Some extensions to the WDD statistic
So for background, I created a statistic (with Jerry Ratcliffe1), the Weighted Displacement Difference (WDD) test. This is meant to evaluate place based crime prevention interventions. Such as you have a treated and control hotspot. You look at the difference pre to post for both those areas, and can calculate how much the intervention reduced crime, as well as a standard error around that estimate.
For example, if you had:
- Control area, pre=100, post=90
- Treated area, pre=100, post=80
The treated area dropped by 20 crimes, but the control area dropped by 10. So the extra reduction due to the treatment is 10 crimes. I have an online calculator, you can put this information in and get an estimate of the standard error for the statistic. In this scenario the standard error is 19, and the site calculates a confidence interval around the crime reduction.
The reason to do the test like this, crime rates are often going up and down for other reasons outside the control of the police department doing the intervention. You want to control for that temporal variation. Crime can technically go up in the treated area, but by less than than the control area. Hence still doing a good job, even if crime is rising overall.
Over the years I have been asked for slight variations of the WDD for different scenarios. So I am putting them all here in a single place for reference. You can download this Excel spreadsheet to conduct the different tests, or use my R package ptools.
Different Temporal Time Periods
So imagine you have 3 years of pre-data in the treated and control areas, but only 1 year of data after the treatment started. You do not need to throw away those 3 years, you can use weights to say the pre and post are different time periods. Here is the same example above, except the prior time period is 5 years, so baseline values are 500 crimes. In cells B8 and B9 I change the pre-time period to 5, and post-time period to 1.
This shows the same reduction of 10 crimes, but the standard error is slightly reduced, from 18.6 to 14.5. So using more data will reduce the standard error, even if it is just pre intervention data. The inferences are at a single time period. So in this example if the 1 post is a year, the -10 is the estimate for crime reductions for an entire year. You can put in multiple time period weights for both pre and post, so say you wanted to estimate the monthly decline, but had 60 months pre and 12 months post. It does not change the inferences any, it just puts the estimates on the monthly scale instead of the yearly in the prior example.
It is good to use as much data as possible. You just want to make sure the treated and control areas follow similar trends over time before the intervention. So e.g. plot the data on the monthly level and make sure they are going up and down approximately the same over time.
Using Rates per Area
Another ask was how to use the WDD when comparing different sized areas. It is similar to the scenario with different time periods, you just use the weights differently than the time period scenario. Here is an example that also includes displacement and a control displacement area.
Again, the inferences are in whatever unit a 1 is on this scale. So if 1 here is a square mile, this would estimate a crime reduction of -100 per square mile.
I say area in the spreadsheet, but these denominators can be anything. Say you wanted to calculate per capita rates of crime reduction, just put in your estimates of the population. All of these examples I have discussed as well use place based counts, but there is no reason you cannot count up crimes for other units, such as people or just comparing crime types.
For people, imagine you had an intervention for chronic offenders, and you had a sample of chronic offenders vs the rest of the city crime. Or you could compare one crime type vs another. Such as say you worked with stores to reduce shoplifting, you may have a control crime that includes other types of larceny. All that matters is they follow the same trends over time.
Combining Multiple WDD Stats with Harm Weights
Many place based crime interventions would be expected to reduce all crime types, and not just be limited to specific crimes. You often do not want to just count up property crimes and more serious violent crimes the same way. One way to do that is to use harm weights, and the spreadsheet also has an example of doing that.
What you do here is estimate the original WDD estimate for each specific crime type. Then combine them in this spreadsheet, along with their harm weight. (Harm weights are arbitrary, but will often be highly correlated.)
Pooling multiple results together, the same as pooling longer time periods, will result in more power to estimate crime reduction effects. Or put another way, the standard error will be smaller, so it is easier to say if your intervention is working!
Continuous Monitoring Over Time
The last example I am going to show is using these statistics in a continuous monitoring framework. Often people do interventions over too short a period of time. If the baselin is 20 crimes, you need to almost elimate all crimes to know if the intervention worked. Whereas if the baseline is 100 crimes, you need to eliminate 30.
It can be frustrating to say to the Chief “we need to do this for 10 months before we know it works”. In response to that, I have created an example of continuous monitoring. It is just doing the same WDD stat, but looking at it cumulatively over time.
If you create the typical line charts just comparing treated vs control, it may look like it is not working, you can see the line charts are not consistently higher/lower, but the cumulative effects are more clear over time.
It will still take as long for the error bars + a reasonable crime reduction to not cover the 0 line in that chart as it would in the planning scenario. But at least you have short term feedback you can give the chief in a CompStat meeting on a regular basis.
Although I do not recommend it typically, but that sheet also has a way to estimate percent changes in a similar manner over time with standard errors. So if you really want something similar for percent changes, just scroll a bit to the right in the spreadsheet.
Reach Out
All of these extensions I developed, as well as the initial WDD statistic, were based off of requests. If there is some scenario you need help with for your department, always feel free to reach out with a question. While I want to develop larger paid projects with departments, you can often nerd snipe me and get the results for free if you have a tricky mathematical question, especially if it is related to my prior work.
The math though is not really the hard part, planning and boots on the ground implementation is. So if you need help with longer term operations, it would make sense to get in touch with me and see if Crime De-Coder can help.
It was Jerry’s original idea. He wanted to know if you could create a standard error estimate around the weighted displacement quotient. I suggested the alternative WDD and worked out the math to do the statistic. You can read the paper and the math behind the work here.↩︎