### Comparing Event Counts with the Poisson Means Test

#### by Joseph Ryan Glover

An exceedingly common task for crime analysts is to count the number of events that occur in one time period (e.g. day, week, month, YTD) and compare them to a prior time period. The change between these two numbers if often the source of much strategy and project-making in law enforcement. What does it mean that we had more motor vehicle collisions this week compared to last? What should we do about the increased number of mental health apprehensions YTD? What are we doing right to make robberies go down this month? These types of questions, and the actions that are taken from the discussions they generate, are the daily bread of law enforcement. Unfortunately, one of the things that is seldom done is to check if the change in the event counts is statistically significant. That is, is there really a difference between these numbers or can we just chalk it up to randomness?

This is where the Poisson Means Test comes in to play. It is a statistical test that can be used to determine if the difference between two counts is statistically significant or not. In prior posts I have written about statistical significance being about deciding between two hypotheses. The first (or null) hypothesis says that there is no difference between the two sets of events. The second hypothesis (the exciting one) says that, yes, these numbers are different enough that we would only expect to see this level of difference by chance every, say, 1 out of 20 times (when dealing with a 95% confidence interval). In this case, what we want to be able to do is pick a confidence interval (I’m sticking with 95%) and look at two different event counts (let’s say robberies YTD this year and last) and say that ‘yes, something is meaningfully different about these two numbers’, according to the statistical test. And we want to be able to do this in Excel.

But before we get into Excel a little background on Poisson. A Poisson process is a process that generates independent events over a period of time. Because the events are independent the time between consecutive events is considered random. It’s a perfect process for modelling things like criminal occurrences because notionally they are unrelated and we can’t predict with specificity when the phone is next going to ring. I’m not going to get into any math in this post as it’s not super important for understanding how to use the test so just know that Poisson is about counting the number of times something happens in a time period.

In order to compare two sets of events we need to define a couple of terms. The first set of events has a total count C1 that occurred over time intervals T1. The count C1 is the total number of events but there is some flexibility in defining the time intervals and that flexibility has to do with the other event count you want to compare. In our example I have the number of robberies in Jan-Feb-Mar of 2015 and 2014. I could say that T1=1 quarter, or I could say T1=3 months or I could say that T1=90 days, it really doesn’t matter because T2, the time for the events counted in C2, is exactly the same, since we’re doing a YTD comparison between two years. However, not all the use cases are like that, it is very possible to compare Jan-Feb-Mar 2015 to April 2015 in which case you’d want T1 to be 3 (months) and T2 to be 1 (month). Basically, you pick the largest common time period you have between the two data sets and used that as the basis for T.

Enough chit-chat, an example will make it things clearer.

I am sure this looks common to everyone: two year’s worth of crime data, the difference in the events and the percent increase. Can you just hear the operations inspector demanding to know why robberies are up 21% and the detectives scrambling for answers? The problem is that we don’t know if that change is because of anything real or is just due to the randomness of the data. But we can use the Poisson Means Test to help us figure it out and in this example, using the nomenclature above, we have C1=44, T1=1 (because it’s one quarter), C2=56 and T2=1. What we need to do next is send those numbers through a function to tell us whether the difference is statistically significant or not. I sure hope Excel has a function for that … but it does not. However, thanks to some talented programmers I found a solution.

I first learned about the Poisson Means Test from the blog of Evan Miller, a programmer and statistician who has developed a great piece of statistical software for Mac named Wizard. Evan has a blog post on the subject of event counting where he explicitly tells readers to never make decisions on plus/minus differences without first checking the Poisson Means Test. To help facilitate this he has created a great online tool for checking two event counts to see if there is a meaningful difference between the counts.

The online tool works great but I need to be able to run these kinds of tests in bulk in Excel (where all crime data resides) and I wanted to see if I could make Excel do the test. Following Evan’s notes in his blog post and peeking under the hood at his tool’s javascript I discovered that most of the heavy lifting for the tool is performed by a javascript library known as jstat. Jstat provides numerous special functions including the, ahem, ‘Inverse of the Lower Regularized Incomplete Gamma Function’. I took the time to figure out what that means so you don’t have to and what I learned is that Excel doesn’t offer a function to do it. But, Excel does offer VBA, and the jstat source code is available on GitHub, so I ported all the javascript functions I needed into VBA user defined functions and made them available through the following GIST.

Now, I don’t expect you to understand this code because I don’t understand a lot of it. I know what it is doing but there is a lot of code in here that I suspect comes from some flavor of a Numerical Recipes book. The point is, we don’t need to know how it works, the geniuses at jstat already figured it out, I just need to make sure that it works in VBA and I didn’t mess it up during the translation. I’m pretty confident in my coding after a lot of testing and you can test it if you want by trying Evan’s tool and my Excel macro together and seeing if you get the same results (you will). You can also, if you are adventurous, try out the R library zipfR, which has a function named Rgamma.inv, which does the same thing.

So how do we use the VBA to do something useful? Check the screen shot below where I have added a fifth column to my table named P-value. This is your usual statistical p-value that, for a 95% confidence interval, needs to have a value less than 0.05 in order to indicate statistical significance. As you can see in the image, I have called the custom function PoissonPValue from the VBA code and passed in the four values I identified above: C1, T1, C2, T2. The output of the function call is the value is 0.271, which means that, statistically speaking, there is no difference between 44 and 56, so that 21% increase is nothing to get worked up about (certainly not something to change strategy over).

I know that this is a hard truth to swallow because it runs counter to the thinking of most law enforcement personnel. They want to believe that if a number goes up then something is to blame and that something can be handled to make the number go back down. However, from personal experience I can attest, much of the week-to-week and month-to-month differences are not statistically significant. The reality is, randomness happens, and unless we want to be chasing our tails running after statistical ghosts, we would be well advised to concentrate our efforts on differences that the stats indicate are real.

To wrap up, the Poisson Means Test is an incredibly valuable tool that you should put in your toolbox. It’s not just for your weekly/monthly/annual crime report either. You can use it to test for meaningful differences between any two sets of events. So, for example, say someone ran a project over the last two months and wants to know if you can see an impact in the numbers. If you test the events from a period before against the 2-month project period you can see if there is a stat-sig difference between the counts. Now, that’s not to say that the project is responsible but at least you can demonstrate statistically there is a difference. Anecdotally, I have found new applications for the test every week and I like it because it adds a further layer of rigour to our analysis and I think that’s important. I hope you do too.