Friday, November 16, 2012

25,982 Reasons Why Pedestrian Deaths On 5th St W Are "Statistically Significant."

"Searching for Answers on Fifth Street," Sonoma Index-Tribune
November 16, 2012
available at
Two people have been killed at the same intersection near my home in the last seven years. My city government believes that two deaths in that time period at the same intersection are not statistically significant.  My local paper instinctively senses they're something amiss despite the city's assertion. Guess what, Sonoma Index-Tribune? I think you're right, and pro bono publico, here's what I think the problem is with the city's argument.
"Busy" Intersections in Sonoma.

To set the scene for non-local readers, I live in a relatively small town, Sonoma, California, with about 10,000 people (10,741, according to Google). Per the National Highway Transportation Safety Administration, there are approximately 1.73 pedestrian deaths per 100,000 population in the United States per year.  21.2% of these deaths happen at intersections.  I've looked over a map of Sonoma, and there are a lot of intersections; but I've tried to count only the substantial ones -- I think there are 29 (the list is on the right).  Please note that if I were more conservative, and counted each intersection, it would only make the chances of a second fatality at the same intersection less likely.

Thus, I think the chance of a pedestrian fatality at any given intersection, if the intersections are roughly equally dangerous, in any given year, to be a relatively straightforward application of the multiplication rule -- it's (1.73/100000) * 10,741 * 212/1000 * 1/29, or 1 in 736.  Long odds - you'd have a better chance of drawing a full house in a single draw of the cards at poker.

OK, but what are the chances of getting another pedestrian fatality at the same intersection within seven years? My old statistics book from Berkeley came in handy here -- it's an application of the binomial formula.  The formula is on the right; the binomial function from Excel made calculation pretty straightforward.  The chances of another pedestrian accident happening at the same intersection, if the intersections are equally dangerouswithin seven years, is 1 in 18,518 25,982.  It's not quite as hard as drawing a straight flush, but it's pretty close.
Freedman, Pisani, Purves & Adhikari
"Statistics, Second Edition," p.241.

It's unlikely that Sonoma was so unlucky. Instead, it's more probable that the intersection in question is vastly more dangerous than normal. Indeed, 1 in 25,982 is somewhere between a 4σ and 5σ event; mere "statistical significance" usually requires only 2σ (95%), and anything beyond 3σ is typically "highly significant."

But of course, I am no statistician, and this is all the work of an amateur. The problem is that the City staff aren't either, and I suspect they're even worse at it than me. The City shouldn't be saying something is statistically insignificant without talking to someone who has the education and experience necessary to determine that fact. This isn't a $30,000 study, it's something a grad student from UCB can handle in an afternoon. The City needs to do the work to prove this is merely bad luck, and judging by the staff report, they simply haven't.

Spreadsheet with formulas.
The I-T knows there's an issue here--for instance, they have been raising hue and cry about installing sidewalks in the Boyes Hot Springs area, based on the argument that pedestrians aren't safe (and they're right).  The hard question, though, is whether the I-T, given the economic vise the newspaper industry has been placed in, still has the resources to challenge arguments like those advanced by the City, that in incidents of these types that "the pedestrian or bicyclist was the party most at fault."  Personally, I think the I-T is on the right track, and I say, please keep pushing, because the voters are depending on you to do so, to keep us informed.  And public safety depends upon you making sure our government isn't just hand-waving in response to citizen concerns -- our officials need to do the math to prove their points, and need to show us the results.

Updated 4:55 PM Saturday, November 17:  The odds of two deaths in the same intersection in 7 years were updated to reflect the 21.2% NHTSA figure, rather than 25%.  Further, John Capone, the writer for the Index-Tribune, pointed out in his article that Beatriz Villanueva was killed in the same intersection in 1996.  The chances of three pedestrian fatalities in 17 years occurring at random at the same intersection under the assumptions detailed above is 1 in 597,956. By way of comparison, the chance of drawing a royal flush in a single hand of poker is 1 in 649,739.