Use circles instead of choropleth for MSAs

We are homeschooling the kiddo at the moment (the plunge was reading by Bryan Caplan’s approach, and seeing with online schooling just how poor middle school education was). Wife is going through AP biology at the moment, and we looked up various job info on biomedical careers. Subsequently came across this gem of a map of MSA estimates from the Bureau of Labor Stats (BLS) Occupational Employment and Wage Stats series (OES).

I was actually mapping some metro stat areas (MSAs) at work the other day, and these are just terrifically bad geo areas to show via a choropleth map. All choropleth maps have the issue of varying size areas, but I never realized having somewhat regular borders (more straight lines) makes the state and county maps not so bad – these MSA areas though are tough to look at. (Wife says it scintillates for her if she looks too closely.)

There are various incredibly tiny MSAs next to giant ones that you will just never see in these maps (no matter what color scheme you use). Nevada confused for me quite a bit, until I zoomed in to see that there are 4 areas, Reno is just a tiny squib.

Another example is Boulder above Denver. (Look closely at the BLS map I linked, you can just make out Boulder if you squint, but I cannot tell what color it corresponds to in the legend.) The outline heavy OES maps, which are mostly missing data, are just hopeless to display like this effectively. Reno could be the hottest market for whatever job, and it will always be lost in this map if you show employment via the choropleth approach. So of course I spent the weekend hacking together some maps in python and folium.

The BLS has a public API, but I was not able to find the OES stats in that. But if you go through the motions of querying the data and muck around in the source code for those queries, you can see they have an undocumented API call to generate json to fill the tables. Then using this tool to convert the json calls to python (thank you Hacker News), I was able to get those tables into python.

I have these functions saved on github, so check out that source for the nitty gritty. But just here quickly, here is a replicated choropleth map, showing the total employees for bio jobs (you can go to here to look up the codes, or run my function bls_maps.ocodes() to get a pandas dataframe of those fields).

# Creating example bls maps
from bls_geo import *

# can check out
bio = '172031'
bio_stats = oes_geo(bio)
areas = get_areas() # this takes a few minutes
state = state_albers()
geo_bio = merge_occgeo(bio_stats,areas)

ax = geo_bio.plot(column='Employment',cmap='inferno',legend=True,zorder=2)
ax.set_xlim(-0.3*1e7,0.3*1e7)   # lower 48 focus (for Albers proj)

And that is not much better than BLSs version. For this data, if you are just interested in looking up or seeing the top metro areas, just doing a table, e.g. above geo_bio.to_excel('biojobs.xlsx'), works just as well as a map.

So I was surprised to see Minneapolis pop up at the top of that list (and also surprised Raleigh doesn’t make the list at all, but Durham has a few jobs). But if you insist on seeing spatial trends, I prefer to go the approach of mapping proportion or graduate circles, placing the points at the centroid of the MSA:

att = ['areaName','Employment','Location Quotient','Employment per 1,000 jobs','Annual mean wage']
form = ['',',.0f','.2f','.2f',',.0f']

map_bio = fol_map(geo_bio,'Employment',['lat', 'lon'],att,form)'biomap.html')
map_bio #if in jupyter can render like this

I am too lazy to make a legend, you can check out nbviewer to see an interactive Folium map, which I have tool tips (similar to the hover for the BLS maps).

Forgive my CSS/HTML skills, not sure how to make nicer popups. So you lose the exact areas these MSA cover in this approach, but I really only expect a general sense from these maps anyway.

These functions are general enough for whatever wage series you want (although these functions will likely break when the 2021 data comes out). So here is the OES table for data science jobs:

I feel going for the 90th percentile (mapping that to the 10 times programmer) is a bit too over the top. But I can see myself reasonably justifying 75th percentile. (Unfortunately these agg tables don’t have a way to adjust for years of experience, if you know of a BLS micro product I could do that with let me know!). So you can see here the somewhat inflated salaries for the SanFran Bay area, but not as inflated as many might have you think (and to be clear, these are for 2020 survey estimates).

If we look at map of data science jobs, varying the circles by that 75th annual wage percentile, it looks quite uniform. What happens is we have some real low outliers (wages under 70k), resulting in tiny circles (such as Athen’s GA). Most of the other metro regions though are well over 100k.

In more somber news, those interactive maps are built using Leaflet as the backend, which was create by a Ukranian citizen, Vladimir Agafonkin. We can do amazing things with open source code, but we should always remember it is on the backs of someones labor we are able to do those things.

Simulating data with numpy and scipy

I’ve answered a few different questions on forums recently about simulating data:

I figured it would be worth my time to put some of my notes down in a blog post. It is not just academic, not too infrequently I use simulations at work to generate estimated return on investment estimates for putting a machine learning model into production. For social scientists this is pretty much equivalent to doing cost/benefit policy simulations.

Simulations are also useful for testing the behavior of different statistical tests, see prior examples on my blog for mixture trajectories or random runs.

Here I will be showing how to mostly use the python libraries numpy and scipy to do various simulation tasks. For some upfront (note I set the seed in numpy, important for reproducing your simulations):

import itertools as it
import numpy as np
import pandas as pd

# total simulations for different examples
n = 10000

# helper function to pretty print values
def pu(x,ax=1):
  te1,te2 = np.unique(x.sum(axis=ax),return_counts=True)
  te3 = 100*te2/te2.sum()
  res = pd.DataFrame(zip(te1,te2,te3),columns=['Unique','Count','Percent'])
  return res

Sampling from discrete choice sets

First, in the linked data science post, the individual had a very idiosyncratic set of simulations. Simulate a binary vector of length 10, that had 2,3,4 or 5 ones in that vector, e.g. [1,1,0,0,0,0,0,0,0,0] is a valid solution, but [0,0,0,0,1,0,0,0,0,0] is not, since the latter only has 1 1. One way to approach problems like these is to realize that the valid outcomes are a finite number of discrete solutions. Here I use itertools to generate all of the possible permutations (which can easily fit into memory, only 627). Then I sample from that set of 627 possibilities:

# Lets create the whole sets of possible permutation lists
res = []
zr = np.zeros(10)
for i in range(2,6):
    for p in it.combinations(range(10),i):
        on = zr.copy()
        on[list(p)] = 1

resnp = np.stack(res,axis=0)

# Now lets sample 1000 from this list
total_perms = resnp.shape[0]
samp = np.random.choice(total_perms,n)
res_samp = resnp[samp]

# Check to make sure this is OK

Ultimately you want the simulation to represent reality. So pretend this scenario was we randomly pick a number out of the set {2,3,4,5}, and then randomly distribute the 1s in that length 10 vector. In that case, this sampling procedure does not reflect reality, because 2’s have fewer potential permutations than do 5’s. You can see this in the simulated proportions of rows with 2 (7.25%) vs rows with 5 (39.94%) in the above simulation.

We can fix that though by adjusting the sampling probabilities from the big set of 627 possibilities though. Pretty much all of the np.random methods an option to specify different marginal probabilities, where in choice it defaults to equal probability.

# If you want the different subsets to have equal proportions
sum_pop = resnp.sum(axis=1)
tot_pop = np.unique(sum_pop,return_counts=True)
equal_prop = 1/tot_pop[1].shape[0]
pop_prob = pd.Series(equal_prop/tot_pop[1],index=tot_pop[0])
long_prob = pop_prob[sum_pop]

samp_equal = np.random.choice(total_perms,n,p=long_prob)
res_samp_equal = resnp[samp_equal]

So now we can see that each of those sets of results have similar marginal proportions in the simulation.

You can often figure out exact distributions for your simulations, for an example of similar discrete statistics, see my work on small sample Benford tests. But I often use simulations to check my math even if I do know how to figure out the exact distribution.

Another trick that I commonly use in other applications that don’t have access to something equivalent to np.random.choice, but most applications have a random uniform generator. Basically you can generate random numbers on whatever interval, chunk those up into bits, and turn those bits into your resulting categories. This is what I did in that SPSS post at the beginning.

unif = np.floor(np.random.uniform(1,33,(32*n,1)))/2

But again this is not strictly necessary in python because we can generate the set/list of items and sample from that directly.

# Same as using random choice from that set of values
half_vals = np.arange(1,33)/2
unif2 = np.random.choice(half_vals,(32*n,1))

But if limited in your libraries that is a good trick to know. Note that most random number generators operate over (0,1), so open intervals and will never generate an exact 0 or 1. To get a continuous distribution over whatever range, you can just multiply the 0/1 random number generator (and subtract if you need negative values) to match the range you want. But again most programs let you input min/max in a uniform number generator.

Sampling different stat distributions

So real data often can be reasonably approximated via continuous distributions. If you want to generate different continuous distribtions with approximate correlations, one approach is to:

  • generate multi-variate standard normal data (mean 0, variance 1, and correlations between those variables)
  • turn that matrix into a standard uniform via the CDF function
  • then for each column, use the inverse CDF function for the distribution of choice

Here is an example generating a uniform 0/1 and a Poisson with a mean of 3 variable using this approach.

from scipy.stats import norm, poisson

# Here generate multivariate standard normal with correlation 0.2
# then transform both to uniform
# Then transform 2nd column to Poisson mean 3

mu = [0,0]
cv = [[1,0.2],[0.2,1]] #needs to be symmetric
mv = np.random.multivariate_normal([0,0],cov=cv,size=n)
umv = pd.DataFrame(norm.cdf(mv),columns=['Uniform','Poisson'])
umv['Poisson'] = poisson.ppf(umv['Poisson'],3)

This of course doesn’t guarantee the transformed data has the exact same correlation as the original specified multi-variate normal. (If interested in more complicated scenarios, it will probably make sense to check out copulas.)

Like I mentioned in the beginning, I am often dealing with processes of multiple continuous model predictions. E.g. one model that predicts a binary ‘this claim is bad’, and then a second model predicting ‘how bad is this claim in terms of $$$’. So chaining together simulations of complicated data (which can be mixtures of different things) can be useful to see overall behavior of a hypothetical system.

Here I chain together a predicted probability for 3 claims (20%,50%,90%), and mean/standard deviations of (1000,100),(100,20),(50,3). So pretend we can choose 1 of these three claims to audit. We have the predicted probability that the claim is wrong, as well as an estimate of how much money we would make if the claim is wrong (how wrong is the dollar value).

The higher dollar values have higher variances, so do you pick the safe one, or do you pick the more risky audit with higher upside. We can do a simulation to see the overall distribution:

pred_probs = np.array([0.2,0.5,0.9])
bin_out = np.random.binomial(n=1,p=pred_probs,size=(n,3))
print( bin_out.mean(axis=0) )

# Pretend have both predicted prob
# and distribution of values
# can do a simulation of both

val_pred = np.array([1000,100,50])
val_std = np.array([100,20,3])
val_sim = np.random.normal(val_pred,val_std,size=(n,3))

revenue = val_sim*bin_out

I could see a reasonably risk averse person picking the lowest dollar claim here to audit. Like I said in the discrete case, we can often figure out exact distributions. Here the expected value is easy, prob*val, the standard deviation is alittle more tricky to calculate in your head (it is a mixture of a spike at 0 and then the rest of the typical distribution):

# Theoretical mean/variance
expected = pred_probs*val_pred
low_var = (expected**2)*(1-pred_probs)
hig_var = ((val_pred - expected)**2)*pred_probs
std_exp = np.sqrt(low_var + hig_var)

But it still may be useful to do the simulation here anyway, as the distribution is lumpy (so just looking at mean/variance may be misleading).

Other common continuous distributions I use are beta, to simulate between particular endpoints but not have them uniform. And negative binomial is a good fit to not only many count distributions, but things that are more hypothetically continuous (e.g. for distances instead of gamma, or for money instead of log-normal).

Here is an example generating a beta distribution between 0/1, with more of the distribution towards 0 than 1:

# mean probability 0.2
a,b = 1,5
beta_prob = beta.rvs(a, b, size=n)
plt.hist(beta_prob, bins=100,density=True,label='Sim')
# theoretical PDF
x = np.arange(0,1.01,0.01)
plt.plot(x,beta.pdf(x,a,b),label=f'beta({a},{b}) PDF')

For beta, the parameters a,b, the mean is a/b. To make the distribution more spread out, you have larger overall values of a/b (I don’t have the conversion to variance memorized offhand). But if you have real data, you can plop that into scipy to fit the parameters, here I fix the location and scale parameters.

# If you want to fit beta based on observed data
fitbeta =,floc=0,fscale=1)

We can do similar for negative binomial, I often think of these in terms of regression dispersion parameters, and I have previously written about translating mean/dispersion to n/p notation:

# Ditto for negative binomial
mean_nb = 2
disp_nb = 4

def trans_np(mu,disp):
    x = mu**2/(1 - mu + disp*mu)
    p = x/(x + mu)
    n = mu*p/(1-p)
    return n,p

nb_n,nb_p = trans_np(mean_nb,disp_nb)
nb_sim = nbinom.rvs(nb_n,nb_p,size=(n,1))
nb_bars = pu(nb_sim)['Unique'],nb_bars['Percent'],label='Sim')

x = np.arange(0,nb_sim.max()+1)
         label=f'PMF Negative Binomial')

Downloading geo files from Census FTP using python

I was working with some health data that only has MSA identifiers the other day. Not many people seem to know about the US Census’s FTP data site. Over the years they have had various terrible GUI’s to download data, but I almost always just go to the FTP site directly.

For geo data, check out for example. Python for pandas/geopandas also has the nicety that you can point to a url (even a url of a zip file), and load in the data in memory. So to get the MSA areas was very simple:

# Example download MSA
import geopandas as gpd
from matplotlib import pyplot as plt

url_msa = r''
msa = gpd.read_file(url_msa)

Sometimes the census has files spread across multiple states. So here is an example of doing some simple scraping to get all of the census tracts in the US. You can combine the geopandas dataframes the same as pandas dataframes using pd.concat:

# Example scraping all of the zip urls on a page
from bs4 import BeautifulSoup
import pandas as pd
import re
import requests

def get_zip(url):
    front_page = requests.get(url,verify=False)
    soup = BeautifulSoup(front_page.content,'html.parser')
    zf = soup.find_all("a",href=re.compile(r"zip"))
    # Maybe should use href 
    zl = [os.path.join(url,i['href']) for i in zf]
    return zl

base_url = r''
res = get_zip(base_url)

geo_tract = []
for surl in res:

geo_full = pd.concat(geo_tract)

# See State FIPS codes

geo_full[geo_full['STATEFP'] == '01'].plot()

Unfortunately for the census data tables, such as, those zip files contain two files (an estimate file and a margin of error file), so you cannot just do pd.read_csv(url) for those tables. But for the shapefile zip files this appears to work just fine and dandy.

I am currently working on a project at work (but Gainwell has given me the thumbs up to open source it) to build tables to create the CDC’s Social Vulnerability Index, which I can build for multiple geographies in combo with the census data. So hopefully in the next few weeks will be able to share that work.

Running files locally in SPSS

Say I made alittle python script for a friend to scrape data from a website whenever they wanted updates. I write my python script, say, and a run_scrape.bat file for my friend on their windows machine (or on Mac/Unix). And inside the bat file has the command:


I tell my friend save those two files in whatever folder you want, you just need to double click the bat file and it will save the scraped data into the same folder. Here the bat file is run locally, it sets the current directory to wherever the bat file is located.

This is how the majority of code is packaged – can clone from github or email a zip file, and it will just work no matter where the local user saves those scripts. I need to have my friend have their python environment set up correctly, but most of the stuff I do I can say download Anaconda and click yes to setting python on the path and they are golden.

SPSS makes things more painful, say I added SPSS to my environment variable in my windows machine, and I run from the command prompt an SPSS production job:

cd "C:\Users\Andrew"
spss print_dir.spj" -production silent

And say the spj file, all it does is call a syntax show.sps which has as the only command SHOW DIR. This still prints out wherever SPSS is installed as the current working directory inside of the SPSS session. On my machine currently C:\Program Files\IBM\SPSS Statistics. So SPSS takes over the location of the current directory. Also we can open up the spj file (it is just a plain text xml file). Here is what a current spj file looks like for me (note it is all on one line as well!):

And that file also has several hard coded file locations. So to get the same behavior as python earlier, we need to do dynamically set the paths in the production job as well, not just alter the command line scripts. This can be done with a little command line magic in windows, dynamically replacing the right text in the spj file. So in a bat file, you can do something like:

@echo on
set "base=%cd%"
:: code to define SPJ (SPSS production file)
echo ^<?xml version=^"1.0^" encoding=^"UTF-8^" standalone=^"no^"?^>^<job xmlns=^"^" codepageSyntaxFiles=^"false^" print=^"false^" syntaxErrorHandling=^"continue^" syntaxFormat=^"interactive^" unicode=^"true^" xmlns:xsi=^"^" xsi:schemaLocation=^"^"^>^<locale charset=^"UTF-8^" country=^"US^" language=^"en^"/^>^<output imageFormat=^"jpg^" imageSize=^"100^" outputFormat=^"text-codepage^" outputPath=^"%base%\job_output.txt^" tableColumnAutofit=^"true^" tableColumnBorder=^"^|^" tableColumnSeparator=^"space^" tableRowBorder=^"-^"/^>^<syntax syntaxPath=^"%base%\show.sps^"/^>^<symbol name=^"setdir^" quote=^"true^"/^>^</job^> > transfer_job.spj
"C:\Program Files\IBM\SPSS Statistics\stats.exe" "%base%\transfer_job.spj" -production silent -symbol @setdir "%base%"

It would be easier to use sed to find/replace the text for the spj file instead of the superlong one-liner on echo, but I don’t know if Window’s always has sed installed. Also note the escape characters (it is crazy how windows parses this single long line, apparently the max length is around 32k characters though).

You can see in the call to the production job, I pass a parameter, @setdir, and expand it out in the shell using %base%. In show.sps, I now have this line:

CD @setdir.

And now SPSS has set the current directory to wherever you have the .bat file and .sps syntax file saved. So now everything is dynamic, and runs wherever you have all the files saved. The only thing that is not dynamic in this setup is the location of the SPSS executable, stats.exe. So if you are sharing SPSS code like this, you will need to either tell your friend to add C:\Program Files\IBM\SPSS Statistics to their environment path, or edit the .bat file to the correct path, but otherwise this is dynamically run in the local folder with all the materials.

An update on the WaPo Officer Involved Shooting Stats

Marisa Iati interviewed me for a few clips in a recent update of the WaPo data on officer involved fatal police shootings. I’ve written in the past the data are very consistent with a Poisson process, and this continues to be true.

So first thing Marisa said was that shootings in 2021 are at 1055 (up from 1021 in 2020). Is this a significant increase? I said no off the cuff – I knew the average over the time period WaPo has been collecting data is around 1000 fatal shootings per year, so given a Poisson distribution mean=variance, we know the standard deviation of the series is close to sqrt(1000), which approximately equals 60. So anything 1000 plus/minus 60 (i.e. 940-1060) is within the typical range you would expect.

In every interview I do, I struggle to describe frequentist concepts to journalists (and this is no different). This is not a critique of Marisa, this paragraph is certainly not how I would write it down on paper, but likely was the jumble that came out of my mouth when I talked to her over the phone:

Despite setting a record, experts said the 2021 total was within expected bounds. Police have fatally shot roughly 1,000 people in each of the past seven years, ranging from 958 in 2016 to last year’s high. Mathematicians say this stability may be explained by Poisson’s random variable, a principle of probability theory that holds that the number of independent, uncommon events in a large population will remain fairly stagnant absent major societal changes.

So this sort of mixes up two concepts. One, the distribution of fatal officer shootings (a random variable) can be very well approximated via a Poisson process. Which I will show below still holds true with the newest data. Second, what does this say about potential hypotheses we have about things that we think might influence police behavior? I will come back to this at the end of the post,

R Analysis at the Daily Level

So my current ptools R package can do a simple analysis to show that this data is very consistent with a Poisson process. First, install the most recent version of the package via devtools, then you can read in the WaPo data directly via the Github URL:


url <- ''
oid <- read.csv(url,stringsAsFactors = F)

Looking at the yearly statistics (clipping off events recorded so far in 2022), you can see that they are hypothetically very close to a Poisson distribution with a mean/variance of 1000, although perhaps have a slow upward trend over the years.

# Year Stats
oid$year <- as.integer(substr(oid$date,1,4))
year_stats <- table(oid$year)
mean(year_stats[1:7]) # average of 1000 per year
var(year_stats[1:7])  # variance just under 1000

We can also look at the distribution at shorter time intervals, here per day. First I aggregat the data to the daily level (including 0 days), second I use my check_pois function to get the comparison distributions:

#Now aggregating to count per day
oid$date_val <- as.Date(oid$date)
date_range <- paste0(seq(as.Date('2015-01-01'),max(oid$date_val),by='days'))
day_counts <-$date,levels=date_range)))

pfit <- check_pois(day_counts$Freq, 0, 10, mean(day_counts$Freq))

The way to read this, for a mean of 2.7 fatal OIS per day (and given this many days), we would expect 169.7 0 fatality days in the sample (PoisF), but we actually observed 179 0 fatality days, so a residual of 9.3 in the total count. The trailing rows show the same in percentage terms, so we expect 6.5% of the days in the sample to have 0 fatalities according to the Poisson distribution, and in the actual data we have 6.9%.

You can read the same for the rest of the rows, but it is mostly the same. It is only very slight deviations from the baseline Poisson expected Poisson distribution. This data is the closest I have ever seen to real life, social behavioral data to follow a Poisson process.

For comparison, lets compare to the NYC shootings data I have saved in the ptools package.

# Lets check against NYC Shootings
date_range <- paste0(seq(as.Date('2006-01-01'),max(nyc_shoot$OCCUR_DATE),by='days'))
shoot_counts <-$OCCUR_DATE,levels=date_range)))

sfit <- check_pois(shoot_counts$Freq,0,max(shoot_counts$Freq),mean(shoot_counts$Freq))

This is much more typical of crime data I have analyzed over my career (in that it deviates from a Poisson process by quite a bit). The mean is 4.4 shootings per day, but the variance is over 13. There are many more 0 days than expected (433 observed vs 73 expected). And there are many more high crime shooting days than expected (tail of the distribution even cut off). For example there are 27 days with 18 shootings, whereas a Poisson process would only expect 0.1 days in a sample of this size.

My experience though is that when the data is overdispersed, a negative binomial distribution will fit quite well. (Many people default to a zero-inflated, like Paul Allison I think that is a mistake unless you have a structural reason for the excess zeroes you want to model.)

So here is an example of fitting a negative binomial to the shooting data:

# Lets fit a negative binomial and check out
fnb <- fitdist(shoot_counts$Freq,"nbinom")

sfit$nb <- 100*mapply(dnbinom, x=sfit$Int, size=fnb$estimate[1], mu=fnb$estimate[2])
round(sfit[,c('Prop','nb')],1) # Much better overall fit

And this compares the percentages. So you can see observed 7.5% 0 shooting days, and expected 8.6% according to this negative binomial distribution. Much closer than before. And the tails are fit much closer as well, for example, days with 18 shootings are expected 0.2% of the time, and are observed 0.4% of the time.

So What Inferences Can We Make?

In social sciences, we are rarely afforded the ability to falsify any particular hypothesis – or in more lay-terms we can’t really ever prove something to be false beyond a reasonable doubt. We can however show whether empirical data is consistent or inconsistent with any particular hypothesis. In terms of Fatal OIS, several ready hypotheses ones may be interested in are Does increased police scrutiny result in fewer OIS?, or Did the recent increase in violence increase OIS?.

While these two processes are certainly plausible, the data collected by WaPo are not consistent with either hypothesis. It is possible both mechanisms are operating at the same time, and so cancel each other out, to result in a very consistent 1000 Fatal OIS per year. A simpler explanation though is that the baseline rate has not changed over time (Occam’s razor).

Again though we are limited in our ability to falsify these particular hypotheses. For example, say there was a very small upward trend, on the order of something like +10 Fatal OIS per year. Given the underlying variance of Poisson variables, even with 7+ years of data it would be very difficult to identify that small of an upward trend. Andrew Gelman likens it to measuring the weight of a feather carried by a Kangaroo jumping on the scale.

So really we could only detect big changes that swing OIS by around 100 events per year I would say offhand. Anything smaller than that is likely very difficult to detect in this data. And so I think it is unlikely any of the recent widespread impacts on policing (BLM, Ferguson, Covid, increased violence rates, whatever) ultimately impacted fatal OIS in any substantive way on that order of magnitude (although they may have had tiny impacts at the margins).

Given that police departments are independent, this suggests the data on fatal OIS are likely independent as well (e.g. one fatal OIS does not cause more fatal OIS, nor the opposite one fatal OIS does not deter more fatal OIS). Because of the independence of police departments, I am not sure there is a real great way to have federal intervention to reduce the number of fatal OIS. I think individual police departments can increase oversight, and maybe state attorney general offices can be in a better place to use data driven approaches to oversee individual departments (like ProPublica did in New Jersey). I wouldn’t bet money though on large deviations from that fatal 1000 OIS anytime soon though.

Cointegration analysis of Ethereum and BitCoin

So a friend recently has heavily encouraged investment into Ethereum and NFTs. Part of the motivation of these cryptocurrencies is to be independent of fiat currency. So that lends itself to a hypothesis – are cryptocurrency prices and more typical securities independent? Or are we simply seeing similar trends in these different securities over time? This is a job for cointegration analysis. The python code is simple enough to follow along in a blog post.

So first I import the libraries I am using – it leverages the Yahoo finance API to download ticker data (here I analyze closing prices), and statsmodels to conduct the various analyses in python.

from datetime import datetime
import numpy as np
import pandas as pd
import yfinance as yf
import matplotlib.pyplot as plt

from statsmodels.tsa.stattools import adfuller, ccf
from statsmodels.tsa.api import VAR
from statsmodels.tsa.vector_ar import vecm

Now we can download the ticker data, here I will analyze BitCoin and Ethereum, along with Gold prices and the S&P 500 index fund.

# BTC-USD : Bitcoin
# ETH-USD : Ethereum
# ^GSPC ; S&P 500
# GC=F : Gold

end_date ="%Y-%m-%d")
print(end_date) #running as of 2/9/2022

tick_str = 'BTC-USD ETH-USD ^GSPC GC=F'
dat =,start='2017-01-01',end=end_date)

Now for data prep – I am going to interpolate missing data (for when the market was closed). Then I only subset out Fridays at close to conduct a weekly analysis. Even weekly is too short for me to bother with rebalancing if I do decide to invest.

# Fill in missing data before sub-sampling to once a week
dat2 = dat.interpolate()

# Only Fridays close post 11/9/2017
after = pd.to_datetime('2017-11-09')
sel = (dat2.Date >= after) & (dat2.Date.dt.weekday == 4) #Friday
sdat = dat2.loc[sel,['Date','Close']]
sdat.columns = ['Date'] + ['BitCoin','Eth','S&P 500','Gold']

Now lets look at the overall trends by superimposing these four stocks on the same graph. Just min-max normalizing to range from 0 to 1.

# Time Series Graphs of Each
# Normalized to be 0/1
snorm = sdat.copy()
for v in sdat:
    mi,ma = sdat[v].min(),sdat[v].max()
    snorm[v] = (sdat[v] - mi)/(ma-mi)

# All four series on the same graph

So you can see these all appear to follow a similar upward trajectory after Covid hit in 2020, although crypto has way more volatility recently. If we subset out just the crypto’s, we can see how they trend with each other more easily.

s2 = sdat.copy()
s2['BitCoin/10'] = s2['BitCoin']/10

So based on this, I would say that maybe Bitcoin is a leading indicator of Ethereum (increases in BitCoin precede increases in Eth with maybe just a week lag).

Typically with any time series analysis like this, we are concerned with whether the series are stationary. Just going off of my Ender’s Applied Econometric Time Series book, we typically look at the Adjusted Dickey-Fuller test for the levels:

# Integration analysis
adfuller(sdat['BitCoin'], regression='ct', maxlag=5, autolag='t-stat', regresults=True)

And we can see this fails to reject the null, so we would conclude the series is integrated. Since we have a fairly large sample here (over 200 weeks), the test should be reasonably powered. If we then take the first differences and conduct the same test, we then reject the null of an integrated series (here for Bitcoin).

# Create differenced data
sdiff = sdat.diff().dropna()

# All appear 1st order integrated!
adfuller(sdiff['BitCoin'], regression='ct', maxlag=5, autolag='t-stat', regresults=True)

So this reasonably suggests Bitcoin is an I(1) process. Doing the same for all of the other securities in this example you come to the same inference, all integrated of order 1 (which is very typical for stock data).

Using the differenced data, we can see the cross-correlations between different securities. In this example, it appears BitCoin/Ethereum just have a large 1 positive lag, and close to 0 after that.

# Only 1 lag positive in differenced data

So based on this, I subsequent only look at 1 lag in subsequent models. (Prior week impacts current week, since we are analyzing weekly data.)

So you need to be careful here – typically we want to avoid doing regression analysis of integrated time series, as that can lead to spurious correlations. But in the case a series is co-integrated, it is ok to conduct analysis on the levels. So here we do the analysis of the levels for each of the securities to assess our hypothesis. (Including temporal trends results in different coefficients, but similar overall inferences.)

mod = VAR(sdat)
res = #trend='ctt'

So we can see here that contrary to the graphs, Ethereum has a negative relationship with BitCoin – when Ethereum goes up a dollar, the following week BitCoin goes down $1.7. For BitCoin the relationships with S&P is negative (but weaker), and Gold it is positive.

# Ethereum causes BitCoin to go down
irf = res.irf(4)

For Ethereum the converse is not true though – BitCoin + increases Ethereum (although given that BitCoin is currently 10x the value of Eth the magnitude is smaller).

# Ethereum causes BitCoin to go down
irf = res.irf(4)

There are more formal tests to look at Granger causality and cointegration with error correction models, but looking at the VAR of the levels I think is the easiest to Grok here.

Do not take this as investment advice, looking at the volatility of these securities makes me very hesistant to invest even a small sum.

# Granger causality test
gc = res.test_causality('Eth', 'BitCoin', kind='f').summary()

# Cointegration test
ecm = vecm.coint_johansen(sdat[['BitCoin','Eth']], 1, 1)

ecm = vecm.VECM(sdat[['BitCoin','Eth']],deterministic='co')
est =


Based on this analysis it might make sense to include BitCoin as a portfolio diversification relative to traditional stocks – if willing to assume quite a bit of risk. If you are a gambler it may make sense to do some type of pairs trading strategy between Eth/Bitcoin on a short term basis. (If I had some real magic low risk money making strategy I would not put it in a blog post!)

Gambling is fun (and it is fun to think damn if I invested in Eth in 2019 I would be up 10x) – but I don’t think I am going onto the crypto roller-coaster at the moment.

Prediction Intervals for Random Forests

I previously knew about generating prediction intervals via random forests by calculating the quantiles over the forest. (See this prior python post of mine for getting the individual trees). A recent set of answers on StackExchange show a different approach – apparently the individual tree approach tends to be too conservative (coverage rates higher than you would expect). Those Cross Validated posts have R code, figured it would be good to illustrate in python code how to generate these prediction intervals using random forests.

So first what is a prediction interval? I imagine folks are more familiar with confidence intervals, say we have a regression equation y = B1*x + e, you often generate a confidence interval around B1. Imagine we use that equation to make a prediction though, y_hat = B1*(x=10), here prediction intervals are errors around y_hat, the predicted value. They are actually easier to interpret than confidence intervals, you expect the prediction interval to cover the observations a set percentage of the time (whereas for confidence intervals you have to define some hypothetical population of multiple measures).

Prediction intervals are often of more interest for predictive modeling, say I am predicting future home sale value for flipping houses. I may want to generate prediction intervals that cover the value 90% of the time, and only base my decisions to buy based on the much lower value (if you are more risk averse). Imagine I give you the choice of buy a home valuated at 150k - 300k after flipped vs a home valuated at 230k-250k, the upside for the first is higher, but it is more risky.

In short, this approach to generate prediction intervals from random forests relies on out of bag error metrics (it is sort of like a for free hold out sample based on the bootstrapping approach random forest uses). And based on the residual distribution, one can generate forecast intervals (very similar to Duan’s smearing).

To illustrate, I will use a dataset of emergency room visits and time it took to see a MD/RN/PA, the NHAMCS data. I have code to follow along here, but I will walk through it in this post (that code has some nice functions for data definitions for the NHAMCS data).

At work I am working on a project related to unnecessary emergency room visits, and I actually went to the emergency room in December (for a Kidney stone). So I am interested here in generating prediction intervals for the typical time it takes to be served in an ER to see if my visit was normal or outlying.

Example Python Code

First for some set up, I import the libraries I am using, and read in the emergency room use data:

import numpy as np
import pandas as pd
from nhanes_vardef import * #variable definitions
from sklearn.ensemble import RandomForestRegressor
from sklearn.model_selection import train_test_split

# Reading in fixed width data
# Can download this data from 
nh2019 = pd.read_fwf('ED2019',colspecs=csp,header=None)
nh2019.columns = list(fw.keys())

Here I am only going to work with a small set of the potential variables. Much of the information wouldn’t make sense to use as predictors of time to first being seen (such as subsequent tests run). One thing I was curious about though was if I changed my pain scale estimate would I have been seen sooner!

# PAINSCALE [- missing]
# VDAYR [Day of Week]
# VMONTH [Month of Visit]
# ARRTIME [Arrival time of day]
# AGE [top coded at 95]
# SEX [1 female, 2 male]
# IMMEDR [triage]
#  9 = Blank
#  -8 = Unknown
#  0 = ‘No triage’ reported for this visit but ESA does conduct nursing triage
#  1 = Immediate
#  2 = Emergent
#  3 = Urgent
#  4 = Semi-urgent
#  5 = Nonurgent
#  7 = Visit occurred in ESA that does not conduct nursing triage 

nh2019 = nh2019[keep_vars].copy()

Many of the variables encode negative values as missing data, so here I throw out visits with a missing waittime. I am lazy though and the rest I keep as is, with enough data random forests should sort out all the non-linear effects no matter how you encode the data. I then create a test split to evaluate the coverage of my prediction intervals out of sample for 2k test samples (over 13k training samples).

# Only keep wait times that are positive
mw = nh2019['WAITTIME'] >= 0
print(nh2019.shape[0] - mw.sum()) #total number missing
nh2019 = nh2019[mw].copy()

# Test hold out sample to show
# If coverage is correct
train, test = train_test_split(nh2019, test_size=2000, random_state=10)
x = keep_vars[1:]
y = keep_vars[0]

Now we can fit our random forest model, telling python to keep the out of bag estimates.

# Fitting the model on training data
regr = RandomForestRegressor(n_estimators=1000,max_depth=7,
  random_state=10,oob_score=True,min_samples_leaf=50)[x], train[y])

Now we can use these out of bag estimates to generate error intervals around our predictions based on the test oob error distribution. Here I generate 50% prediction intervals.

# Generating the error distribution
resid = train[y] - regr.oob_prediction_
# 50% interval
lowq = resid.quantile(0.25)
higq = resid.quantile(0.75)
# negative much larger
# so tends to overpredict time

Even 50% here are quite wide (which could be a function of both the data has a wide variance as well as the model is not very good). But we can test whether our prediction intervals are working correctly by seeing the coverage on the out of sample test data:

# Generating predictions on out of sample data
test_y = regr.predict(test[x])
lowt = (test_y + lowq).clip(0) #cant have negative numbers
higt = (test_y + higq)

cover = (test[y] >= lowt) & (test[y] <= higt)

Pretty much spot on. So lets see what the model predicts my referent 50% prediction interval would be (I code myself a 2 on the IMMEDR scale, as I was billed a CPT code 99284, which those should line up pretty well I would think):

# Seeing what my referent time would be
myt = np.array([[6,4,12,930,36,2,6]])
mp = regr.predict(myt)
print( (mp+lowq).clip(0), (mp+higq) )

So a predicted mean of 35 minutes, and a prediction interval of 4 to 38 minutes. (These intervals based on the residual quantiles are basically non-parametric, and don’t have any strong assumptions about the distribution of the underlying data.)

To first see the triage nurse it probably took me around 30 minutes, but to actually be treated it was several hours long. (I don’t think you can do that breakdown in this dataset though.)

We can do wider intervals, here is a screenshot for 80% intervals:

You can see that they are quite wide, so probably not very effective in identifying outlying cases. It is possible to make them thinner with a better model, but it may just be the variance is quite wide. For folks monitoring time it takes for things (whether time to respond to calls for service for police, or here be served in the ER), it probably makes sense to build models focusing on quantiles, e.g. look at median time served instead of mean.

Optimal and Fair Spatial Siting

A bit of a belated MLK day post. Much of the popular news on predictive or machine learning algorithms has a negative connotation, often that they are racially biased. I tend to think about algorithms though in almost the exact opposite way – we can adjust them to suit our objectives. We just need to articulate what exactly we mean by fair. This goes for predictive policing (Circo & Wheeler, 2021; Liberatore et al., 2021; Mohler et al., 2018; Wheeler, 2020) as much as it does for any application.

I have been reading a bit about spatial fairness in siting health resources recently, one example is the Urban Institutes Equity Data tool. For this tool, you put in where your resources are currently located, and it tells you whether those locations are located in areas that have demographic breakdowns like the overall city. So this uses the container approach (not distance to the resources), which distance traveled to resources is probably a more typical way to evaluate fair spatial access to resources (Hassler & Ceccato, 2021; Koschinsky et al., 2021).

Here what I am going to show is instead of ex-ante saying whether the siting of resources is fair, I construct an integer linear program to site resources in a way we define to be fair. So imagine that we are siting 3 different locations to do rapid Covid testing around a city. Well, we do the typical optimization and minimize the distance traveled for everyone in the city on average to those 3 locations – on average 2 miles. But then we see that white people on average travel 1.9 miles, and minorities travel 2.2 miles. So it that does not seem so fair does it.

I created an integer linear program to take this difference into account, so instead of minimizing average distance, it minimizes:

White_distance + Minority_distance + |White_distance - Minority_distance|

So in our example above, if we had a solution that was white travel 2.1 and minority 2.1, this would be a lower objective value than (4.2), than the original minimize overall travel (1.9 + 2.2 + 0.3 = 4.4). So this gives each minority groups equal weight, as well as penalizes if one group (either whites or minorities) has much larger differences.

I am not going to go into all the details. I have python code that has the functions (it is very similar to my P-median model, Wheeler, 2018). The codes shows an example of siting 5 locations in Dallas (and uses census block group centroids for the demographic data). Here is a map of the results (it has points outside of the city, since block groups don’t perfectly line up with the city boundaries).

In this example, if we choose 5 locations in the city to minimize the overall distance, the average travel is just shy of 3.5 miles. The average travel for white people (not including Hispanics) is 3.25 miles, and for minorities is 3.6 miles. When I use my fair algorithm, the white average distance is 3.5 miles, and the minority average distance is 3.6 miles (minority on average travels under 200 more feet on average than white).

So this is ultimately a trade off – it ends up pushing up the average distance a white person will travel, and only slightly pushes down the minority travel, to balance the overall distances between the two groups. It is often the case though that one can be somewhat more fair, but in only results in slight trade-offs though in the overall objective function (Rodolfa et al., 2021). So that trade off is probably worth it here.


Buffalo shootings paper published

My article examining spatial shifts in shootings in Buffalo pre/post Covid, in collaboration with several of my Buffalo colleagues, is now published in the Journal of Experimental Criminology (Drake et al., 2022).

If you do not have access to that journal, you can always just email, or check out the open access pre-print. About the only difference is a supplement we added in response to reviewers, including maps of different grid cell areas, here is a hex grid version of the changes:

The idea behind this paper was to see if given the dramatic increase in shootings in Buffalo after Covid started (Kim & Phillips, 2021), they about doubled (similar to NYC), did spatial hot spots change? The answer is basically no (and I did a similar analysis in NYC as well).

While other papers have pointed out that crime increases disproportionately impact minority communities (Schleimer et al., 2022), which is true, it stands to be very specific what the differences in my work and this are saying. Imagine we have two neighborhoods:

Neighborhood A, Disadvantaged/Minority, Pre 100 crimes, Post 200 crimes
Neighborhood B,    Advantaged/Majority, Pre   1 crimes, Post   2 crimes

The work that I have done has pointed to these increases due to Covid being that relative proportions/rates are about the same (shootings ~doubled in both Buffalo/NYC). And that doubling was spread out pretty much everywhere. It is certainly reasonable to interpret this as an increased burden in minority communities, even if proportional trends are the same everywhere.

This proportional change tends to occur when crime declines as well (e.g. Weisburd & Zastrow, 2022; Wheeler et al., 2016). And this just speaks to the stickiness of hot spots of crime. Even with large macro changes in temporal crime trends, crime hot spots are very durable over time. So I really think it makes the most sense for police departments to have long term strategies to deal with hot spots of crime, and they don’t need to change targeted areas very often.


Power and bias in logistic regression

Michael Sierra-Arévalo, Justin Nix and Bradley O’Guinn have a recent article about examining officer fatalities following gunshot assaults (Sierra-Arévalo, Nix, & O-Guinn). They do not find that distance to a Level 1/2 trauma ERs make a difference in the survival probabilities, which conflicts with prior work of mine with Gio Circo (Circo & Wheeler, 2021). Justin writes this as a potential explanation for the results:

The results of our multivariable analysis indicated that proximity to trauma care was not significantly associated with the odds of officers surviving a gunshot wound (see Table 2 on p. 9 of the post-print). On the one hand, this was somewhat surprising given that proximity to trauma care predicts survival of gunshot wounds among the general public.1 On the other hand, police have specialized equipment, such as ballistic vests and tourniquets, that reduce the severity of gunshot wounds or allow them to be treated immediately.

I think it is pretty common when results do not pan out, people turn to theoretical (or sociological) reasons why their hypothesis may be invalid. While these alternatives are often plausible, often equally plausible are simpler data based reasons. Here I was concerned about two factors, 1) power and 2) omitted severity of gun shot wound factors. I did a quick simulation in R to show power seems to be OK, but the omitted severity confounders may be more problematic in this design, although only bias the effect towards 0 (it would not cause the negative effect estimate MJB find).

Power In Logistic Regression

First, MJB’s sample size is just under 1,800 cases. You would think offhand this is plenty of power for whatever analysis right? Well, power just depends on the relevant effect size, a small effect and you need a bigger sample. My work with Gio found a linear effect in the logistic equation of 0.02 (per minute driving increases the logit). We had 5,500 observations, and our effect had a p-value just below 0.05, hence why a first thought was power. Also logistic regression is asymptotic, it is common to have small sample biases in situations even up to 1000 observations (Bergtold et al., 2018). So lets see in a simple example ignoring the other covariates:

# Some upfront work
logistic <- function(x){1/(1+exp(-x))}

# Scenario 1, no covariates omitted
n <- 2000; 
de <- 0.02
dist <- runif(n,5,200)
p <- logistic(-2.5 + de*dist)
y <- rbinom(n,1,p)

# Variance is small enough, seems reasonably powered
summary(glm(y ~ dist, family = "binomial"))

Here with 2000 cases, taking the intercept from MJB’s estimates and the 0.02 from my paper, we see 2000 observations is plenty enough well powered to detect that same 0.02 effect in mine and Gio’s paper. Note when doing post-hoc power analysis, you don’t take the observed effect (the -0.001 in Justin’s paper), but a hypothetical effect size you think is reasonable (Gelman, 2019), which I just take from mine and Gio’s paper. Essentially saying “Is Justin’s analysis well powered to detect an effect of the same size I found in the Philly data”.

One thing that helps MJB’s design here is more variance in the distance parameter, looking intra city the drive time distances are smaller, which will increase the standard error of the estimate. If we pretend to limit the distances to 30 minutes, this study is more on the fence as to being well enough powered (but meets the threshold in this single simulation):

# Limited distance makes the effect have a higher variance
n <- 2000; 
de <- 0.02
dist <- runif(n,1,30)
p <- logistic(-2.5 + de*dist)
y <- rbinom(n,1,p)

# Not as much variation in distance, less power
summary(glm(y ~ dist, family = "binomial"))

For a more serious set of analysis you would want to do these simulations multiple times and see the typical result (since they are stochastic), but this is good enough for me to say power is not an issue in this design. If people are planning on replications though, intra-city with only 1000 observations is really pushing it with this design though.

Omitted Confounders

One thing that is special about logistic regression, unlike linear regression, even if an omitted confounder is uncorrelated with the effect of interest, it can still bias the estimates (Mood, 2010). So even if you do a randomized experiment your effects could be biased if there is some large omitted effect from the regression equation. Several people interpret this as logistic regression is fucked, but like that linked Westfall article I think that is a bit of an over-reaction. Odds ratios are very tricky, but logistic regression as a method to estimate conditional means is not so bad.

In my paper with Gio, the largest effect on whether someone would survive was based on the location of the bullet wound. Drive time distances then only marginal pushed up/down that probability. Here are conditional mean estimates from our paper:

So you can see that being shot in the head, drive time can make an appreciable difference over these ranges, from ~45% to 55% probability of death. Even if the location of the wound is independent of drive time (which seems quite plausible, people don’t shoot at your legs because you are far away from a hospital), it can still be an issue with this research design. I take Justin’s comment about ballistic vests as reducing death as essentially taking the people in the middle of my graph (torso and multiple injuries) and pushing them into the purple line at the bottom (extremities). But people shot in the head are not impacted by the vests.

So lets see what happens to our effect estimates when we generate the data with the extremities and head effects (here I pulled the estimates all from my article, baseline reference is shot in head and negative effect is reduction in baseline probability when shot in extremity):

# Scenario 3, wound covariate omitted
dist <- runif(n,5,200)
ext_wound <- rbinom(n,1,0.8)
ef <- -4.8
pm <- logistic(0.2 + de*dist + ef*ext_wound)
ym <- rbinom(n,1,pm)

# Biased downward (but not negative)
summary(glm(ym ~ dist, family = "binomial"))

You can see here the effect estimate is biased downward by a decent margin (less than half the size of the true effect). If we estimate the correct equation, we are on the money in this simulation run:

What happens if we up the sample size? Does this bias go away? Unfortunately it does not, here is an example with 10,000 observations:

# Scenario 3, wound covariate ommitted larger sample
n2 <- 10000
dist <- runif(n2,5,200)
ext_wound <- rbinom(n2,1,0.8)
ef <- -4.8
pm <- logistic(0.2 + de*dist + ef*ext_wound)
ym <- rbinom(n2,1,pm)

# Still a problem
summary(glm(ym ~ dist, family = "binomial"))

So this omission is potentially a bigger deal – but not in the way Justin states in his conclusion. The quote earlier suggests the true effect is 0 due to vests, I am saying here the effect in MJB’s sample is biased towards 0 due to this large omitted confounder on the severity of the wound. These are both plausible, there is no way based just on MJB’s data to determine if one interpretation is right and the other is wrong.

This would not explain the negative effect estimate MJB finds though in their paper, it would only bias towards 0. To be fair, Jessica Beard critiqued mine and Gio’s paper in a similar vein (saying the police wound location data had errors), this would make our drive time estimates be biased towards 0 as well, so if that factor may be even larger than me and Gio even estimated.

Potential robustness checks here are to simply do a linear regression instead of logistic with the same data (my graph above shows a linear regression would be fine for the data if I included interaction effects with wound location). And another would be to look at the unconditional marginal distribution of distance vs probability of death. If that is highly non-linear, it is likely due to omitted confounders in the data (I suspect it may plateau as well, eg the first 30 minutes make a big difference, but after that it flattens out, you’ve either stabilized someone or they are gone at that point).


In the case of intra-city public violence, the policy implication of drive times on survival are relevant when people are determining whether to keep open or close trauma centers. I did not publish this in my paper with Gio (you can see the estimates in the replication code), but we actually estimated counter-factual increased deaths by taking away facilities. Its marginal effect is around 10~20 homicides over the 4.5 years if you take away one of the facilities in Philadelphia. I don’t know if reducing 5 homicides per year is sufficient justification to keep a trauma facility open, but officer shootings are themselves much less frequent, and so the marginal effects are very unlikely to justify keeping a trauma facility open/closed by themselves.

You could technically figure out the optimal location to site a new trauma facility from mine and Gio’s paper, but probably a more reasonable response would be to site resources to get people to the ER faster. Philly already does scoop and run (Winter et al., 2021), where officers don’t wait for an ambulance. Another possibility though is to proactively locate ambulances to get to scenes faster (Hosler et al., 2019). Again though it just isn’t as relevant/feasible outside of major urban areas though to do that.

Often times social science authors do an analysis, and then in the policy section say things that are totally reasonable on their face, but are not supported by the empirical analysis. Here the suggestion that officers should increase their use of vests by MJB is totally reasonable, but nothing in their analysis supports that conclusion (ditto with the tourniquets statement). You would need to measure those incidents that had those factors, and see its effect on officer survival to make that inference. MJB could have made the opposite statement, since drive time doesn’t matter, maybe those things don’t make a difference in survival, and be equally supported by the analysis.

I suspect MJB’s interest in the analysis was simply to see if survival rates were potential causes of differential officer deaths across states (Sierra-Arévalo & Nix, 2020). Which is fine to look at by itself, even if it has no obviously direct policy implications. Talking back and forth with Justin before posting this, he did mention it was a bit of prodding from a reviewer to add in the policy implications. Which it goes for both (reviewers or original writers), I don’t think we should pad papers with policy recommendations (or ditto for theoretical musings) that aren’t directly supported by the empirical analysis we conduct.


  • Bergtold, J. S., Yeager, E. A., & Featherstone, A. M. (2018). Inferences from logistic regression models in the presence of small samples, rare events, nonlinearity, and multicollinearity with observational data. Journal of Applied Statistics, 45(3), 528-546.
  • Circo, G. M., & Wheeler, A. P. (2021). Trauma Center Drive Time Distances and Fatal Outcomes among Gunshot Wound Victims. Applied Spatial Analysis and Policy, 14(2), 379-393.
  • Gelman, A. (2019). Don’t calculate post-hoc power using observed estimate of effect size. Annals of Surgery, 269(1), e9-e10.
  • Hosler, R., Liu, X., Carter, J., & Saper, M. (2019). RaspBary: Hawkes Point Process Wasserstein Barycenters as a Service.
  • Mood, C. (2010). Logistic regression: Why we cannot do what we think we can do, and what we can do about it. European Sociological Review, 26(1), 67-82.
  • Sierra-Arévalo, M., & Nix, J. (2020). Gun victimization in the line of duty: Fatal and nonfatal firearm assaults on police officers in the United States, 2014–2019. Criminology & Public Policy, 19(3), 1041-1066.
  • Sierra-Arévalo, Michael, Justin Nix, & Bradley O’Guinn (2022). A National Analysis of Trauma Care Proximity and Firearm Assault Survival among U.S. Police. Forthcoming in Police Practice and Research. Post-print available at
  • Winter, E., Hynes, A. M., Shultz, K., Holena, D. N., Malhotra, N. R., & Cannon, J. W. (2021). Association of police transport with survival among patients with penetrating trauma in Philadelphia, Pennsylvania. JAMA network open, 4(1), e2034868-e2034868.