The very first Good Math/Bad Math post ever was about an idiotic bit of antivaccine rubbish. I haven't dealt with antivaccine stuff much since then, because the bulk of the antivaccine idiocy has nothing to do with math. But the other day, a reader sent me a really interesting link from what my friend Orac calls a "wretched hive of scum and quackery", naturalnews.com, in which they try to argue that the whooping cough vaccine is an epic failure:

(NaturalNews) The utter failure of the whooping cough (pertussis) vaccine to provide any real protection against disease is once again on display for the world to see, as yet another major outbreak of the condition has spread primarily throughout the vaccinated community. As it turns out, 90 percent of those affected by an ongoing whooping cough epidemic that was officially declared in the state of Vermont on December 13, 2012, were vaccinated against the condition -- and some of these were vaccinated two or more times in accordance with official government recommendations.

As reported by the Burlington Free Press, at least 522 cases of whooping cough were confirmed by Vermont authorities last month, which was about 10 times the normal amount from previous years. Since that time, nearly 100 more cases have been confirmed, bringing the official total as of January 15, 2013, to 612 cases. The majority of those affected, according to Vermont state epidemiologist Patsy Kelso, are in the 10-14-year-old age group, and 90 percent of those confirmed have already been vaccinated one or more times for pertussis.

Even so, Kelso and others are still urging both adults and children to get a free pertussis shot at one of the free clinics set up throughout the state, insisting that both the vaccine and the Tdap booster for adults "are 80 to 90 percent effective." Clearly this is not the case, as evidenced by the fact that those most affected in the outbreak have already been vaccinated, but officials are apparently hoping that the public is too naive or disengaged to notice this glaring disparity between what is being said and what is actually occurring.

It continues in that vein. The gist of the argument is:

- We say everyone needs to be vaccinated, which will protect them from getting the whooping cough.
- The whooping cough vaccine is, allagedly, 80 to 90% effective.
- 90% of the people who caught whooping cough were properly vaccinated.
- Therefore the vaccine can't possibly work.

What they *want* you to do is look at that 80 to 90 percent effective rate, and see that only 10-20% of vaccinated people should be succeptible to the whooping cough, and compare that 10-20% to the 90% of actual infected people that were vaccinated. 20% (the upper bound of the succeptible portion of vaccinated people according to the quoted statistic) is clearly much smaller than 90% - therefore it's obvious that the vaccine doesn't work.

Of course, this is rubbish. It's a classic apple to orange-grove comparison. You're comparing percentages, when those percentages are measuring different groups - groups with wildly difference sizes.

Take a pool of 1000 people, and suppose that 95% are properly vaccinated (the current DTAP vaccination rate in the US is around 95%). That gives you 950 vaccinated people and 50 unvaccinated people who are unvaccinated.

In the vaccinated pool, let's assume that the vaccine was fully effective on 90% of them (that's the highest estimate of effectiveness, which will result in the lowest number of succeptible vaccinated - aka the best possible scenario for the anti-vaxers). That gives us 95 vaccinated people who are succeptible to the whooping cough.

There's the root of the problem. Using numbers that are ridiculously friendly to the anti-vaxers, we've still got a population of twice as many succeptible vaccinated people as unvaccinated. so we'd expect, right out of the box, that better than 2/3rds of the cases of whooping cough would be among the vaccinated people.

In reality, the numbers are much worse for the antivax case. The percentage of people who were ever vaccinated is around 95%, because you need the vaccination to go to school. But that's just the childhood dose. DTAP is a vaccination that needs to be periodically boosted or the immunity wanes. And the percentage of people who've had boosters is extremely low. Among adolescents, according to the CDC, only a bit more than half have had DTAP boosters; among adults, less that 10% have had a booster within the last 5 years.

What's your succeptibility if you've gone more than 5 years without vaccination? Somewhere 40% of people who didn't have boosters in the last five years are succeptible.

So let's just play with those numbers a bit. Assume, for simplicity, than 50% of the people are adults, and 50% children, and assume that all of the children are fully up-to-date on the vaccine. Then you've got 10% of the children (10% of 475), 10% of the adults that are up-to-date (10% of 10% of 475), and 40% of the adults that aren't up-to-date (40% of 90% of 475) is the succeptible population. That works out to 266 succeptible people among the vaccinated, which is 85%: so you'd expect 85% of the actual cases of whooping cough to be among people who'd been vaccinated. Suddenly, the antivaxers case doesn't look so good, does it?

Consider, for a moment, what you'd expect among a non-vaccinated population. Pertussis is *highly* contagious. If someone in your household has pertussis, and you're succeptible, you've got a better than 90% chance of catching it. It's *that* contagious. Routine exposure - not sharing a household, but going to work, to the store, etc., with people who are infected still gives you about a 50% chance of infection if you're succeptible.

In the state of Vermont, where NaturalNews is claiming that the evidence shows that the vaccine doesn't work, how many cases of Pertussis have they seen? Around 600, out of a state population of 600,000 - an infection rate of one tenth of one percent. 0.1 percent, from a virulently contagious disease.

That's the highest level of Pertussis that we've seen in the US in a long time. But at the same time, it's really a very low number for something so contagious. To compare for a moment: there's been a huge outbreak of Norovirus in the UK this year. Overall, more than one million people have caught it so far this winter, out of a total population of 62 million, for a rate of about 1.6% or *sixteen times* the rate of infection of pertussis.

Why is the rate of infection with this virulently contagious disease so different from the rate of infection with that other virulently contagious disease? Vaccines are a big part of it.

Good points all. Our brains are not, by and large, good at parsing this kind of problem. That's true for me, too, but I know it, and have to consciously go through Bayes' theorem every time the question is posed: "OK, let's see, my chances of getting pertussis given that I'm vaccinated are equal to the probability of my being vaccinated given that I have pertussis times the probability that I have pertussis, all divided by the probability that I'm vaccinated." Whew. Much easier just to assume that public health doctors know what they're doing, and to go in and get the damn Tdap booster. Which I do and I did.

A side note: one reason we have a resurgence of pertussis now is that the acellular vaccine is somewhat less effective and shorter-lasting than the old whole-cell pertussis vaccine. Which had a small but significant rate of adverse events, and so was phased out. When we had an effective vaccine, Barbara Loe Fisher and Mike Adams and the rest of the wackaloon menagerie warned of horrible side effects. When we switched to a less effective but safer vaccine, they switched to complaining that it's not effective enough.

Nits: "groups with wildly difference sizes" -> "groups with wildly different sizes." and "50 unvaccinated people who are unvaccinated"

I should be able to comment without WordPress giving me a 34-character hex nickname. Let's see ...

Another way to look at this is that the anti-vaxers are making the classic conditional probability mistake. To gauge the effectiveness of a vaccine, you want to compare P(disease given vaccination) and P(disease given no vaccination). When the anti-vaxers point out that 90% of the infected were vaccinated, what they're actually noting is that P(vaccination given disease) = 90%, and P(no vaccination given disease) = 10%. They're trying to naively reverse the conditional probabilities and claim that P(disease given vaccination) = 90%. But Bayes's theorem tells us that in order to reverse conditional probabilities, you need to account for the base rate, which they neglected to do.

Of course, this is the same argument you made, it's just a rephrasing.

Oh, those comments on the "interesting link"...

Once again Mark, thank you for this. I've seen this refutation numerous times (and I dare say the antivaxxers have seen it too), but this twisting of statistics to "demonstrate" vaccine ineffectiveness keeps happening. Thank you for fighting the good fight.

Where does the five-year figure come from? As far as I can tell, the CDC says that adults who were properly vaccinated as children should have a one-time booster some time after turning 19, most easily by substituting it for the every-ten-year tetanus booster.

(I'm not immediately concerned for myself, since I had a TDaP in November 2009, but if I need another next year instead of in 2019, I want to know.)

I'm pretty sure it's mentioned in the document I linked.

The older vaccine, which contained the attenuated pertussis bacteria, needed a one-time booster. The newer, acellular vaccine has a lower risk of complications, but it needs more frequent boosters.

A possibly-simpler example: Let's say that on a rainy day, using an umbrella gives you a 95% chance of being dry.

EVERYONE used an umbrella. Of the subset of people who managed to get wet instead of dry, what percentage used an umbrella?

Obviously, 100%.

Ergo, the 95% figure must somehow be wrong!

Or, we've just mixed things up, as explained by many on this page.

Fantastic analogy. Very well done.