Is county-level racial bias in police shootings unrelated to race-specific crime rates?

Yesterday, I suggested that the overrepresentation of blacks among the victims of police shootings may not be primarily attributable to racial animus on the parts of police officers. In response, a paper by Ross (2015; PLOS ONE), which claims the following, was brought to my attention:

There is no relationship between county-level racial bias in police shootings and crime rates (even race-specific crime rates), meaning that the racial bias observed in police shootings in this data set is not explainable as a response to local-level crime rates.

To his credit, Ross made his dataset publicly available (see S1_File.zip in the Supporting Information), so I carried out a few analyses on it myself. 

The total number of victims in the dataset with known race is 647. There are 260 counties with at least one police shooting of a victim with known race (8% of total counties). 23% of counties have 100% black victims, 55% of counties have 100% white victims, and the remaining 23% have some mix of black and white victims. The mean percentage of victims who are black is 33%. Furthermore, 90% of counties have five or fewer victims, 80% have three or fewer, and 57% have exactly one victim; the mean number of victims per county is 2.5. 

The low number of victims per county seems to me to be a serious limitation of Ross's analysis. How can one reliably model an aggregate-level variable (such as the ratio of black to white victims, or the proportion of victims who are black) that, for most counties, is based on only one or two observations? 

I calculated the 95% confidence interval for the proportion of victims who are black in all 260 counties. (Thanks to Emil Kirkegaard for calculating confidence intervals for cases where p = 1 using the prop.test in R.) This confidence interval overlapped with p = .13 (the percentage of blacks in the general population) in 87% of counties. And when restricting the analysis to unarmed victims, the confidence interval for the proportion of victims who are black overlapped with p = .13 in 89% of counties. In other words, for >85% of counties, one cannot rule out that the true proportion of victims who are black is equal to the proportion of blacks in the general population.

The limitation of low numbers of cases per county was not lost on Ross. In the Methods section he notes that:

County-level police shooting rates are estimated using binomial probabilities, and a prior, estimated from the data, under hierarchical partial pooling. Hierarchical pooling allows information collected in other counties within the United States to partially inform the parameter estimates in a focal county, which improves out-of-sample predictive inference globally... Prior to the introduction of multi-level modeling methods, relative risk ratios at local levels were very hard to infer... The multi-level Bayesian methods used here, partially (rather than fully) pool information across counties, allowing for more stable estimates in relative risk ratios

Not being familiar with these methods myself, I am not in a particularly strong position to judge their efficacy. But it seems to me that Ross's dependent variable will have been subject to considerable measurement error. Therefore, I'm not sure he can really be confident that there is no relationship between racial bias in police shootings and race-specific crime rates at the county level. This is not to say that there is a relationship, but simply that the evidence Ross presents for there not being one may not be very compelling. I would interested to hear others' perspectives. (Stata .do file available upon request.)

Does racial animus explain killings of black civilians by US police?

This post examines the distribution of victims of police shootings by race, and by sex. Data on individuals killed by police were taken from the Washington Post database for 2015 and 2016. According to these data, over the last two years, 27% of those killed by police were black, and 39% of those killed by police while unarmed were black. Insofar as blacks represent only 13% of the US population, this implies that blacks are overrepresented among the victims of police shootings.

However, blacks are––for whatever reason––also overrepresented among the perpetrators of violent crime. It is possible that blacks are more likely to be killed by police because they are more likely to get into violent confrontations with police, or because police officers practice statistical discrimination. The chart below shows, from left to right, the racial distribution of: the US population (taken from the US Census Bureau); those killed by police; those killed by police while unarmed; murder offenders (taken from the FBI); and alleged police killers (taken from the FBI––averaging over the last five years of available data was done to obviate sampling error). 

Blacks represent 13% of the US population, 27% of those killed by police, 39% of those killed by police while unarmed, 53% of murder offenders, and 39% of alleged police killers. These figures suggest that blacks may not be overrepresented among the victims of police shootings once involvement in murder or police killings is adjusted for. By way of comparison, consider the corresponding distributions by sex, which are shown in the chart below. Men are much more likely to be killed by police than women. But they are about as likely to be killed by police as one would expect on the basis of their involvement in murder or police killings. 

Given the highly sensitive nature of the subject matter, some caveats are in order. First, I am not arguing that blacks deserve to be killed more by police than members of other races. Rather, I am simply pointing out that the overrepresentation of blacks among the victims of police shootings may not be primarily attributable to racial animus on the parts of police officers. Second, I am not denying that there are cases of racially motivated violence against blacks by police officers. Such cases are of course deplorable. Third, I am not denying that there is a problem with police violence in the United States. The rate of police shootings in the US seems disproportionate even to the US's comparatively high homicide rate.

Finally, I am happy to send the dataset I have assembled to anyone who wants it.