July 23, 2012

From the Herald this morning

More people have died in fewer car smashes since January 1 than at this time last year, prompting a Government reminder about the responsibility drivers hold over others’ lives.

“The message for drivers is clear,” Associate Transport Minister Simon Bridges said yesterday of a spate of multi-fatality crashes that have boosted the road toll to 161.

The number of fatal crashes is 133, compared to 144 last year at this time, and the number of deaths is 161, compared to 155 last year.

How do we calculate how much random variation would be expected in counts such as these?  It’s not sampling error in the sense of opinion polls, since these really are all the crashes in New Zealand.  We need a mathematical model for how much the numbers would vary if nothing much had changed.

The simplest mathematical model for counts is the Poisson process.  If dying in a car crash is independent for any two people in NZ, and the chance is small for any person (but not necessarily the same for different people) then number of deaths over any specified time period will follow a Poisson distribution.    The model cannot be exactly right — multiple fatalities would be much rarer if it were — but it is a good approximation, and any more detailed model would lead to more random variation in the road toll than the Poisson process does.

There’s a simple trick to calculate a 95% confidence interval for a Poisson distribution, analogous to the margin of error in opinion polls.  Take the square root of the count, add and subtract 1 to get upper and lower bounds, and square them: a count of 144  is consistent with underlying averages rates from 121 to 169.   And, as with opinion polls, when you look at differences between two years the range of random variation is about 1.4 times larger.

Last year we had an unusually low road toll, well below what could be attributed to random variation.  It still isn’t clear why, not that anyone’s complaining.  The numbers this year look about as different from last year’s as you would expect purely by chance.  If the message for drivers is clear, it’s only because the basic message is always the same:

Thomas Lumley (@tslumley) is Professor of Biostatistics at the University of Auckland. His research interests include semiparametric models, survey sampling, statistical computing, foundations of statistics, and whatever methodological problems his medical collaborators come up with. He also blogs at Biased and Inefficient