For more than a century scholars, from a variety of disciplines, have
debated the wisdom of using ratio variables when conducting correlational
or regression analyses of macro-social data. Beginning with Pearson (1897),
a number of statisticians have cautioned that correlations between ratio
variables that contain common terms can appear to reveal the presence
of statistically significant associations when, in fact, none are present
(Bollen and Ward 1979; Kronmal 1993; Logan 1982; Schuessler 1974). Yet
others reject, with comparable enthusiasm, the assertion that some portion
of the correlation among ratio variables is inherently spurious (Firebaugh
and Gibbs 1985, 1986; Long 1979; MacMillan and Daft 1980).
Conspicuous in its absence from the ongoing dispute concerning the consequences
of utilizing ratio variables in macro-social research is any discussion
about whether or not we should be using ratio variables in the first
place.1 This omission is particularly
evident with respect to macro-level analyses of crime. Specifically, there
is a broad consensus among criminologists that the population size of
the social unit under investigation should be used to deflate raw counts
of crime. To be sure, there is some disagreement about which population
measure should be used to create crime rates (cf. Chamlin and Cochran
1996; Gibbs and Erickson 1976; Harries 1981; Stafford and Gibbs 1980).
However, the fundamental notion that one must control for the effects
of population by creating crime rates prior to the estimation of model
specifications goes virtually unchallenged.2
The purpose of this exercise to twofold. First, we seek to ascertain why
macro-criminologists, seemingly without exception, prefer to control for
the effects of population size on crime by the process of deflation (the
ratio variable approach) rather than by including population size among
the other structural predictors of interest and estimating their partial
effects on raw counts of crime (the components approach). Second, and
more importantly, we seek to better understand the substantive implications
of these competing techniques for assessing the influence of population
size on the level of crime across macro-social units.
The practice of dividing raw counts of crime by the population size of
the unit of analysis under investigation prior to model estimation is
so accepted among macro-criminologists that one is hard pressed to find
any justification for it in the empirical literature. Indeed, the only
reference to this matter that we could find appears in Gibbs and Erickson's
(1976) discussion of the relative merits of deflating a city's crime figures
by its population size or that of the larger social aggregation within
which it is located. According to their view, macro-criminologists use
rates in lieu of raw numbers because more populous social units contain
a greater number of potential victims and offenders. Hence, "[w]ithout
such control, the incidence of crime is virtually certain to be greater
for California than for, say, Wyoming." (Gibbs and Erickson 1976:606).
There is little doubt that if one wants to make meaningful comparisons
about the risk of victimization across social aggregates, or within a
social aggregate over time, one must standardize raw counts of crime in
some manner. Consider two hypothetical cities. Each experiences 100 felonies
within a given time period. Without information about the number of people
residing within these cities one could make no reasonable inferences about
their relative safety. This, of course, is intuitively obvious and requires
little explication. Indeed, the insight that the concentration of large
numbers of individuals in one location fosters higher levels of social
associations, including criminal victimizations, can be traced to Durkheim's
(1933) discussion of the relationship between dynamic density and the
division of labor and Spencer's (1972) discourse on the impact of population
growth on societal evolution.
What is less clear, however, is why it is necessary, or even advisable,
to deflate crime counts by population size when one is trying to estimate
the relative effects of population size and other structural predictors
on the level of crime. To the best of our knowledge we do this with no
other structural variables. For example, not unlike population size, poverty
is often hypothesized to be positively related to crime. Yet, in contrast
to how we control for the causal influence of population size on crime,
we do not deflate crime by poverty and then include poverty among the
predictors of a crime-poverty ratio.
Admittedly, we are somewhat at a loss to understand the predilection among
macro-criminologists for "controlling" for differences in the
population size by the calculation of crime rates when conducting multivariate
analyses. As has been addressed elsewhere, there are no statistical reasons
for rejecting, a priori, the practice of including population size, along
with the standard array of macro-level predictors, in model specifications
to determine its relative impact on the level of crime (Bollen and Ward
1979; Firebaugh and Gibbs 1986; Schuessler 1974).3
The only compelling justification that we could envision for opting to
study crime rates in lieu of crime counts would be if one could establish
that the effect of population size upon crime is either trivial or spurious.
If such were the case, then it would make sense to create population-based
crime rates to remove the "confounding" effects of structured
opportunities for offending before one can assess the impact of more theoretically
interesting variables on various forms of crime (Gibbs and Erickson 1976;
Harries 1981; Mayhew and Levinger 1976). However, as we explicate in the
next section, we doubt that this is so.
THEORETICAL LINKAGES FROM POPULATION SIZE TO CRIME
The potential deleterious effects of living in large social aggregates
has long been a subject of concern among urban sociologists (Park and
Burgess 1925; Weber 1958; Wirth 1938). At the risk of oversimplification,
one can distinguish among three broad intervening causal processes by
which population size promotes criminality.
The first, the social control perspective, emphasizes that urbanization
and population growth weaken informal mechanisms of social control which,
in turn, result in more crime and delinquency. This occurs, in part, because
large numbers, regardless of individual proclivities, constrain the quality
of social interactions. Ostensibly, population growth increases the frequency
of more predatory secondary contacts while simultaneously decreasing the
frequency of more affective primary ones. Hence, the bonds of solidarity
that were once produced by intimate social associations no longer function
to inhibit social deviation (Kornhauser 1978; Shaw and McKay 1972; Simmel
1955; Wirth 1938).
The second, the structuralist perspective, focuses on how the size and
distribution of population groups within geographic units delimits opportunities
for social interactions (see Simmel's  seminal exposition of the
relationship between group size and forms of association). Interestingly,
little, if any, attention is paid to the content of interpersonal relations.
Thus, in contrast to the social control perspective, the structuralist
approach centers on the causal impact of population size on the quantity
of social contacts.
Mayhew and Levinger's (1976) explication of the relationship between population
size and generic forms of social interactions typifies the structuralist
approach to theory construction. Based on series of observations about
the mathematical properties of population elements and the number of potential
contacts among population elements within a geographic area, they derive
a formal, structural model of social interaction. Specifically, Mayhew
and Levinger (1976) postulate that as the population size increases, the
frequency of social interactions increases, at an increasing rate. However,
Mayhew and Levinger's (1976:94) theoretical presentation offers no clues
concerning the likely purpose of any structurally induced associations:
"[i]t predicts for example, that as the aggregate size increases
additively, both the number of phone calls and the number of homicides
will increase multiplicatively."
Blau (1977:160-162), based on the simple assumption that social associations
require opportunities for social contacts, also posits a positive relationship
between population size and crime. Yet unlike Mayhew and Levinger (1976),
Blau (1977) hypothesizes that the population size-crime relationship is
likely to be linear. The causal linkages from population size to crime
are rather straightforward. Accordingly, population size promotes various
social interactions, including criminal victimizations, by reducing the
physical distances among members of a community while it simultaneously
increases the number of potential associates within a community. However,
since the opportunity assumption is silent with respect to the content
of structurally induced social contacts, one can deduce that population
growth is equally likely to lead to an increase in integrative, as well
as conflictive, social interactions (Blau 1977:163).
In short, while there may be some debate concerning the functional form
of the population size-crime relationship, formal structural theory unequivocally
asserts that population growth facilitates all sorts of social contacts
- from the most benign to the most pernicious (Mayhew and Levinger 1976;
Third, the subcultural perspective suggests that the concentration of
relatively large numbers of individuals within macro-social units fosters
the creation and expansion of deviant subcultures. Urbanization, through
the complementary processes of structural differentiation and value diffusion,
promotes social support for a multiplicity of behavioral choices. Further,
it engenders greater tolerance for nonconformity among the more conventional
members of the community. As a consequence, more populous urban areas
are expected to experience more criminal activity than less populous ones
(Fischer 1975; 1995; Simmel 1955; Tittle 1989).
The implications of the preceding review of the various causal linkages
among urbanization, population size, and crime are clear. There are both
diverse and compelling theoretical grounds for including population size
in any model specification seeking to determine the relative impact of
macro-structural conditions on the level of crime. Stated in the context
of the present discussion, one cannot conclude that it is preferable to
control for the impact of population size on crime by dividing crime figures
by population counts because there is reason to believe that the effect
of population size is likely to be trivial or spurious.
By way of compromise, one might be tempted to address this matter by deciding
to employ population size as a predictor variable and as the denominator
in a crime rate measure (see, for example, Blau and Blau 1982; Parker
and Pruitt 2000; Sampson 1987). However, this strategy, though normative,
may be less expedient than one might think.
Recall that the ratio variable approach contends that by dividing the
number of crimes within a social aggregate by the number of people residing
within that social aggregate one can effectively remove the linear effect
of population size on crime prior to conducting statistical analyses of
interest (Bollen and Ward 1979; Mayhew and Levinger 1976). Let us assume,
for the sake of discussion, that this procedure is efficacious. Let us
further assume that the relationship between population size and crime
is linear. If both conjectures are valid, then one cannot simultaneously
control for the effects of population size by the process of deflation
and the inclusion of population counts among a set of predictors.
One would invariably find that population size has no impact on crime.
There can be no doubt that subjecting crime counts to the arithmetic operation
of division by population counts reduces its variance. What remains to
be seen, however, is the extent to which the decision to deflate crime
figures by population counts prior to the estimation of multivariate equations
tempers the impact population size on the level of crime. The proceeding
analyses are designed to address this matter.
In order to discern the effects of deflating crime counts by the number
of inhabitants residing within macro-social units, we compare the relative
influence of linear and non-linear measures of population size on crime
and crime rates.
The sample is part of a sample of 354 U. S. cities, originally selected
for another project because it contained information concerning charitable
contributions. Missing data, primarily for the crime measures, yield a
final sample size of 271. These cities are distributed with a mean population
of approximately 211,000 and a standard deviation of 53,000.
We decided to employ this sample for two reasons. First, larger social
aggregations, such as metropolitan areas or states, are probably too heterogenous
to allow for an assessment of the differential effects of population size
on crime. Second, the selection of a large sample of cities assuages several
statistical concerns. That is to say, it increases the likelihood that
there will be substantial variation in population size, decreases the
likelihood of encountering harmful collinearity, and allows one to specify
the requisite control variables without worrying about the loss of degrees
of freedom (Hanushek and Jackson 1977).
Crime counts and ratios.
The present investigation examines the effects of alternative measures
of population size, as well as a number of other structural predictors
on property crime rates, violent crime rates, the simple count of property
crimes, and the simple count of violent crimes. Following convention,
burglaries, larcenies, and motor vehicle thefts are designated as property
offenses (1990), while homicides, robberies, aggravated assaults, and
forcible rapes are classified as violent offenses (1990). The rate measures
are deflated by the city population in units of 100,000.
To foster a comparison of the ratio and components approaches to crime
measurement, population size is estimated using three alternative functional
forms of the total number of inhabitants in each city (1990): the simple
count (a linear model), the simple count and the count squared (a quadratic
model), and the natural log transformation of the simple count (a semi-log
Following previous research (Blau an Blau 1982; Liska and Bellair 1995;
Sampson 1987), nine control variables are included in the model specifications
to account for the predictions of motivational, opportunity, and subcultural
macro-level theories of crime.
A number of motivational theories contend that economic deprivation has
a substantial impact on the level of crime across macro-social units.
For example, traditional Marxist theory (Bonger 1916) and anomie theory
(Merton 1938) suggest that blocked opportunities produce frustration and
thereby motivate the disadvantaged to engage in crime to satisfy their
material needs. Given the ongoing debate concerning the relative importance
of absolute and relative deprivation as predictors of crime (Kovandzic,
Vieraitis, and Yeisley 1998), the models contain measures of both dimensions
of economic deprivation. Absolute deprivation is measured as the percentage
of families below the poverty level (1990). Relative economic deprivation
is measured by the Gini index of economic concentration (1989).
Opportunity theories of crime focus on the relationships among the physical
and social structures of ecological units, informal social control, and
crime. For instance, urbanism theory, including the social disorganization
approach, suggests that structural conditions that impede communication
and the formation of affective interpersonal relationships foster high
rates of crime. Neighborhoods, as well as larger social areas, that have
large, heterogenous populations and that possess few economic resources
have difficulties creating and maintaining social institutions that discourage
criminal victimizations (Bursik and Grasmick 1993; Fischer 1975; Kornhauser
1972; Sampson, Raudenbush, and Earls 1997; Shaw and McKay 1972; Wirth
1938). To take into account the predictions of urbanism theory, the model
specifications include three measures of population heterogeneity and
an estimate of residential mobility. The first indicator of population
heterogeneity, racial heterogeneity, is measured as the percentage of
the population that is black (1990). The second, ethnic heterogeneity,
is measured as the percentage of the population that is foreign born (1990).
The third, age structure, is measured as the percentage of the population
aged 18 to 24 (1990). Lastly, residential mobility is measured as the
percentage of persons five years of age and older living in different
locations in 1990 than in 1985.
Another variant of opportunity theory, the routine activity approach,
suggests that household structure affects levels of capable guardianship
and target suitability. Specifically, household structure is hypothesized
to simultaneously decrease guardianship, but increase target attractiveness,
thereby increasing rates of crime, especially those involving theft (Cohen
and Felson 1979). The models include two indicators of household structure:
the percentage of single-person households (1990) and the percentage of
persons 15 and older who are divorced (1990).
Lastly, the subculture of violence thesis maintains that the South, as
a result of idiosyncratic historical processes, has developed a value
system that condones the use of violence to settle interpersonal disputes
(Gastil 1971; Hackney 1969). To control for the impact of regional variations
in normative orientations on the level of crime, especially that of a
violent nature, a measure of southern location is included in the model
specifications. The regional dummy variable is coded 1 for cities located
in the South and 0 for those located elsewhere.
Information concerning the official count of violent and property offenses
was obtained from the Uniform Crime Reports (Federal Bureau of
Investigation 1991). With the exception of residential mobility and the
percentage of divorcees, data for each of the structural predictors, as
well as the income distributions used to calculate the Gini index, were
obtained from the County and City Data Book (Bureau of the Census
1994). Residential mobility was calculated from data ascertained from
Table 172 of the Census of Population: Social and Economic Characteristics
(Bureau of the Census 1993). The percentage of divorcees was calculated
from data gathered from Table 64 of The Census of Population: General
Population Characteristics (Bureau of the Census 1992).
Preliminary Data Analyses
While the specification of a comprehensive array of statistical controls
assuages fears about omitted variable bias, it simultaneously increases
the risk of multicollinearity among the predictors. And while the analysis
of raw counts of crime is essential if we are to evaluate our contention
that the use of crime rate measures obscures the nature of the relationship
between population size and crime, it simultaneously increases the likelihood
that the disturbance terms will be heteroskedastic. Hence, we performed
a number of diagnostic tests to determine whether or not the final equations
are affected by either of these potential problems.
Although high zero-order correlations between predictor variables do not,
in and of themselves, indicate that multicollinearity is present (Hanushek
and Jackson 1977:90), the zero-order correlations between percent black
and poverty (.64), percent black and the Gini index (.52), poverty and
the Gini Index (.60), and percent 18-24 and residential mobility (.61)
are large enough to warrant concerns about this issue.4
To assess the extent to which collinearity among the exogenous variables
affects the parameter estimates collinearity diagnostics were examined
(Belsley, Kuh, and Welsch 1980). Experiments conducted by Belsley et al.
(1980) reveal that a condition index threshold of 30 is indicative of
potentially harmful collinearity and a variance-decomposition proportion
threshold of 0.5 should be used to identify dependencies among the predictor
variables. As expected, inspection of the collinearity diagnostics for
the each of the equations suggests that there are strong linear dependencies
among the Gini index, poverty, and percent black.
One solution to the problem of muliticollinearity that has emerged from
the literature is the use of principal components analysis to reduce the
number of predictors prior to the estimation of any regression equations
(Land, McCall, and Lawrence 1990; Morenoff and Sampson 1997). However,
both Greene (1993:273) and Maddala (1992:285) warn that this approach
has a number of limitations, chief among them is that it often combines
predictors into principal components that possess little, if any, substantive
meaning. Consequently, we decided to modify the principal components procedure
by rotating the final solution using the varimax option to make the factors
more coherent and easier to interpret.
Table 1 presents the results of the final principal component factor analysis
with varimax rotation. Two of the nine control variables, the dummy variable
for southern location and percent foreign born, were excluded from the
analyses. The former was excluded because it is a nominal variable, while
the latter was excluded because it did not load on any of the factors.
As is clear from inspection of Table 1, the remaining seven control variables
can be combined into three latent constructs. The first appears to capture
the racial and economic structure of the sample. Three variables: the
Gini index, poverty, and percent black load on this factor. The second,
which is not as readily interpretable as the first, seems to tap a dimension
of the population structure. Two variables, the percentage of the population
aged 18 to 24 and residential mobility, load on this factor. The final
principal component measures household structure. Two variables, the percentage
of single person households and percent divorced, load on this factor.
In sum, the three factor solution reduces the number of control variables
from nine to five. More importantly, the revised models reveal no evidence
We also explored the possibility that the error terms are heteroskedastic.
We calculated the Breusch-Pagan test statistic, which is distributed as
chi-square, to evaluate the null hypothesis that the model residuals are
homoskedastic. Unfortunately, the residual analyses clearly indicate that
the disturbance terms produced by each of the violent crime rate, the
number of violent crimes, and the number of property crimes, equations
Various remedies have been proffered in the literature to correct for
the problems that arise from the presence of heteroskedastic errors. We
decided to employ White's (1980) correction of the standard error to the
OLS regression solutions for the violent and property crime equations
to address this matter. Unlike other approaches, which entail making assumptions
about the underlying processes that are responsible for the production
of heteroskedastic errors (e.g., Weighted Least Squares regression, log
transformations of the data), White's correction requires no such conjectures
to generate unbiased estimates of the variance of the least squares estimator
Violent and property crime rates.
Tables 2 and 3 present the OLS regression estimates of the effects of
the structural predictors and the alternative measures of population size
on the violent and property crime rates, respectively. Each table contains
the results of three analyses, which differ only with respect to how we
estimate the functional form of population size-crime relationship. The
first equation estimates the linear effect of population size on crime.
The second equation also includes a quadratic term to capture a change,
if any, in the slope of the relationship between population size and crime.
Lastly, the third equation includes the natural logarithmic transformation
of the population size in lieu of the original measure to estimate the
semi-log effects of population size on crime.
Two patterns of interest emerge from these analyses. First, the structural
predictors, both the composite factors and individual variables, have
a substantial impact on violent and property crime rates. Moreover, with
the exception of the effects of southern location, these effects are virtually
identical across all six equations. Consistent with a variety of theoretical
perspectives, the racial and economic composition factor, the household
structure factor, and percent foreign born positively affect violent and
property crime rates. Contrary to the predictions of subcultural theory,
southern location has no appreciable impact on rates of violent crime,
but is positively related to the rate of property crime. Second, the effect
of population size on each of the crime rate measures is small and insignificant.
This result holds for both the linear and non-linear models.
In sum, our conventional, multivariate analyses of violent and property
crime rates are fairly consistent with prior city-level research (cf.
Carroll and Jackson, 1983; Loftin and Parker, 1985). We find that variations
in violent and property crime are best accounted for by the structural
antecedents of intergroup conflict and/or disorganization. In contrast,
the effects structured opportunities for social interaction, as approximated
by population size, are negligible.
The number of violent and property crimes.
Tables 4 and 5 contain OLS regression estimates of the effects of the
structural predictors and the alternative measures of population size
on the number of violent and property crimes, respectively. As with Tables
2 and 3, each table contains the results of three analyses, which differ
only with respect to how we estimate the functional form of population
size-crime relationship. The first equation estimates the linear effect
of population size on crime. The second equation also includes a quadratic
term to capture a change, if any, in the slope of the relationship between
population size and crime. Lastly, the third equation includes the natural
logarithmic transformation of the population size in lieu of the original
measure to estimate the semi-log effects of population size on crime.
Inspection of tables 4 and 5 leaves little doubt as to the influence
that a priori measurement decisions can have on subsequent data analyses.
In stark contrast to analyses of the rate measures, the examination of
the count measures suggests that variations in the level of crime across
cities are primarily determined by opportunities for social contacts.
This is not to say, of course, that the more substantive macro-level
predictors have no influence on crime counts. Somewhat akin to what we
found for the rate measure equations, the racial and economic structure,
the household structure, and percent foreign born positively affect both
the number of violent and property crimes. However, unlike what we reported
above, these effects are not invariant across model specifications (compare
tables 2 and 3 with table 4 and 5). Similarly, southern location continues
to exhibit positive partial effects on property crime in the linear and
quadratic equations, but not in the semi-log equation. Nonetheless, compared
to the impact of the population size, the magnitudes of these effects
Indeed, regardless of the functional form specified, population size is,
by far and away, the strongest predictor of the number of violent and
property crimes. One should not infer, of course, that there are no important
differences across equations. Based upon a comparison of the variance
explained by the competing models, it is clear that the quadratic equations
provide the best fit to the counts of violent and property offenses.
Consistent with Mayhew and Levinger's (1976) thesis, the effect of population
size on the number of violent crimes is positive, with an increasing slope.
However, contrary to their expectations, the shape of the population size-property
crime relationship approximates that of an inverted U. Admittedly, we
did not anticipate that the functional form of the population-crime relationship
would vary across offense types. Nonetheless, we are reluctant to attribute
these findings to some methodological deficiency. As we discussed above,
the collinearity and other diagnostics reveal no statistical problems
with the final equations. Moreover, we are not attempting to analyze rare
events that occur within small social aggregates. Hence, there is no reason
to believe that Gaussian-based regression models are inappropriate (Osgood
2001; Osgood and Chambers 2000).
Therefore, if we can conclude that these findings do not arise from some
statistical error, then we are forced to deduce that the effects of racial,
economic, and social structural characteristics of cities pale before
the influence of population size on variations in the number of crimes.
Put in the larger context of this exercise, it would appear that the decision
whether or not to measure crime as a rate or as a simple count has substantial
ramifications for the appraisal of macro-social theory.
In the introduction to this manuscript we posited two questions concerning
the manner in which macro-criminologists study the relationship between
population size and crime. First, we sought to discover why the vast majority
of macro-criminologists have come to accept, with virtually no debate,
the conventional practice of accounting for the influence of population
size on crime through the process of deflation. Second, we sought to discern
the consequences of following this procedure for the evaluation of competing
To be frank, we are still uncomfortable about making any strong statements
concerning why the ratio variable approach has come to dominate the empirical
literature. Initially, we speculated that the penchant for deflating the
number of crimes by the number of inhabitants of a geographic unit of
interest arises from the perception that the effects of population are
spurious or trivial. We now think that this is unlikely. As we explicated
above, there is ample theory to support the contention that the influence
of population size on crime is substantively interesting. Urban (Wirth,
1938), formal macro-structural (Blau 1977; Mayhew and Levinger 1976),
and subcultural (Fischer 1975; Tittle 1989) theories, albeit for different
reasons, posit a causal relationship between population size and crime.
To be sure, part of the answer probably rests with the desire to control
for victimization risk. Indeed, we are in total agreement with Gibbs and
Erickson's (1976:606) observation that the incidence of crime is going
to be greater in more, rather than in less, populated communities. Nonetheless,
the recognition that cross-jurisdictional comparisons should take into
account opportunities for criminal events does not necessarily require
investigators to study crime rates. As has been demonstrated elsewhere,
one can just as easily account for the risks of victimization associated
with differences in the number of potential offenders and victims by including
population size as an additional predictor in models of crime counts (Bollen
and Ward 1979; Firebaugh and Gibbs 1986; Schuessler 1974). Moreover, this
method of addressing the issue, the components approach, has the advantage
of allowing one to assess the relative impact of population size on crime
without having to worry whether or not some portion of this effect has
been removed by the process of deflation (Chamlin and Cochran 1996).
Admittedly, we can offer no definitive explanation for the overwhelming
preference among macro-criminologists for modeling the structured opportunities
for criminal victimizations associated with population size by the process
of deflation. Regardless, we suspect that there is a relatively simple
answer to our question. At the risk of appearing naive, we speculate that
most criminologists study crime rates, in lieu of crime counts, because
their teachers, colleagues, and peer reviewers study crime rates. That
this to say, this 'convention' has become so reified that it no longer
invites much scholarly interest or debate. Independent of how the practice
of deflation became normative, it is abundantly clear that it can affect
the analysis of the population size-crime relationship. In an effort to
delineate the consequences of a priori measurement decisions for assessing
the influence of population size on crime, we conducted two, complementary
analyses. The first set of equations estimated the linear and non-linear
partial effects of population size on violent and property crime rates
(per 100,000 population); while the second set of equations estimated
the linear and non-linear partial effects of population size on the number
of violent and property crimes, respectively. As we reported above, two
patterns of interest emerge from these analyses. First, population size
exhibits null effects in each of the crime rate equations. However, regardless
of the functional form examined, population size significantly affects
the number of violent and property crimes. Second, the results from the
crime rate equations indicate that the racial and economic structure composite
variable and, to a lesser extent, the percentage of foreign-born have
the largest impact on the level of crime. In contrast, the results from
the count equations indicate that population size is, by far and away,
the single best predictor of the level of violent and property crime.
What are we to infer from all this? At a minimum, our analyses reveal
that how one decides to "control" for the influence of population
size on the level of crime across macro-social units has a substantial
impact on the findings one is likely to generate. Consequently, we think
it is time that macro-criminologists revisit the how best to model the
influence of population size on crime.
To the extent that one wants to determine the relative impact of various
macro-level variables, once the "opportunity" effects of the
population size have been removed from the amount of crime, then one should
probably examine crime rates. However, careful attention should be given
to who is included, and excluded, from the denominator of a particular
rate of crime. As demographers have long recognized, for comparisons across
time and space to be meaningful they must take into account the risk of
experiencing the behavioral outcome of interest (Shryock and Siegel 1976).
For example, population of origin is often used to control for the number
of people that can move from one place to another in the calculation of
migration rates (Haenszel 1967), while sex- and age-specific population
distributions are typically used to calculate marriage rates (Hajnal 1953).5
We encourage those who decide to control for the "opportunity"
effects of population size by the calculation of rate measures to explicitly
consider which individuals are likely to comprise the pool of victims
and offenders for the crime category under investigation. One can envision
a number of situations where the gross population of a place might over-
or under-estimate the number of potential victims or offenders.
Consider, for purposes of illustration, the problem of inanimate victims
of crime. The supposition that total population size accurately measures
the number of potential victims assumes that only humans can be the targets
of crime. Clearly, for property crimes this is not the case. For example,
it appears self-evident that the appropriate risk denominator for burglary
rates should be the number of commercial and residential buildings (Boggs
1965). Similarly, the number of motor vehicles is likely to be a better
deflator for motor vehicle theft than total population size. To be sure,
the quantity of physical targets in a social aggregate is likely to be
highly correlated with the quantity of individuals. However, to the extent
that the use of population-based and target-based denominators produce
inconsistent rankings of crime rates within, and across, political units
(Phillips 1973; Boggs 1965; Harries 1981), grounding the selection of
the denominator in either theory or logic becomes critical.
Alternatively, if one is interested in ascertaining the relative partial
effects of population size, we advocate abandoning the conventional methodology,
because we believe that it underestimates the partial effects of population
size on the level of crime among macro-social units. Deflating the number
of crimes by the population counts removes a substantial portion of the
variance in the level of crime, which would be attributable to the size
of the populace, prior to the estimation of the multivariate models. In
the present case, the process of deflation removed approximately 92% of
the variance in the amount of violent crimes and approximately 94% of
the variance in property crimes.
In short, we believe that the process of deflation, by partially controlling
for the effects of population size on crime prior to the estimation of
any multivariate models, misspecifies the causal relationship between
population size and macro-level indicators of crime. It tends to overestimate
the effects of the social, economic, and political conditions, while it
simultaneously underestimates the importance of opportunities for social
contacts (the number people in a geographic area) on variations in the
level of crime. We recognize, as we discussed above, that the analysis
of crime counts (the components approach) is not without its problems
(e.g., heteroskedasticity). However, as we also discussed above, these
limitations are not fatal and, depending on the source of the problem,
can be addressed in a number of ways (Greene 1993; Osgood 2000; White
1. Some proponents of ratio variable approach do consider whether or not
the proportions are "theoretically meaningful." However, the
interest here is not so much with the thinking that led to the creation
of a particular rate, but rather with the belief that "theoretically
meaningful" ratios that contain common terms are less likely to be
spuriously related to one another (Kasarda and Nolan 1979; MacMillan and
2. Recently, concern about the use of ratio measures of crime has expanded
to consider the relative efficacy of OLS, Poisson, and negative binomial
regression analyses of crime rates among small social aggregates. In brief,
this exchange focuses on the limitations associated with the use of OLS
techniques for the purpose of studying rare events (Gardner, Mulvey, and
Shaw 1995; Osgood 2000). For example, Osgood (2000) provides persuasive
evidence that supports the conclusion that Poisson-based regression models
of count data can (and should) be used in lieu of OLS regression techniques
to analyze per capita offense rates when the number of crimes approaches
zero. Interestingly, this statistically-motivated discussion of the benefits
accrued from the use of Poisson-based regression to model crime rates
in sparsely populated places (not unlike the statistically-motivated discussion
of the correlation between ratios with common terms) fails to consider
the theoretical issues that inform the decision whether or not to deflate
raw counts of crime. In contrast, our investigation focuses on the substantive,
rather than the statistical, implications of how one elects to account
for the influence of population size on the level of crime across macro-social
3. It is true that statistical models of crime count data are more likely
than similar analyses of crime rate data to produce heteroskedastic disturbance
terms. It should be recognized, however, that rate models are not immune
to this problem (see, for example, Sampson and Groves 1989).back
4. By construction, the various population measures are, of course, highly
collinear. However, the addition of a quadratic term to an equation that
includes a linear term (the only situation where the collinear population
variables will appear in the same equation) has no effect on the unstandardized
coefficients for the linear or quadratic terms or their respective significance
tests (Allison 1977).back
5. We would like to thank an anonymous reviewer for calling our attention
to the contributions of demographers with respect to discerning and estimating
populations at risk. back
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