More on the racial disparity in incarceration rates

I argued in another post that, despite what most American liberals believe, racism can only explain a relatively small part of the white/black gap in incarceration rates. (If you haven’t read it yet, you should probably do so before you read the rest of this post.) As I anticipated, many people were totally unconvinced, because they thought that my back-of-the-envelope calculation rested on assumptions that were far too optimistic. However, as I also pointed out in that post, I think exactly the opposite is true. So, in order to convince the skeptics, I decided to write another post, where I’m going to use recent meta-analyses about bias in the criminal justice system to estimate the value of the parameters in the model on which my back-of-the-envelope calculation rests.

According to a meta-analysis of Tammy Kochel, David Wilson and Stephen Mastrofski published in 2011, the best estimate of the effect-size of race on arrest, expressed as an odds ratio, was 1.38 for the studies they examined.  If we assume a probability of arrest of 20% for white people, as the authors of that meta-analysis do, it means that a black person has a probability to be arrested of 26%. In other words, according to this meta-analysis, black people are approximately 28% more likely than white people to be arrested. For reasons I will explain shortly, this is actually a lot more than the value we should use in a more realistic model, but since I want to be pessimistic it’s the one I will use in my back-of-the-envelope calculation. In fact, if this can help me convince the skeptics, let’s make that a round number and assume that blacks are 30% more likely to be arrested than whites, keeping all the relevant variables equal.

Now we turn to to the next parameter in the model, namely the bias in the decision to prosecute. According to a meta-analysis published by Jawjeong Wu in 2016, the mean effect-size of race on the decision to prosecute was 1.093. Again, this is an odds ratio, so we need to convert that into a ratio of probabilities for my back-of-the-envelope calculation. According to Wu, when you do that, you find that blacks are approximately 9% more likely than whites to be prosecuted in similar circumstances. Again, since I want to err on the side of pessimism, let’s make that 10%. Note that, as with the effect-size of race on arrests derived from Kochel et al.’s meta-analysis, this estimate is based on the best studies available in the literature. Surprisingly, in both cases, using only studies with adequate controls actually produced a larger effect-size.

Next, we turn to bias in the decision to convict, for which I will use the meta-analysis published by Ojmarrh Mitchell in 2005. He reports a mean odds ratio of 1.38 for the effect of race on the decision to convict. Assuming a conviction rate of 50% for white defendants, this means that, according to that meta-analysis, black defendants are 16% more likely to be convicted. Note that by assuming a conviction rate of 50% for white defendants, I’m significantly exaggerating the anti-black bias, since the conviction rate in the US is apparently much higher. (The figure reported by the BJS includes both white and black defendants, but presumably it would still be true even if we only included white defendants.) But since I’m deliberately making pessimistic assumptions to convince the skeptics, itI will assume a bias of 16% against black defendants anyway. In fact, to be even more pessimistic, let’s make that 20%.

Mitchell’s meta-analysis also gives a mean odds ratio of 1.23 for the effect of race on sentence length. If we convert that odds ratio to a Cohen’s d effect-size, we find that black defendants received sentences whose length was approximately 0.114 standard deviations longer than white defendants. According to the BJS, in 2006 (the last year for which I was able to find data), the average sentence length in prison for people convicted by a state court was 59 months. (This excludes people who were sentenced by federal courts and those who served time in local jails, but they are a relatively small minority, so it shouldn’t matter. It also includes both white and black convicts, so it’s probably somewhat lower for white convicts alone, but I can’t tell by how much.) I don’t know what the standard deviation is, but let’s assume that it’s 20 months, which is probably way more than the actual figure. Under those assumptions, blacks receive sentences that are on average 3.8% longer than whites, but let’s be pessimistic again and make that 10% since I had to make a pretty random assumption about the standard deviation.

If we use these values for the parameters of the model on which my back-of-the-envelope calculation rests, we find that bias in the criminal justice system only explains approximately 16% of the white/black gap in incarceration rates. Now, to be sure, the model on which this back-of-the-envelope calculation rests is simplistic in more ways than I care to explain. But I think it’s pretty clear that, if we did things more seriously, we’d find that 16% actually overestimates how much of the gap bias explains. To be sure, as I pointed out in my original post about this, there are many ways in which a more complicated model could increase the part of the gap that bias can explain somewhat. For instance, the racial disparity in incarceration rates could also be explained by bias in the decision to grant parole, since it would mean that black prisoners serve a longer part of their sentence other things being equal. But since bias in the decision to grant parole also seems to be relatively small, I doubt it would make a big difference if we modified the model to take into account the possibility of bias at that stage of the legal process. In general, while I can think of many ways to make the model more realistic that could increase the part of the white/black gap in incarceration rates that is explained by bias in the criminal justice system, I can’t think of any that is likely to make a big difference in the end.

On the other hand, there are many things about the model and the way in which I estimated the value of its parameters that, if done more seriously, would probably lead to a much lower estimate of the part of the black/white gap in incarceration rates that can be explained by racism in the criminal justice system. For instance, as I noted last time, if blacks are disadvantaged relative to whites other things being equal at one stage of the legal process, it means things are not equal at the next stage, which my back-of-the-envelope calculation doesn’t take into account. Perhaps more importantly, as I sometimes pointed out above, I have been very pessimistic in using the recent literature to estimate the value of the parameters in the model. In particular, whenever giving more weight to study with more controls led to a higher estimate for the effect of bias, that’s the estimate I used. (This was the case for bias in the decision to arrest and prosecute, where studies that controlled for more variables generally found more bias, which is surprising.) But when controlling for more variables led to a lower estimate for the effect of bias, instead of using the estimate based on the best studies, I used the mean effect-size for all studies regardless of how good they were. (In only used a best estimate in the case of Kochel et al.’s meta-analysis about the effect of race on arrest, because they explicitly reported one, but it wouldn’t make any meaningful difference if I used the mean odds ratio.) If we did things more seriously, we should presumably use only the best studies, which according to Mitchell in his meta-analysis would lead to effect-sizes of bias about half as small.

Moreover, the model only uses the mean bias at each stage of the process, but we should disaggregate since there is no reason to assume that bias is the same for every type of crime. Now, if you have a look at the meta-analyses I used to estimate the value of the parameters, you will see that bias seems to be a lot stronger for crimes that account for a relatively small share of the carceral population. (In particular, bias seems stronger for drug-related crimes, but despite the nonsense made popular by Michelle Alexander’s book,  the vast majority of people aren’t in prison for drug-related crimes.) For instance, if I derived a bias of 30% from Kochel et al.’s meta-analysis, it’s only because the vast majority of the studies on which that estimate is based analyzed police stops. This kind of studies typically find a lot of bias, but presumably only a tiny proportion of the people in prison were arrested in that way, so we should give the studies in question less weight in estimating the bias for arrests or use a more complicated model that disaggregates between different types of crime at each stage of the legal process. (It’s also worth noting that, as I argued in another post, despite a widespread fallacy that purports to show otherwise, bias in police stops could just be statistical discrimination and not the result of prejudice.)

If I had to bet, I’d say that if we did things more rigorously, we’d find that bias can’t explain more than 10% of the white/black gap in incarceration rates. I certainly think that anyone who believes it’s more than 20% is completely irrational. Again, if you disagree, that’s fine with me. But you can’t just say that I’m wrong, on the ground that my back-of-the-envelope rests on a simplistic model, which it certainly does. You have to show that it’s simplistic in a way that invalidates my conclusion, which is not the same thing. In order to do that, you have to construct a more realistic model, use the literature on bias in the criminal justice system to estimate the value of its parameters and show that it can reasonably explain a larger part of the gap. Somehow, I don’t think it’s going to happen, but I guess we’ll see…

Now, even if bias in the criminal justice system can’t explain more than 10% of the white/black gap in incarceration rates, the gap is so ridiculous that it could still be that blacks are seriously disadvantaged relative to whites. (However, some of that is no doubt statistical discrimination, which may be justified to some extent on consequentialist grounds and, in any case, is probably inevitable to a large extent as long as blacks commit significantly more crimes than whites.) But the belief that, relative to other factors, bias in the criminal justice system plays a major role in the racial disparity in incarceration strikes me as completely irrational. The disparity in incarceration rates is so ridiculously large that anyone who lives in the real world can tell that without having to read any meta-analysis. Nobody reasonable would deny that racism exists in the criminal justice system, but despite what American liberals seem to think, magic does not.

P. S. I already mentioned it recently, but in case you missed it, I can’t recommend enough this piece by Adolph Reed. This guy single-handedly convinced me that perhaps the American left was not completely fucked.

EDIT: I said above that, given how large the white/black gap in incarceration rates was, even if bias in the criminal justice system only explained 10% of that gap, it could still be that blacks face a substantial disadvantage. To give you a sense of how large this disadvantage might be, if bias in the criminal justice system explained 10% of the gap, using the same model on which my back-of-the-envelope calculation above is based, it would mean that the expected length of the sentence received by a black person who commits a crime is approximately 54% greater than for a white person who commits the same crime and is otherwise identical as far as the legally relevant variables are concerned. That’s hardly insignificant and would obviously be morally concerning, but it’s clear that even if we could somehow completely eliminate bias, the main problem would remain as blacks would still be overwhelming more likely than whites to be in prison. Moreover, for the kind of reasons that Reed gives and many others he doesn’t even mention, focusing on bias is politically unwise, but I guess that’s a topic for another post.