There's No U

 The New York Times had a story about the health effects of coffee, based on a paper in the Annals of Internal Medicine. The study found that coffee drinkers had substantially lower mortality rates compared to people who didn't drink coffee, even after adjusting for a large number of variables involving health, diet, and exercise.  The point that caught my attention was that the story emphasized that the results involved moderate coffee consumption, and concluded with a warning: 

 "The study showed that the benefits of coffee tapered off for people who drank more than 4.5 cups of coffee each day.  Past studies have shown that consuming 'extreme amounts' — over seven cups per day — can take a toll, she [Christina Wee, professor at Harvard medical school and deputy editor of the Annals] said.

'Moderation is good,' Dr. Goldberg [professor at NYU medical school] said. 'But too much of a good thing isn’t necessarily more of a good thing.'"  

I like coffee, and probably fall into the "extreme amounts" group, so I wanted to see how strong the evidence was.  

The estimated relationship between coffee consumption and mortality rates is shown in this figure from the paper:

The first column is people who drink unsweetened coffee, the second people who drink coffee with sugar, and the third is people who drink it with artificial sweetener--all are compared to people who didn't drink coffee.  For unsweetened coffee, it looks like the reduction in mortality is about the same for all of the range of 3-8 cups, and for coffee with artificial sweeteners, it gets larger the more you drink.   It's only for coffee with sugar that there's evidence that high amounts are worse than moderate amounts.  The confidence intervals are pretty wide in all cases, so they don't rule out the possibility that the reduction in mortality gets smaller with larger amounts of unsweetened coffee, but they don't support it either.   It seemed strange that the Times story emphasized a point that wasn't really supported by the analysis, but on reading the article more closely, I can see why it did.  It was just following the paper, which mentioned a "U-shaped association with mortality" several times.  Moreover, the paper was accompanied by an editorial (written by Dr Wee) which said "they found a U-shaped association."  

The main focus of the paper wasn't the shape of the relationship, but whether there was a difference between sweetened and unsweetened coffee--specifically, whether the (apparent) benefits were smaller for sweetened coffee.  In fact, the sample wasn't very useful for estimating the shape, because there weren't many people who drank large amounts.  The paper didn't describe the distribution in detail, but the mean number of cups per day for those who drank coffee was 2.4, with a standard deviation of 1.3, so 7 cups would be about 3.5 standard deviations above the mean.*  

So why did the paper and the editorial emphasize a secondary point and mischaracterize it?  I think it's a combination of two things:  

1.  Traditionally people modeled non-linear relationships using polynomial regressions, starting (and often ending) by adding x squared.  A regression with x and x-squared will always be symmetrical around the minimum/maximum.  So people got into the habit of  thinking in terms of U or inverted U shapes, and that has persisted even after more flexible methods of modelling non-linear relationships have come into use.  

2.  Medical people are inclined to start from the assumption that extreme consumption of anything is bad.

2a.  When communicating with the public, they are not purely concerned with conveying information, but also with getting people to do the right thing, or not encouraging them to do the wrong thing.  Since some people tend to excess, emphasizing moderation is part of that.

*The data were from Britain.