Event dependence in U.S. executions
Event dependence in U.S. executions
Frank R. Baumgartner 0 1
Janet M. Box-Steffensmeier 1
Benjamin W. Campbell 1
0 Department of Political Science , UNC-Chapel Hill, 313 Hamilton Hall, Chapel Hill, NC, 27599-3265 , United States of America, 2 Department of Political Science, The Ohio State University , 230 North Oval Mall, Columbus, OH 43210 , United States of America
1 Editor: LuÂõs A. Nunes Amaral, Northwestern University , UNITED STATES
Since 1976, the United States has seen over 1,400 judicial executions, and these have been highly concentrated in only a few states and counties. The number of executions across counties appears to fit a stretched distribution. These distributions are typically reflective of self-reinforcing processes where the probability of observing an event increases for each previous event. To examine these processes, we employ two-pronged empirical strategy. First, we utilize bootstrapped Kolmogorov-Smirnov tests to determine whether the pattern of executions reflect a stretched distribution, and confirm that they do. Second, we test for event-dependence using the Conditional Frailty Model. Our tests estimate the monthly hazard of an execution in a given county, accounting for the number of previous executions, homicides, poverty, and population demographics. Controlling for other factors, we find that the number of prior executions in a county increases the probability of the next execution and accelerates its timing. Once a jurisdiction goes down a given path, the path becomes self-reinforcing, causing the counties to separate out into those never executing (the vast majority of counties) and those which use the punishment frequently. This finding is of great legal and normative concern, and ultimately, may not be consistent with the equal protection clause of the U.S. Constitution.
The U.S. Supreme Court validated the modern death penalty in its Gregg v. Georgia decision in
1976. Since then, over 1,400 judicial executions have followed, through the end of 2015. These
executions have been highly concentrated in a small number of jurisdictions, however: Texas
had 513 where the next closest state (Oklahoma) had just 112, and the average across all death
penalty states is fewer than ten. Across the 3,000+ counties in the U.S., the vast majority have
executed not a single prisoner, whereas Harris County, Texas (home to Houston) has executed
125. Just twenty U.S. counties have executed 10 or more individuals in the 40 years since the
Gregg decision. Here, we refer to the county where the death sentence was handed down.
County-level prosecutors make the decision of whether to seek death, and trials are typically in
the county of the crime. Therefore, the county is of interest even through states, not counties,
typically carry out the executions. Fig 1A shows the concentration of murders; Fig 1B presents
an equivalent presentation of executions.
Fig 1. Average murders and executions by county per year per 100,000 residents, 1977±2014. Maps present the average number of
murders and executions experienced per year per 100,000 residents by each county between 1977 and 2014.
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Fig 1 makes two things apparent: executions are highly concentrated, and the counties with
the greatest number of executions are not the same as those with the homicides, controlling
for population size. New Orleans has the greatest number of homicides per year per 100,000
residents, with Washington DC, St. Louis, Richmond, and Baltimore following. Wayne
County, Michigan (home to Detroit) is the only one of the high-homicide locations in a state
without the death penalty. This poses an interesting puzzle that runs counter to the prior
expectations of many [1±4]Ðif murders and the opportunity for utilizing the death penalty
appear to be unrelated, what explains this clustering of executions? This question motivates our
research. We hold that executions reflect a self-reinforcing process. We present empirical
support for this theory through a series of analyses that demonstrate that executions reflect a
power law or log normal self-reinforcing process. We do so by utilizing bootstrapped
Kolmogorov-Smirnov tests [
] and further use conditional frailty models to test for event dependence
]. Our finding that executions are conditional on the number of previous executions both
explains their high concentration in just a few jurisdictions and raises important substantive
concerns. After all, the historical track record of a given county in carrying out previous death
sentences should not be a ªlegally relevant factorº in determining whether the next inmate
deserves the ultimate punishment. That should relate solely to the nature of the crime and the
characteristics of the offender.
Correlated v. random local variation
As described in [
], with further analyses available in [9±11] no single actor determines
whether an execution will take place. Homicides occur with great regularity across the US,
some of which meet the statutory requirements to be eligible for a capital prosecution.
Prosecutors may vary in their proclivities to seek death. Capital defenders, typically part of a public
defender's office or appointed by the Court, may be more or less well funded, active, and
successful. Juries, judges and appellate courts may be more or less supportive of the death penalty.
While some degree of local variation is to be expected because of random fluctuations in any
variable, if the many actors involved in capital punishment showed independent random
fluctuation, the Central Limit Theorem would dictate a Gaussian distribution of outcomes. If, on
the other hand, variation is correlated, then, as [
] explains, such a process ªtends to produce
momentum in favor or against sustained capital activityº, producing ªa separating equilibrium
for death penalties and executions: a few capitally active localities, and many more than tend
towards abstentionº (p. 264). If a prosecutor bases their decision to seek death on the basis of
previous experience in the county and an assessment of the likelihood that other actors will go
along with the decision, and other actors do the same, then event dependency can ensue. The
same crime, in two different counties (or at two different times) would have dramatically
different odds of leading to execution. Our test is designed to capture these dynamics, which is
exactly what we uncover.
Materials and methods
To demonstrate support for the proposition that judicial executions reflect a self-reinforcing
processes, we adopt a two-pronged modeling approach. The first prong involves using
bootstrapped Kolmogorov-Smirnov (KS) tests to infer the functional form of the frequency
distribution of executions across counties. Should the distribution of executions follow a stretched
distribution, such as the power law or exponential, there would be suggestive evidence of event
]. The second prong includes the use of survival analysis, and in particular
the conditional frailty model [
], to infer whether this self-reinforcing process holds once
accounting for relevant confounding variables. All analyses were conducted using the R
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Statistical Computing Environment. All code and source data files for reproduction will be
made available publicly through the Harvard Dataverse and the Interuniversity Consortium
for Political and Social Research (ICPSR) data repository. In this section, we describe the data
used for analyses and this two-pronged modeling approach.
Bootstrapped Kolmogorov-Smirnov (KS) tests to infer distributional form. To infer
the functional form of a particular set of outcomes, [
] recommend the use of bootstrapped
Kolmogorov-Smirnov (KS) tests. This strategy allows one to compare the distribution of
observed data to data simulated from a distribution with parameters determined by the
observed data. These two distributions are then compared using KS tests. Should the test yield
a p-value greater than α = 0.05, then the null that the two distributions are of the same
functional form cannot be rejected. We conduct four tests to determine the distributional form
that best fits the observed dataÐpower law, log normal, Poisson, or exponentialÐwith 100,000
bootstrap replications per form. Each of these distributions, save the Poisson, is referred to a
ªstretchedº distribution and is reflective of a self-reinforcing process whereby the probability
of observing an event increases for each previous observation of an event. As such, we would
expect bootstrapped KS test p-values greater than α = 0.05 for the power law, log normal, and
exponential distributions and p-values less than α = 0.05 for the Poisson.
Using the conditional frailty model to test for event dependence. While bootstrapped
KS tests are useful in providing suggestive evidence that phenomena appear to reflect a
selfreinforcing process, they cannot account for observed or unobserved confounding variables
that may produce spurious evidence of event dependence. To account for these confounding
variables our model of choice, we use the the conditional frailty model. This model can be
understood as a special case of the Cox proportional hazard model. The conditional frailty
model allows scholars to explicitly account for unobserved unit-level heterogeneity, while also
incorporating event dependence that may pose inferential challenges [
]. These extensions
allow for a model that minimizes bias while increasing efficiency relative to alternative models
that may be used to test theories of event dependence [
]. In addition, the conditional frailty
model performs better under conditions of event dependence and heterogeneity, making this
model the safest choice [
The model presented in the manuscript includes the variables described in the next
subsection in addition to a state-level frailty term to account for any unobserved heterogeneity at the
state level. This term is said to increase estimation efficiency while accounting for any
confounding effects that may exist at the state level, including, for example, a culture tolerant of
the death penalty. This frailty term is estimated using the gamma distribution with variance
estimated according to an expectation-maximization algorithm.
The execution data used for both analyses draw from data developed by [
]. This data consists
of all 1,422 US executions measured across 474 counties in 34 different states from 1977 to
2014, containing information on the date of execution and the county of conviction. The list of
executions was originally sourced from the widely available ªEspy fileº of known executions
], confirmed and supplemented with further follow up by [
]. This provides the distribution
of executions that is examined through both analyses. For the conditional frailty model
presented in the manuscript, observations are modeled as the gap time between executions with
three levels of event dependence stratification, the number of homicides in the county, racial
threat in the county, poverty rate, and the population of the county. To clarify, the county-unit
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of analysis refers to the county in which a sentence was handed down, not the county in which
the execution takes place. Executions are typically carried out in a single location in each state,
but the legal actors involved relate to the county of conviction or to the state capital, not the
location of the execution chamber. The homicides control variable is sourced from [
provide the number of homicides per year, by county. This was compiled from annual
Supplemental Homicide Reports (that is, FBI data). The inclusion of this control is essential, as it
seems intuitive that the number of executions that a county experiences may be correlated
with the number of homicides [
]. The second control, racial threat, is an important control
as theory tells us that as the members of a minority group increase to a certain point, the
majority race will seek to impose higher levels of social control. As such, the death penalty
represents one such example of a punitive measure for law breaking. This variable is coded
according to the conventionally used routine: 100 − |70−percentage of population white| with
data sourced from the census [
]. A similar story could be told of povertyÐthe amount of
poverty in a county may drive executions and the overall prevalence of murder. This variable
is coded as the percentage of the population of the county that is in poverty, with data sourced
from the census. The final control included, the population of a county, could be said to
influence executions as counties with large populations may be more likely to experience large
numbers of murder, and as a function of prosecutions, more executions. Large counties also
have larger and more professionalized district attorney's offices. Data for this variable is taken
from the census.
We start by presenting naive evidence suggesting support for our hypothesis in Fig 2, which
demonstrates that the concentration of executions appears to approximate a power law or
lognormal distribution. That is to say, the distribution appears to be linear or approaching
linearity on a log-log scale. The high concentration of executions in such a small number of judicial
jurisdictions can be observed whether looking across the 50 states, across the counties as done
in Fig 2, within the largest executing states (e.g., across the counties of within Texas or
Oklahoma, for example), or across countries of the world. These patterns have already been alluded
to by the senior author in other publications [
]. However, this evidence is highly descriptive
and does not rule out the potential for these data to be distributed according to another
nonstretched distribution, such as the Poisson, or confounded by the population of or murders
within a county. Thus, this paper asks not just about the patterns of these executions, but the
substantive meaning of these patterns and the relative robustness of the pattern uncovered by
. We turn to this question in the following sections.
Distributional form of death penalty executions
It is often suggested that the distribution of many events appear to fit a power law (or another
stretched) distribution on the log-log scale, and that such visual checks are not confirmatory
]. We agree with this criticism, and to provide additional evidence, we utilize
bootstrapped Kolmogorov-Smirnov tests using the routine described by [
] to rule out the
possibility that the distribution of executions does not appear to be be distributed according to a
power law or some other stretched distribution like the log-normal or exponential. Each of
these tests were conducted on a bootstrapped sample which included 100,000 replications.
Results from these analyses are presented in Table 1.
Overall, there appears to be significant support for our proposition. The distribution of
executions is distributed according to some self-reinforcing process. This is demonstrated by
the support for the power law, log normal, and exponential, and the lack of support for the
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Fig 2. Log-log plot of executions by county. Each colored line represents a distribution fitted to the observed data.
Poisson. This leads us to conclude that the data appear to be distributed either exponential, log
normal, power law. Any of these stretched distributions reflects a variation on a themeÐa
selfreinforcing and fat-tailed distribution. This is naive support for our proposition of event
dependence in death penalty executions. These tests, however, cannot account for the
population of a county, or the number of murders it experiences, which may influence its number of
executions. To resolve this concern, we move towards a more proper identification strategy in
the following section.
Identifying event dependence in executions
Given that we know the distribution of executions follows a stretched distribution, the
question is can we demonstrate that a self-reinforcing dynamic produced that empirical
distribution even after accounting for relevant confounders? Our empirical strategy is to estimate the
monthly hazard rate for each county in the United States over the entire modern period of the
death penalty, controlling for population size, poverty rate, homicides, ªminority threatº [
and crucially, the number of previous executions the jurisdiction has carried out. This includes
3,607 county-gap time observations, reflecting a census of all relevant cases of counties with
the death penalty. We use state-level fixed effects as well in the form of a shared-state frailty
term that may account for unobserved state-level confounders. In addition, we exclude
counties in states that do not have the death penalty. Nine states abolished the death penalty during
the 1977±2015 period, and we exclude counties in those states for the years after abolition.
We are interested in the effects of previous executions on the likelihood of observing a
subsequent execution within a fixed time period. But first, let us assess the control variables in the
model. Population size and homicides are perhaps obvious: large jurisdictions, and those with
many homicides, will naturally give rise to the opportunity for more executions. Given that the
distribution of counties in the US is extremely skewed with respect to population, with many
counties with small populations and just a few with large populations, we want to ensure that
the self-reinforcing aspect of executions cannot be accounted for by these variables; it cannot.
We control for poverty and racial dynamics because of the possible effect of poverty on crime
rates and for racial threat because of the possible effect of the use of punitive criminal justice
sanctions in areas with particular racial dynamics. Crucially, we are not interested in these
effects (and indeed, once we control for population, neither homicides nor poverty are
significant in the statistical results, and racial threat has only a small bearing). Rather, we need them
in the model to assure that our event-dependence modeling be accurate.
Within the context of event history modeling, we assess event dependence by examining
the relative hazard functions across different event strata, defined by the number of executions
previously carried out. The three strata are clustered as: 0, 1, and 2 or more executions. This
clustering routine was chosen to ensure a sufficient number of observations at each stratum
to ensure that one or two choice counties did not drive results and lead to Type 1 errors, and
is fairly standard across the Event History literature. Models estimated with varying strata
revealed similar results. We estimated a fully specified conditional frailty model with a
statelevel frailty term, event stratification, and the relevant confounding variables including
county-level homicides, poverty rate, racial tension, and population [
]. The frailty term
accounts for any unobserved heterogeneity across states and thus any unobserved state-level
confounders. For example, all counties in the same state would be subject to the same decisions
informed by that state's Supreme Court decisions, Governor's scheduling decisions, or
Corrections Department protocols covering the administration of executions and availability of lethal
injection drugs. Similarly, our state-level controls provide assurance that our results cannot be
explained by questions of legal precedence. All localities are bound by decisions of the US
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Supreme Court, but decisions by US Circuit Courts, US District Courts, or state supreme
courts would apply only within the relevant jurisdictions. None of these would vary, however,
within the boundaries of a given state. Our state-level frailty term is therefore essential to an
appropriate modeling strategy accounting for different state laws as well as different legal
precedents. While county-level frailty terms may capture the most fine-grained degree of variation
that could occur between groups of observations, such terms could not be successfully
estimated according to an acceptable convergence criteria.
To account for sparsity in higher-levels of the strata, we utilize a bootstrapping routine to
generate the cumulative hazards presented, but rely upon a non-bootstrapped model for all
other discussions. The conditional frailty model estimated according to the specification
previously outlined is consistent with theoretical expectations. Fig 3 demonstrates a large amount of
event dependence in the frequency of executions as the number of executions in the series
increases, even once accounting for the number of murders within the county, or its
population. In particular, it demonstrates that as time progresses, counties are increasingly likely to
conduct another execution, but that the hazard rate depends crucially on how many previous
executions have taken place. The three curves are placed as expected, and become increasingly
steep, suggesting more rapid increases in the probability of the next execution for those with
more previous executions. In addition, their 95% bootstrapped confidence intervals do not
overlap at the higher levels, demonstrating statistical significance of event dependence at the
α = 0.05 level. Among the counties that have never executed an inmate, the light blue curve
in Fig 3, probabilities of execution remain low, regardless of time. While one may expect the
never-executing counties to have estimated cumulative hazards of zero, a value greater than
zero is expected because these counties have values on the predictor variables whose
coefficients are estimated with other strata, and as such, inevitably contribute to the fitted
cumulative hazard values. In fact, diagnostics presented in the S1 Appendix demonstrate that the
model accounts for these never-executers quite well. The curve for those counties that have
one prior execution, the orange curve, increases more rapidly over time and is statistically
distinguishable from the baseline curve. This trend continues for the highest-order event
dependence (green curve) which becomes statistically distinguishable from all previous strata as time
A clear substantive interpretation of this is shown at the dashed line representing month
18 in Fig 3 which shows the relative conditional probabilities of observing an execution at 18
months across the full range of strata. At 18 months, the probability of observing an execution
for counties in the three groups specified increase from 0.04 to 0.11, to 0.19. In other words,
the probability of an execution for a county with previous executions relative to those that
have never experienced the executions is approximately five times higher. Note that the model
controls for homicides, population, poverty, minority population, and state. Additional model
diagnostics and robustness checks are provided in the S1 Appendix.
Of course, some prosecutors seek the death penalty more than others. Some counties are
more violent than others. Racial tensions or poverty, combined with crime, may make the
local political culture different from place to place. The effects of these controls are presented
in Table 2. Only the effect of two variables are robust at any conventional threshold. First, the
racial threat variable is significant and in the expected direction. As the white population of a
county decreases, which is to say, racial threat increases, there is an increase in the cumulative
hazard, or likelihood, of observing an event. This is consistent with previous findings about
the enforcement of disproportionate punishments in racially polarized counties and is of
particular normative importance. To date, this relationship has not been established for
punishments as extreme as the death penalty [
]. This result is exceptionally problematic, it
demonstrates that the death penalty may be used in some cases as a means to control communities
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Fig 3. Cumulative hazard functions for fully specified conditional frailty model. Specified with state-level frailty term and three levels of event
dependence. 1000 bootstrap replications were used to generate hazards.
of color. The same effect holds for the population variable, as the population of a county
increases, so does the likelihood of observing another execution. This seems intuitive, and is
not particularly novel; as the number of individuals living in a county increases, naively, the
number of potential murders and convictions that would occur should increase
proportionately. Neither poverty nor homicides produce significant effects. Note that homicides are
closely related to population size.
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The model presented in Table 2 seems to fare quite well with respect to model fit. The
inclusion of the frailty term and event dependence maximizes the R-squared of the model to 0.96
from 0.30. Relative to a baseline model that mirrors the model specification in Table 2
excluding event dependence, there is a significant improvement in model fit. Incorporating event
dependence, our primary quantity of interest, reduces the Bayesian Information Criterion
(BIC) by 15.2% from 10862.33 to 9212.73. This is demonstrative of superior model fit and the
added explanatory power of event dependence when considering executions. Unfortunately,
further model fit assessments are unavailable for conditional frailty models. In particular, there
does not currently exist a cross-validation approach to the conditional frailty model, nor any
other out-of-sample model prediction techniques. However, conventional Cox proportional
hazards diagnostics are typically considered sufficient to demonstrate model fit. These model
diagnostics are presented in the S1 Appendix.
One may note that we do not model dependence in sentencing and as such, are unable to
examine dependence between sentences as fewer than 20 percent of death sentences result
in an execution, as explained in [
]. Therefore, we cannot examine whether a death sentence
increases the likelihood that more cases will be raised and brought to trial by prosecutors
seeking the death sentence, but such questions are beyond the scope of this particular paper and
are unanswerable by the data currently available. As such, we leave this question open to future
It is worth noting that our study is not without its limitations. We have tried our best to
acknowledge the relatively dynamic landscape that is the American experience with the death
penalty; between 1977 and 2014 there were many changes affecting states in potentially
idiosyncratic ways. We have explicitly accounted for changes that would legally preclude a state
from using the death penalty. There are, however, a variety of immeasurable changes. These
changes might include (but are not limited to) states considering abolition without formally
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outlawing the death penalty, federal or state moratoria halting executions, and changes in the
conditions allowing for death sentences or in the availability of execution drugs. We have
attempted to account for these changes by using a conditional frailty model with a
state-specific frailty term. This modeling approach should account for these immeasurable factors as
they only vary across states, but not within states.
Nevertheless, immeasurable variables driving event dependence that vary within states at
the county-level represent a set of limitations. For example, we should note that we do not
have data on the number of capital-eligible homicides which would more directly explain a
county's decision to use the death penalty than the number of total homicides experienced.
Reliable data on the number of capital-eligible homicides by county is unavailable but
ultimately likely to be very highly correlated with our measure of all total homicides.
Finally, we should note that we do not model event dependence in death sentences but
rather executions. While modeling sentences is desirable as it is a more direct test of our
proposed theoretical mechanism and is less likely to be affected by political decisions or drug
shortages, such a study is impossible as data on death sentences are presently unavailable. Our
work on dependence in executions is the most comprehensive study to date, and we save an
extension to sentencing for future research.
We find evidence of a self-reinforcing mechanism for executions, even when accounting for
important covariates such as population, racial threat, homicides, state, or poverty. Previously
this has been suspected, only considered through examining naive evidence. As counties have
more and more experience with the death penalty, they are increasingly likely to use the death
penalty and do so at quicker rates. Not only is this made evident by the different slopes of
hazard curves in Fig 3, but the different probabilities of observing an execution at a fixed point in
time like 18 months.
The United States Constitution puts crime squarely in the realm of the states, making
some degree of local variation inevitable, and this is made stronger by locally elected district
attorneys and the powerful role for citizen-based juries. There is no reason, in other words,
to expect uniformity in the use of the death penalty across jurisdictions. What we observe,
however, is not consistent with a random-fluctuation model. A stretched distribution is
consistent only with some form of self-reinforcing mechanism, and in this paper we have tested
a model whereby individual jurisdictions adapt to their own histories, over and above such
legally or socially relevant factors as changing crime rates, poverty, population, state judicial
elections, or even homicides. The extreme concentration of cases in just a few jurisdictions
implies that the death penalty is not an option in most American communities. The vast
majority have none at all, even across 40 years of experience, and often after thousands of
homicides. Rather, the counties separate out into high- and low-execution users based on
A self-reinforcing process suggests that decision-makers may base their actions on their
expectations of the actions of others. If prosecutors believe that juries will never support a
death sentence, because none ever has, they may decide that seeking it is a poor use of limited
resources. If they believe, on the other hand, that local history shows that other actors will
carry the process all the way through to completion, they may feel compelled to go along.
There is nothing wrong per se in these expectations affecting local decision makers. However,
the US Supreme Court has never held that the number of previous executions in a county
should be a relevant consideration in deciding who gets death and who lives. This should relate
to the nature of the crime and the characteristics of the defendant, according to the Court.
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Local history should neither be an aggravating nor a mitigating factor in determining who gets
death. But it is.
S1 Appendix. Supplementary information appendix. This document contains additional
robustness checks and diagnostics associated with the manuscript ªEvent Dependence in U.S.
Executionsº. In particular, it contains an assessment of the proportional hazards assumption
and residuals (Cox-Snell, Martingale, Deviance) for the presented Conditional Frailty Model.
In addition, we assess the possibility that influential observations may influence the results
presented. Overall, the Conditional Frailty Model presented appears to fit particularly well and
perform well with respect to essential diagnostics.
The authors thank those who assisted in the early staged of this project, including William
Massengill for his research assistance, and Gary King, Skyler Cranmer, and David Darmofal
for their comments on early drafts. FRB acknowledges the work of his former student Woody
Gram, who compiled much of the homicides data used here and explored some of the ideas
formalized here as part of his senior thesis, as well as law professors Rob Smith, Lee Kovarsky,
Brandon Garrett who have also explored the geographic distribution of executions, and Dick
Dieter and Robert Dunham of the Death Penalty Information Center whose public dataset on
executions has been invaluable.
Conceptualization: Frank R. Baumgartner.
Data curation: Frank R. Baumgartner, Benjamin W. Campbell.
Formal analysis: Janet M. Box-Steffensmeier, Benjamin W. Campbell.
Investigation: Frank R. Baumgartner, Janet M. Box-Steffensmeier, Benjamin W. Campbell.
Methodology: Janet M. Box-Steffensmeier, Benjamin W. Campbell.
Project administration: Frank R. Baumgartner, Janet M. Box-Steffensmeier, Benjamin W.
Resources: Frank R. Baumgartner.
Software: Benjamin W. Campbell.
Supervision: Frank R. Baumgartner, Janet M. Box-Steffensmeier.
Validation: Janet M. Box-Steffensmeier.
Visualization: Benjamin W. Campbell.
Writing ± original draft: Frank R. Baumgartner, Janet M. Box-Steffensmeier, Benjamin W.
Writing ± review & editing: Frank R. Baumgartner, Janet M. Box-Steffensmeier, Benjamin W.
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