Clusters of Jail Incarcerations in US Counties: 2010-2018

Abstract

This study investigates the spatiotemporal variations in jail incarcerations in addition to associations with several risk factors and jail incarceration counts at the county level for the period 2010-2018 in the contiguous USA. The disease surveillance software SaTScan^TM was utilized to identify and test purely spatial and spatiotemporal variations in jail incarceration. Significant spatial and space-time clusters with elevated relative risk for jail incarceration were found in analysis. Additionally, a negative binomial regression model was used to predict jail incarcerations counts based on several covariates and found significant and non-random spatial clusters of jail incarceration that are explained after adjusting for these covariates. The results in this study provide useful information on possible associations in geographical areas where jail incarceration rates are higher than expected and demonstrate significant correlations between jail incarceration counts and several covariates. The study and its conclusions provide an epidemiological framework for identifying and addressing geographic patterns of unusually high jail incarceration rates in the U.S. and provide evidence of appropriate locations to further investigate underlying causes of disproportionate incarceration.

Keywords:

• spatial analysis
• spatial scan statistic
• incarceration rate
• jail population
• regression analysis
• predictive modeling

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1. Introduction

1.1 The recent history of incarceration

Incarceration in the United States has a long and varied history that changes significantly depending on the lens through which it is viewed. Local, state, and federal policies surrounding laws and their enforcement blend together and create different systems of incarceration dependent on location. These unique and localized approaches can lead to differences in incarceration influenced by the social makeup of each town, city, or large geographic region under consideration. From a global perspective, the United States has jailed and imprisoned more people than any other nation in the world on a per capita basis and has nearly doubled its prison population between 1990 and 2019 (Davis 2021). Over two million people were incarcerated in US jails and prisons by the end of the 2019, nearly 50 years after what is widely considered the beginning of the mass incarceration era in America, and incarceration rates have not fallen as predicted with the contemporaneous decrease in crime trends (Beckett et al. 2018).

1.2 Factors related to incarceration

There are many factors which influence these high incarceration rates in the US, the most direct being the legislation which leads to incarceration. Research often points to the beginning of the era of mass incarceration in the 1970s, when US public policy shifted towards a “tough on crime” agenda that led to stricter sentencing for what had previously been less serious crimes. Campell and Vogel (2017) suggested an association between the strength of the Republican Party in a location and higher rates of incarceration based on the political party’s effectiveness in connecting crime to individual moral failings. For example, states with certain demographic divides such as aging white populations and younger Black populations had harsher criminal justice policies, leading to higher incarceration rates and suggesting a relationship between incarceration and voting trends. There are also links between incarceration rates and health and wellbeing. Weidner and Schultz (2019) used a multivariate model that estimated the population health outcome as a function of predicted incarceration rate and found that higher incarceration rates were associated with greater morbidity and mortality. In terms of socioeconomic factors, a special report from the Bureau of Justice Statistics (James 2004) looked at survey responses from 465 jails and found that over half of surveyed inmates grew up in single-parent households or with a non-parent guardian, more than 60% of inmates were racial or ethnic minorities, and 69% of inmates were regular users of drugs or illicit substances. Although these factors represent just a few of those linked to incarceration in the literature, they serve to illustrate how such factors could be useful in contextualizing spatial trends in incarceration throughout the United States.

2. Data and Methods

2.1 Choosing factors as predictive covariates

Based on the literature and prior research by the Vera Institute of Justice (Peirce et al. 2022), we made an initial selection of factors to be used in a county-level covariate adjusted spatial analysis. This included voting partisanship; poverty rates; rates of single-parent households; rates of death from opioid drug usage; rates of violent crime; low birth weights; excessive drinking; physical and mental health; smoking rates; teen birth rates; obesity rates, and suicide rates. Several were discarded based on missingness of data for several years or across a large number of counties. Some were removed after testing within a regression model due to small or non-significant effects or due to high correlations with other more significant covariates. As an example, the rate of children in poverty was removed due to a high correlation with the rate of poverty. Based on sources of complete and relevant data it was decided to use the following factors for the covariate analysis: The population per square mile, representing population density; the Black population (Aged 15 - 64), representing race as a variable by the percentage of Black residents; the Latinx population (Ages 15 - 64), representing ethnicity as a variable by the number of Latino and Latina residents; the percentage of medically uninsured persons; the percentage of unemployed persons; the percentage of single parent households; the violent crime rate; the rate of poverty; the rate of deaths due to opioid usage, and the local Republican partisanship measured by vote counts in national elections.

2.2 Data sourcing and collection

This study sourced incarceration data from the Incarceration Trends Dataset (ITD) by the Vera Institute of Justice (Vera 2020), which draws from the Census of Jails, the Annual Survey of Jails, and the National Corrections Reporting Program from the Bureau of Justice Statistics (2023). The dataset includes observations at the county and jurisdiction levels for jails between the years 1970 to 2018, and for prisons between the years 1983 to 2016. This study is interested in the count of jailed persons, or total jail population, in each county of the United States due to the preponderance of county-level observations from 2010 onward compared to the prison data; many counties lack prisons and therefore lack prison data. The jail incarcerations count is obtained from the average daily jail population, or ADP, measured by the Bureau of Justice and Statistics. In cases where the ADP was not available, the BJS used a measure of the confined population on a given day, typically falling on the last weekday in June (Kang-Brown et al. 2019). The study period of 2010-2018 was chosen to provide results from recent and relevant data, and which also offers a robust selection of county-level covariate data from other sources. This 9-year period reflects the most recent trends in incarceration in the United States, with 27,936 observations included in the purely spatial analysis, and 26,512 county-level observations included in the final covariate-adjusted analysis. The corresponding covariate data was sourced from the County Health Rankings & Roadmaps (CHR&R), a program from the University of Wisconsin Population Health Institute (2023); the U.S. Census Bureau’s Small Area Income and Poverty Estimates (SAIPE) program (U.S. Census Bureau 2023); the CDC’s Wide-ranging Online Data for Epidemiologic Research (WONDER) system for multiple causes of death between 1990 and 2020 (CDC WONDER 2023); the National Neighborhood Data Archive (NaNDA) hosted at OPENICPSR (2020); the US Census website (2022), and the National Weather Service (2023).

2.3 Study design

The analysis was conducted in three general stages. First, SaTScan^TM (Kulldorff 2006) was used to conduct a spatial and a space-time (retrospective and prospective) analysis of the data with a discrete Poisson model, identifying significant geographical clusters with high relative risk (RR) for jailed populations and ArcMap (2021) was used with these results to create a choropleth map of counties in the conterminous United States based on relative risk. Second, a negative binomial regression model was used to adjust for the effects of covariates on incarceration counts. Predicted counts from the model were contrasted with observed incarceration counts to test for and identify clusters which were explained or not explained by the covariates, and a 3-color map was used to show which clusters were affected by the adjustment of these covariates. Third, a multivariate spatial analysis was performed using three of the selected factors as variables alongside jail incarceration counts to find geographical clusters where two or more of the variables have jointly higher rates than the rest of the country.

2.4 SatScan^TM

SaTScan^TM is a disease surveillance software which takes spatial, temporal, or space-time data and generates scan windows of circular or elliptical shaped regions called clusters. For valid inferences it is important that the cluster shape is chosen a priori before starting the data analysis to avoid pre-selection bias in the analysis. Neither model shape is better than the other; the differences in the power are slight, regardless of what the shape of the true underlying cluster is, and the exact borders of the clusters are unknown (Kulldorff et al. 2006). With either shape, cluster analysis in SaTScan^TM can be described as a moving window which scans across pre-defined geographical regions of interest. Within this window, expected and observed counts of a variable are compared to identify US counties with higher-than-expected counts among all counties based on the discrete Poisson distribution. The counties serve as the centers of circles in which clusters are identified based on a likelihood function maximized across all locations: Cluster 1 is the cluster with the largest likelihood function and is least likely to have occurred by chance; cluster 2 is calculated in the same way, followed by the calculation of cluster 3 and so on, where clusters are non-intersecting and geographically distinct. In the case of this study such a test was against the null hypothesis that the rate of jailed persons is equal to the rate in all counties. This process extends into a multi-dimensional application which allows for a dynamic scanning window for both the purely spatial and space-time analyses in SaTScan^TM where the space-time analysis extends the window from a circular base to a cylindrical field with time as the height (Kulldorff 1997). We defined such clusters as circles capturing at most 25% of the population at risk, and space-time clusters were defined to capture a period up to 50% of the study period, or between 1 and 4 years. We also set a minimum cluster relative risk of 1.3, and a p-value smaller than 0.05 in order to search only for clusters with a high and statistically significant risk compared to the rest of the country.

2.5 SAS and negative binomial regression

The Statistical Analysis System Software (SAS 2021) was used for data preprocessing and analysis and to create a negative binomial regression model. The negative binomial regression is a multiple linear regression model implemented using the maximum likelihood estimation with observed count data that follows the negative binomial distribution (Zwilling 2013). Traditionally, the negative binomial has been derived from the Poisson model, specifically a Poisson-gamma mixture model. It can also be considered part of the group of single parameter exponential distributions, also known as generalized linear models (Hilbe 2007). The advantage of the negative binomial over the Poisson distribution is the inclusion of an extra dispersion parameter when the assumption of an equal mean and variance in the Poisson distribution is not met, loosening this assumption by modeling Poisson heterogeneity via a gamma distribution (Hilbe 2014). The jail count data in this analysis suffers from overdispersion which made the negative binomial regression an ideal candidate for modeling the covariate adjusted data, allowing the significance of each covariate to be judged based on the Wald Chi-Square value and p-value of each estimate. This model generated predicted jail incarceration counts that were used in SaTScan^TM to perform the covariate adjusted analysis and produce a descriptive 3-color map of the covariate effects. One possible limitation of using this fixed-effects regression model is that it does not account for spatial auto-correlation. However, SaTScan^TM does not assume that there is no spatial auto-correlation in the data and is instead testing whether there is spatial auto-correlation or other divergences from the null hypothesis. If the interest is in spatial auto-correlation, then other tests for spatial auto-correlation which have higher power can be used instead. This study used the negative binomial regression to complement the spatial scan statistic rather than methods which adjust for spatial auto-correlation due to the focus on the detection of statistically significant local clusters.

2.6 Multivariate cluster analysis

The multivariate spatial analysis was performed using SaTScan^TM by employing the multivariate scan statistic (Kulldorff, 2007) to determine when geographic clusters occur simultaneously in more than one data set where each data set represents a different count variable. This multivariate approach creates a single likelihood function that adds the log likelihood ratios for each individual data set and finds the maximum values of the sums of the individual log likelihood ratios. The maximum value is the multivariate scan statistic that signals clusters of higher or lower relative risk occurring in multiple data sets simultaneously. This study presents the high-rate multivariate likelihood of clusters occurring simultaneously in three important variable datasets and the jail incarceration dataset.

3. Results

The results of the analysis indicate significant, non-random clusters of jail incarceration after adjusting for significant related risk factors at the county level. In each SaTScan^TM analysis the jail population count is measured against the county population, aged 15 to 64. This population age range was chosen as it is rare that anyone under the age of 15 is jailed with adult sentencing (Griffin 2011) or that anyone over 64 is jailed for an extended period (Kang-Brown et al. 2019).

3.1 Purely Spatial Analysis

A purely spatial Poisson analysis was performed using SaTScan^TM and identified 136 significant ( $p<0.05$) clusters which are displayed in Figure 1. These clusters identify areas with disproportionately high rates of incarcerated persons relative to the baseline rate of incarceration calculated across the entire country which is equal to a relative risk of 1 (100%). For example, orange counties in Figure 1 represent locations that have a relative risk of incarceration 1.3 to 2 times greater than the rest of the study area. Similarly, counties in red represent a relative risk of incarceration at least double the baseline rate. The significant clusters of these counties are ranked by the likelihood-ratio of the spatial scan statistic with the twenty clusters of the highest likelihood-ratios indicated in Figure 1.

Table 1 gives a description of each cluster including the number of counties included in each cluster and the population within the cluster. Some clusters such as 6, 7, 12 and 16 include only one county, indicating locations of particular concern.

3.2 Retrospective Space-Time Analysis

Retrospective analysis of data is derived from events that have occurred over a period of time to create a baseline analysis (Kulldorff et al. 1998). Retrospective clusters can help determine the impact of events at a given point in time which might be related to higher rates of jail incarceration counts (e.g. changes in local, state, or federal criminal policy, financial crises, widespread job upheaval). The retrospective analysis also helps determine the ability to create a prospective model for predicting future clusters based on which times and areas are identified as significant in the past. The same cluster identification is used in Figure 2 as in the purely spatial analysis with the time period of each cluster noted in Table 2.

Table 2 shows that many significant clusters formed towards the beginning or end of the study period. Clusters which border the beginning of the period may represent those which were more significant before the start of the study period (2010) while those bordering the end may represent areas that are increasingly significant based on the most recent observations. These combinations of geography and time could produce further insights on factors relating to high rates of incarceration.

3.3 Prospective Space-Time Analysis

Prospective analysis is based on forward analysis from a baseline (Kulldorff 2001); in most prospective studies, a baseline is established using the initial 1-4 years of data against which yearly data is collected and analyzed for trends. This technique can be used to attempt to predict high-rate clusters of incarceration in the future based on past data. Most clusters identified in Table 3 are based on the period of 2015-2018 strongly suggesting that these clusters are active and present clusters of high incarceration rates. Clusters from 2015-2018 which were identified in both the retrospective and prospective analysis are significant areas of concern based on the high log-likelihood ratio obtained in both past and forward analysis frameworks.

3.4 Covariate Analysis

The negative binomial regression model with covariates listed in Table 4 was used to adjust jail incarceration counts for spatial analysis. The table shows the parameter estimates for predicting jail incarceration counts along with the associated test statistics. Positive parameter estimates indicate a positive relationship between rising levels of the covariates and jail incarceration counts; all covariate parameter values were positive except for the estimate for population density, linking a rising population density with decreasing jail incarceration counts.

The parameters are listed in decreasing order using the Wald’s Chi-Square statistic which considers the effect of the covariate when it is added last to the model. The Black population percentage in a county had the largest Chi-Square value of 1191.41, implying it has the largest effect on incarceration counts when the other covariates are held constant in the model. Counties with a high ratio of Republican party votes to Democratic party votes was the second most important covariate with a Chi-Square value of 980.13. The poverty rate was the third most important covariate in the model with a Chi-Square value of 608.14.

The adjusted incarceration counts from the model were used in SaTScan^TM to show the effects of the covariate adjustment against the original clusters from the purely spatial analysis using the 3-color ArcMap in Figure 4. Orange counties represent the original spatial clusters of jail incarceration that remained in place after using all covariates, which we consider persistent clusters where the jail incarceration count and covariates are not associated with each other. The red counties are locations in which the covariates explain the high rate of jail incarceration and no longer appear in any clusters after adjusting for the covariates. In other words, the high incarceration rates can be explained by the levels of the covariates in these counties. The yellow counties represent new cluster areas that appeared only after adjusting for all covariates and were not inside any initial spatial clusters in Figure 1. These areas are those where jail incarceration counts are higher than average after adjusting for the covariates. The area of red counties compared to orange counties is a general indicator of how well the covariates predict high incarceration counts at the national scale; the results in Figure 4 indicate that the chosen covariates are good predictors of incarceration in a country-wide model. The orange areas remain as particular areas of interest where the chosen covariates do not explain the abnormally high rate of incarceration.

3.5 Multivariate Cluster Analysis

A multivariate cluster analysis was done with jail incarceration counts and three of the factors selected for the study (poverty rate, Republican partisanship, and Black population percent) to discover clusters with joint high rates of more than one of these variables. The multivariate scan in SaTScan^TM calculates the log likelihood ratios (LLR) for each data set and for data sets with larger than expected counts, the log likelihood ratios are added together to get the likelihood for a specific window (Kulldorff et al. 2007). The windows are compared and the window with the largest likelihood ratio is the most likely cluster. In order to fit the count requirement of the Poisson distribution, the poverty rate (as a percentage) and Black population percentage were changed to counts by multiplying each observation by the population count. Additionally, the Republican vote counts for presidential elections in 2008, 2012, and 2016 were used to fill the lagging years (2010-2011, 2013-2015, 2017-2018) in order to balance counts for years that would otherwise have only senate vote counts tallied in the total.

The analysis identified 12 significant clusters with cluster 1 having the highest log likelihood-ratio, listed in descending order in Table 5 with the relative risk for each of the variables also shown. Only cluster 1, the most likely cluster covering the entire southeastern United States, had significant clusters in all the four variables. Clusters are listed only if they represent joint clusters of jail population and at least one other variable; these clusters are cluster 1 in the southeast, cluster 3 with only one county, Wayne County in southern Michigan, cluster 4 in the mid-west, and cluster 8 in southern Wisconsin, Milwaukee County, with only one county (overlapped by the circle of cluster 11).

Table 5 demonstrates the most significant geographical relationships between incarceration and these variables across the United States. For example, we see the Black population variable being a major factor and the only factor to appear of the three in cluster 3. Cluster 4 shows the effect of poverty and Republican voting, but not of race. Cluster 8 shows the absence of political partisanship as a factor with poverty and race being present.

Conclusion and Discussion

This surveillance study provides a spatiotemporal analysis of jail incarcerations in the United States at the county level in addition to identifying which risk factors of covariates are the best predictors of the jail populations for the period 2010-2018. Jail incarceration rates vary greatly throughout the United States and the study identifies clusters with the highest risk of incarceration. The spatial clusters show where jail incarcerations are unusually high when compared to the rest of the country, while space-time clusters also identify time periods during which jail incarceration was highest historically or as emerging clusters are observed towards the end of the study period.

In the first three tables, it is shown that the cluster with the highest relative risk of incarceration was in Toole County, MT. The relative risk of this cluster is nearly double that of any other cluster and roughly 50 times the risk of incarceration compared to the baseline of the study area from the spatial scan statistic. The cluster that covers the largest area and population of the county for the spatial analysis and both space-time analysis is the cluster that covers the southeast of the United States. The relative risk of this cluster is less than 2, which allows for comparative interpretations of the results; we might say that the relative risk of jail incarceration in Toole County is 25 times greater than the relative risk of the southeastern United States. It is important to interpret such comparative results cautiously: Single-county clusters could be influenced by factors such as a jail serving multiple surrounding counties, or a jail making up most of the population in an otherwise empty county. Possibilities such as these require post hoc investigations and explanations if effective geographic comparisons are to be made. With such considerations in mind, focusing both on broad regions identified as spatially significant, such as the south and west, and individual counties such as Toole, point to starting locations for further research on the current state of high incarceration rates in jails.

Table 4 presents the results from the negative binomial regression model of the different covariates. These covariates may have different effects for different regions across the country. The three-color map displays the differences in the spatial map after adjusting for all the covariates where persistent clusters, or clusters that still have high incarceration counts after adjusting for the covariates, are shown in orange in Figure 4. These regions are of particular interest because they were not fully explained by the covariates chosen in the study, implying that the abnormally high rates of incarceration have some explanation other than these factors. Further investigations of explanatory variables for these regions would be of high interest. The clusters in the southwest and southeast of the country are mostly in red, meaning the combination of chosen covariates explain these high-risk regions. The yellow areas are those which are expected to have high rates based on the covariates and could indicate potential areas of concern for high incarceration rates. These yellow clusters could be used in conjunction with the prospective clusters to locate potential “hotspots” in this context.

The analysis also identified societal and cultural links to high incarceration rates such as the percentage of Black Americans living in a location, Republican voting strength, and poverty rates. Sociological and economic research can offer further explanations for how these factors are tied to the jail incarceration rates and to geographic regions of the United States in combination with our results. Looking at the multivariate analysis in Figure 5, cluster 1 is located in the southeastern United States and has a significant 4-dimensional effect of jail incarceration, race, political leaning, and poverty. This is the only cluster in this analysis with all four variables jointly having high relative risk in a region of the country greatly transformed after the American Civil War and Reconstruction era. Using the results of this study in an interdisciplinary context may help further extend our understanding of how the history of this region continues to impact incarceration today. Research which expands on these results could delve deeper into the regional dynamics of spatial clusters, the temporal aspects of past and emerging clusters, and the complex relationships between social and economic factors and jail incarceration rates. This study does not directly demonstrate any causal effects between single covariates and high incarceration rates; however, the covariate and multivariate analyses suggests an association with higher incarceration rates in regions with a higher percentage of Black American residents, higher vote margins of Republican votes, and higher rates of poverty. Each of these covariates demands unique attention in the geographical study of incarceration. Finally, analyzing the impacts of county and state laws surrounding incarceration would provide invaluable insights into this topic and is recommended for future spatial research on incarceration trends.

Table 1: The top 20 significant purely spatial clusters of jail incarceration for 2010-2018 based on the likelihood-ratio of the purely spatial scan statistic ( $p<0.0001)$.

CLUSTER	LOCATIONS IN CLUSTER	LIKELIHOOD-RATIO	OBSERVED TO EXPECTED RATIO	RELATIVE RISK	POPULATION
1	774	256536.25	1.72	1.97	30,048,640
2	125	26673.79	2.14	2.16	1,669,095
3	2	25812.62	2.21	2.22	1,444,831
4	89	25254.90	1.86	1.88	2,712,077
5	53	11710.70	1.84	1.85	1,269,218
6	1	9602.25	33.45	33.47	3,442
7	1	7273.03	25.40	25.41	3,835
8	1	6017.53	4.63	4.64	52,765
9	1	4832.46	2.17	2.17	289,325
10	15	3941.37	1.88	1.88	406,748
11	1	3669.13	1.80	1.81	429,875
12	1	2982.16	16.50	16.50	2,954
13	2	2678.23	5.45	5.45	17,011
14	57	2259.86	1.31	1.32	1,552,997
15	17	2210.63	1.31	1.31	1,561,339
16	1	1876.87	18.17	18.18	1,609
17	22	1863.44	1.37	1.37	924,728
18	1	1819.33	1.78	1.78	223,532
19	22	1776.07	2.18	2.18	125,571
20	1	1669.58	31.83	31.83	641

Table 2: The top 20 significant retrospective space-time clusters of jail incarceration for 2010-2018 based on the likelihood-ratio of the retrospective space-time scan statistic ( $p<0.0001)$.

CLUSTER	TIME	LOCATIONS IN CLUSTER	LIKELIHOOD-RATIO	OBSERVED TO EXPECTED RATIO	RELATIVE RISK	POPULATION
1	2010-2013	724	105866.36	1.78	1.87	27,568,335
2	2012-2015	2	13503.12	2.33	2.33	1,444,831
3	2011-2014	196	12606.83	1.74	1.75	3,865,574
4	2011-2014	68	11715.34	1.89	1.90	2,673,430
5	2015-2018	1	7470.92	52.44	52.45	3,442
6	2015-2018	90	7382.72	1.81	1.82	1,981,474
7	2013-2016	1	3445.27	26.68	26.69	3,835
8	2010-2013	1	3433.90	2.21	2.21	429,875
9	2013-2016	1	3140.57	5.02	5.02	52,765
10	2011-2014	8	1910.09	1.60	1.61	847,414
11	2014-2017	1	1372.78	16.88	16.89	2,954
12	2013-2015	8	1349.83	1.30	1.30	2,935,392
13	2012-2015	1	1325.79	1.64	1.64	531,510
14	2015-2018	10	1263.38	1.85	1.85	303,430
15	2011-2014	1	1229.11	1.98	1.98	223,532
16	2013-2016	37	1210.74	1.33	1.33	1,714,337
17	2010-2012	1	1184.63	1.76	1.76	482,939
18	2014-2017	3	1128.80	1.99	1.99	207,140
19	2015-2018	4	1038.59	3.19	3.19	47,119
20	2015-2018	22	964.10	2.32	2.32	125,571

Table 3: The top 20 significant prospective space-time clusters of jail incarceration for 2010-2018 based on the likelihood-ratio of the prospective space-time scan statistic ( $p<0.0001)$.

CLUSTER	TIME	LOCATIONS IN CLUSTER	LIKELIHOOD-RATIO	OBSERVED TO EXPECTED RATIO	RELATIVE RISK	POPULATION
1	2015-2018	774	98263.16	1.71	1.80	30,048,640
2	2015-2018	152	11455.18	1.97	1.98	2,134,341
3	2015-2018	89	11058.92	1.86	1.87	2,712,077
4	2015-2018	2	8857.35	2.04	2.05	1,444,831
5	2015-2018	1	7470.92	52.44	52.45	3,442
6	2015-2018	53	6346.86	1.95	1.96	1,269,218
7	2015-2018	1	3397.68	26.53	26.54	3,835
8	2015-2018	1	2420.33	4.40	4.40	52,765
9	2015-2018	1	1858.33	2.08	2.08	289,325
10	2015-2018	2	1379.15	5.99	6.00	17,011
11	2015-2018	1	1369.85	16.81	16.81	2,954
12	2015-2018	10	1263.38	1.85	1.85	303,430
13	2015-2018	17	1057.02	1.32	1.32	1,561,339
14	2015-2018	4	1038.59	3.19	3.19	47,119
15	2015-2018	2	1028.66	3.08	3.08	50,201
16	2015-2018	22	964.10	2.32	2.32	125,571
17	2015-2018	22	963.31	1.40	1.40	924,728
18	2015-2018	1	886.92	19.38	19.38	1,609
19	2015-2018	1	818.14	2.61	2.61	61,625
20	2015-2018	5	758.64	1.32	1.32	1,120,060

Table 4: Parameter estimates from the negative binomial regression model.

PARAMETER	ESTIMATE	STANDARD ERROR	WALD’S 95% CONFIDENCE LIMITS	WALD’S ${\mathit{\chi }}^2$	$\mathit{p}$-VALUE
Black population percent Ages 15 to 64 years	0.2075	0.0060	0.1958, 0.2193	1191.41	<0.0001
Republican voting percent	0.1772	0.0057	0.1661, 0.1883	980.13	<0.0001
Rate of poverty	0.1195	0.0048	0.1100, 0.1290	608.14	<0.0001
Latinx population percent Ages 15 - 64 years	0.1167	0.0052	0.1065, 0.1269	503.11	<0.0001
Rate of uninsured	0.1282	0.0060	0.1164, 0.1400	455.90	<0.0001
Percent single parent homes	0.0873	0.0056	0.0763, 0.0982	244.57	<0.0001
Rate of violent crime	0.0713	0.0053	0.0608, 0.0818	178.29	<0.0001
Rate of unemployment	0.0391	0.0056	0.0281, 0.0501	48.33	<0.0001
Rate of opioid mortality	0.0251	0.0048	0.0156, 0.0345	26.81	<0.0001
Population density	−0.0144	0.0041	−0.0225, -0.0064	12.28	0.0005

Table 5: Multivariate clusters of jail incarceration, Republican vote, poverty, and Black population %. ( $p<0.0001$)

CLUSTER	NUMBER OF LOCATIONS IN CLUSTER	RELATIVE RISK
		Jail population	Poverty	Republican vote	Black population %
1	965	1.75	1.13	1.17	2.42
3	1	1.33	–	–	3.10
4	1044	1.03	1.00	1.25	–
8	1	1.08	1.19	–	2.03

Figure 1: Purely spatial clusters calculated with county-level jail incarceration counts and the population of each county using the discrete scan statistic with Poisson distribution. The clusters are overlaid onto a choropleth map of the relative risk of jail incarceration in each county. The 20 clusters with the highest likelihood-ratios are labeled.

Figure 2: Retrospective space-time clusters calculated using the discrete scan statistic with Poisson distribution. The 20 clusters with the highest likelihood-ratios are labeled.

Figure 3: Prospective space-time clusters calculated using the discrete scan statistic with Poisson distribution. The 20 clusters with the highest likelihood-ratios are labeled.

Figure 4: A comparison map of locations in the original unadjusted spatial clusters and locations in covariate-adjusted spatial clusters. Red areas represent those which appeared only in the original significant spatial clusters. Yellow areas are clusters which appeared only after adjusting for the covariates. Orange areas or those which appeared in clusters both before and after covariate adjustment.

Figure 5: Purely spatial multivariate clusters of jail population, poverty, Republican vote, and Black population.