Bayes, Race, and Police Killing

Much has already been written about the draft paper by Roland Fryer, An Empirical Analysis of Racial Differences in Police Use of Force. Fryer makes several findings in his paper, but says the “startling” finding is that “Blacks are 23.8 percent less likely to be shot by police, relative to whites.” (p24—though Fryer observes that non-lethal force is much more likely to be used against blacks than whites). What he means by that is not that more white people in his study were shot at than blacks—indeed, in his data (Table 1C), 46% of total police shootings were of black people (as opposed to 24% non-black, non-Hispanic) and 52% were black in Houston, where he did much of his detailed analysis (compared to 14% non-black, non-Hispanic). The analysis he performs says for any given encounter between cops and citizens, whites are more likely to be shot. But controlling for police encounters removes much of the disparity of interest, as others have said (see Michelle Phelps’ take here, and another take here), and as Fryer himself acknowledged in a Times Q&A).   I’m going to add a slight methodological wrinkle based on my sabbatical project: learning Bayesian data analysis, but know that the definitive Bayesian analysis has already been written by Cody Ross, A Multi-Level Bayesian Analysis of Racial Bias in Police Shootings at the County-Level in the United States, 2011–2014. This is just the Cliff’s Notes version. But I figure that’s suitable for a blog. So what could a Bayesian approach help us with?

Bayesianism, in my view, helps surface some common problems about how to analyze results. Bayesian analysis is particularly good for conjoined probabilities: the probability that, given X, Y is true (written, reading right to left, as p(Y|X), leading some to translate it to plain English as the probability of Y given X). The initial thing to note is that p(Y|X) is not the same as p(X|Y). Given that I can see the sun, the probability that it is daytime is high (unless I’m in outer space). But it’s less likely that, given that it’s daytime, I can see the sun (I might be indoors, it might be cloudy or foggy, the sun might be behind a building, etc.). But Bayesianism is also particularly good at expressly taking into account how we should read evidence like Fryer’s, and why “controlling” for the number of stops obscures more than it reveals.

Let me give an example. Let’s say there’s a fatal, but rare, disease, where the probability of any individual having the disease is 1 percent (in equation form, p(D)=.01). We have a test for it, but our tests aren’t that great. If you have the disease, your likelihood of testing positive for it is 80 percent (p(Pos|D)=.80). If you don’t have it, you are likely to get a false positive about 60 percent of the time (p(Pos|not D)=.6). Here’s the pre-Bayesian, “blacks and whites are stopped roughly equally” question: if you get a positive test, is it more likely than not that you have the disease? The answer is no. It’s more likely that you have the disease than it was before you took the test, but, remember, the chance that you had the disease before you took the test was only one percent. Your chances increase, but only so far. This is what Bayesianism is useful for. The prior probability of having the disease was low (the “Bayesian prior” or baserate probability, among other terms), which bakes in to the analysis how you should revise your results.

Screen Shot 2016-07-14 at 7.33.10 AM

The top line of the equation tells you the number of true positives: the likelihood that you have the disease, and the likelihood that, if you have the disease, you will test positive. The bottom divides this by all the positive results, both true and false positives. Since the disease population is so small, the numerator is also small, relative to the large denominator.

Plugging in the data, we get:

Screen Shot 2016-07-14 at 7.35.45 AM

 

For a total of 1.3 percent. So it’s slightly more likely that you have the disease, but not by much, since the disease is still really rare (and our tests aren’t that accurate). The false positives swamp the true positives.

Fryer’s work is extremely helpful for its data collection, and there is much to admire in it. But when we get to the “startling” finding, we can map it onto the above equations like this: Fryer also only looked at the probability of lethality given contact with a black suspect—or p(lethal|black encounter) and p(lethal|white encounter). But without understanding the likelihood of those encounters, that’s like looking at disease likelihood alone. Lethal police encounters are rare—about one percent according to this study by Goff, et. al (see Table 1). But stops are not. What I think we really want to know is not necessarily whether a given encounter is more likely to be lethal. What I think we are interested in is p(black stops|lethal): that is, given that someone has been shot, what is the likelihood that it came from a black person being stopped? (Heather MacDonald, whose work I have previously criticized, asks a still different (irrelevant) question: p(cops|black homicides)—the percentage of killings of black people assigned to police.) What we ultimately care about—including Fryer, in his article—is racial disparity.

What do we know? Based on the New York City stop and frisk data, much of the stopping in New York of all races wasn’t fruitful: about 6 percent of all stops (across races) resulted in an arrest, and about 3 percent turned up weapons or contraband. But black people were stopped much more often: about 58 percent of the total, more than twice their percentage of the population. Whites were stopped about 10 percent of the total. Fryer uses this data, but, unfortunately, it doesn’t include lethal events. For those, he uses the figures I quoted earlier, with an emphasis on Houston. I happen to agree with Fryer (p4) that there is too little good data. Here is what I think a good equation would look like, assuming that there are only two races for simplicity:

Screen Shot 2016-07-13 at 11.44.45 PM

So let’s take Fryer’s estimates. The probability of a black stop turning lethal is about 75 percent of the probability of a white stop turning lethal. That, in fact, is the same ratio I used in my disease example. But lethal events are still much more likely to involve black people because black people have many more encounters with the police.  The huge disparity in p(black stop) relative to p(white stop) overwhelms any difference in the relative lethality of any given encounter.  This explains the results Cody Ross and many, many others have found—Ross, in particular, found (among many other interesting things) that the probability of being black, unarmed, and shot by police was about 3.49 times the probability of being white, unarmed, and shot by police. Base rates explain the difference.

Again, there is nothing wrong with Fryer’s answer about the percentage of stops that are lethal—it’s just not a very interesting question, and it lends itself to misunderstanding. Saying blacks are less likely to be shot really depends on the question you’re asking, and I don’t think Fryer’s is the right one. Ignoring base rates of stopping, choosing to start the measurement at the point of police contact, distorts what those who are concerned about racial disparities—including me—care about the most: why so much police activity, including lethality, is directed at black people.

There can be some legitimate debate about what baseline to use. This analysis looks at the rate of lethal encounters per arrest. In those terms, white lethality is more likely than black. But this is where Bayesianism has an advantage: it wears its assumptions on its sleeve. I think that arrests are the wrong baseline to measure, and that’s where the argument needs to take place: not just hand-waving at difficult math (much as I do it myself), but at the model’s assumptions. Many stops don’t result in arrests—in New York, black people were stopped more often and police still didn’t uncover wrongdoing more often. Using arrests, again, starts the clock at the wrong place.

In his paper, Fryer has suggested that the higher incidence of physical (but non-lethal force) is also a cause for concern, given that he found that on a given stop, police were more likely to use force on blacks than whites. When you couple that with the higher likelihood that black people are going to be stopped in the first place, that means the problem is perhaps even greater than he realized.

None of this is to say that any of this is particularly easy. There are legitimate arguments about what goes into the police stopping of black people in the first place. Fryer has, I think, also been unjustly criticized on methodological grounds, even though he acknowledges these deficiencies in the paper. He noted the problems with relying on police department participation (“It is possible that these departments only supplied the data because they are either enlightened or were not concerned about what the analysis would reveal,” p7), potential misrepresentation by police departments, and how representative the cities he chose were. He even acknowledges—without incorporating— “the possibility that there are important racial differences in whether or not these police-civilian interactions occur at all.” (p.25). He does not deal with the critiques that certain crimes are endogenous, the result of prior run-ins with the law (bench warrants for failure to pay fines, felon-in-possession laws).

My analysis is a simple model; Ross’s article is much more sophisticated. But the insights that this very basic analysis gives (from a very basic analyst just getting started) will, I hope, suggest why I think Bayesian analysis is so interesting in the first place.

[Note: I updated the post to correct an error in the first equation.]

Author: W. David Ball

W. David Ball is an Associate Professor at Santa Clara School of Law. He writes and teaches primarily in the fields of criminal law and criminal procedure, with a special focus on sentencing and corrections. He also serves as the Co-Chair of the Corrections Committee of the American Bar Association.

44 thoughts on “Bayes, Race, and Police Killing”

  1. This is a sort of public-health look at the data. Also useful might be a person-level view. If you're a black person, what's your 10-year probability of being shot and killed by a police officer versus a similarly situated white person? (Ditto for being the object of various uses of force). And that, to a first approximation, is pretty much the product of p(stop) and p(shooting) or p(other use of force). Those probabilities are, of course, very small, but they still act to inform opinions and behavior.

    1. The big question, of course, is what goes into two people being "similarly situated". As is usually noted in these discussions, blacks have a rather higher offending and victimization rate, too. And tend to live in neighborhoods with much higher crime rates than whites.

      So, all things considered, they SHOULD have a considerably higher rate of encounters with police. Ideally, criminals should end up encountering police. Police should spend more time in areas where crime is high.

      Normalizing the statistics for these factors isn't going to be easy. But it needs to be done.

      1. We don't really know if blacks have a higher rate of offending. We know that they have a higher rate of being caught and then getting convicted or pleading out, and we use that as a proxy for offending, and we imagine that it's at least a reasonably good proxy. But we don't really know how good it is.

        As a young white kid growing up in the '70s is an almost-entirely-white town, I got up to a bit of stupid stuff here and there. Mostly I just kept to myself and my books. But summers are long. So there was a fair amount of underage drinking. Here and there a bit of possession and/or use of controlled substances. Shit-tons of trespassing and a some goofing off which could probably be construed as vandalism. And some (in hindsight frightfully dangerous and irresponsible) arsing around with a saturday night special.

        Nobody got hurt, and nobody got arrested. Our few encounters with police were unremarkable. ("I'm confiscating all your fireworks and quit driving like idiots. Now run along, you scamps!") But we weren't all that heavily-policed. (Like I said–white kids, white town, '70s.) So my youthful misbehavior–and when you tot it all up, it looks like a remarkable criminal career–never got into the system, and doesn't contribute to white offending rates.

        There was a nearby city with a population that was largely African-American. If those kids were more heavily-policed, then maybe they'd have gotten picked up outside the liquor store for trying to arrange a straw purchase through that wino. Maybe if they'd driven down broadway throwing lit bottle rockets out the window and one of them flew way farther than you'd reasonably expect and exploded on the windshield of a nearby police car, maybe those officers wouldn't have let those kids go with nothing but a brief talking-to. Maybe they'd have asked questions about the (stolen) fire extinguisher. Maybe the drugs would have been found on a routine stop-and-frisk. Maybe the gun would have been.

        Or maybe I'm wrong? Maybe social science is farther along than I think, and we really do have a strong sense of what the offending rate is, and not just the getting-caught rate?

        1. Similarly. When I think of some of the things I and my friends did that could have resulted in long prison sentences. Assaults with a weapon, B&E, blowing up a house…

        2. Sorry, unless blacks are lying about the race of their assailants in victimization surveys, they really do have a high offending rate.

          1. Relatedly, I offer "Ron's Law": Anyone who begins a sentence with "Sorry" is not actually sorry at all, but very much enjoys saying what they're saying.

          2. Brett's law: "No fact that reflects badly on a favored minority can be admitted to be real, unless some way can be found to blame it on a disfavored group."

            Look, blacks on average have a high crime rate, and this inevitably is reflected in their interactions with the police. Given disparate underlying facts, only actual racial discrimination could prevent a disparity in things that are dependent on those underlying facts.

            That doesn't mean the entire disparity is innocent on the part of the police. It does mean that disentangling the causes will be difficult and intensely fact bound, in a way global statistics won't illuminate.

          3. Brett: Blacks have a high crime rate.
            Ron: Well…maybe? We know they have a high get-caught rate, but I'm not sure we really know how well that corresponds to the actual offend rate? Anyone know?
            Brett: *fingers in ears* Blacks have a high crime rate.

          4. Okay, I'm looking. To recap: I've wondered if the get-caught rate is a good proxy for the offending rate, since it depends both on the offending rate and on the ubiquity and effectiveness of policing. You've suggested that reported victimization rates can confirm that the get-caught rate is in fact a good proxy for the offending rate.

            Since blacks are about 3.5 times as likely as whites to be shot by police when unarmed, and your position is that this is solely because blacks are more likely to commit violent crimes, I'm looking for some indication in the victimization rates that blacks are 3.5 times as likely as whites to offend.

            On page one I see a graph which…okay, nevermind. That's about the race of victims, not the race of offenders. Those are surely related, but in any event this graph doesn't get us there because it shows black victimization as maybe a third higher than white victimization. If we're going to get to that 3.5, we'll need more than this.

            You specifically called out page 5? So let's look there. Okay…I see some stuff about most black crime victims report black offenders. About what you'd expect. And some stuff about gang affiliation and then drugs, neither of which are directly on point.

            Okay, I give up. Can you help me out here: Blacks are X times as likely as whites to commit violent crimes. What's your best guess as to the value of X in that statement, and in what way does your link support this estimate?

            (And since I've asked for your answer you might reasonably ask for mine. Here's my answer: I have no idea what the value of X is in that statement. You seem sure you know. Will you teach me?)

          5. " and your position is that this is solely because blacks are more likely to commit violent crimes, "

            No, my proposition is that you can't tell the extent to which the disparity is due to police misbehavior unless you know the extent to which its due to a lot of other factors, of which blacks being more likely to commit violent crime is surely significant.

            I object to this whole, "Disparate impact = racism" idiocy. Like people on the wrong end of a disparity can't have any responsibility for it, it has to be all imposed from the outside.

          6. "…of which blacks being more likely to commit violent crime is surely significant"

            It surely is. Have you got any idea just how big that factor is? A minute ago you sounded sure that you did. I'm not. So: Can you walk me through it?

            "I object to this whole…"

            For the record, you are literally the first person to use the word "racism" in this thread. I'm talking about disparate impact. If you'd prefer to talk about racism, go find someone who wants to have that conversation.

            (Let me guess: Next you're going to tell me what it "seems" or "feels" like I'm saying. Pro tip: When people say that it "seems" like you're saying X, or "feels" like you're saying X, what it really means is that they really wish you'd said X, and are going to just go ahead and pretend that you did.)

          7. Going to have to side with Mr. Bellmore here. He and I are likely to differ is in our explanations for why African Americans commit much more violent crime… but between the victimization surveys, the ethnicities of murder victims (combined with the strong intra-ethnic tendency in homicides), the ethnicities of people who shoot at the police, etc. It is impossible to avoid the conclusion that African Americans are much more likely to commit violent crimes. Just under 15 percent of the population, just over 50 percent of the murders, if I recall Kleiman's class correctly.

            As a liberal you almost have to believe this. You would agree with me, I would think, that evil racist forces have conspired over hundreds of years to cause our current racial distribution of the worst concentrated poverty. As a liberal, you have to believe that concentrated poverty causes violent crime (I certainly do) or you run into really really big problems. If you combine those two beliefs, a liberal would actually predict much higher violent crime rates from liberal first principles. Also, yes, this is very well studied and yes, nobody doubts that there is much more violent crime in African American neighborhoods.

            While it is true that African American communities are more heavily policed, the get-caught rate per felony crime committed is actually lower in bad neighborhoods (e.g., homicide clearance rates). The fact that it is safer to commit felonies in neighborhoods already full of felonies is one of the horrible feedback loops that one encounters in crime control policy.
            http://www.bjs.gov/content/pub/pdf/htus8008.pdf See table 1, page 3.

            My whole philosophy of the criminal justice system in this country is essentially: Evil forces ghettoized blacks and left them out of the great wealth creation (increase in land value in decent-good neighborhoods (which were kept white in various ways)) in the 20th century (things weren't exactly great before the 20th century for black folks either). The result is that the worst neighborhoods are black. Ghettoized people always behave poorly; blacks have been ghettoized, and are now behaving poorly (I don't need any lessons about base rates, I am perfectly well aware of the fact that vast majority of African Americans are not behaving poorly). The police, who are not the ones that originally ghettoized black people because they weren't alive yet, become biased because of the black quasi-monopoly on really bad violent behavior; especially violence against police, where African Americans punch considerably beyond their weight. Police are human beings, they notice patterns and are more than usually susceptible to tribal impulses (because of the nature of their job). The majority of black military-age males, who are not criminals, end up being the targets of this bias from time to time. Naturally and understandably, they resent this. This is what I meant in the other thread when I said that I thought the tension between black communities and the police was inevitable. I blame powerful dead white men. If you could create a White Man's Burden with John Travolta world, you would get the same result except the police bias would be against military-age white males.

          8. Okay, to recap: We're all clear that black people are more (let's say X times as) likely to commit violent crimes than white people. And we're all clear that black people are more likely (Y times) to be victims of police violence than white people.

            Bellmore's claim is that X is roughly equal to Y. So I begin by asking "How sure are we that we've got a reasonably solid value of X?" Bellmore says "Real certain, check out page 5 of BJS." I say "Okay, I've looked, and I'm not seeing it. Help me out?"

            Leroy argues that liberals would surely expect that X is significantly greater than 1.0, because very good reasons.

            I find this to be entirely persuasive, but I'm still not clear on the value of X. Which, recall, is important because the key issue is Bellmore's claim that X≈Y.

            So you've convinced me: X surely >> 1.0. Can we nail it down any further than that?

          9. Oh I see, you are after the ratio.

            Uhhh, interesting question. IIRC, the ratio for violence exchanged between blacks and the police is pretty close to 1… by which I mean, police shoot black people at a rate commensurate with the rate that black people shoot at police. It was a stat that got some major media play just after the Ferguson shooting: I believe it was, blacks are 5 times more likely to be shot per capita than whites (by police), and people who shoot at police are 6 times more likely (per capita) to be black rather than white. I can't do the math right now, but I think the murders in general (not specifically against police) stat breaks down to a similar effect size, something like blacks are 5 times more likely per capita to commit murder relative to whites.

          10. My tentative explanation is, roughly: Slavery and Jim Crow left blacks in America with a diminished cultural capital, and disproportionately poor. They were, however, overcoming this, when along came the Great Society and the war on poverty, which, though well intended, did horrible damage to the poor, and thus disproportionately to blacks.

            Welfare programs subsidized single motherhood, and encouraged people to stay in places where there was little prospect of lawful, gainful employment. So a couple of generations of blacks were raised without good role models in single parent families.

            The war on drugs provided inner city gangs with a lucrative income, allowing them to grow and fill the void.

            The end result was that, at least in the inner cities, a very dysfunctional culture was created an anti culture, even. Which is now even being exported to the rest of society.

            The rest of society reacts badly to this, but, what are we to do, embrace it? The barbarians are inside the gates, and we made them ourselves. How do we unmake them? I haven't a clue.

          11. You really think the Great Society did more damage than redlining, restrictive covenants, block busting, predatory lending, and Jim Crow, all of which basically remained in effect right up until the beginning of the Great Society programs? I won't disagree that there were some perverse incentives in Great Society programs, although I suspect I disagree with you about the ratio of costs to benefits in those programs.

          12. Yes, I do. You can look at indicators like the rate of unwed motherhood, for evidence of this.

            It isn't as though this view of the problem is out of the blue.

          13. Well, we already know that white people often lie about the races of *their* assailants, blaming them on fake black or Hispanic people, which causes the police to go victimize more black and Hispanic people. They even blame fake black people for crimes they commit themselves!

          14. "They even blame fake black people for crimes they commit themselves!"

            Gasp! When people lie to the police and give false profiles of criminals, they pick the most common profile of a criminal!??? Shocking!

            If I were going to lie to cover a crime of my own, I would blame an elderly asian female, because I am PC bro. Totally PC.

      1. Thanks. On reading it, I was particularly struck by two points:
        1) the fact that in the median county the odds of being shot by police when black and unarmed were roughly the same as when white and armed (with some counties having a much high relative risk for black and unarmed).
        2) the difference in the relative risks for white suspects depending on whether they were armed and the relative risk for black suspects. This is probably not good for police in the long run, because if you're black and worried about being shot by the police, you're not necessarily going to reduce your risk substantially by eschewing a gun.

        1. Good point! I also think that there's another way in which this is bad for police in the long run: all of Tom Tyler's work on procedural justice. If people see the legal system as being unfair, they are both less likely to turn to it when they are victimized (which is bad for public safety, since (I hope) it's obvious that victims of crime are equally important, regardless of race), and because if people trust cops, they cooperate in investigations, give leads, etc., all of which aids in law enforcement.

          Put more succinctly, you don't win hearts and minds by hassling people, treating them all with suspicion, and using force. Maybe we'll eventually learn this lesson.

  2. The equation you use has p(pos|D) twice. I assume the one on the left should be p(D|pos) instead?

    1. Thanks for the keen eyes! Actually, the problem was on the left side of the equation. Fixed!

  3. If people are describing this study accurately, it seems to be indicating that the police are *not* more likely to shoot black people than white people, in similar situations. You seem to be saying this is not news. I think it's kind of huge.

    Which isn't at all to discount your very important points. This is a very helpful post and I hope a ton of people read it.

    One study can't answer all our questions and it doesn't sound as if the author ever claimed that.

    When you ask, what's the real question we want answered… you know, I'm going to have to think about that. While it might be progress if we someday say, Americans' likelihood of being shot by police is now perfectly equal *might* be an improvement, from a certain point of view… I could see that also going South. Certainly, we don't want to *increase* it for non-African Americans, right? ; ) Here I use an emoticon, since we know from the past that I am pretty much embarrassment-proof! Well, anyway, your question about What is the Question is a good one. I will have to think some more.

      1. Hmm. Well, you know I can't say graphs are my forte. The column on the left is in hundred thousands – okay. The second column is in ______ ?

        I'm not sure what I'm looking at, though I see that there appears to be data indicating that black suspects? (but are they even suspects? something about that doesn't sound right… maybe it seems unfair since they never get a trial…) are getting shot by police more often, compared to the reported race of assailants from crime surveys, many/most of whom are not ever caught, presumably… so, how sure are we about this data anyway, and what does it mean?

        As far as biases, I guess I would have expected that there would be a difference, as your data show, but I'm not sure I think this data is as good as the data of the study described in this post, which seemed possibly more rigorously compiled.

        I was running around all day today and didn't get to figuring out what the question is… but I think a piece of it is, is the amount of police attention given to a group proportional to … the amount of help the other people in that group ask for? But then again, allowing a group of people to decide for themselves how much policing they want relies heavily on residential segregation, which I believe we are all supposed to be against. Which isn't to sound like one of those people who want to bang on about black-on-black crime, since being more focused on police makes perfect sense to me, since the police officer is supposed to care about you, and a gangbanger isn't (expected to). F.e., when Alicia Garza says she wants to see police budgets cut in half, I think, fine… in her neighborhood maybe, if the people there vote on it. (No idea where she lives actually. Just a thought experiment.) *My* city is under-policed. People can send their police over to my neighborhood if they want, bc we are having a crime wave. No joke.

        Or maybe we just need more Latino and Black officers, more community policing, and a lot more talking. I just caught a bit of the President's town hall. What a nice man he is to do this. Oh, also…. jobs.

    1. David isn't saying it's not news; he's saying that it doesn't indicate "that the police are *not* more likely to shoot black people than white people, in similar situations." It only does so because the authors used an improper conditional probability by only looking at the situation after the police have already made a stop. Since there is such a large disparity in who gets stopped and for what reasons, the population of police stops of white people is not similar to that of black people being stopped.

      1. All due respect… something isn't "improper" bc you don't like it. Ball is correct to point out when a study is being misinterpreted and I think it's very useful.

        Having said that, I think you're mischaracterizing Ball's post. He's not attacking the study *as I described it.* He's saying there are such and such larger important questions which it does not answer. But saying that officers are *not* more likely to shoot black people than white people, in similar situations – bearing in mind that there are unfairness questions related to an individual's risk of *being put in,* or *getting into* such situations – I think is still pretty big effing news. Now, is that not what the study says, Professor?

        1. He can reply if he wants, but I really have no idea how you can read the post and come to the conclusion you have. How you get from, "Again, there is nothing wrong with Fryer’s answer about the percentage of stops that are lethal—it’s just not a very interesting question, and it lends itself to misunderstanding," to, "He's not attacking the study *as I described it.*," eludes me. The whole point of the discussion of conditional probabilities is about the fact that, just because you can correctly derive an answer, it doesn't mean that the question you asked isn't flawed.

          In this case, the "startling" answer comes about only because the questioning starts in the wrong place. I'll offer a hypothesis as to how this is so, without claiming that is correct, merely possible: whites stopped by police are more likely to get shot because they are more likely to be actually violent criminals, whereas a lot of the black people stopped aren't criminals and had no inclination to react violently. If that is true, then the stops of white people are not similar situations to the stops of blacks. There is some hint of that in the data that white people stopped and searched by police are more likely to be carrying illegal items than are black people stopped, but there are other hypotheses one could come up with as to why the discrepancy exists. The assumption you make that they are similar situations is just that: an assumption, and that is what we are questioning, and if it doesn't hold, then the study isn't very useful.

          1. How is there no context if he's using Stop and Frisk data? and measuring the levels of force? Let me put it this way – having skimmed the first few pages of the study draft, I see someone who tried to get the best data he could, and was able to distinguish between the police interaction experiences of thousands of black and white New Yorkers. You don't like the other things he came up with? Fine. But if you think your answer to me was some kind of slam dunk, well I don't think so.

            Then again, I'm *not* a statistitian, so I am probably missing the finer points. Too bad that *nobody* on this blog can make them relatable. I am not loving the attitude, from either of you. And I don't find snotty people persuasive as a general rule. Have a nice day, I'm kinda done with you.

          2. Sorry, but I tried doing it the polite way, and I got, "All due respect… something isn't "improper" bc you don't like it." As far as I can tell, no one ever uses, "All due respect . . ." unless they mean nothing of the sort. So, you need to clean up your own act before going after other people.

            As for trying to explain it, I don't know how I could be any clearer than I was with my hypothetical. When you have such a huge disparity among races when it comes to police stops, you cannot use "people stopped by the police" as the population for your study unless you can explain and control for that disparity. If you don't control for it, the reason(s) for the disparity almost certainly mean that you aren't measuring the same thing when you look at whites as you are when you look at blacks. It's likely that the kinds of people being stopped, on average, aren't the same, and/or that the police doing the stopping aren't the same, and/or that what they were stopped for isn't the same, and/or any number of other things that differ and mean that you are looking at two different things.

            As I said, I don't know how to make it any clearer than that.

          3. I came up with another way to try to explain this.

            When you construct correlations, of which this is a very simple example, you have an independent variable and a dependent variable.* The independent variable is your input; in this case, the independent variable is the race of the individual who is stopped. The dependent variable is your output; in this case, the outcome of the stop, whether it be no violence, the use of non-lethal violence, the use of lethal violence, and whatever other categories were used.

            In order for the correlations you generate to have any validity, the data within one category of the independent variable can't differ from the data within the other categories except in the ways you are actually measuring.** So, to be valid, all of the possible factors that could influence the outcome other than the race of the person stopped have to be the same, on average and variance, for the pool of blacks stopped as it is for whites stopped, other than the race and the outcome. If they're divergent, then you can't have confidence in the correlations you generate. All of the other variables that you didn't include in your calculations have a high probability of being more important on the divergent outcomes of each stop than the one you did include.

            In this case, the data on how frequently each race is stopped strongly suggests that there is other information buried in that divergence that is influencing the outcomes. If the authors don't attempt to figure out what that information is and control for it***, then their results aren't valid. Simple, two-factor analysis has to be justified, because, as liberals are prone to saying, the universe is more complicated than that.

            *You can have multiple independent variables, but let's not get into that right now; as I said, this is a very simple example.

            **Again, the full picture is more complicated than this. There are rules for just how much they can differ and still be valid, and complicated mathematical tests for which independent variables you should discard.

            ***"Control for it" means including those factors as additional independent variables in the early stages of your analysis and running those mathematical tests to see if it's appropriate to discard them.

  4. No, I am not, I was trying to make a point about obviousness and unproductive comments.

  5. These are not really the right analysis techniques. What we're really interested in here is a particular causal pathway: police prejudice–> shooting blacks. The techniques for extracting such causal information do not amount to just giving conditional probabilities P(A|B)for some obvious choices of possible B. Usually you need at least some sort of inverse propensity weighting or more sophisticated techniques.

    1. Why are we only interested in one causal pathway? Don't dead bodies matter even if you can't blame them on police racism?

      If we could do something about the "Blacks being criminals leads to their being shot." pathway, the impact would be hugely greater than that of eliminating the last vestiges of police racism. In fact, it would help to eliminate them!

      Really, isn't police racism just a distraction here, diverting our concern from the real problem?

      1. Of course we're interested in all those pathways. Obviously I meant that the current studies were intended to evaluate one of them. Others can also be evaluated. Unbiased estimates of effects for each path require using the right techniques, usually not simple conditional probabilities. My comment was technical, not political.

  6. The business about encounters could be even more important, I think, than David says. The result looks like a possible example of diminishing marginal "returns."

    Let's assume that the probability of there being a shooting depends, in part, on the seriousness of what induced the encounter. That seems reasonable to me. An apparent drug deal is more likely to lead to trouble than jaywalking, or some guys just drinking beer. Now, if the police are more likely to stop blacks for minor reasons, then a higher percentage of such stops will end without violence, leading to the result Fryer reports.

    That paragraph is not altogether clear, even to me, so let's try an example. Imagine two communities of equal size, one 100% white and the other 100% black. In the white community police make 25 stops, of which 20 are based on something serious and five are for minor reasons. In the black community the numbers are 50 stops, 30 serious, 20 minor.

    Now if a serious top has a 10% chance of leading to a shooting, while a minor stop has 0%, then three blacks and two whites will be shot. But 8% (2/25) of encounters with whites will lead to shootings while only 6% (3/50) of encounters with blacks will.

    What this means is that Fryer's result is exactly what we would expect if police were in the habit of hassling blacks for things they don't hassle whites for, and if such encounters are much less likely to lead to shootings than more justified ones.

    Concavity is important.

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