Friday, July 18, 2014

The Guardian Gives Us a Chance to Play With Relative Risks

A post in The Guardian about black imprisonment in Britain offers this shocking, headline-grabbing statistic. To quote,
Evidence of this damaged relationship can be found in the commission's report. On the streets, black people were subjected to what the report describes as an "excess" of 145,000 stop and searches in 2008. It notes that black people constitute less than 3% of the population, yet made up 15% of people stopped by police.
The commission found that five times more black people than white people per head of population in England and Wales are imprisoned. The ethnic minority prison population has doubled in a decade – from 11,332 in 1998 to 22,421 in 2008. Over a similar period, the overall number of prisoners rose by less than two thirds. The commission says that the total number of people behind bars accelerated in the last decade despite "a similar number of crimes being reported to the police as in the early 1990s … the volume of indictable offences has fallen over this time".
If you were willing to skip over the correction at the beginning of the article, this can lead someone to quickly conclude that "The United Kingdom incarcerates 75% more black people proportionally than the United States." What are we to make of that? What are we to make of The Guardian's retraction?


One way to talk about this comparison is through the framework of conditional probabilities, and the ratio of these probabilities, which we'll call relative risk here. Let Wikipedia pull you up to speed.

To start, lets organize our data into the following style of table.

Risk Disease status
Present Absent
Smoker a b
Non-smoker c d

We can get the relative risk through the following formula:


As mentioned above, this is the ratio of conditional probabilities. In the smoking example shown above, you are taking the probability of having a disease, given that you are a smoker, and dividing it by the probability of having the disease, given that you are not a smoker.

The odds ratio is another way of assessing this kind of relationship. It's found through the following:


Odds ratios and relative risks have similar values when the probabilities are small, but there are some important technical differences. It's worth referring to the Wikipedia article if you're interested.

So, what do we make of the The Guardian's numbers? Well, there is some good and some bad. To start, just getting good data on race in prison in the US is really hard. The UK (here to mean just England and Wales) is much more transparent in this regard, and the Infowars fan in me can't help but smell a conspiracy. For that reason, a lot of my estimates should be considered with a pretty broad margin of error. I wouldn't be surprised if The Guardian has similar problems.

Keeping the data problems in mind, the relative risk of being in prison if your black (given that you're either in the US or in the UK) is about 2. In other words, black Americans are about as twice as likely and black people in the UK to end up in prison. Comparing white Americans and Britons helps explaining the an important confounding factor: the US imprisons a lot more people. Being white in the US means that your risk is of ending up in prison is 3.3 times greater than if you were in England or Wales.

Ouch.

This means that The Guardian's little factoid isn't nearly as misleading as the original article and correction would have you believe. Given that you are in the UK, being black raises the risk of being in prison over 9 times the risk for white people. In the US, it's only 5.5. You could say, somewhat abstrusely, that the risk in the UK is 70 percent higher. Given the problems with getting normal data, I'd say that The Guardian is within my margin of error for this little project.

But. BUT. BUT.

There are other issues to consider. First and foremost, the US criminal justice system extends far beyond prison walls, and we deprive felons of many fundamental rights for much longer than just their "time." Staying away from things like felon labels, denied public services and lack of a vote, we can capture some of this by looking at the broader population of people under "correctional supervision." This includes people on probation as well as those still forced to visit a parole officer.

The risks become a lot more evened out. Being black in the UK (as opposed to white) raises your risk of being under supervision by about 4.5, in the US it goes up by about 4. I'd say that the risks are about tied.

Still, you can't say this without pointing out a much more salient fact. While there are 25 times as many black people in the US as in the UK, there are 100 times as many black people under correctional supervision here. Let's be honest, that's insane.

It poses an important counter-factual. If there population of black Britons was large as that of black Americans and if they length of there history in the country was similar, would you expect the same huge differences in rates of imprisonment? In Britain, you have a very small group of mostly recent immigrants. Sadly enough, they are easy for a system to exploit. In the US, you have a very large population that is repressed by institutions meant to target them for their race. These institutions were built on a terrible legacy of apartheid and slavery.

I know I'm preaching to the choir here, and I know that British racism is no joke. I'll never forget the fact that Margaret Thatcher spent most of her career calling Mandela a terrorist. But at the very least, it's a bit of a treat to try and refute a headline grabbing statistic published by The Guardian, especially when I knew that they already had to backtrack on it quite a bit.

All of this is summarized in the spreadsheet below.

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