Previously in this series:
N=1: Introduction
N=1: Single-Subject Research
N=1: Hidden Variables and Superstition

I. 

Many chronic illnesses are much more common in women than in men. IBS is about 2-2.5 times more common in women than in men; migraines are about 2-3 times more common; chronic fatigue is about 4 times more common. 

This is pretty weird, and more than a little mysterious. And it’s doubly weird that the ratio is pretty similar — each of these examples is about 3 times more common in women than in men.

Normally this gender gap, if it is addressed at all, is written off as a biochemical difference (e.g. here). But another possibility is that gender is just a proxy for body size (e.g. here). If some chronic illnesses are caused by exposure to irritants, heavy metals, or other contaminants, smaller people will generally have more of a response to the same level of exposure, and women on average are smaller than men.

If this is the case, it should be possible to detect if gender is a proxy for body size in some chronic illnesses. If body size is what really matters and gender is just a proxy, larger-than-average women will be underrepresented and smaller-than-average men will be overrepresented. Basically, once you know someone’s height and weight (and maybe % body fat), their gender shouldn’t give you any further information about their likelihood of getting sick.

II. 

We can show this with some simulations.

Here’s a simulation of 10,000 men and 10,000 women. The men have an average height of 69 inches with a standard deviation of 3 inches, and the women have an average height of 64 inches with a standard deviation of 3 inches. 

Let’s start by seeing what things look like if the greater prevalence of women is the result of something like hormone levels, and body size has nothing to do with it. In this case, the men all have a 1% chance of getting the illness, and the women have a 3% chance. Height doesn’t factor in at all. So when you look at the distribution of heights of men and women in the group of people with the chronic illness, it looks something like this:

As you can see, three times as many women have the illness as men do, but otherwise the distributions are quite generic. These are basically just subsets of the distributions for each gender. They should be normally distributed and should generally look similar to one another, except that there are more women than men and the two groups have different average heights.

Now in comparison, we can consider what the data would look like if gender is just acting as a proxy for height, and there are more women with chronic illness only because they are shorter on average. 

Here’s another simulation of 10,000 men and 10,000 women, with the same distributions for height. Without getting into the exact model,[1] this is what it looks like if height is the only thing that determines if you get sick, and shorter people are much more likely to get sick: 

Again we see that there are about three times more women than men, even though this time, gender doesn’t have a direct effect. In this simulation, height is the only thing influencing who gets the illness, but the difference in average height is enough to make it so that there are three times as many women as men. 

While it’s not clear from just eyeballing the distributions, there are signs in the data that height is driving this difference. For example, about 1% of women are 70 inches or taller in the height-based simulation (compared to about 2.2% in general) and about 9% of men are 63 inches or shorter (compared to about 2.2% in general). This seems like a clear sign that height is the actual thing that determines who gets sick.

III.

Since we don’t know what the real-world dynamics would look like, it’s not clear what you would see in real-world data. It could just be that people with the chronic illness would be shorter on average than people without — American women are about 64 inches tall on average, so it would be interesting if the average height on a chronic illness subreddit was just 61 inches (though you might want to account for age and ethnicity). If the effect was strong or nonlinear enough, there might be a noticeable skew in the data instead. Or you might see the underrepresentation of larger-than-average women and overrepresentation of smaller-than-average men that we describe above.

You could conceivably detect this kind of difference with normal survey methods, as long as you got a large enough sample size. To our mind, evidence that height (or possibly weight, you would want to collect both) explains why women are much more likely to have a chronic illness would be evidence that the chronic illness in question is caused by some kind of contaminant, since other causes shouldn’t be so sensitive to body size. If anyone wants to help collect this data for their community, please contact us.

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[1]: The probability of a simulated person getting sick was proportional to 82 inches minus their height in inches, cubed. That is to say, in this model someone who is 56 inches tall was 17,576 times more likely to get sick than someone who is 81 inches tall. These numbers mean nothing, we pulled them out of our ass.