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Conclusions from the history of research into the effects of police force size on crime—1968 through 2013: a historical systematic review

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Objectives We describe and explain how the findings from nonexperimental studies of the relationship between police force size and crime have changed over time. Methods We conduct a systematic review of 62 studies and 229 findings of police force size and crime, from 1971 through 2013. Only studies of U.S. policing and containing standard errors of estimates were included. Using the robust variance estimation technique for meta-analysis, we show the history of study findings and effect sizes. We look at the influence of statistical methods and units of analysis, and time period of studies’ data, as well as variation in police force size over time. ResultsFindings vary considerably over time. However, compared to research standards and in comparison to effect sizes calculated for police practices in other meta-analyses, the overall effect size for police force size on crime is negative, small, and not statistically significant. Changes in research methods and units of analysis cannot account for fluctuations in findings. Finally, there is extremely little variation in police force size per capita over time, making it difficult to estimate the relationship with reliability. Conclusions This line of research has exhausted its utility. Changing policing strategy is likely to have a greater impact on crime than adding more police.
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Conclusions from the history of research into the effects
ofpoliceforcesizeoncrime1968 through 2013:
a historical systematic review
Yo n g J e i L e e
1
&John E. Eck
1
&Nicholas Corsaro
1
#Springer Science+Business Media Dordrecht 2016
Abstract
Objectives We describe and explain how the findings from nonexperimental studies of
the relationship between police force size and crime have changed over time.
Methods We conduct a systematic review of 62 studies and 229 findings of police force
size and crime, from 1971 through 2013. Only studies of U.S. policing and containing
standard errors of estimates were included. Using the robust variance estimation
technique for meta-analysis, we show the history of study findings and effect sizes.
We look at the influence of statistical methods and units of analysis, and time period of
studiesdata, as well as variation in police force size over time.
Results Findings vary considerably over time. However, compared to research stan-
dards and in comparison to effect sizes calculated for police practices in other meta-
analyses, the overall effect size for police force size on crime is negative, small, and
not statistically significant. Changes in research methods and units of analysis cannot
account for fluctuations in findings. Finally, there is extremely little variation in police
force size per capita over time, making it difficult to estimate the relationship with
reliability.
Conclusions This line of research has exhausted its utility. Changing policing strategy
is likely to have a greater impact on crime than adding more police.
J Exp Criminol
DOI 10.1007/s11292-016-9269-8
Electronic supplementary material The online version of this article (doi:10.1007/s11292-016-9269-8)
contains supplementary material, which is available to authorized users.
*Yong Je i L e e
lee2yj@mail.uc.edu
John E. Eck
john.eck@uc.edu
Nicholas Corsaro
nicholas.corsaro@uc.edu
1
School of Criminal Justice, University of Cincinnati, Cincinnati, OH 45221, USA
Keywords Police force size .Systematic review .Meta-analysis .Policing strategy
Introduction
Social scientists have demonstrated that some police practices can be effective at
reducing crime or disorder. Systematic reviews of hot spots policing (Braga et al.
2014), focused deterrence (Braga and Weisburd 2012), problem-oriented policing
(Weisburd et al. 2010), and third-party policing (Mazerolle and Ransley 2006), for
example, provide evidence for the relative effectiveness of these strategies. Long before
researchers and practitioners conceived of these strategies, police and researchers were
asking a far more basic effectiveness question: does hiring more police reduce crime.
Two systematic reviews of the impact of police force size on crime have been recently
published. Lim and colleagues (2010) count the number of findings that demonstrate a
statistically significant crime reduction effect and find that such studies were in the
minority of published findings. Even more recently, Carriaga and Worrall (2015)
conducted a systematic review of 24 studies and meta-analysis of 12 studies. They
found a small but significant crime reduction impact of police force size on crime.
In this paper, we are interested in the development of research over time. We use
methods developed for systematic reviews and meta-analysis for this purpose. We are
not simply concerned with drawing a conclusion about whether adding police reduces
crime, though this is important. We are also interested in how the findings change over
time and in uncovering possible reasons for the findings: do changes in methods and
data correspond with differences over time, for example.
Our paper builds on prior findings in three specific ways. First, we show how
findings have changed over time. These changes are critical for understanding what we
know about the police force sizecrime relationship. Second, we include all relevant
studies identified in our analysis (over 60 studies with over 200 separate findings). This
provides a more complete picture of what the research shows. Third, we examine likely
explanations for the cumulative research findings. Our inquiry provides insights into
how researchers produce knowledge and the impact of their evolving research methods
on their findings. In the end, there appears to be no impact on crime in general of hiring
more police, and advances in research methods do not seem to have helped to produce
stronger conclusions on this important issue.
We organize our paper as follows. Following this introduction, we describe the
economic theory of the police production of less crime, and outline the principle
difficulties researchers face when trying to test it. In the third section, we describe
the methods we use to identify the empirical research linking police force size to crime.
The fourth section shows how the findings from this research have fluctuated over time.
We first illustrate this change using descriptive findings over time and then we show
effect sizes over time. These show contradictory results. The descriptive findings
indicate that results have fluctuated over time, but recently the addition of police seems
to have had a greater impact on crime. In contrast, the meta-analysis shows that, at least
for the last four decades, findings seem to be constant and have no impact on crime
reduction. In the fifth section, we test for possible explanations related to these findings:
changes in the statistical methods and units of analysis, and the lack of variation in the
principle independent variable (i.e., police size stability over time). Based on the tests
Y. J . L e e e t a l .
of these explanations, we conclude that there is little reason to continue conducting
research examining the police force sizecrime relationship. However, for cost-
conscious mayors, city managers, and city councils, there is a silver lining: modest
planned reductions in police force size are unlikely to have a consequential impact with
regard to overall crime.
The hypothesis that more police reduces crime
The hypothesis that increasing police force size reduces crime is relatively simple. It
treats a police agency as a Bfirm^with a single homogeneous input, labor, and set of
outcomes that are various forms of crime (Becker 1968). Using the typical ceteris
paribus argument, as the number of police officers increases, crimes should decline, for
a variety of reasons. More police should increase arrests of offenders, some of whom
end up incarcerated for their crimes. Thus, one mechanism driving this relationship is
incapacitation: what Nagin (2013) refers to as the apprehension role of police. General
deterrence is a second mechanism: with more police, potential offenders perceive that
their risk of being caught after offending is higher, so they too cut back on their crimes.
Nagin (2013) calls this the sentinel role of police. Another possible mechanism is
specific deterrence: more police officers allows tracking of specific offenders, who then
cut back on their misdeeds. This is a special case of the sentinel role. Nagin, Solow, and
Lum (2015) present a theoretical model of policing that suggests that the sentinel role is
more effective than the apprehension role, though the two roles are intertwined.
Becker (1968) focuses his attention on the apprehension role; BThe more that is
spent on policemen, court personnel, and specialized equipment, the easier it is to
discover offenses and convict offenders. One can postulate a relation between the
output of police and court activity (A)and various inputs of manpower (m), materials
(r), and capital (c), as in A = f(m, r, c), where f is a production function summarizing the
state of the arts.^(p. 174). Though Becker focuses on apprehension, his argument is
not dependent on the precise mechanism by which numbers of police produce crime
reductions.
Becker (1968) also assumes, following standard economic theory, that there are
diminishing marginal returns to policing. That is, as inputs increase, the outputs would
increase, but at a declining rate: the benefit of the first 100 officers deployed would be
greater than the benefits of the next 100 deployed, which would be greater than the next
100, and so on. We can summarize BeckersideasinFig.1. The horizontal axis is the
number of police, and the vertical axis is the number of crimes. The curves represent
the theoretical relationship between these two variables, assuming all other factors
influencing crime have been controlled for. The shapes of the curves show the declining
marginal utility of police officers.
Historical examples provide evidence that removing police can spark a large
increase in crime (the equivalent of moving from the right to the extreme left on the
curve). Andenaes (1974) gives two examplesthe Liverpool Police strike of 1919, and
the Nazi arrest of the Danish police in 1944where the removal of policing preceded a
dramatic surge in crime. Russell (1975 [1930]) describes a similar result from the 1919
Boston police strike (the Finnish police strike of 1976 produced only a small increase in
crime and may provide a counter example [Makinen and Takala 1980]). From these
The effects of police force size on crime
examples, it is reasonable to conclude that moving from zero police to a modest number
of police is likely to reduce crime substantially.
The research we are reviewing does not address this. Rather, it examines whether a
small (marginal) change in the size of an existing police force has an impact on crime.
In short, it looks at the right part of the curve, rather than the extreme left part where
crime drops rapidly with foundational (baseline) changes in police force size.
Becker (1968) notes that Bit would be cheaper to achieve any given level of
activity... the more highly developed the state of the arts, as determined by technologies
like fingerprinting, wiretapping, computer control, and lie-detecting^(p. 174). This too
is in keeping with standard economic theory, which assumes that at any given time, a
firm is using a particular technology. An improvement in technology shifts the curve
downward. In Fig. 1, strategy A is the Bolder^technology, while strategy B is a
Bnewer^technology. For any given level of police force size, there is less crime with
strategy B than strategy A, though both curves show a similar relationship between
police force size and crime. Becker (1968) reflects the thinking of the 1960s by
focusing on physical technology. Today, we can think of Bstate of the art^as reflecting
the operational strategy of a police department. We will come back to this extremely
important point later in the paper.
The first empirical attempt to estimate the relationship between police force size and
crime was by Morris and Tweeten in 1971. Controlling for other confounders, they
found that an increase in police force size results in an increase in crime. Since this
study, there have been numerous other efforts. Two recent systematic reviews suggest
contrasting conclusions. Lim and colleagues (2010) claimed there is more evidence
contradicting a crime reduction effect of hiring more police than there is evidence
supporting it. In contrast, Carriaga and Worrall (2015) suggested there is a small but
significant crime reduction effect, on average. This contrast may be due to the criteria
for selecting studies and the methods they used to analyze them. Lim, Lee, and
Fig. 1 Theory of police force size and crime reduction
Y. J . L e e e t a l .
Cuvelier (2010) examined 58 studies. They counted the number of findings supporting
and contradicting the crime reduction hypothesis: a vote count analysis. Carriaga and
Worr a l l ( 2015) restricted the studies they examined to those that analyzed the relation-
ship between police force size and crime over time. They included 24 studies in their
systematic review and 12 in their meta-analysis. Nevertheless, reconciling their differ-
ent conclusions is important. Given that researchers have been examining this topic for
over 40 years, it is also important to examine how findings have changed over the past
four decades.
Methods
To systematically assess the relationship between police force size and crime, we
followed established systematic review methods (Cook et al. 1997; Higgins and
Green 2011; Mulrow and Oxman 1997) and used an advanced meta-regression tech-
nique. Our target studies are those that describe an empirical relationship between
police force size (or a proxy for this independent variable) and crime, for police
agencies in the United States. We restricted the inquiry of our study to the U.S. because
most of the studies of this type describe U.S. policing, thus including the small number
of non-U.S. studies would increase the heterogeneity in the findings without shedding
additional light on the subject. Consequently, we do not make any assertions about the
generalizability of our findings to police forces outside the U.S., nor do our findings
apply to U.S. Federal law enforcement agencies, as well as county, or state police
forces.
Search strategy
We examined the English-written literature using both electronic and manual searches.
Our electronic search of various databases
1
used these keywords: police force,police
employment,police level,police expenditure,police budget,police effectiveness,police
hiring,COPS and GAO (Government Accountability Office),and police deterrence.
Because the literature on this topic prior to Beckers(1968)economictheoryofcrime
offered few empirical findings, we retrieved studies from 1968 through March 2015.
We also conducted manual reference checks of relevant literature, including the two
recent systematic reviews described above. Finally, we contacted scholars in this field
and asked them about any study we may have missed. We presented an early version of
this study at the Annual Conference of the American Society of Criminology at Atlanta,
Georgia in 2013, and asked attendees if they knew of gaps in our literature. We relied
upon an iterative approach rather than a sequential approach. Once we found studies
that met the keywords criteria from a search of online databases, we searched the
bibliographies in the new studies for other papers. If new keywords became apparent,
1
Sociological Abstracts, Social Science Abstracts (SocialSciAbs), Social Science Citation Index, Arts and
Humanities Search (AHSearch), Criminal Justice Abstracts, National Criminal Justice Reference Service
(NCJRS) Abstracts, Educational Resources Information Clearinghouse (ERIC), Legal Resource Index, Dis-
sertation Abstracts, Government Publications Office, Monthly Catalog (GPO Monthly), Google Scholar,
Online Computer Library Center (OCLC) SearchFirst, CINCH data search, and C2 SPECTR (The Campbell
Collaboration Social, Psychological, Educational and Criminological Trials Register).
The effects of police force size on crime
we conducted another on-line search. This was particularly important because the
online databases have limited access to studies prior to the 1980s, and because the
terminology used has changed over the decades.
Literature screening and data extraction
We used a three-step screening process to select studies. First, we reviewed both
abstracts and tables with empirical findings to determine if a study contained an
empirical assessment of the relationship between police force size and crime. We then
examined the entire article for all studies that met the step-one criteria. Second, we
dropped the studies that did not provide the standard errors of estimated coefficients
showing the effect of police force size on some form of crime. Six studies failed to
provide this information. Finally, we eliminated any study of policing outside the U.S.
Tab le 1summarizes these steps. By step three, we had included 62 studies and 229
findings (most studies reported on multiple crime types resulting in multiple findings).
A comparison with two other recently published systematic reviews by Lim, Lee, and
Cuvelier (2010) and Carriaga and Worrall (2015) suggests that we drew upon a similar
but slightly more comprehensive body of studies, despite using slightly different
selection criteria (see Table 1).
Coding protocol
Though the search period for literature was between 1968 and 2015, we found relevant
studies between 1971 and 2013. For each finding, we coded whether or not it supported
a crime reduction hypothesis (a statistically significant negative relationship between
police force size and crime). To meta-analyze the overall effect size of 229 findings
across 62 studies, we recorded the standard errors and relevant statistics (e.g., sample
size, t-statistics, confidence interval, and standard deviations) for each estimated
coefficient linking police force size to crime.
We also coded statistical modeling techniques of the 62 studies over this 43-year
period: Ordinary least squares, 2-stage least squares, 3-stage least squares, hierarchical
linear modeling, and first-difference generalized method of moments. We also accounted
Tab l e 1 Number of studies and findings considered for systematic review
Number
of articles
Percent
of articles
Number
of findings
Percent
of findings
Step 1 empirical studies 70 100 258 100
Step 2 include standard error 64 91 236 91
Step 3 U.S. police agencies 62
a
89 229 89
Lim et al. 2010 (includes five
non-US studies)
58 256
Carriaga and Worrall 2015 (includes
one non-US study)
24
b
(12) Not reported
a
Appendix A. provides a full list of these studies
b
Reports 68 relevant studies found
Y. J . L e e e t a l .
for the year of publication, geographic unit of analysis (city, county, metropolitan area, or
state) focused on, and the years from which the data were collected. Table 2shows the
characteristics of the studies reviewed in this paper.
Descriptive findings over time
Our first interest is to determine whether findings appear to change over time. We are
interested in showing how a diligent reader of the police force size literature might view
the conclusions of studies as the cumulative findings evolve. Such a reader will not be
conducting a progressive meta-analysis, updated with every new study. Rather, she or
Tab l e 2 Characteristics of the studies analyzed
Characteristics Number of studies % Number of
findings
%
Year of publication
19711979 16 25.8 50 21.8
19801989 9 14.5 14 6.1
19901999 12 19.4 31 13.5
20002009 21 33.9 116 50.7
20102013 4 6.5 18 7.9
Geographic unit
City, district, precinct 33 53.2 121 52.8
County 6 9.7 27 11.8
Juris, Sub/SMSA, area 10 16.1 39 17.0
State, nation 13 21.0 42 18.3
Measure of police
a
Dollar 24 38.7 82 35.8
Number 37 59.7 143 62.4
Typ e of a n al y si s
OLS, BA, VAR 35 56.5 113 49.3
2SLS, 3SLS 22 35.5 91 39.7
HLM, GMM 5 8.1 25 10.9
Control for simultaneity
Yes 32 51.6 143 62.4
No 30 48.4 86 37.6
Crime reduction hypothesis
Supportive 32 51.6 83 36.2
Non-supportive 30 48.4 146 63.8
Total 62 100 229 100
BA bivariate analysis, GMM (first differenced) generalized method of moments, OLS ordinary least squares, 2
(or 3) SLS two (or three) stage least squares, VAR vector auto regression, HLM hierarchical linear modeling
a
One study (and corresponding four findings) used number of days as a measure of police force
The effects of police force size on crime
he is likely to count the number of studies that seem to support the hypothesis that more
police reduce crime, relative to the studies that fail to support this hypothesis.
There are several serious limitations to such an approach (which we describe
shortly), but this approach is probably a reasonable approximation of how researchers
and practitioners might alter their views of the hypothesis over time. We began by
sorting the 229 findings into two categories: those that support the crime reduction
hypothesis, and those that do not. We classified a finding as supportive if there was at
least one significant negative coefficient for police force size. Otherwise, we classified
it as contradictory. If a study had at least one supportive finding, we coded the study as
supportive. Otherwise, we coded it as contradictory. We then, for each year, subtracted
the contradictory studies from the supportive studies.
Figure 2displays the differences between the supporting and contradicting studies
over time. It is apparent that the net conclusions have fluctuated a great deal across the
study period. Specifically, we find that an equal number of studies had supported and
contradicted during the 1970s. However, during the 1980s and 1990s, the number of
contradicting studies was two times greater than the number of supporting studies.
After 2000, it is apparent that the crime reduction hypothesis became more predomi-
nant. Over the entire 43-year period, more studies support the crime reduction hypoth-
esis than contradict it. In conclusion, a diligent reader who had followed this literature
over 43 years might change her or his mind several times, but conclude in 2015 that on
balance hiring more police seems to have an impact on crime reduction. Not only are
there a few more studies supporting this conclusion, since 2000 the research seems to
pointinthisdirection.
There are two reasons this analysis may be misleading. First, changing how we
count changes our conclusions. If we count findings rather than studies, we find that
about 36 % of findings across studies are supportive, while 64 % of findings are not.
Fig. 2 The fluctuating findings for studies of police force sizecrime relationships: 1971 through 2013
Y. J . L e e e t a l .
This is consistent with Lim and colleagues (2010) who reported 21 % of the findings in
their database supported a crime reduction effect of police strength (the other 79 % of
the findings were either nonsignificant or significant positive relationships). Second,
comparing studies or findings in this way assumes each study (or finding) is equally
weighted, thus we are accepting the null hypothesis for nonsignificant findings, rather
than concluding we are uncertain of the true finding. In fact, our confidence in findings
is dependent on its standard error. We should, therefore, weight findings by their
standard errors and compare effect sizes. We next turn to this more precise analysis.
Statistical analysis
Effect size analysis
To estimate effect sizes, we used each studys standardized regression coefficient (and
standard errors) between police force size and crime variables (Higgins and Green
2011; Wilson 2001). Because the impact of police force size on crime is often measured
using different methods and metrics across the studies, the direct pooling of regression
coefficients is not meaningful (Nieminen et al. 2013). In such a case, standardized
regression coefficient may offer a solution. Standardized regression coefficients are the
estimates from an analysis carried out on variables that have been standardized to their
variances equal to one (Vittinghoff et al. 2005). Therefore, in the context of the police
crime relationship, standardized coefficients show how many standard deviations in the
number of crimes will change per a standard deviation increase (or decrease) in the
police force size variable. For most of the studies, researchers supplied standardized
coefficients, or relevant statistics (e.g., raw coefficient estimate, standard deviations of
each variable in the model specification) in tables, figures, or in the body of the text, so
we were able to calculate the standardized coefficient estimates of each study.
2
We
excluded studies that fail to provide one or more of these components from our
systematic review. We then used these standardized coefficients of all 229 findings
from 62 studies to estimate the overall effect size in the Stata 14 statistical package.
Overall effect size
There are two different methods for estimating the overall effect size of police force
strength on crime: fixed-effects and random-effects models. Both models rely on the
inverse variance weight, so findings with smaller variances (larger studies) contribute
more to the weighted average than studies with larger variances (smaller studies)
(Helfenstein 2002). However, the final weight in a random-effects model is the inverse
2
If beta, the standardized regression coefficient, is reported in the estimated regression model, we use beta as
the effect size. Standard errors of betas were calculated by applying formula: SE βðÞ¼1r2
ffiffiffiffiffiffi
n1
p, where ris
Pearson or Spearman correlation coefficient. However, if a study reports the raw coefficient estimate (= b)asa
result of a linear regression, we calculate the standardized coefficient by applying formula: β¼SD XðÞ
SD YðÞ
b, where
SD(Y) is the standard deviation of the outcome variable while SD(X) is the standard deviation of the police
force size variable used in the study. The standard error for beta is obtained by applying formula:
SE βðÞ¼
SD XðÞ
SD YðÞ
SE bðÞ. If the standard error was not reported, we obtained it from the confidence interval.
The effects of police force size on crime
of both study level variance and the estimated between study variance. This gives
random-effects models precise type I error rates, so it yields more conservative effect
size estimates compared to the fixed-effects model (Lipsey and Wilson 2001). Further,
the fixed-effects model assumes that all findings come from the same population,
controlled for same variables, use the same outcome definitions, and are otherwise
similar to each other with regard to factors that influence analytical findings. This is
often an invalid assumption (Higgins et al. 2003; Wilson 2001), and the assumption is
unlikely to be valid for the 62 studies and 229 findings we examined. For example,
researchers measure police force size using the number of police officers or the dollar
amount spent for hiring additional police officers (e.g., GAO 2005). They also fre-
quently rely on a different geographic unit of analysis (e.g., city, county, metropolitan
area, state, and nationwide), and analyze different periods. As we have noted, the
studies used several different statistical modeling techniques. Due to the presence of
heterogeneity across different studies, a random effects model seems to be an appro-
priate method to estimate the overall effect size.
However, a conventional random effects model relies on the assumption that the
effect sizes from different studies are independent of one another. This assumption is
not valid where multiple findings are nested within the same study. Without considering
this dependent nature among multiple findings, a conventional random effects model
would incorrectly estimate the overall effect size. To remedy the hierarchical nature of
multiple findings per study, we estimated the overall effect size using the robust
variance estimation (RVE) technique in meta-analysis (Hedberg 2014; Hedges et al.
2010). The RVE method provides a robust method for estimating standard errors in
meta-regression, particularly when there are dependent effects among multiple findings
within the same study. However, multiple findings per study can further cluster by the
same research group. For example, effect sizes reported in Boba and Lilley (2009)and
Lilley and Boba (2008) are likely to cluster at a higher level if the researchers used
similar data and statistical method to estimate their effect sizes in several studies.
Therefore, it makes sense to assume that the observed effect sizes (findings) are nested
within studies, which are further, nested within hierarchically higher-level clusters. So,
we used the RVE method with Bhierarchical^model weight in Stata to operationalize
the hierarchical nature across findings, studies, and research group levels.
We estimated a mean effect size of police force size on crime of about .030 (with a
95 % confidence interval between .078 and .019).
3
The effect size from the RVE
method is not statistically significant. The corresponding tau-squared value (= 0.0007;
an estimate of the between-clusters variance component) shows that there is a between-
clusters variance among the 229 findings nested with 62 studies. The nonsignificant
and tiny mean effect size between police force size and crime
4
suggests that simply
increasing police force size may not help reduce crime, and if it does, it does not reduce
crime by much.
It is instructive to compare this effect size to the effect sizes from other meta-
analyses of police strategies, as this provides a set of standards by which to judge the
importance of hiring more police. In Fig. 3, we compare this effect size of police force
3
This is consistent with the results from conventional random-effects model we ran as an exploratory analysis
for comparison.
4
We provide a full list of effect sizes with forest plot for 229 findings in Appendix B.
Y. J . L e e e t a l .
size to the effect sizes reported in meta-analyses of crime hot spots (Braga et al. 2014),
problem-oriented policing (Weisburd et al. 2010), neighborhood watch (Bennett et al.
2006), and focused-deterrence (Braga and Weisburd 2012). Though all effect sizes for
these programs are negatively related to crime, we recalibrate their negative signs into
positive (crime prevention gain) signs so that the height of bars corresponds to a bigger
impact on crime reduction. It is apparent that the effect size for adding police is
miniscule compared to the other effect sizes. Thus, it appears that cities might reduce
more crime by using specific strategies to reduce crime than by hiring more police.
One possible explanation for the nonsignificant tiny effect size is that effect size
might change over time. If recent studies demonstrate significant effect sizes while older
studies report nonsignificant effect sizes, then the nonsignificant effect sizes from older
studies might be masking the significant (and more valid) effect sizes of recent studies.
Period effect sizes
To test whether the effect sizes for police force size and crime have changed over time,
we divided our 43-year study period into four overlapping 20-year time periods. This
provides sufficient years in each period to show the change in effect sizes over four
decades. We assigned the studies (and their findings) to one of the four periods where
the dataset of the study was located. For example, Kovandzic and Sloan (2002)
analyzed data for the years from 1980 to 1998. The authors found three significant
negative relationships and five nonsignificant relationships between police force size
and crime. We assigned these findings to 19801999 but not to 19902013, because the
authorsdataset only fit in the first period. This assignment process did force us to drop
ten studies and 70 findings from the analysis because these studies overlapped two or
more periods.
We calculated the mean effect sizes for each period using the RVE technique. Our
findings are shown in Table 3. The effect sizes for four overlapping decades fluctuate
from positive, to highly negative, to around zero, to small and negative, but no period
Fig. 3 Effect sizes from systematic review studies
The effects of police force size on crime
effect size is significantly different from zero. One persistent finding is clear. The
nonsignificant overall effect size is not due to older studies; indeed, the effect size
between police strength and crime has been consistently nonsignificant for the past four
decades.
However, another possibility for the nonsignificant overall effect size might be due
to the methods used. Studies that use weaker methods might be driving the overall
nonsignificant effect size. If so, then looking at only strong studies would give a more
valid estimate of the average effect size. We look at that next.
Methods used
Researchers prefer advanced statistical methods because they assume these methods
produce more valid findings than more basic correlational techniques. For instance,
since Levitt introduced it in 1997, researchers have increasingly employed the instru-
mental variable (IV) technique to control for the endogenous relationships between
police force and crime. Recently, Kovandzic and his colleagues (2016)havesuggested
that applying generalized method of moments (GMM) provides more advanced spec-
ification tests for instruments rather than conventional IV technique. They reanalyzed
Levitts two studies (1997,2002) and showed that when using GMM they get different
results than Levitt. One way of thinking about the history of research on this topic is as
an arms race between those supporting and those skeptical of the crime reduction
hypothesis, where the arms are statistical methods.
Though newer methods are introduced over time, researchers may not switch to
newer methods all at once. We attempt to unravel the context and the history of the use
of statistical methods in the study of relationships between police force size and crime.
Figure 4shows that statistical methods change over time. Specifically, prior to 2000,
OLS regression was the most used statistical method, but after 2000 other methods
became dominant. Studies that use lagged variable models to control the endogenous
relationship between police force size and crime also decreased after 2000. Two- and
three-stage least squares have not appeared in this literature since 2010.
If the advanced methods consistently produce more valid results, we should see
evidence that changes in research methods have changed the size or the sign of effect
sizes, regardless of the decade the research was conducted. To test this hypothesis, we
used the RVE method with beta as outcome variable, and dummy variables for the
methods and measures used. We coded both two-stage least-squared and three-stage-
least-squared models as 1 if a study used either statistical model; otherwise 0. We coded
Tab l e 3 Effect sizes between police force size and crime by decades of dataset
Period Effect size
(random)
pvalue Number
of findings
Number
of studies
Between-clusters
differences (τ2)
From 1960 to 1979 .092 .563 59 22 .0281
From 1970 to 1989 .173 .382 30 17 .1681
From 1980 to 1999 .003 .646 43 15 .0001
From 1990 to 2013 .020 .532 71 14 .0026
Y. J . L e e e t a l .
both generalized method of moments and hierarchical linear model as 1 if a study used
either method; otherwise 0. We also considered simultaneity bias between police force
size and crime. Levitt (1997) notes that while more police may drive crime down, more
crime might prompt elected officials to hire more police. Failing to control for this loop
(simultaneity) between the two variables may result in a biased estimate of the
relationship between police size and crime. Therefore, we coded the studies that used
longitudinal data to control for simultaneity as 1, otherwise 0.
Since choice of statistical method may be linked to other methods choices, we also
controlled for the way the independent variable was measured, and the units of analysis
used. If a study measures police force size using the amount of dollars spent for hiring
additional police officers, we coded the dummy variable as 1. Consequently, any study
that measures police force size using the number of police officers or other measures is
coded as 0. We also created dummy variables for the geographic unit in which the
researchers conducted their study: city or county were coded as 1 and larger units were
coded as 0 and used as the reference value.
Tab le 4shows no significant relationships between the statistical methods used and
effect sizes controlling other factors fixed. This is consistent with the recent findings of
Carriaga and Worrall (2015) that neither unit of analysis nor study design has signif-
icant impact on the outcome variable. Specifically, using the city or county as unit of
analysis does not increase the chances of discovering a crime reduction effect relative to
other possible units. We also find that the choice of a geographic unit of analysis does
not significantly influence the effect size. Measuring police force size by police budget
(rather than by number of officers) does not seem to influence the effect size. These
results hint at the possibility that neither statistical method nor measures of police force
size and geographic unit of the study have significant impacts on the main relationship
between police force and crime. For now, we must conclude that variation in the
statistical methods used is not related to effect sizes. While we cannot assert that the
next advance in such methods will be unproductive, these results suggest we should not
be overly optimistic that newer methods would give us different findings.
0
2
4
6
8
10
12
14
16
18
20
1970s1980s1990s2000s2010s
Number of statistical methods
OLS 2 or 3 SLS Advanced (FD-GMM, HLM) Simultaneity (Lagged-Variable)
Fig. 4 Statistical methods employed in police force size and crime studies: 1971 through 2013
The effects of police force size on crime
Lack of variation in police force size
Finally, we examine another factor that might explain the tiny and nonsignificant effect
size: lack of variation in police force size. In an experiment, the amount of an
intervention should exceed a certain dosage, relative to a control, to assure sufficient
variation of the intervention variable. If this does not occur, then the independent
variable approximates a constant and it will be impossible to measure the impact of
the intervention. Something similar to insufficient dosage could influence our estimates
of the relationship between police force size and crime study. Specifically, if there is
very little change in police force size over time, then whatever impact hiring more cops
has on crime may be difficult to detect.
How much change has occurred in police strength over time? To help answer this
question we analyzed the temporal change in police force size for different groups of
cities based on their populations. The data come from the FBI Table 78, police force
size by city agency (Federal Bureau of Investigation [FBI], 2010). The data describe all
U.S. cities with population greater than 100,000 between 1990 and 2010 with complete
data (about 77 % of all U.S. cities). We divided cities into groups of different
populations. In Fig. 5, we display the temporal variation in police per capita of each
group over time. Within each group police force size is stable for most of these cities.
Only large cities (about 5 % of all cities) with population greater than 1,000,000 show
some changes in police force size. There is a growth from 1990 through 1999, and a
large decline in 2003. This decline is due to the large decrease in the police force of a
single city, New York: the six other cities in the same group did not experience any
change in police force size. Table 5provides further evidence of the stability of police
force size over time: standard deviations of the average number of officers are very
small relative to their mean values. For the most part, it appears that cities and counties
maintain a constant level of police per capita. Large variation in police per capita is not
created by hiring and attrition (or firing), but by different size jurisdictions having different
levels of policing: bigger jurisdictions have more police per capita than smaller ones.
These findings are very important for four reasons. First, police force size
behaves more like a constant than an independent variable, and without much
Tab l e 4 Test of advanced method hypothesis in police force size and crime studies
Model variables Robust variance estimation method with hierarchical
model weight schema (DV = effect size)
Bpvalue
(Constant) .042 (.063) .516
Categorical variables City .153 (.139) .295
County .110 (.080) .253
2-SLS or 3-SLS .057 (.156) .721
HLM or GMM .237 (.110) .151
Longitudinal analysis .041 (.088) .649
Police number (by dollar) .061 (.071) .401
Level-1 (Finding level) N= 229; Level-2 (study level) N=62;τ2=.0093
Y. J . L e e e t a l .
variation in an independent variable, it is difficult to find a connection to a
dependent variable. Second, the tiny variation in police force size makes any
empirical findings about the link to crime highly sensitive to model specification.
This would help explain the variation in findings across studies. Third, with high
sensitivity, both positive and negative relationships between police force size and
crime can be found (as shown in Fig. 2). Finally, the lack of temporal variation
also helps explain why studies that made use of temporal changes in police and
crime did not produce different findings from those that relied on cross-sectional
comparisons only. In conclusion, like the Dudes rug, this lack of variation
brings together the various findings in our systematic review.
Discussion and conclusions
In review, here are our findings. First, the overall effect size of police force size is
negative, small, and statistically not significant. Compared to the effect sizes of
150
200
250
300
350
400
450
Number of Police Officers per Capita (x 100,000)
Year
1,000 <
(5%)
500 - 1,000
(13%)
200 -500
(29%)
150 - 200
(19%)
< 150
(34%)
Group of Cities with population (x1,000)
(percentageof cities among all cities)
Fig. 5 Police force size rate for U.S. cities grouped by population: 1990 through 2010
Tab l e 5 Police officers per capita for U.S. cities with populations greater than 100,000: 1990 through 2010
Group of cities
with
pop (x1000)
Number
of cities
Percentage
of cities
Mean
officers/
capita
Standard
deviation
Overall
change
(%)
1000 < 7 4.8 % 384.3 15.8 2.2
5001,000 19 13.1 % 276.1 6.6 7.7
200500 42 29.0 % 246.7 3.1 0.6
150200 28 19.3 % 222.4 6.9 6.6
<150 49 33.8 % 200.0 5.2 3.7
Overall 145 100.0 % 291.4 7.0 1.6
The effects of police force size on crime
strategies that are (randomized and quasi-experimentally) tested in the policing litera-
ture, we conclude that merely increasing police force size does nothing to reduce crime.
This finding is different from the most recent meta-analysis of this topic (Carriaga and
Worr a l l 2015).
Second, the reported effect of police force size on crime has been constant and not
significant for more than 40 years. This is contrary to what we showed in the
descriptive analysis (Fig. 2). There is no good reason to suspect that marginally more
police reduce crime at a meaningful level and that more recent research is more
supportive of the crime reduction hypothesis.
Third, effect sizes do not appear to be related to the research methods or statistical
techniques used in the 62 studies. This conclusion is consistent with Carriaga and
Worr a l l s(2015)analysisof24studies.
Fourth, the little temporal variation in police force size from 1990 through 2010
helps explain the tiny and statistically insignificant effect sizes we found. It also might
explain why researchers have produced so many contradictory findings over 43 years.
Reconsider Fig. 1. The most optimistic summary of the evidence to date is that
police services in the U.S. are located along the right hand end of the curve: any crime
control utility from adding more officers will be difficult to detect, particularly given
that police agencies are unlikely to make substantial increases in their force sizes.
Further, greater progress might be made in police crime-control effectiveness by
changing the policing strategy from curve A to curve B. The substantially larger (and
statistically significant) effect sizes from meta-analyses of police strategies indicate that
policy makers who want police to have an impact on crime would be better suited
investing resources in new evidence-based strategies than funding surges in police
hiring.
For researchers interested in policing, these findings should be sobering. After more
than 40 years of study, during which researchers have used increasingly sophisticated
statistical models, we can state that police force size is unlikely to make much
difference in crime, on average. However, the increased sophistication of research
methods over these four decades has contributed little to improve our understanding
of this topic. Unlike Lim and colleagues (2010) and Kovandzic and coauthors (2016),
we are not optimistic that further research in this area using statistical modeling is likely
to be productive. It is possible that police force size might influence some specific
crimes more so than others. Several studies suggest this (e.g., Carriaga and Worrall
2015;GAO2005; Lim et al. 2010). Unfortunately, we cannot reliably address this
question with 62 studies and 229 findings: researchers have not consistently studied the
same crime types so we have too few studies to assess this thesis.
Further, the lack of variation in the principle independent variable suggests that any
result will be highly sensitive to data and model specification. This is true of crime
specific results as well. This explains why findings have appeared to randomly fluctuate
over time even when researchers used similar methods and datasets to decipher the
relationship between police force size and crime.
We should put the weak, and possibly nonexistent, connection between police force
size and crime in context. This finding is part of a larger literature that indicates that the
types of methods used to answer this type of question may not be up to the task. After
reviewing the research on the impact of the death penalty and crime Daniel Nagin
(2013, p. 92), for example, concludes that BThis unpredictability calls into question the
Y. J . L e e e t a l .
usefulness of prior data on the death penalty when calculating present and future risk.^
Looking outside of the criminal justice policy literature, we find a similar conclusion in
education policy research. In the conclusions of an article on the effects of classroom
size on future earnings of students, Dustmann et al. (2003:F118-19)state,BThe main
result of our analysis is that class size effects on wages are present, but very small. They
are unlikely to be detected in simple reduced form regressions when using data sets of
moderate size.^Hanushek (2002) comes to the same conclusion.
Within this context, our results seem unremarkable. However, because of this
context the major implication of our systematic review is clear: researchers should
reconsider efforts to link simple police outcomes to simple inputs using nonexperi-
mental statistical modeling. In contrast to the quasi- and randomized controlled exper-
imental literature, nonexperimental studies are easy to implement, but their collective
findings maybe too weak and unreliable to inform policy. Here we break with the
traditional way of concluding a research paper. We anticipate little benefit to pursuing
this line of enquiry, and suggest that it is time to end it.
Acknowledgments We are grateful to Francis Cullen, Alfred Blumstein, Christopher Sullivan, John
Wooldredge, and Aaron Chalfin for their suggestions on our earlier research leading to the current paper.
We also wish to thank SooHyun O and Natalie Martinez for helping us with various aspects and comments on
this topic. Finally, we owe our gratitude to the referees and editors for their many insightful suggestions. We
are particularly grateful to David Wilson for his tough and important comments; they greatly improved the
paper.
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Yo ng J e i L e e is a PhD candidate in the School of Criminal Justice, University of Cincinnati. He received a
masters degree in public policy and management from Carnegie Mellon University. His current research
examines police effectiveness, crime hot spots detection and forecasting algorithm, and the concentration of
crime at places, offenders, and victims. For this paper, his role was to gather the studies using a systematic
review process, conduct the statistical analysis, and write the paper.
Y. J . L e e e t a l .
John E. Eck specializes in police effectiveness and crime prevention. He has been active in research in these
areas since 1977, and with the University of Cincinnati since 1998. Eck has studied investigations effective-
ness and problem-oriented policing. Since the mid-1990s he has focused much of his attention to examining
why crime and disorder is highly concentrated at specific addresses and what police and others can do about
this. Having monitored research findings on police force size and crime for his entire professional career, he
became curious as to whether decades of study had resulted in anything useful. In this paper, his principle
contribution was to launch the investigation, supervise the research, and edit drafts.
Nicholas Corsaro is an Associate Professor at the School of Criminal Justice at the University of Cincinnati.
His research focuses on the role of the police in crime prevention with a particular emphasis on the use of
strategies, tactics, and organizational policies. For this paper, he provided assistance in structuring the
statistical analysis, interpreting the findings, and provided editorial feedback.
The effects of police force size on crime

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... For that reason, "Changing policing strategy is likely to have a greater impact on crime than adding more police". 31 However, it is known that most executives cannot articulate their business's objective, scope, and advantage in a simple statement. If they can't, neither can anyone else. ...
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Country-specific knowledge offered by police science is a necessary precondition for successful policing. However, cursory data on police science in the Baltic countries raises many questions about the real use of scientific knowledge opportunities in these countries.The paper elaborates on a framework to advance a science-oriented mindset in the police. Principles of narratives and framing were used to overcome a possible conflict between scientists and practitioners, and bridge all the parties related to policing. In this framework, scientists should take the lead. External, internal, and strategic factors, as Alisson’s three key management functions, were used to specify factors of the framework in the research. The framing process that holds a focusing function allows the use of discovered narratives and presents a variety of possible future directions. Creating a metanarrative with beginning, middle, and end grasps all the agents and ties into future developments.
... The paper by Yong Jei Lee and collaborators [40] is an example from criminology. The aim of their work was to show how many standard deviations in the number of crimes will change per a standard-deviation increase (or decrease) in the police-force size variable in the USA. ...
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The lack of consistent presentation of results in published studies on the association between a quantitative explanatory variable and a quantitative dependent variable has been a long-term issue in evaluating the reported findings. Studies are analyzed and reported in a variety of ways. The main purpose of this review is to illustrate the procedures in summarizing and synthesizing research results from multivariate models with a quantitative outcome variable. The review summarizes the application of the standardized regression coefficient as an effect size index in the context of meta-analysis and describe how it can be estimated and converted from data presented in original research articles. An example of synthesis is provided using research articles on the association between childhood body mass index and carotid intima-media thickness in adult life. Finally, the paper shares practical recommendations for meta-analysts wanting to use the standardized regression coefficient in pooling findings.
... Somewhat paradoxically, research indicates a negative correlation, where countries with high levels of police per capita on average display lower levels of trust in the police (Nägel and Vera 2021;Kääriäinen 2016). One explanation for this is that having more police does not seem to lead to lower crime rates (Lee, Eck, and Corsaro 2016) or changes in perceptions of police effectiveness (Schaap 2018). Moreover, the police force tends to be larger in countries that are more corrupt (Nägel and Vera 2021). ...
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The relationship between police and ethnic minorities has been the subject of increasing interest in many Western societies in recent years. We examine first-generation immigrants' trust in the police in Europe from a comparative and longitudinal perspective. Based on roughly 20,000 immigrants observed in 22 countries over 13 years in the European Social Survey, results show that initially high levels of trust in the police among immigrants tend to erode with the length of their stay in the host country. We show that two simultaneous processes drive this pattern: a fading reference effect (downward assimilation) and an increasing discrimination effect. Cross-national comparisons show that, on average, immigrants in countries with more police trust the police less. However, there is no effect of police size within countries, mostly because police numbers hardly change over time. We discuss implications for future research and policy development based on our findings.
... The potential effect of the more drastic proposals of police defunding, which call for the abolition or large-scale downsizing of the police institution, is arguably more elusive. The most recent systematic review and meta-analysis of research on changes to police force size found a small, nonsignificant effect size on crime (Lee et al., 2016). Studies conducted since Lee et al.'s (2016) review have found significant negative relationships between police force size and crime across a range of crime types, inclusive of murder, rape, robbery, and part-1 property crime (Chalfin & McCrary, 2018;Kaplan & Chalfin, 2019;Mello, 2019). ...
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... First, a keyword search[3] was conducted on eleven literature databases [4], with texts and abstract screening [5]. Second, references from reviews that focused on police programs, police practices, and patrol strategies were consulted (Famega 2005, Bradford 2011, Lee et al. 2013, Carriaga and Worrall 2015, Braga and Welsh 2016, Lee et al. 2016, Chalfin and McCrary 2017, Braga et al. 2019b, Braga et al. 2019a, Braga et al. 2019c, Kounadi et al. 2020). Third, a cross-reference search was conducted on the preliminary selection to identify relevant publications, which were not yielded during the search (e.g., Thaler 1977, Richards et al. 1985, Draca et al. 2007, Hinkle and Weisburd 2008, Rosenfeld et al. 2014, Blattman et al. 2017, Mitchell 2017. ...
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Third party policing represents a major shift in contemporary crime control practices. As the lines blur between criminal and civil law, responsibility for crime control no longer rests with state agencies but is shared between a wide range of organisations, institutions or individuals. The first comprehensive book of its kind, Third Party Policing examines this growing phenomenon, arguing that it is the legal basis of third party policing that defines it as a unique strategy. Opening up the debate surrounding this controversial topic, the authors examine civil and regulatory controls necessary to this strategy and explore the historical, legal, political and organizational environment that shape its adoption. This innovative book combines original research with a theoretical framework that reaches far beyond criminology into politics and economics. It offers an important addition to the world-wide debate about the nature and future of policing and will prove invaluable to scholars and policy makers.
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