Currently in the two countries closest to my heart – New Zealand and Norway – there is an ongoing debate about arming the police. The arguments for this are a little different in each country, and are complex. However without too much effort they can be simplified (over-simplified?) to
“there are more armed incidents (of whatever type), therefore the police need to be armed to protect themselves and the public.”
My aim here is not to argue the merits (or otherwise) of arming the police – that is far too much for a mere Optimeering blog entry. Instead I want to examine one of the pillars of the argument: that an increased number of incidents means the world is a more dangerous place. That is, can we say “things are worse than before”, simply because the number of armed incidents over the past couple of months/years/whatever time interval has increased?
The “it’s getting worse” argument is not just used when discussing arming the police. It appears often in other crime statistics (e.g. burglary), accident statistics, or similar. In all cases, we need to ask a simple question – is what we are seeing a real change, or is it just random variation (with the corollary that things are not actually getting worse)?
As this is an Optimeering blog, it’s clear what I am going to say – let’s look at what the numbers (and models) can tell us. Let us assume for a moment that the world is not getting worse – that is, that the expected number of armed incidents in any given year is not changing, but is the same year-to-year. How would the number of incidents year-on-year then vary? Would we see, for example, periods of upwards trend (or downwards trend) that may make it look like things are getting worse (or better)?
To do this, I built a small model of armed incident occurrence. In the model, I have assumed that we have a fixed average number of armed incidents per year, and that each armed incident is independent (meaning, for example, that the number of armed incidents next year is unrelated to the number that occurred this year, or that an incident occurring last week does not change the chance of one occurring next week). The expected time to the next armed incident does not depend on when the last one was, but just on the rate (e.g. if the average were 6 per year, we would expect one incident to occur in the next 2 months). For those of you into this sort of thing, I’ve assumed what we have here is a (stationary) Poisson process.
This may seem wrong – you may say, for example, that more armed incidents encourage copy-cat acts, and thus one incident is not independent of another. In fact, many processes we see day-to-day actually do often behave like this – including the goals scored in football games, the number of calls received by a call-centre per hour, and the incidents of the flu in different towns and cities. And in any case, what I am doing here is saying: wait, lets assume a world that is not getting more violent. In this world, we have the same expected number of armed incidents per year, although the actual number in a given year and their timing will vary (randomly). Each incident is independent, and the number occurring next year is independent of the number that occurred this year. That is, things are not getting any worse, but staying the same. In such a world, what could the pattern of armed incidents look like?
Running the model through once for a 10 year period and assuming an average of 6 armed incidents per year gives the results below. The first graph shows when each incident occurs over the 10 year period, and the second graph shows how many incidents occur per year.
Now, remember that this is all random – we have the same chance of events occurring in each year, and each event occurs randomly and independently of the previous event. In the model things are not getting worse, and one event does not cause (or inspire) more events. Even given this, we see what look like patterns – both trends (more incidents over time) and clusters (e.g. lots of events in years 8 and 9 very close to each other – it’s a crime wave!). In this example the number of incidents really do look like they are increasing – and the calls for a response (tougher laws, more surveillance, armed police, or whatever) could be expected to grow alongside. However, thinking that things are getting worse is actually wrong – in this example world, things are just the same as they’ve always been, and any variation is just down to plain chance.
Running the model for the next 10 years shows this clearly:
The number of incidents in falls from 9 in year 10 to 4 in year 11 and stays pretty low for 5 years afterwards, before a (completely random) uptick in year 17, 18 and 19.
So, a completely random process, where events (the armed incidents) are unrelated to each other, and where the chance of an event occurring does not change, produces something that looks like it has patterns. Where we see patterns, we tend to see meaning, and tend to try to produce a story to explain why. However, often there is no underlying reason – its all just chance. I could easily imagine a politician in year 9 or 10 campaigning to arm police (now!) to stop this dangerous development, and then to claim victory a couple of years later because, well, the number of armed incidents has of course gone down. And of course it is just rubbish – the changes were just due to luck, not the actions of the politician.
And in that case, making policy decisions, and especially fundamental decisions such as whether or not we should arm our police, on seemingly convincing patterns in the data record – such as a few years of increases in armed incidents – is misguided at best. It can be very tempting to try to find a reason why something is occurring, or to belive a reason you are given, but always, always be sceptical. It is often very, very hard to determine whether what you are seeing in the data is real or not (and just due to random chance). And its not just policy where we see this – it occurs in many walks of life, from picking prices in the oil or stock market, to justifying high wealth because I’m better in some way (rather than just lucky).
The point is not per se that it is random, its just that it is extremely hard to tell if it is or not. In that case, we should be very careful about basing big decisions on something that may simply just be an illusion.