Swimming standards

I love Thomas Lumley's explanation of the new river water quality standards.

Suppose you wanted a target for river water quality so that people didn't get sick while swimming. River catchments vary but so too will water quality in any given river depending on whether there's runoff from recent rain.

The standard they're settling on is that the risk involved in swimming in a swimmable river should be no more than one chance in twenty of getting sick, 95% of the time that you might swim in that river.

Suppose we imagine a slightly implausible extreme sports facility that sends 100 backpackers on one-day swimming parties each day. On 95% of days (347 days per year), they’d expect fewer than 5 to get infected. On 5% of days (18 days per year) they’d expect more than 5 to get infected, but it couldn’t possibly be more than 100. So the total number of infections across the year is less than 5*347+100*18, or 10% of swimmers. That sounds bad, but it’s an extremely conservative upper bound. In fact, when the risk is less than 5% it’s often much less, and when it’s greater than 5% it’s usually nowhere near 100%. To say more, though, you’d need to know more about how the risk varies over time.

There are statistical models for all of this, and since everyone seems to be using the same models we can just stipulate that they’re reasonable. The detailed report is here (PDF), and Jonathan Marshall, who’s a statistician who knows about this sort of thing, has scripts to reproduce some calculations here.

Using those models, a `yellow’ river, with risk less than 1/20 95% of the time actually has risk less than 1/1000 about half the time, but occasionally has risks well over 10%. Our imaginary extreme sports facility will have about 3 infections per 100 customers, averaged over the year. About half these infections will happen on the worst 5% of days.

So, the 1/20 of 1/20 level doesn’t by itself guarantee anything better than 10% infection risk for people swimming on randomly chosen days, but combined with knowledge of the actual bacteria distribution in NZ rivers, seems to work out at about a 3% risk averaged over all days. Also, if you can detect and avoid the worst few days each year, your risk will be reduced quite a lot.

There's been the usual Twitter snark about how a restaurant that made one in twenty people sick would be shut down. Better to think of it this way: imagine that the restaurant runs every day, but sometimes the power goes out and some of the food in the fridge goes bad. One in twenty, over the course of the year, could be almost entirely on those bad days. Isn't it better then just to put a sign on the door saying "Sorry, the power was out, we're closed today"?