How to lie about flu vaccines in one easy step

I wrote a news article on flu the other day, and a reader emailed me asking, "How many people who get the flu got a flu shot?" I did a couple basic calculations and discovered something interesting (well, interesting to me anyway): The more people get the flu shot, the more easily the resulting statistics can be used deceptively to imply the vaccine doesn't work!

Why? I believe it's known as "conditional probability." Anyway, here are the numbers I ran:

According to the Centers for Disease Control and Prevention, the flu shot cuts people's chances of getting the flu by about 60%. (It depends on your age and health status, the accuracy of the vaccine that year, and so on, but we're talking back-of-the-envelope numbers here.) WebMD says your odds of getting the flu are 5% to 20%. I used 10% as my baseline (i.e., pre-vaccine) risk. For population, I chose 500,000, roughly the number of people in my county.

CASE A: Suppose 50% of the population gets a flu vaccine. (The actual 2013-14 rate was 46.2%.) That means 250,000 people are unvaccinated, while 250,000 get the shot. Of the first half, 10%, or 25,000, will get the flu. Of the others, 25,000 would have gotten the flu, but they have protection that's 60% effective, so only 40%, or 10,000, actually come down with it.  That still gives us 10,000 people complaining that the vaccine didn't do them a lick of good, but our population of 25,000 people who didn't get vaccinated and subsequently got sick is 2 1/2 times larger. Clearly, vaccination is doing its job.

CASE B: Now suppose that 90% of the population, 450,000 people, get a flu vaccine, and only 50,000 don't. We're protecting a lot more people: The vaccine will really show its worth now, right? Let's see.

Among our 50,000 unvaccinated people, 5,000 (10%) will get sick. As for the vaccinated majority, 10% of 450,000 is 45,000, but they have 60% protection, so only 40% get sick. Unfortunately, 45,000 times 0.4 is 18,000. Horrors! Among or population of sick people, the number who got vaccinated vastly exceeds those who didn't. To be precise, the vaccinated group accounts for 78% of the total. Clearly, vaccines are dangerous and make you more likely to get flu!*

CASE C: If the CDC could achieve 99 percent vaccination compliance, then in my hypothetical population of 500,000, there would be 20,300** flu cases, and 19,800 of them - over 97%!! - would be people who got the shot.

When you work it out step by step, it's easy to see what's happening here - a smaller portion of a large number can be (much) bigger than a larger portion of a small number. But if you're innumerate, a writer of clickbait headlines, an anti-vaxxer, or some combination of the three, you write PATIENTS SICK WITH FLU 97% MORE LIKELY TO BE VACCINATED THAN NOT and all hell breaks loose.

The moral: The worse the stats look, the better off we are! 

The follow-up moral: Get a flu shot.

*In case it isn't sufficiently obvious, this "inference" is 100% wrong.

**Note that our total case numbers drop as vaccination spreads: 35,000 to 23,000 to 20,300. Even at 60% effectiveness, vaccination works.