	10		Y	Y
Nutrisystem	11		Y	Y
Vegan	12		Y	Y
South Beach	13	Y	Y	Y
Eco Atkins	14	Y	Y	Y
Zone	14	Y	Y	Y
Glycemic Index	16
Medifast	16		Y	Y
Raw food	18	Y	Y	Y
Atkins	19	Y
Paleo	20	Y

Methodology

For the most part, I just read the descriptions of each diet. The most controversial thing I did was including "collateral damage" - e.g. I said that the vegan diet is calorie-restricted since the dietitians said that vegans eat fewer calories. I think this is a better way to go about it, since we want to find out the nutritionist's opinions about these commandments, not about how some diet theoretically could be followed. If a diet had less than 1,500 calories per day for the average person, I considered it calorie-restricted.

In the ranks of "collateral damage" I also included a "low sodium" requirement as a restriction on processed foods. Four diets (Slim Fast, Nutrisystem, Zone, Medifast) limit saturated fat (i.e. their limit is significantly below the recommended daily max) so I considered these as having restrictions on meat, since a saturated fat restriction usually (but not always) results in eating less meat.

My notes and sources that weren't within the USN pages can be found here.

Discussion

I'd say Pollan's rules hold up pretty damn well. The top 5 diets all follow all three of them, and the most anti-Pollan diets are ranked 16th, 19th and 20th.

It seems like the no-processed-food rule is the least followed, but I think that's just because the rankings include a number of companies whose sole mission is to sell processed foods.

I wish they had included a "control" or "do nothing" diet to tell whether any of these diets are actually harmful, or whether the bottom of the list is good but not great. Here's what they said about the worst diet:

Experts took issue with the Paleo diet on every measure. Regardless of what a dieter's goal is—weight loss, heart health, or finding a diet that's easy to follow—most experts concluded he or she is better off looking elsewhere.

Does "elsewhere" include "not going on a diet"? I don't know, but if it does this might be a mark against Pollan, since the paleo diet excludes processed foods. (An alternative explanation: since the paleo diet tends to be meat-heavy, maybe the "mostly plants" maxim is stronger than the "eat food" one.)

Quantification

I scored the diets 0-3 on how many of the rules they followed and tested how well this score correlated with their ranks. Because there are a lot of ties, I used the Pearson product-moment metric. This yielded a correlation of -.52, significant at p=0.018. So it appears unlikely that the relationship between rank and extent to which they follow Pollan's criteria is random chance. However, the relationship isn't incredibly strong.

Kendall's tau and Spearman's rho tests were significant at p=.035 and .033 respectively, but keep in mind that ties are excluded, and there are lots of ties.

Conclusions

Go ahead: eat food, not too much, mostly plants.

A clever proof of Euler's Theorem

I've been learning abstract algebra for the last six months or so - and I have learned a lot - yet I get the distinct impression that if Euclid were to appear before me and demand to know why he should care about groups or rings or what have you I wouldn't have much to say. So I was happy to learn of a simple proof of Euler's theorem, which relies solely on Lagrange's theorem, a basic result of group theory.

Euler's theorem deals with $\phi$, the Euler totient function, where $\phi(n)$ is the number of integers less than $n$ that are coprime to $n$. For example, $\phi(6)=2$, since 1 and 5 are the only integers less than 6 that are coprime to 6. If $p$ is prime, $\phi(p)=p-1$ since every number less than $p$ is relatively prime to $p$.

Upon first hearing about this function, I thought it was too insanely complex to know anything about. How the hell should I know what $\phi(n)$ is? There's no easy formula. But it turns out that we can prove the following theorem quite easily:

Euler's theorem: Suppose $a,n$ are coprime. Then $a^{\phi(n)}\equiv 1 \pmod{n}$.

Note that, since $\phi(p)=p-1$, this is just a generalization of Fermat's little theorem. Here's a three-step proof:

1. An integer $a$ is invertible modulo $n$ if and only if $gcd(a,n)=1$ ("invertible" means there's some $a^{-1}$ such that $aa^{-1}\equiv 1 \bmod n$). This is the Linear Congruence Theorem.
2. Consider the group $G$ of all positive integers less than $n$ which are invertible modulo $n$ (formally, $G=\left(\mathbb{Z}/n\mathbb{Z}\right)^{\times}$). $\phi(n)$ simply counts the number of positive integers $a$ where $gcd(a,n)=1$, so by (1) there are $\phi(n)$ elements in $G$ (the order of $G$ is $\phi(n)$). You can satisfy yourself that $G$ follows the group axioms quickly.
3. Lagrange's theorem says that there must be some $k$ such that $a^k=1$ where $k$ divides the order of the group, namely $\phi(n)$; let's say $km=\phi(n)$. Then $a^{\phi(n)}=a^{km}=\left(a^k\right)^m=1^m=1$ which proves our theorem!

Isn't that cool? I went from thinking that $\phi$ was an impossibly complex function to knowing a foundational result about it within 5 minutes, all because of what seemed like an esoteric result in group theory. 90% of this short post is explaining the terms, the proof itself is only a paragraph!