Wednesday, November 11, 2009

Smoke Detector Stats

Busy, busy, but I have to take up an obvious piece of faulty reasoning here. An opinionator in the Swedish daily paper Dagens Nyheter by the name of Malin Siwe has a short piece arguing against excessive safety regulations. It's not really unusually bad; indeed, that's my problem with it. This is rather typical for newspaper opinion pieces, in fact.

She is trying to make a perfectly good point: when safety rules become too numerous and too detailed, people end up trying to comply with the rules rather than actually improving safety. The problem with the piece is that her main example is an argument in favour of more rules, not against it. And her lack of statistical understanding means she is completely oblivious to that fact. Here's the relevant passages (roughly translated, emphases mine):

All apartment buildings built since 1999 must have smoke detectors. Many local governments rule that all homes must have them installed. Nine out of ten households have such smoke detectors today.


Most cases [of deadly home fires] lacked a smoke detector, but they were there in a large minority of cases - and some worked.

She's arguing that smoke detectors don't help, and that a law mandating them everywhere wouldn't do any good. But she is innumerate enough that her example is actually arguing against her position. Now, can anybody spot the problem above? The information in bold is enough to figure it out.

90% of homes have smoke detectors and 10% do not. Now, if smoke detectors really were useless, then 90% of the fatal fires would also happen in smoke detector-equipped homes while 10% would happen in detector-less ones. But they don't. As she says, a "large minority" of fatal cases happened when a smoke detector was present. Even with a generous interpretation of "large minority", less than half of fatal fires happened when there was a smoke detector in the home.

Let's be very generous, in fact, and say that half - 50% - of fatal fires happened in homes with smoke detectors. In that case, you're nine times less likely to die if you have a smoke detector than if you don't. If "large minority had smoke detectors" actually is something like a third then the number is closer to 20 times. That, to me, is some very persuasive data in favour of mandating smoke detectors.

I don't have a background in statistics1. I suck at this kind of reasoning, so I always need to draw a figure or two to really understand what is going on. Just to make it clear, take a look at the figures below:

diagram 1

The total area of the rectangle above is all the homes, smoke detector or not. The blue part - 90% - is the homes with a smoke detector. The small orange part at 10% are the homes without. We want to figure out how much more likely you are to have a fatal fire if you're in a home without a smoke detector.

diagram 2

Now we add the fires; that's the small rectangle straddling the two home areas. We don't know how many fires there are compared to the overall number of homes, but it doesn't matter. We're only interested in the difference in risk between smoke detector and detector-less homes.

We were generous above, and said that half the fires where in homes with smoke detectors. So half of the "fire" area is on the "smoke detector" side, and half on the "no detector" side. Again, the size of the "fire" area depends on how many fires there are, but the proportions don't change.

Now, note that the "fire"-part is a much bigger part of the "no detector" area than it is of the "smoke detector" area. That is the point - a much larger proportion of no-detector homes will have a fire than the proportion of smoke detector ones. You're much more likely to have a fatal fire without a smoke detector than when you do.

Let's make it clearer still: Here's the "no detector" area at the top, and the "smoke detector" area at the bottom. We've scaled them so we can compare them directly. Again, the "fire" area - the risk of having a fatal fire - is much larger for the detector-less homes than for those with a detector.

How large the risk is depends on the total number of homes and the number of fatal fires. But the difference in risk depends only on the proportion of smoke detectors, and the proportion of fires in those homes. With our numbers - 90% of homes have smoke detectors, and have 50% of the fires - it turns out you're nine times more likely to have a fatal fire when you don't have a detector than when you do.

Now, she points out that only some smoke detectors actually work. True enough - but that means a larger proportion of fires should be counted among households without smoke detectors too, and becomes an argument in favour of mandating smoke detectors and periodic checks, rather than against. Even with the most pessimistic estimate of working smoke detectors, you're still a couple of times more likely to avoid a fatal fire when you have one in your home2.

For those who can't read Swedish she also has a rather confused line of argument around the total number of fatal fires, but it's so incomplete and specious that the only thing it shows is that she doesn't understand high-school level statistics.

Now, I don't blame her. She's not good with statistics - but then, most people are not. Nobody is good at everything. Where she goes wrong is not in misunderstanding statistics; it's in not asking somebody who does understand it for feedback and help.

This is why I despair over the future of the newspaper. After all, the defining difference between professionally staffed newspapers and bloggers is supposed to be that newspapers have the time, the resources and the know-how to check original sources, ask experts for second opinions, fact-check every word and edit the finished text. This opinion piece could have been sanity-checked in ten minutes by anybody with a basic mathematical background. No need to even go outside; somebody from the newspaper's Accounting or IT department would have enough math background to catch the faulty reasoning.

But it wasn't fact checked. As it is, the columnist has published a blog post an opinion piece arguing against the (otherwise quite reasonable) standpoint it was meant to support. When newspaper opinion and editorials ends up reading like second-rate blog posts, I'd rather just skip the papers and read first-rate posts on the net instead.

#1 To the occasional frustration of my coworkers.

#2 I don't know about you, but given the choice I'd rather not burn to death if I can help it. Fire safety is cheap compared to the consequences.

We have legally required smoke detectors, central fire alarm and sprinkler system throughout the apartment, a fire door in the corridor and an emergency ladder on the balcony. It is tested and checked once a year by a couple of fire-safety inspectors; we even get to activate the ladder and climb onto it if we want so we really know how it works. In addition we've added a small fire extinguisher in the kitchen near the gas stove, and we plan to add another one in the bedroom.

Living right on top of an active fault in an earthquake-prone country tends to focus your mind in these matters.


  1. The situation here is also a nice example for the so-called "prosecutor's fallacy", where the misunderstanding occurs in conditional and prior probabilities. Realizing the increasing frequency of this misunderstanding, one might start questioning not only the quality, but also the way math is being taught at schools. At least, I do :)

    - Taha

  2. True. It's really a variation of Bayesian inference. But I can't really fault the schools for not teaching that - I wasn't kidding when I wrote above that I need a diagram to understand this. It really is quite difficult and counterintuitive to come to grips with and if people (like me) can't internalize it after years of using it, what are the chances of teaching it over a week or two in high school?

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